MULTIFUNCTION FLOATING POINT ADDITION/SUBTRACTION PIPELINE AND BIPARTITE LOOK-UP TABLE

Abstract: An add/subtract pipeline has far and close data paths. The far data path handles effective addition operations, and effective subtraction operations for operands having an absolute exponent difference greater than one. The close data path handles all other effective subtraction operations. Selection of the output value in the close data path effectuates the round-to-nearest operation. Floating point-to-integer conversion may be executed in the far data path integer-to-floating point instructions in the close data path. The execution unit may include a pluraliy of add/subtract pipelines, allowing vectored add, subtract, and integer/floating point conversion instructions to be performed. Additional arithmetic instructions (such as reverse subtract and accumulate functions minimum/maximum and comparison) may also be implemented. A method for generating entries for a bipartite look-up table having base and difference table portions is also disclosed. So is a multi-function look-up table.

Description: TITLE: Multifunction Floating Point Addition/Subtraction Pipeline And Bipartite Look-up Table BACKGROUND OF THE INVENTION 1 Field of the Invention This invention relates to floating point arithmetic within microprocessors, and more particularly to an add/subtract pipeline and multifunction bipartite look-up table withm a floatmg pomt arithmetic unit 2 Descπption of the Related Art Add/Subtract Pipeline Numbers may be represented withm computer systems m a variety of ways. In an integer format, for example, a 32-bit register may store numbers ranging from 0 to 2 32 -l (The same register may also signed numbers by givmg up one order of magnitude m range) This format is limiting, however, smce it is incapable of representmg numbers which are not integers (the binary pomt in integer format may be thought of as bemg to the πght of the least significant bit m the register). To accommodate non-mteger numbers, a fixed point representation may be used. In this form of representation, the binary point is considered to be somewhere other than to the right of the least significant bit. For example, a 32-bit register may be used to store values from 0 (inclusive) to 2 (exclusive) by processing register values as though the binary pomt is located to the πght of the most significant register bit. Such a representation allows (in this example) 31 registers bit to represent fractional values. In another embodiment, one bit may be used as a sign bit so that a register can store values between -2 and +2. Because the binary pomt is fixed withm a register or storage location durmg fixed point arithmetic operations, numbers with differing orders of magnitude may not be represented with equal precision without scaling. For example, it is not possible to represent both 1101b (13 m decimal) and .1101 (.8125 m decimal) usmg the same fixed pomt representation. While fixed point representation schemes are still quite useful, many applications require a larger dynamic range (the ratio of the largest number representation to the smallest, nonzero, number representation in a given format). In order to solve this problem of dynamic range, floatmg point representation and arithmetic is widely used. Generally speaking, floating point numeric representations include three parts: a sign bit. an unsigned fractional number, and an exponent value. The most widespread floating point format in use today, IEEE standard 754 (single precision), is depicted in Fig 1. Turning now to Fig. 1, floating pomt format 2 is shown Format 2 includes a sign bit 4 (denoted as S), an exponent portion 6 (E), and a mantissa portion 8 (F) Floatmg pomt values represented in this format have a value V, where V is given by: V = (-\) s ■ 2 ε - b,as - (I. F). (1) Sign bit S represents the sign of the entire number, while mantissa portion F is a 23-bit number with an implied leading 1 bit (values with a leadmg one bit are said to be "normalized") In other embodiments, the leading one bit may be explicit Exponent portion E is an 8-bit value which represents the true exponent of the number V offset by a predetermined bias A bias is used so that both positive and negative true exponents of floatmg pomt numbers may be easily compared The number 127 is used as the bias in IEEE standard 754 Format 2 may thus accommodate numbers having exponents from -127 to +128 Floating pomt format 2 advantageously allows 24 bits of representation within each of these orders of magnitude Floatmg point addition is an extremely common operation in numerically-intensive applications (Floating pomt subtraction is accomplished by inverting one of the mputs and performing addition) Although floating point addition is related to fixed point addition, two differences cause complications First, an exponent value of the result must be determined from the mput operands Secondly, roundmg must be performed The IEEE standard specifies that the result of an operation should be the same as if the result were computed exactly, and then rounded (to a predetermined number of digits) using the cuπent roundmg mode IEEE standard 754 specifies four roundmg modes round to nearest, round to zero, round to +oo, and round to -oo The default mode, round to nearest, chooses the even number m the event of a tie Turning now to Fig 2, a pπor art floating pomt addition pipeline 10 is depicted All steps in pipelme 10 are not performed for all possible additions (That is, some steps are optional for vaπous cases of puts) The stages of pipelme 10 are descnbed below with reference to mput values A and B Input value A has a sign bit A s , an exponent value A E , and a mantissa value A F Input value B, similarly, has a sign bit B s , exponent value B E , and mantissa value B F Pipelme 10 first includes a stage 12, m which an exponent difference E ώff is calculated between A E and B E . In one embodiment, if E dlff is calculated to be negative, operands A and B are swapped such that A is now the larger operand In the embodiment shown in Fig 2, the operands are swapped such that E dlff is always positive In stage 14, operands A and B are aligned This is accomplished by shifting operand B E dlff bits to the πght In this manner, the mantissa portions of both operands are scaled to the same order of magnitude If A E =B E , no shiftmg is performed, consequently, no roundmg is needed If E ώf » 0, however, information must be mamtamed with respect to the bits which are shifted πghtward (and are thus no longer representable withm the predetermined number of bits) In order to perform IEEE rounding, information is mamtamed relative to 3 bits the guard bit (G), the round bit (R),and the sticky bit (S) The guard bit is one bit less significant than the least significant bit (L) of the shifted value, while the round bit is one bit less significant the guard bit The sticky bit is the logical-OR of all bits less significant than R For certain cases of addition, only the G and S bits are needed In stage 16, the shifted version of operand B is inverted, if needed, to perform subtraction In some embodiments, the signs of the input operands and the desired operation (either add or subtract) are examined m order to determine whether effective addition or effective subtraction is occurring In one embodiment, effective addition is given by the equation EA = A S @ B S @ op, (2) where op is 0 for addition and 1 for subtraction For example, the operation A minus B, where B is negative, is equivalent to A plus B (ignoring the sign bit of B) Therefore, effective addition is performed The mversion in stage 16 may be either of the one's complement or two's complement variety In stage 18, the addition of operand A and operand B is performed As descnbed above, operand B may be shifted and may be inverted as needed Next, m stage 20, the result of stage 18 may be recomplemented, meanmg that the value is returned to sign-magnitude form (as opposed to one's or two's complement form) Subsequently, in stage 22, the result of stage 20 is normalized This includes left-shifting the result of stage 20 until the most significant bit is a 1 The bits which are shifted m are calculated according to the values ofG, R, and S In stage 24, the normalized value is rounded according to nearest rounding mode If S includes the R bit OR'ed in, round to nearest (even) is given by the equation RTN = G(L + S). (3) If the roundmg performed in stage 24 produces an overflow, the result is post-normalized (nght- shifted) in stage 26 As can be seen from the descnption of pipelme 10, floatmg pomt addition is quite complicated This operation is quite time-consuming, also, if performed as shown m Fig 2 stage 14 (alignment) requires a shift, stage 18 requires a full add, stage 20 (recomplementation) requires a full add, stage 22 requires a shift, and stage 24 (roundmg) requires a full add Consequently, performing floatmg point addition usmg pipelme 10 would cause add/subtract operations to have a similar latency to floatmg pomt multiplication Because of the frequency of floatmg point addition, higher performance is typically desired Accordingly, most actual floatmg pomt add pipeline include optimizations to pipeline 10 Turning now to Fig 3, a pnor art floating pomt pipelme 30 is depicted which is optimized with respect to pipelme 10 Broadly speaking, pipeline 30 includes two paths which operate concurrently, far path 31A and close path 3 IB Far path 31A is configured to perform all effective additions Far path 31A is additionally configured to perform effective subtractions for which E d , ff >1 Close path 3 IB, conversely is configured to perform effective subtractions for which E d , ff <1 As with Fig 2, the operation of pipelme 30 is described with respect to input values A and B Pipelme 30 first includes stage 32, in which operands A and B are received The operands are conveyed to both far path 31A and close path 3 IB Results are then computed for both paths, with the final result selected m accordance with the actual exponent difference The operation of far path 31A is descnbed first In stage 34 of far path 31 A, exponent difference E dlff is computed for operands A and B In one embodiment, the operands are swapped if A E >B E If E dlff is computed to be 0 or 1 , execution m far path 31 A is cancelled, since this case is handled by close path 3 IB as will be described below Next, m stage 36, the mput values are aligned by right shifting operand B as needed In stage 38, operand B is conditionally inverted in the case of effective subtraction (operand B is not inverted in the case of effective addition) Subsequently, m stage 40, the actual addition is performed Because of the restrictions placed on far path (E dlff >l), the result of stage 40 is always positive Thus, no recomplementation step is needed The result of stage 40 is mstead rounded and post-normalized m stages 42 and 44, respectively The result of far path 31 A is then conveyed to stage 58 In stage 46 of close path 3 IB, exponent difference E dlff is calculated m stage 46 If E dlff is computed to less than equal to 1, execution contmues m close path 31B with stage 48 In one embodiment, operands A and B are swapped (as in one embodiment of far path 31 A) so that A E ≥B E In stage 48, operand B is inverted to set up the subtraction which is performed in stage 50 In one embodiment, the smaller operand is also shifted by at most one bit Smce the possible shift amount is low, however, this operation may be accomplished with greatly reduced hardware The output of stage 50 is then recomplemented if needed in stage 52, and then normalized m stage 54 This result is rounded m stage 56, with the rounded result conveyed to stage 58 In stage 58, either the far path or close path result is selected accordmg to the value of E dlff It is noted that in close path 3 IB, stage 52 (recomplementation) and stage 56 (roundmg) are mutually exclusive A negative result may only be obtamed in close path 3 IB m the case where A E =B E and A F <B F In such a case, however, no bits of precision are lost, and hence no roundmg is performed Conversely, when shifting occurs (givmg πse to the possibility of roundmg), the result of stage 50 is always positive, eliminating the need for recomplementation m stage 52 The configuration of pipelme 30 allows each path 31 to exclude unneeded hardware For example, far path 31A does not require an additional adder for recomplementation as descnbed above Close path 3 IB eliminates the need for a full shift operation before stage 50, and also reduces the number of add operations required (due to the exclusivity of roundmg and recomplementation described above) Pipelme 30 offers improved performance over pipelme 10 Because of the frequency of floatmg pomt add/subtract operations, however, a floatmg point addition pipelme is desired which exhibits unproved performance over pipelme 30 Improved performance is particularly desired with respect to close path 3 IB Multifunction Bipartite Look-Up Table Floating-point instructions are used within microprocessors to perform high-precision mathematical operations for a variety of numeπcally-intensive applications Floating-point arithmetic is particularly important withm applications that perform the rendering of three-dimensional graphical images Accordingly, as graphics processmg techniques grow more sophisticated, a corresponding increase in floating-point performance is required Graphics processmg operations within computer systems are typically performed m a senes of steps referred to collectively as the graphics pipelme Broadly speakmg, the graphics pipelme may be considered as having a front end and a back end The front end of receives a set of vertices and associated parameters which define a graphical object m model space coordinates Through a number of steps m the front end of the pipeline, these vertices are assembled into graphical primitives (such as triangles) which are converted mto screen space coordmates One distinguishing feature of these front-end operations (which include view transformation, clipping, and perspective division) is that they are primarily performed usmg floating-point numbers The back end of the pipelme, on the other hand, is typically integer-intensive and mvolves the rasterization (drawing on a display device) of geometnc primitives produced by the front end of the pipelme High-end graphics systems typically include graphics accelerators coupled to the microprocessor via the system bus These graphics accelerators mclude dedicated hardware specifically designed for efficiently performing operations of the graphics pipelme Most consumer-level graphics cards, however, only accelerate the rasterization stages of the graphics pipelme. In these systems, the microprocessor is responsible for performing the floating-point calculations in the initial stages of the graphics pipelme The microprocessor then conveys the graphics primitives produced from these calculations to the graphics card for rasteπzmg For such systems, it is clear that mcreased microprocessor floating-point performance may result m increased graphics processmg capability. One manner in which floating-point performance may be mcreased is by optimizing the divide operation Although studies have shown that division represents less than 1% of all instructions in typical floating-point code sequences (such as SPECfp benchmarks), these instructions occupy a relatively large portion of execution time. (For more information on the division operation withm floating-point code sequences, please refer to "Design Issues m Division and Other Floating-Point Operations", by Stuart F. Oberman and Michael J. Flynn, published m IEEE Transactions on Computers, Vol. 46, No. 2, February 1997, pp. 154-161) With regard to the front-end stages of the graphics pipelme, division (or, equivalently, the reciprocal operation) is particularly cntical during the perspective correction operation A low-latency divide operation may thus prevent a potential bottleneck and result m mcreased graphics processmg performance. Additional floating-point performance may be gamed by optimization of the reciprocal square root operation (l/sqrt(x)). Most square roots in graphics processmg occur m the denominators of fractions, so it is accordmgly advantageous to provide a function which directly computes the reciprocal of the square root. Smce the reciprocal square root operation is performed duπng the common procedures of vector normalization and viewmg transformations, optimization of this function represents a significant potential performance enhancement One means of increasing performance of the reciprocal and reciprocal square root operations is through the use of dedicated floating-point hardware Because floating-point hardware is relatively large as compared to comparable fixed-pomt hardware, however, such an implementation may use a significant portion of the hardware real estate allocated to the floating-point unit. An alternate approach is to utilize an existmg floating- pomt element (such as a multiplier) to implement division based on iterative techniques like the Goldschmidt or Newton-Raphson algoπthms. Iterative algorithms for division require a starting approximation for the reciprocal of the divisor A predetermined equation is then evaluated usmg this starting approximation. The result of this evaluation is then used for a subsequent evaluation of the predetermined equation. This process is repeated until a result of the desired accuracy is reached. In order to achieve a low-latency divide operation, the number of iterations needed to achieve the final result must be small One means to decrease the number of iterations m the division operation is to mcrease the accuracy of the starting approximation. The more accurately the first approximation is determined, then, the more quickly the division may be performed. Starting approximations for floatmg-pomt operations such as the reciprocal function are typically obtamed through the use of a look-up table A look-up table is a read-only memory (ROM) which stores a predetermmed output value for each of a number of regions withm a given mput range. For floatmg-pomt functions such as the division operation, the look-up table is located withm the microprocessor's floatmg-pomt unit. An mput range for a floatmg-pomt function is typically bounded by a smgle bmade of floatmg pomt values (a "binade" refers to a range of numbers between consecutive powers of 2). Input ranges for other floatmg-pomt functions, however, may span more than one bmade. Because a single output value is assigned for each region withm a function's input range, some amount of error is inherently introduced mto the result provided by the table look-up operation. One means of reducing this error is to mcrease the number of entnes m the look-up table. This limits the error in any given entry by decreasing the range of mput arguments. Often times, however, the number of entnes required to achieve a satisfactory degree of accuracy m this manner is prohibitively large. Large tables have the unfortunate properties of occupymg too much space and slowmg down the table look-up (large tables take longer to mdex mto than relatively smaller tables) In order to decrease table size while still mamtammg accuracy, "bipartite" look-up tables are utilized. Bipartite look-up tables actually mclude two separate tables: a base value table and a difference value table. The base ta"ble mcludes function output values (or "nodes") for vanous regions of the mput range. The values m the difference table are then used to calculate function output values located between nodes m the base table. This calculation may be performed by linear interpolation or vanous other techniques. Dependmg on the slope of the function for which the bipartite look-up table is bemg constructed, table storage requirements may be dramatically reduced while mamtammg a high level of accuracy. If the function changes slowly, for example, the number of bits required for difference table entnes is much less than the number of bits m the base table entnes. This allows the bipartite table to be implemented with fewer bits than a comparable naive table (one which does not employ interpolation). Pnor art bipartite look-up tables provide output values havmg a minimal amount of maximum relative error over a given mput interval This use of relative error to measure the accuracy of the look-up table output values is questionable, however, because of a problem known as "wobbling precision". Wobblmg precision refers to the fact that a difference m the least significant bit of an mput value to the look-up table has twice the relative error at the end of a binade than it has at the start of the bmade. A look-up table constructed m this manner is thus not as accurate as possible. It would therefore be desirable to have a bipartite look-up table having output values with improved accuracy. As described above, increasmg the efficiency of the reciprocal and reciprocal square root functions may lead to increased floating-point performance (and thus, mcreased graphics processmg performance). While pnor art systems have implemented a smgle function (such as the reciprocal function) usmg a look-up table, this does not take advantage of the potential savings of optimizing both the reciprocal and reciprocal square root functions usmg look-up tables. This potential performance gam is outweighed by additional overhead required by the separate look-up table. It would therefore be desirable to have a multi-function look-up table which implements both the reciprocal and reciprocal square root functions with minimal overhead It would further be desirable for the multi-function look-up table to be a bipartite look-up table SUMMARY OF THE INVENTION Add/Subtract Pipelme The problems outlmed above are in large part solved by an execution unit in accordance with the present invention In one embodiment, an execution unit is provided which is usable to perform effective addition or subtraction upon a given parr of floatmg point mput values The execution unit mcludes an add/subtract pipeline having a far data path and a close data path each coupled to receive the given pair of floatmg point mput values The far data path is configured to perform effective addition as well as effective subtraction upon operands having an absolute exponent difference greater than one The close data path, on the other hand, is configured to perform effective subtraction upon operands having an absolute exponent difference less than or equal to one The add/subtract pipelme further mcludes a result multiplexer unit coupled to receive a result from both the far data path and the close data path A final output of the result multiplexer unit is selected from the far path result and the close path result accordmg to the actual calculated absolute exponent difference value In one embodiment, the far data path mcludes a pair of πght shift units coupled to receive mantissa portions of each of the given parr of floatmg pomt mput values The πght shift units each receive a shift amount from a corresponding exponent difference unit The first nght shift unit conveys a shift amount equal to the second exponent value minus the first exponent value, while the second πght shift unit conveys a shift amount equal to the first exponent value minus the second exponent value The outputs of the πght shift units are then conveyed to a multiplexer-mverter unit, which also receives unshifted versions of the mantissa portions of each of the given parr of floating point input values The multiplexer-mverter unit is configured to select one of the unshifted mantissa portions and one of the shifted mantissa portions to be conveyed as mputs to an adder unit The adder mputs conveyed by the multiplexer-mverter unit are aligned m order to facilitate the addition operation The multiplexer-mverter unit is further configured to invert the second adder mput if the effective operation to be performed is subtraction The adder unit is configured to add the first and second adder inputs, thereby generating first and second adder outputs The first adder output is equal to the sum of the two inputs, while the second adder output is equal to the first adder output plus one One of the two adder outputs is selected accordmg to a far path selection signal generated by a far path selection unit The far path selection unit is configured to generate a plurality of preliminary far path selection signals Each of these preliminary far path selection signals corresponds to a different possible normalization of the first adder output For example, one of the preliminary far path selection signals corresponds to a prediction that the first adder output is properly normalized Another preliminary far path selection signal corresponds to a prediction that the first adder output is not normalized, while still another select signal indicates that said first adder output has an overflow bit set One of these prelimmary far path selection signals is selected to be conveyed as the final far path selection signal based on which of these predictions actually occurs The far data path further mcludes a multiplexer-shift unit configured to receive the first and second adder outputs as well as the final far path selection signal The appropriate adder output is selected, and a one- bit left or nght shift may also be performed to properly normalize the result In the case of a left shift, a guard bit previously shifted out of one of the mantissa values by a right shift unit may be shifted back mto the final result The selected value is conveyed as a mantissa portion of the far data path result value The exponent portion of the far path result is calculated by a exponent adjustment unit The exponent adjustment unit is configured to receive the oπgmal larger exponent value along with the amount of shifting required for proper normalization (which may be no shift, a one-bit left shift, or a one-bit nght shift) In contrast to a generic floatmg pomt addition subtraction pipelme, the far data path is optimized to perform effective additions The far data path is additionally optimized to perform effective subtractions on operands havmg an absolute exponent difference greater than one This configuration allows the recomplementation step to be avoided, smce all operations produce positive results Furthermore, smce adder outputs require at most a one-bit shift, only one full-size shifter is needed m the far data path This results m improved floatmg pomt addition and subtraction performance for the far data path In one embodiment, the close data path is coupled to receive mantissa portions of the given parr of floatmg pomt mput values, as well as two least significant bits of each of the exponent values The mantissa values are conveyed to a shift-swap unit, which also receives an exponent difference prediction from an exponent prediction unit The exponent difference prediction is indicative of whether the absolute exponent difference is 0 or 1 It is used to align and swap (if needed) the mput mantissa values for conveyance to a close path adder unit The mantissa values are swapped such that the exponent value associated with the first adder mput is greater than or equal to the exponent value associated with the second adder mput The first adder mput is not guaranteed to be greater than the second adder mput if the exponent values are equal, however The shift- swap unit is also configured to invert the second adder input since the adder unit within the close data path performs subtraction It is further noted that the exponent difference value generated by the exponent prediction unit may be incorrect This is true smce the exponent prediction is based only on a subset of the total number of bits The result produced by the close data path is thus speculative The actual exponent difference calculated m the far data path is used to determine whether the result produced by the close data path is valid The adder unit within the close data path produces a first and second output value The first output value is equal to the first adder input plus the second adder mput, which is effectively equivalent to the first mantissa portion minus the second mantissa portion The second output value, on the other hand, is equal to the first output value plus one Both values are conveyed to a multiplexer-mverter unit A close path selection signal provided by a close path selection unit is usable to select either the first adder output or the second adder output as a prelimmary close path result The selection unit includes a plurality of logic sub-blocks, each of which is configured to generate a prelimmary close path selection signal indicative of either the first adder output value or the second adder output value Each of the prelimmary close path selection signals corresponds to a different predictions scenario For example, a first logic sub-block generates a prelimmary close path select signal for the case in which the exponent values are equal and the first mantissa value is greater than the second mantissa value A second logic sub-block generates a select signal for the case in which the exponent values are equal and the first mantissa value is less than the second mantissa value A third logic sub-block corresponds to the case m which the first exponent value is greater than the second exponent value and the first adder output is not normalized The last sub-block corresponds to the case m which the first exponent value is greater than the second exponent value and the first adder output is normalized Each of the preliminary selection signals is conveyed to a close path selection multiplexer, the output of which is used to select either the first or second adder output as the prelimmary close path subtraction result The output for the close path selection multiplexer is determined by which of the various predicted cases actually occurs Accordmgly, the close path selection multiplexer receives as control signals the exponent prediction value (mdicatmg whether the exponents are equal or not), the sign value of the first adder output (mdicatmg whether a negative result is present), and the MSB of the first adder output (mdicatmg whether the result is properly normalized or not) The sign value and the MSB value are generated concurrently withm both the adder unit and the selection unit This is accomplished usmg a carry cham dπven by C MSB , the carry m signal to the most significant bit position of the adder unit This concurrent generation allows faster selection of either the first or second adder outputs The selection of one of these values effectuates roundmg the close path result to the nearest number (an even number is chosen m the event of a tie) This configuration advantageously eliminates the need for a separate adder unit to perform roundmg If the first adder output is negative, the multiplexer-mverter unit inverts the first adder output to produce the correct result This occurs for the case m which the exponents are equal and the second mantissa value is greater than the first mantissa value In any event, the selected close path prelimmary subtraction result is then conveyed to a left shift unit for normalization The close path prelimmary subtraction result conveyed to the left shift unit is shifted accordmg to a predicted shift amount generated by a shift prediction unit The shift prediction unit mcludes three leading 0/1 detection units The first unit, a leadmg 1 detection unit, generates a first prediction string for the case in which the first exponent value is greater than the second exponent value The second unit, which performs both leadmg 0 and 1 detection, generates a second prediction string for the case m which the first and second exponent values are equal Leadmg 0 and 1 detection is performed because the result may be positive (leadmg 1) or negative (leading 0) Finally, the third unit, a leadmg 1 detection unit, generates a third prediction string for the case in which the second exponent value is greater than the first exponent value The most significant asserted bits withm each of the strings indicates the position of a leadmg 0 or 1 value Each of the three prediction strings are generated concurrently and conveyed to a shift prediction multiplexer The exponent prediction value generated by the exponent prediction unit withm the close data path selects which of the prediction strings is conveyed by the shift prediction multiplexer to a pπority encoder The pπority encoder then converts the selected prediction strmg to a shift amount which is conveyed to the left shift unit withm the close data path The predicted shift amount may in some instances be incorrect by one bit position For such cases, the close path result is left shifted one place during final selection The calculated results of both the far data path and close data path are conveyed to a final result multiplexer, which selects the correct result based upon the calculated actual exponent difference value Within the shift prediction unit, the second leadmg 0/1 detection unit may not be optimized further, smce no assumptions may be made regardmg its mputs. The first and third prediction units, however, may be optimized, smce it is known that the second mantissa to each unit is inverted and shifted one bit πghtward with respect to the first mantissa. This means that the results predicted by the first and third detection units are both positive Hence, only leadmg 1 detection is desired Further optimizations may also be made since it is known that subtraction is being performed Prediction strings may be formed by assignmg a value to each output bit based on the corresponding mputs for that bit position. In standard T-G-Z notation, a T output value represents mput values 10 or 01, a G output value represents mput values 11, and a Z output value represents output values 00. A leadmg 1 may thus be detected whenever the pattern T'GZ * stops matchmg m the generated prediction strmg. The two leadmg 1 detection units withm the shift prediction unit of the close data path may optimized over pnor art designs by recognizing that the MSB of both mput operands is 1. (The MSB of the first operand is a 1 since it is normalized, and the MSB of the second operand is also a 1 smce the second adder operand is nght shifted one place then inverted). This corresponds to an output value of G m the MSB of the prediction strmg With a G in the initial position of the prediction strmg, it may be recognized that the strmg stops matchmg whenever Z' (the complement of Z) is found. This condition is realized whenever at least one of the mputs m a given bit position is set. The optimized leadmg 1 detection unit mcludes a parr of mput registers and an output register for storing the generated prediction strmg. The first mput register is coupled to receive the first (greater) mantissa value, while the second mput register is coupled to receive an inverted version of the second (lesser) mantissa value. The leading 1 detection unit further mcludes a plurality of logic gates coupled to receive bits from each of the mput registers. Each logic gate generates a bit for the final prediction strmg based on whether one of the inputs is set. The most significant asserted bit m the output prediction strmg mdicates the position of the leadmg 1 bit. The add subtract pipelme may also be configured to perform floatmg pomt-to-integer and lnteger-to- floating point conversions In one embodiment, the far data path may be used to perform floating point-to- mteger conversions, while the close data path performs integer-to-floating point conversions. Both data paths are configured to be as wide as the width of the larger format. In order to perform floating pomt-to-integer conversions withm the far data path, a shift amount is generated from the maximum integer exponent value and the exponent value of the floating point number to be converted. The floatmg pomt mantissa to be converted is then right shifted by the calculated shift amount and conveyed to the multiplexer-inverter unit. The multiplexer-mverter unit conveys the converted mantissa value to the adder unit as the second adder input. The first adder mput is set to zero. As with standard far path operation, the adder unit produces two output values, sum and sum+1 These values are conveyed to the multiplexer-shift unit, where the first adder output (sum) is selected by the far path selection signal. The far path selection unit is configured to select the sum output of the adder unit in response to receiving an indication that a floating pomt-to-integer conversion is being performed. The floatmg point number being converted may greater than the maximum representable mteger (or less than the minimum representable integer) Accordingly, compaπsons are performed to determine whether overflow or underflow has occurred. If either condition is present, the mteger result is clamped at the maximum or minimum value. In order to perform integer-to-floating point conversions withm the close data path, a zero value is utilized as the first operand, while the second operand is the mteger value to be converted The second operand is inverted (smce close path performs subtraction) and conveyed along with the zero value to the adder unit The adder unit, as m standard close path operations, produces two outputs, sum and sum+1 If the mput mteger value is positive, the output of the adder unit is negative Accordmgly, the sum output is chosen by the selection unit as the prelimmary close path result. This output is then inverted in the multiplexer-mverter unit to produce the correct result. If, on the other hand, the mput mteger value is negative, the output of the adder unit is positive. The sum+1 output is thus chosen as the preliminary close path result, and the sign of the resulting floatmg pomt number is denoted as bemg negative. The prelimmary close path result is then conveyed to the left shift unit for normalization, which is performed in accordance with a predicted shift amount conveyed from the shift prediction unit For uiteger-to- floating pomt conversion, the prediction string of the second prediction unit (equal exponents) is used. The zero operand and an inverted version of the mteger value are conveyed as mputs to the second prediction umt. The shift amount generated by the shift prediction unit is usable to left align the prelimmary close path result (with a possible one-bit correction needed). With alignment performed, the number bits m the floating pomt mantissa may thus be routed from the output of the left shift unit to form the mantissa portion of the close path result. The exponent portion of the close path result is generated by an exponent adjustment unit. The exponent adjustment unit is configured to subtract the predicted shift amount from the maximum exponent possible m the mteger format The result (which may also be off by 1) becomes the exponent portion of the close path result. If the dynamic range of the floating pomt format is greater than the maximum representable mteger value, overflows do not occur The execution unit may also be configured to include a plurality of add subtract pipelines each havmg a far and close data path. In this manner, vectored instructions may be performed which execute the same operations on multiple sets of operands This is particularly useful for applications such as graphics in which similar operations are performed repeatedly on large sets of data. In addition to performing vectored add and subtract operations, the execution unit may also be configured to perform vectored floatmg pomt-to-integer and integer-to-floatmg point instructions as descnbed above The execution unit may still further be configured to perform additional vectored aπthmetic operations such as reverse subtract and accumulate functions by appropπate multiplexing of input values to the far and close data paths. Other vectored operations such as extreme value functions and companson operations may be implemented through appropriate multiplexing of output values Multifunction Bipartite Look-Up Table The related problems concerning lookup tables outlmed above are in large part solved by a method for generating entnes for a bipartite look-up table which includes a base table portion and a difference table portion In one embodiment, these entries are usable to form output values for a given mathematical function (denoted as f(x)) in response to receiving corresponding input values (x) withm a predetermmed mput range For example, the bipartite look-up table may be used to implement the reciprocal function or the reciprocal square root function, both of which are useful for performing 3-D graphics operations The method first compnses partitioning the input range of the function mto intervals, subrntervals, and sub-submtervals This first mcludes dividing the predetermmed mput range mto a predetermmed number (I) of intervals Next, the I intervals are each divided mto J subrntervals, resultmg m I*J subrntervals for the input range Finally, each of the I*J subrntervals is divided mto K sub-submtervals, for a total of I*J*K sub- submtervals over the mput range The method next mcludes generatmg K difference table entnes for each mterval in the predetermmed input range Each of the K difference table entnes for a given interval corresponds to a particular group of sub- submtervals withm the given mterval In one embodiment, this particular group of sub-subintervals includes one sub-subinterval per submterval of the given mterval Additionally, each sub-submterval m the particular group has the same relative position withm its respective submterval For example, one of the K difference table entnes for the given interval may correspond to a first group of sub-submtervals wherem each sub- submterval is the last sub-submterval withm its respective submterval In order to generate a first difference table entry for a selected mterval (M), a group of sub-submtervals (N) withm mterval M is selected to coπespond to the first entry The calculation of the first entry then begins with a current submterval (P), which is bounded by mput values A and B A midpoint XI is calculated for current submterval P such that f(A)-f(Xl)=f(Xl)-f(B) (By calculating the midpoint in this way, maximum possible absolute error is minimized for all mput values withm the sub-submterval) Next, a midpoint X2 is computed m a similar fashion for a predetermmed reference sub-submterval withm current submterval P (The reference sub-submterval refers to the sub-subinterval withm each submterval that corresponds to the base table entry) A difference value, f(Xl)-f(X2), is then computed for current submterval P In this manner, a difference value is computed for each sub-submterval m group N A running total is mamtamed of each of these difference values The final total is then divided by J, the number of subrntervals m the selected intervals, in order to generate the difference value average for interval M, sub-submterval group N In one embodiment, the difference value average is converted mto an mteger value before being stored to the difference table portion of the bipartite look-up table The above-described steps are usable to calculate a single difference table entry for interval M In order to calculate the remaming difference table entries for the selected interval, each remaining group of sub- submtervals is selected m turn, and a corresponding difference value average is computed In this manner, the additional K-l difference table entries may be generated for mterval M Difference table entnes for any additional intervals in the predetermined input range are calculated in a similar manner The method next includes generatmg J base table entnes for each interval in the predetermmed mput range Each of the J base table entnes for a given mterval coπesponds to a particular submterval withm the given interval For example, one of the J base table entries for the given interval may correspond to the first submterval of the given mterval In a similar manner to the difference table computations, a particular mterval (M) of the predetermmed mput range is selected for which to compute the J base table entries Next, a submterval of mterval M is chosen as a currently selected submterval P Typically, the first submterval is initially chosen as submterval P The method then includes calculating an initial base value, B, where B=f(X2) (As stated above, X2 is the midpomt of the reference sub-submterval of submterval P of mterval M) Subsequently, a difference value, D, is computed, where D=f(X3) (X3 is the midpomt of the sub-submterval withm submterval P which is furthest from the reference sub-subinterval For example, if the reference sub-submterval is the last sub- submterval in submterval P, X3 is computed for the first sub-submterval m P) The actual maximum midpomt difference for submterval P is given by D-B A reference is then made to the previously computed difference table entry for the sub-submterval (or, more appropnately, the sub- submterval group) withm mterval M which corresponds to the sub-submterval for which D is computed Smce this value is computed by difference averaging as descnbed above, the difference average differs from the quantity D-B The difference of the actual difference value and the average difference value is the maximum eπor for submterval P An adjust value is then computed as a fraction of this maximum error value (In one embodiment, the adjust value is half of the maximum error value m order to evenly distnbute the error over the entire submterval) The final base value is calculated by addmg the adjust value (which may be positive or negative) to the initial base value B In one embodiment, this final base value may be converted to an mteger for storage to the base table portion of the bipartite look-up table The steps descnbed above are repeated for the remammg subrntervals in the selected interval, as well as for the subrntervals of the remammg mtervals of the predetermmed input range In one embodiment, the output values of the bipartite look-up table are simply the sum of selected base and difference table entnes If these entnes are calculated as descnbed above, the resultant output values of the table will have a minimal amount of possible absolute eπor Additionally, this minimized absolute error is achieved withm a bipartite table configuration, which allows reduced storage requirements as compared to a naive table of similar accuracy Furthermore, m an embodiment m which the base and difference values are added to generate the table outputs, this allows the interpolation to be implemented with only the cost of a simple addition This mcreases the speed of the table look-up operation, m contrast to pnor art systems which often require lengthy multiply-add or multiply instructions as part of the interpolation process In one embodiment, a multi-function look-up table is provided for determining output values for a first mathematical function and a second mathematical function These output values correspond to input values which fall within predetermmed input ranges which are divided mto mtervals The mtervals are in turn further divided mto subrntervals, with each of the resulting subrntervals subdivided into sub-submtervals In one embodiment, generated output values have minimized possible absolute error In one embodiment, the multi-function look-up table is a bipartite look-up table including a first plurality of storage locations and a second plurality of storage locations These first plurality of storage locations store base values for the first and second mathematical functions, respectively Each base value is an output value (for either the first or second function) corresponding to an mput region which includes the lookup table input value In one embodiment, each base value m the first plurality of storage locations corresponds to one of the subrntervals m the predetermmed mput ranges The second plurality of storage locations, on the other hand, store difference values for both the first and second mathematical functions. These difference values are used for lmear interpolation m conjunction with a coπesponding base value. In one embodiment, each of the difference values coπesponds to one of a group of sub-subintervals in the predetermmed ranges. The selected group of sub-submtervals mcludes one particular sub-submterval which mcludes the look-up table mput value. The multi-function look-up table further mcludes an address control unit coupled to receive a first set of mput signals. This first set of mput signals mcludes a first mput value and a signal which mdicates whether an output value is to be generated for the first or second mathematical function. The address control unit is configured to generate a first address value from the first set of mput signals. This first address value is m turn conveyed to the first plurality of storage locations and the second plurality of storage locations. In response to receiving the first address value, the first plurality of storage locations is configured to output a first base value. Likewise, the second plurality of storage locations is configured to output a first difference value m response to receivmg the first address value The multi-function look-up table finally mcludes an output unit coupled to receive the first base value from the first plurality of storage locations and the first difference value from the second plurality of storage locations The output umt is additionally configured to generate the first output value from the first base value and the first difference value. In one embodiment, the output unit generates the first output value by adding the first difference value to the first base value. By employmg a multi-function look-up table, a microprocessor may enhance the performance of both the reciprocal and reciprocal square root functions. Floatmg-pomt and graphics processmg performance is thus advantageously enhanced. BRIEF DESCRIPTION OF THE DRAWINGS Other objects and advantages of the invenfton will become apparent upon reading the following detailed descnption and upon reference to the accompanying drawmgs m which Fig. 1 depicts the format of a single precision floatmg pomt number accordmg to IEEE standard 754 Fig. 2 depicts a pnor art floatmg pomt addition pipelme; Fig. 3 depicts a pnor art floatmg point addition pipelme havmg far and close data paths, Fig. 4 is a block diagram of a microprocessor accordmg to one embodiment of the present invention; Fig 5 is a block diagram of an execution unit havmg an add/subtract pipelme accordmg to one embodiment of the present invention; Fig 6 is a block diagram of one embodiment of a far data path withm the add subtract pipelme of Fig. 5, Fig 7 is a block diagram of one embodiment of a multiplexer- inverter unit withm the far data path of Fig. 6; Fig. 8 is a block diagram of one embodiment of an adder unit withm the far data path of Fig. 6, Fig 9 is a block diagram of one embodiment of a selection unit withm the far data path of Fig. 6, Figs 10A-H are examples of addition and subtraction performed withm the far data path of Fig 6, Fig 11 is a block diagram of one embodiment of a multiplexer-shift unit within the c data path of Fig. 6, Fig. 12 is a block diagram of one embodiment of a close data path withm the add/subtract pipelme of Fig. 5, Fig. 13 is a block diagram of one embodiment of a shift-swap unit withm the close data path of Fig. 12, Fig 14 is a block diagram of one embodiment of an adder unit withm the close data path of Fig. 12; Fig. 15 is a block diagram of one embodiment of a selection unit 730 withm the close data path of Fig. 12; Figs 16A-G are examples of subtraction performed withm the close data path of Fig. 12, Fig 17 is a block diagram of one embodiment of a multiplexer- inverter unit 740 withm the close data path of Fig. 12, Fig. 18 is a block diagram of one embodiment of a left shift unit 750 withm the close data path of Fig. 12, Fig. 19 is a block diagram of one embodiment of a result multiplexer umt 250 withm the close data path of Fig. 12, Fig. 20 is a block diagram of a pnor art leadmg 0/1 prediction unit 1400, Fig. 21 is a block diagram of a pnor art TGZ generation unit withm prediction unit 1400 of Fig. 20, Figs. 22A-C are examples of how T-G-Z prediction strings may be utilized to perform leadmg 0/1 prediction; Fig. 23 is a logic diagram of a prediction unit configured to form both leadmg 0 and 1 prediction strings; Fig. 24 is a pnor art simplification of a TGZ generation unit for operands A and B, where A > B; Fig. 25 illustrates the derivation of a simplified leadmg 1 prediction units in which exponent E A of a first operand is one greater than exponent E B of a second operand, Fig 26 is a block diagram of one embodiment of an improved leadmg 1 prediction unit for which E A =E B +1; Figs. 27A-B depict floatmg pomt numbers and converted mteger equivalents accordmg to one embodiment of the present mvention; Fig. 28 is a block diagram of one embodiment of a far data path 2300 which is configured to perform floating pomt to integer (f2ι) conversions; Fig 29 is a block diagram of one embodiment of a multiplexer inverter unit 2330 withm far data path 2300 of Fig. 28, Fig 30 is a block diagram of one embodiment of a result multiplexer unit 2500 withm far data path 2300 of Fig. 28, Figs 31A-B depict mteger numbers and converted floatmg pomt equivalents accordmg to one embodiment of the present invention, Fig 32 is a block diagram of one embodiment of a close data path 2600 which is configured to perform lnteger-to-floating point (ι2f) conversions, Fig. 33 is a block diagram of one embodiment of a shift-swap unit 2610 withm close data path 2600 of Fig. 32, Fig. 34 is a block diagram of one embodiment of a multiplexer-mverter unit 2640 withm close data path 2600 of Fig. 32; Fig. 35 is a block diagram of one embodiment of an exponent withm close data path 2600 of Fig. 32, Fig. 36 is a block diagram of one embodiment of an execution umt withm microprocessor 100 which mcludes a plurality of add/subtract pipelines; Fig. 37A depicts the format of a vectored floatmg pomt addition instruction accordmg to one embodiment of the invention; Fig. 37B depicts pseudocode for the vectored floatmg pomt addition instruction of Fig. 37A; Fig. 38A depicts the format of a vectored floating pomt subtraction instruction accordmg to one embodiment of the invention; Fig. 38B depicts pseudocode for the vectored floatmg pomt subtraction instruction of Fig. 38A; Fig. 39A depicts the format of a vectored floatmg pomt-to-mteger conversion instruction accordmg to one embodiment of the invention; Fig. 39B depicts pseudocode for the vectored floatmg pomt-to-mteger conversion instruction of Fig 39A; Fig. 39C is a table listmg output values for vanous mputs to the vectored floatmg pomt-to-mteger conversion instruction of Fig. 39 A; Fig. 40A depicts the format of a vectored floatmg pomt-to-mteger conversion instruction accordmg to an alternate embodiment of the invention; Fig. 40B depicts pseudocode for the vectored floatmg pomt-to-mteger conversion instruction of Fig. 40A; Fig. 40C is a table listmg output values for vanous inputs to the vectored floating pomt-to-integer conversion instruction of Fig. 40A; Fig. 41 A depicts the format of a vectored integer-to-floating pomt conversion instruction accordmg to one embodiment of the invention; Fig. 4 IB depicts pseudocode for the vectored mteger-to-floating pomt conversion instruction of Fig. 41A; Fig. 42A depicts the format of a vectored lnteger-to-floating point conversion instruction accordmg to an alternate embodiment of the invention; Fig. 42B depicts pseudocode for the vectored integer-to-floating point conversion instruction of Fig. 42A, Fig. 43A depicts the format of a vectored floating pomt accumulate instruction accordmg to one embodiment of the invention; Fig. 43B depicts pseudocode for the vectored floatmg pomt accumulate instruction of Fig. 43 A, Fig 44A depicts the format of a vectored floating point reverse subtract instruction according to one embodiment of the invention; Fig 44B depicts pseudocode for the vectored floatmg point reverse subtract instruction of Fig. 44A, Fig. 45A depicts the format of a vectored floatmg point maximum value instruction accordmg to one embodiment of the mvention, Fig. 45B depicts pseudocode for the vectored floatmg pomt maximum value instruction of Fig. 45 A; Fig. 45 C is a table listmg output values for various mputs to the vectored floatmg point maximum value instruction of Fig. 45 A; Fig. 46A depicts the format of a vectored floatmg pomt minimum value instruction accordmg to one embodiment of the mvention; Fig. 46B depicts pseudocode for the vectored floatmg point minimum value instruction of Fig. 46A; Fig. 46C is a table listmg output values for various inputs to the vectored floatmg point minimum value instruction of Fig. 46A; Fig. 47A depicts the format of a vectored floatmg pomt equality compaπson instruction accordmg to one embodiment of the mvention; Fig. 47B depicts pseudocode for the vectored floating pomt equality companson instruction of Fig. 47A; Fig. 47C is a table listmg output values for vanous mputs to the vectored floatmg pomt equality companson instruction of Fig. 47A; Fig. 48A depicts the format of a vectored floating pomt greater than compaπson instruction accordmg to one embodiment of the mvention; Fig. 48B depicts pseudocode for the vectored floatmg pomt greater than compaπson instruction of Fig. 48A; Fig. 48C is a table listmg output values for various mputs to the vectored floatmg pomt greater than comparison instruction of Fig. 48 A; Fig. 49A depicts the format of a vectored floatmg pomt greater than or equal to companson instruction accordmg to one embodiment of the invention; Fig. 49B depicts pseudocode for the vectored floating pomt greater than or equal to compaπson instruction of Fig. 49A; Fig. 49C is a table listing output values for various mputs to the vectored floating point greater than or equal to compaπson instruction of Fig. 49A; Fig. 50 is a block diagram of one embodiment of an execution unit 136C/D accordmg to one embodiment of the invention which is configured to executed the instructions of Figs. 37-49; and Fig. 51 is a block diagram of one embodiment of a computer system which includes microprocessor 100. Fig. 52 is a block diagram of a microprocessor which configured accordmg to one embodiment of the present mvention; Fig. 53 is a graph depictmg a portion of a function f(x) which is partitioned for use with a prior art naive look-up table; Fig 54 is a prior art naive look-up table usable in conjunction with the function partitioned according to Fιg.52, Fig 55 is a graph depictmg a portion of a function f(x) which is partitioned for use with a prior art bipartite look-up table, Fig 56 is a prior art bipartite look-up table usable in conjunction with the function partitioned according to Fig 55, Fig 57 is a graph depicting a portion of a function f(x) which is partitioned for use with a bipartite look-up table accordmg to one embodiment of the present mvention, Fig 58 is a bipartite look-up table usable m conjunction with the function partitioned accordmg to Fig 57, Fig 59 depicts one format for an mput value to a bipartite look-up m accordance with one embodiment of the present mvention, Fig 60A illustrates a look-up table input value accordmg to the format of Fig 59 m one embodiment of the present mvention, Fig 60B depicts the mantissa portion of a look-up table mput value for the reciprocal function, Fig 60C depicts a base table index for a bipartite look-up table for the reciprocal function, accordmg to one embodiment of the present invention, Fig 60D depicts a difference table index for a bipartite look-up table for the reciprocal function, according to one embodiment of the present mvention, Fig 61 A depicts the mantissa portion of a look-up table mput value for the reciprocal square root function, Fig 6 IB depicts a base table index for a bipartite look-up table for the reciprocal square root function, accordmg to one embodiment of the present mvention, Fig 61C depicts a difference table index for a bipartite look-up table for the reciprocal square root function, accordmg to one embodiment of the present mvention, Fig 62 is a bipartite look-up table for the reciprocal and reciprocal square root functions accordmg to one embodiment of the present mvention, Fig 63 is one embodiment of an address control unit withm the bipartite look-up table of Fig 62, Fig 64A is a graph depictmg a pnor art midpomt calculation for a bipartite look-up table, Fig 64B is a graph depicting a midpoint calculation for a bipartite look-up table accordmg to one embodiment of the present invention, Fig 65A is a flowchart depicting a method for computation of difference table entries for a bipartite look-up table according to one embodiment of the present invention, Fig 65B is a graph depicting difference value averagmg over a portion of a function f(x) partitioned for use with a bipartite look-up table according to one embodiment of the present invention, Figs 66A-B are graphs comparing table output values for a portion of a function f(x) to computed midpomt values for the function portion, Figs 66C-D are graphs comparmg table outputs with adjusted base values for a portion of a function f(x) to computed midpomt values for the function portion, and Fig. 67 is a flowchart depictmg a method for computation of base table entnes for a bipartite look-up table accordmg to one embodiment of the present mvention While the mvention is susceptible to vanous modifications and alternative forms, specific embodiments thereof are shown by way of example m the drawings and will herein be described in detail It should be understood, however, that the drawings and detailed description thereto are not intended to limit the invention to the particular form disclosed, but on the contrary, the mtention is to cover all modifications, equivalents and alternatives falling withm the spirit and scope of the present invention as defined by the appended claims DETAILED DESCRIPTION OF THE INVENTION Turning now to Fig. 4, a block diagram of one embodiment of a microprocessor 100 is shown. As depicted, microprocessor 100 mcludes a predecode logic block 112 coupled to an instruction cache 114 and a predecode cache 115. Caches 114 and 115 also mclude an instruction TLB 116 A cache controller 118 is coupled to predecode block 112, instruction cache 114, and predecode cache 115 Controller 118 is additionally coupled to a bus interface unit 124, a level-one data cache 126 (which mcludes a data TLB 128), and an L2 cache 140. Microprocessor 100 further includes a decode unit 120, which receives instructions from instruction cache 114 and predecode data from cache 115. This information is forwarded to execution engme 130 m accordance with mput received from a branch logic unit 122. Execution engme 130 mcludes a scheduler buffer 132 coupled to receive mput from decode unit 120. Scheduler buffer 132 is coupled to convey decoded instructions to a plurality of execution units 136A-E m accordance with mput received from an instruction control unit 134 Execution units 136A-E mclude a load unit 136A, a store unit 136B, an integer/multimedia X unit 136C, an integer/multimedia Y unit 136D, and a floating point unit 136E. Load unit 136A receives input from data cache 126, while store unit 136B interfaces with data cache 126 via a store queue 138 Blocks referred to herem with a reference number followed by a letter will be collectively referred to by the reference number alone. For example, execution units 136A-E will be collectively refeπed to as execution units 136 In one embodiment, instruction cache 114 is organized as sectors, with each sector mcludmg two 32- byte cache lines.. The two cache lines of a sector share a common tag but have separate state bits that track the stams of the Ime. Accordmgly, two forms of cache misses (and associated cache fills) may take place- sector replacement and cache Ime replacement. In the case of sector replacement, the miss is due to a tag mismatch in instruction cache 114, with the required cache line being supplied by external memory via bus interface unit 124. The cache line withm the sector that is not needed is then marked invalid. In the case of a cache line replacement, the tag matches the requested address, but the Ime is marked as invalid The required cache Ime is supplied by external memory, but, unlike the sector replacement case, the cache Ime within the sector that was not requested remams m the same state. In alternate embodiments, other organizations for instruction cache 114 may be utilized, as well as various replacement policies Microprocessor 100 performs prefetching only in the case of sector replacements in one embodiment Duπng sector replacement, the required cache line is filled If this required cache line is in the first half of the sector, the other cache line in the sector is prefetched If this required cache Ime is in the second half of the sector, no prefetchmg is performed. It is noted that other prefetchmg methodologies may be employed m different embodiments of microprocessor 100. When cache lmes of mstruction data are retπeved from external memory by bus interface unit 124, this data is conveyed to predecode logic block 112. In one embodiment, the instructions processed by microprocessor 100 and stored m cache 114 are vaπable-length (e.g., the x86 mstruction set). Because decode of variable-length instructions is particularly complex, predecode logic 112 is configured to provide additional information to be stored in predecode cache 115 to aid during decode In one embodiment, predecode logic 112 generates predecode bits for each byte m mstruction cache 114 which mdicate the number of bytes to the start of the next vanable-length instruction. These predecode bits are stored in predecode cache 115 and are passed to decode unit 120 when mstruction bytes are requested from cache 114. Instruction cache 114 is implemented as a 32Kbyte, two-way set associative, writeback cache m one embodiment of microprocessor 100. The cache Ime size is 32 bytes in this embodiment. Cache 114 also includes a TLB 116, which mcludes 64 entnes used to translate lmear addresses to physical addresses. Many other vaπations of mstruction cache 114 and TLB 116 are possible m other embodiments. Instruction fetch addresses are supplied by cache controller 118 to mstruction cache 114. In one embodiment, up to 16 bytes per clock cycle may be fetched from cache 114. The fetched information is placed mto an rnstrucfton buffer that feeds mto decode unit 120. In one embodiment of microprocessor 100, fetching may occur along a smgle execution stream with seven outstandmg branches taken. In one embodiment, the instruction fetch logic within cache controller 118 is capable of retπevmg any 16 contiguous mstruction bytes withm a 32-byte boundary of cache 114. There is no additional penalty when the 16 bytes cross a cache Ime boundary Instructions are loaded into the mstruction buffer as the current instructions are consumed by decode unit 120. (Predecode data from cache 115 is also loaded mto the instruction buffer as well). Other configurations of cache controller 118 are possible m other embodiments. Decode logic 120 is configured to decode multiple instructions per processor clock cycle. In one embodiment, decode unit 120 accepts instruction and predecode bytes from the mstruction buffer (in x86 format), locates actual instruction boundaries, and generates coπespondrng "RISC ops". RISC ops are fixed- format internal instructions, most of which are executable by microprocessor 100 in a single clock cycle. RISC ops are combmed to form every function of the x86 mstruction set m one embodiment of microprocessor 100. Microprocessor 100 uses a combination of decoders to convert x86 instructions mto RISC ops. The hardware includes three sets of decoders: two parallel short decoders, one long decoder, and one vectormg decoder. The parallel short decoders translate the most commonly-used x86 instructions (moves, shifts, branches, etc.) mto zero, one, or two RISC ops each. The short decoders only operate on x86 instructions that are up to seven bytes long. In addition, they are configured to decode up to two x86 instructions per clock cycle. The commonly-used x86 instructions which are greater than seven bytes long, as well as those semi- commonly-used instructions are up to seven bytes long, are handled by the long decoder. The long decoder in decode unit 120 only performs one decode per clock cycle, and generates up to four RISC ops. All other translations (complex instructions, interrupts, etc.) are handled by a combmation of the vector decoder and RISC op sequences fetched from an on-chip ROM. For complex operations, the vector decoder logic provides the first set of RISC ops and an initial address to a sequence of further RISC ops. The RISC ops fetched from the on-chip ROM are of the same type that are generated by the hardware decoders. In one embodiment, decode unit 120 generates a group of four RISC ops each clock cycle. For clock cycles m which four RISC ops cannot be generated, decode unit 120 places RISC NOP operations m the remammg slots of the grouping These groupmgs of RISC ops (and possible NOPs) are then conveyed to scheduler buffer 132 It is noted that m another embodiment, an mstruction format other than x86 may be stored m mstruction cache 114 and subsequently decoded by decode unit 120. Instruction control unit 134 contains the logic necessary to manage out-of-order execution of instructions stored m scheduler buffer 132 Instruction control unit 134 also manages data forwardmg, register renaming, simultaneous issue and retirement of RISC ops, and speculative execution In one embodiment, scheduler buffer 132 holds up to 24 RISC ops at one time, equatmg to a maximum of 12 x86 mstructions. When possible, instruction control unit 134 may simultaneously issue (from buffer 132) a RISC op to any available one of execution units 136. In total, control unit 134 may issue up to six and retire up to four RISC ops per clock cycle m one embodiment In one embodiment, microprocessor 10 mcludes five execution units (136A-E). Load unit 136A and store unit 136B are two-staged pipelined designs. Store unit 136B performs data memory and register wπtes which are available for loadmg after one clock cycle. Load umt 136A performs memory reads. The data from these reads is available after two clock cycles Load and store units are possible m other embodiments with varying latencies Execution unit 136C is configured, m one embodiment, to perform all fixed point ALU operations, as well as multiplies, divides (both signed and unsigned), shifts, and rotates. Execution unit 136D, m contrast, is configured to perform basic word and double word ALU operations (ADD, AND, CMP, etc.) Additionally, units 136C-D are configured to accelerate performance of software wntten usmg multimedia mstructions. Applications that can take advantage of multimedia mstructions mclude graphics, video and audio compression and decompression, speech recognition, and telephony. Accordmgly, units 136C-D are configured to execute multimedia mstructions m a single clock cycle in one embodiment Many of these instructions are designed to perform the same operation of multiple sets of data at once (vector processmg) In one embodiment, these multimedia mstructions include both vectored fixed pomt and vectored floatmg point mstructions. Execution unit 136E contams an IEEE 754-compatible floatmg pomt unit designed to accelerate the performance of software which utilizes the x86 instruction set Floatmg point software is typically wntten to manipulate numbers that are either very large or small, require a great deal of precision, or result from complex mathematical operations such as transcendentals Floatmg pomt unit mcludes an adder unit, a multiplier unit, and a divide/square root unit In one embodiment, these low-latency units are configured to execute floating point instructions m as few as two clock cycles Branch resolution unit 135 is separate from branch prediction logic 122 m that it resolves conditional branches such as JCC and LOOP after the branch condition has been evaluated. Branch resolution unit 135 allows efficient speculative execution, enablmg microprocessor 100 to execute mstructions beyond conditional branches before knowing whether the branch prediction was coπect As described above, microprocessor 100 is configured to handle up to seven outstandmg branches m one embodiment Branch prediction logic 122, coupled to decode unit 120, is configured to mcrease the accuracy with which conditional branches are predicted m microprocessor 100 Ten to twenty percent of the mstructions m typical applications mclude conditional branches Branch prediction logic 122 is configured to handle this type of program behavior and its negative effects on mstruction execution, such as stalls due to delayed mstruction fetching In one embodiment, branch prediction logic 122 mcludes an 8192-entry branch history table, a 16- entry by 16 byte branch target cache, and a 16-entry return address stack Branch prediction logic 122 implements a two-level adaptive history algoπthm usmg the branch history table This table stores executed branch information, predicts individual branches, and predicts behavior of groups of branches In one embodiment, the branch history table does not store predicted target addresses m order to save space These addresses are mstead calculated on-the-fly during the decode stage To avoid a clock cycle penalty for a cache fetch when a branch is predicted taken, a branch target cache withm branch logic 122 supplies the first 16 bytes at that address directly to the mstruction buffer (if a hit occurs m the branch target cache) In one embodiment, this branch prediction logic achieves branch prediction rates of over 95% Branch logic 122 also mcludes special circuitry designed to optimize the CALL and RET mstructions This circuitry allows the address of the next instruction following the CALL mstruction m memory to be pushed onto a return address stack When microprocessor 100 encounters a RET mstruction, branch logic 22 pops this address from the return stack and begms fetchmg Like mstruction cache 114, LI data cache 126 is also organized as two-way set associative 32Kbyte storage In one embodiment, data TLB 128 mcludes 128 entnes used to translate lmear to physical addresses Like mstruction cache 114, LI data cache 126 is also sectored Data cache 126 implements a MESI (modified- exclusive-shared-inva d) protocol to track cache Ime status, although other vanations are also possible In order to maximize cache hit rates, microprocessor 100 also includes on-chip L2 cache 140 withm the memory sub-system Turning now to Fig 5, a block diagram of a portion of an execution unit 136C/D is depicted The "C/D" denotes that the execution unit shown in Fig 5 is representative of both execution units 136C and 136D This means of reference is also used below to descπbe other embodiments execution units 136C-D As shown, execution unit 136C/D includes an mput unit 210 which receives an add/subtract indication 202 and operands 204A-B Input unit 210 is coupled an add/subtract pipeline 220, which mcludes a far data path 230 and a close data path 240 Far data path 230 and close data path 240 receive mputs from input unit 210 and generate far path result 232 and close path result 242, respectively, which are conveyed to a result multiplexer unit 250 Far data path 230 also conveys a select signal to multiplexer unit 250 m one embodiment In this embodiment, the select signal is usable to select either far path result 232 or close path result 242 to be conveyed as result value 252, which is the output of add/subtract pipelme 220 Input unit 210 receives the operand data, and conveys sufficient information to far data path 230 and close data path 240 to perform the add or subtract operation In one embodiment, add/subtract mdication 202 is indicative of the operation specified by the opcode of a particular floating pomt arithmetic instruction That is, add subtract mdication 202 coπesponds to the opcode of an mstruction bemg processed by unit 136C/D (a logic 0 may mdicate an add opcode and a logic 1 a subtract opcode m one embodiment) Operands 204 are floatmg pomt numbers havmg sign, exponent, and mantissa portions accordmg to a predetermmed floatmg pomt format (such as IEEE standard 754) If add/subtract mdication 202 corresponds to an opcode add subtract value, mput unit 210 may be configured to make a determination whether effective addition or subtraction is occurring (As descnbed above, an subtract opcode value may effectively be an addition operand dependmg on the signs of operands 204) In one embodiment, mput unit 210 determines whether mputs 202 and 204 represent effective addition or subtraction, and conveys outputs to far data path 230 and close data path 240 In an alternate embodiment, the determination of effective addition or subtraction is made pnor to conveyance to unit 136C/D Add subtract mdication 202 is thus reflective of either effective addition or subtraction, and sign bits of incoming operands 204 are adjusted accordmgly In yet another embodiment, the effective addition/subtraction determination may be made separately within far data path 230 and close data path 240 The format of the outputs of input unit 210 depends upon the format of unit 210 mputs and also the configuration of far data path 230 and close data path 240 In one embodiment, unit 210 conveys the full sign, exponent, and mantissa values (S A , S B , E A , E B , M A , and M B ) of operands 204 to far data path 230, while conveying S A , S B , M A , M B , and two least significant bits of both E A and E B to close data path 240 As will be descnbed the two least significant exponents bits are used for speculative determination of exponent difference (mstead of a full subtract) In other embodiments of add subtract pipelme 220, far data path 230 and close data path 240 may receive mput data of varying formats Far data path 230 is configured to perform addition operations, as well as subtraction operations for operands havmg absolute exponent difference E ώ{( which is greater than 1 Close data path 240, on the other hand, is configured to perform subtraction operations on operands for which E^ <1 As will be descnbed below, close data path 240 mcludes a selection unit which is configured to provide improved performance over prior art pipelines such as pipelines 10 and 30 described above Far data path 230 and close data path 240 generate far path result 232 and close path result 242, respectively, which are both conveyed to result multiplexer unit 250 As shown, far data path also generates a select signal for unit 250, which is usable to select either mput 232 or 242 as result value 252 In alternate embodiments of add/subtract pipelme 220, the select for multiplexer unit 250 may generated differently Turning now to Fig 6, a block diagram of far data path 230 is depicted As shown, far data path 230 receives an add/subtract indication, full exponent values (E A and E B ), and full mantissa values (M A and M B ) from input unit 210 in one embodiment In the embodiment shown, data path 230 also receives sign bits S A and S B , although they are not depicted m Fig 6 for simplicity and clanty Far data path 230 includes exponent difference calculation units 310A-B, which receive mput exponent values E A and E B Units 310 are coupled to nght shift units 314A-B, which receive mantissa values M A and M B , respectively Shift units 314 are also coupled to multiplexer-mverter unit 330 and logic unit 320 (refeπed to as "GRS" logic because unit 320 stores the guard (G), round (R), and sticky (S) bits shifted out m units 314) Multiplexer-mverter unit 330, in response to receiving shifted (316A-B) and unshifted versions of M A and M B , conveys a pair of operands (332A-B) to an adder unit 340 Adder unit 340, m turn, generates a pair of outputs 342A and 342B, which are conveyed to multiplexer-shift unit 360 Adder unit 340 is additionally coupled to a selection unit 350, which generates a select signal for multiplexer-shift unit 360 Selection unit 350 also receives mputs from exponent unit 310 and GRS logic unit 320 m addition to values from adder unit 340. In response to select signal 352 conveyed from selection unit 350, multiplexer shift unit 360 conveys a mantissa value which, when coupled with an adjusted exponent value conveyed from an exponent adjust unit 370, is conveyed as far path result 232 to result multiplexer unit 250 Exponent adjust unit 370 receives the largest mput exponent 309 (which is equal to max(E A , E B )) from an exponent comparator unit 308 coupled to receive E A and E B Exponent 309 is additionally conveyed to close data path 240 for exponent calculations as is descnbed below. As shown in Fig. 6, exponent difference unit 310A is coupled to receive full exponent values E A and E B Unit 310A is configured to compute the difference E B -E A and convey the resultmg shift amount, 312A, to right shift unit 314A Exponent difference unit 310B also receives full exponent values E A and E B , but is configured to compute the difference E A -E B , which is conveyed as shift amount 312B to nght shift unit 314B In this embodiment, unless E A =E B , one of result 312 is negative (and therefore ultimately discarded by pipelme 220) An embodiment is also contemplated in which only one πght shift unit 314 is provided; however, additional multiplexer logic may be needed to convey the proper mantissa value to the smgle shift unit. By providmg two shift units 314, the performance of far data path 230 is mcreased. Shift amount 312A, m one embodiment, is conveyed to a final select generation unit 311, along with add/subtract mdication 202. Unit 311, m turn, generates an exponent difference select signal 313 to be conveyed to result multiplexer unit 250 The signal 313 generated by unit 310 is indicative of either far path result 232 or close path result 242. Signal 313 may thus be used by result multiplexer unit 250 to select either result 232 or result 242 as result value 252. If add subtract indication 202 specifies an add operation, signal 313 is generated to be indicative of far path result 232. Similarly, if add/subtract mdication 202 specifies a subtract operation and E ώff (corresponding to the absolute value of shift amount 312A) is greater than one, signal 313 is also generated to be indicative of far path result 232 Conversely, if add/subtract mdication 202 specifies a subtract operation and E dlfr is 0 or 1, signal 313 is generated to be indicative of close path result 242. In one embodiment, signal 313 may be used to cancel the far path result if E dlfr mdicates result 242. E ώff is also conveyed to selection unit 350 m one embodiment, as will be descnbed below. Right shift units 314A-B generate shift outputs 316A-B, respectively, accordmg to shift amounts 312A-B. These shift outputs are then conveyed to multiplexer-mverter unit 330. Unit 330 is also coupled to receive add subtract indication from mput unit 210 and the sign bit of shift amount 312A. In one embodiment, multiplexer-mverter unit 330 is configured to swap operands 316A and 316B if operand 316B is determined to be greater than operand 316A. This determination may be made m one embodiment from the sign bit of shift amount 312A (or 312B). Additionally, unit 330 is configured to invert the smaller operand if subtraction is indicated by mput unit 210. The outputs of unit 330 are conveyed to adder unit 340 as adder mputs 332A-B. GRS logic unit 320 receives values which are πght-shifted out of units 314A-B. After shift amounts 312 are applied to values in shift units 314, GRS logic unit 320 generates guard, round, and sticky bits coπesponding to the smaller mantissa value As shown, these bit values are forwarded to selection unit 350 for the roundmg computation Adder unit 340 receives adder mputs 332A-B and generates a pair of output values 342A-B. Output 342A corresponds to the sum of input values 332 (sum), while output 342B coπesponds to output 342A plus one (sum+1) Adder unit 340 also conveys a plurality of signals to selection unit 350, which generates and conveys select signal 352 to multiplexer-shift unit 360 Select signal 352 is usable to select either adder output 342A-B to be conveyed as the mantissa portion of far path result 232. By selectmg either sum or sum+1 as the output of multiplexer-shift unit 360, the addition result may effectively be rounded accordmg to the IEEE round-to-nearest mode. In one embodiment, the exponent portion of far path result 232 is generated by exponent adjustment unit 370 Unit 370 generates the adjusted exponent from the oπginal larger exponent value (either E A or E B ) and an mdication of whether the adder output is normalized The output of unit 370 is conveyed along with the output of unit 360 as far path result 232. Turning now to Fig. 7, a block diagram of multiplexer-mverter unit 330 is depicted Unit 330 includes a control unit 331 which receives shift amount 312A from exponent difference calculation unit 310A. Multiplexer-mverter unit 330 also includes a pair of input multiplexers 334A-B. Input multiplexer 334A receives unshifted mantissa values M A and M B , while multiplexer 334B receives shifted outputs 316A-B. In one embodiment, the mputs to multiplexers 334 are configured such that control umt 331 may route a smgle control signal 333 to both multiplexer 334A and 334B. Additionally, the output of multiplexer 334B is inverted by an inverter 336 if a subtract operation is mdicated by signal 202. If a subtract is mdicated, a bit-inverted (one's complement) version of the output of multiplexer 334B is conveyed to adder 340 as adder mput 342B. If an add operation is mdicated by signal 202, inverter 336 is not enabled, and the output of multiplexer 334B is conveyed to adder unit 340 m non-inverted form. Turning now to Fig. 8, a block diagram of one embodiment of adder unit 340 is depicted. Adder unit 340 mcludes adders 400A and 400B, each coupled to receive adder mputs 332A-B. Adder 400A is configured to generate adder output 342 A (sum), while adder 400B is configured to generate adder output 342B (sum+1) As shown, adders 400A and 400B are each coupled to receive the sign and mantissa bits of operands 204 A-B. In one embodiment, adders 400 A and 400B are identical except that adder 400B has a carry m (C LSB ) value of 1, while, for adder 400A, C LSB =0. It is contemplated that adders 400 may be implemented usmg a variety of known adder types. For example, adders 400 may be implemented as ripple-carry adders, carry lookahead adders, carry-select adders, etc. Furthermore, adders 400 may combme features of different adder types In one embodiment, adders 400 compute the upper n/2 bits of their respective results in two different ways: that the carry in from the lower n 2 bits is 0, and that the carry m from the lower n/2 bits is 1. The use of Ling-style pseudo-carry may also be utilized in the lower n/2 bits to further reduce fan-m and gate delay. In yet another embodiment, adder unit 340 may be implemented with ]ust a smgle adder. This may be accomplished by recogmzmg that many of the terms computed m adders 400A-B are shared. Accordmgly, both sum and sum+1 may be produced by a smgle adder Although such an adder is larger (m terms of chip real estate) than either of adders 400, the single adder represents a significant space savings vis-a-vis the two adder configuration of Fig. 8 As will be descnbed below, the most significant bit of the output of adder 400A (S MSB ) is used by selection unit 350 to generate select signal 352 The faster select signal 352 is generated, then, the faster result value 252 can be computed Accordmgly, m the embodiment shown m Fig 8, S MSB is generated m selection unit 350 concurrently with the MSB computation performed in adder 400A To facilitate this operation, A MSB , B MSB , and C MSB (the carry m to adder block 402B which generates S MSB ) are all conveyed to selection unit 350 By conveymg the mputs to adder block 402B to selection unit 350 m parallel, the output of selection unit 350 may be generated more quickly, enhancmg the performance of far data path 230 The two least significant bits of adder output 342A (S LSB+1 and S LSB ) are also conveyed to selection unit 350 In one embodiment, these values are not generated in parallel m unit 350 (m the manner of S MSB ) smce the least significant bits are available relatively early in the addition operation (m contrast to more significant bits such as S MSB ) As noted above, adder 400B operates similarly to adder 400A, except that carry in value 404B is a logical one Since the carry in value (404A) for adder 400A is a logical zero, adder 400B generates a result equal to the output of adder 400A plus one As will be descnbed below, by generatmg the values (sum) and (sum+1) for a given pair of operands, the IEEE round to nearest mode may be effectuated by selectmg one of the two values Turning now to Fig 9, a block diagram of selection unit 350 is shown in one embodiment of far data path 230 The general operation of selection unit 350 is descnbed first, followed by examples of far path computations As shown, selection unit 350 receives a plurality of mputs from adder unit 340 These mputs mclude, m one embodiment, the mputs to adder 400A block 402B (A MSB , B MSB , and C MSB ), the next-to-least significant bit (N) of adder output 342A, the least significant bit (L) of adder output 342B, and the guard (G), round (R), and sticky (S) bits from GRS logic unit 320 A logical-OR of the round and sticky bits, S„ is produced by logic gate 502 Bit S, is used for calculations m which R is not explicitly needed Selection unit 350 also mcludes a selection logic block 510 which mcludes selection sub-blocks 510A-D In response to the mputs received from units 320 and 340, sub-blocks 510A-D generate respective select signals 512A-D Select signals 512 are conveyed to a far path multiplexer 520, which also receives control signals including add/subtract mdication 202, S MSB signal 534, and C s signal 536 S MSB signal 534 is conveyed from a multiplexer 530A, while C s is conveyed from a multiplexer 530B In response to these control signals, multiplexer 520 conveys one of select signals 512 as far path select signal 352 to multiplexer-shift unit 360 As descnbed above, adder unit 340 is configured to generate sum and sum+1 for operands 204A and 240B Selection unit 350 is configured to generate far path select signal 352 such that the sum/sum+1 is a) corrected for one's complement subtraction and b) rounded coπectly accordmg to the IEEE round-to-nearest mode In general, a number generated by one's complement subtraction must have 1 added m at the LSB to produce a correct result Dependmg on the state of the G, R, and S bits, however, such coπection may or may not be needed With respect to rounding, sum+1 is selected m some instances to provide a result which is rounded to the next highest number Dependmg on vanous factors (type of operation, normalization of output 342A), sum or sum+1 is selected usmg different selection equations Accordmgly, selection sub-blocks 510A- D speculatively calculate selection values for all possible scenarios These selection values are conveyed to multiplexer 520 as select signals 512A-D Control signals 302, 534, and 536 mdicate which of the predicted select signals 512 is valid, conveymg one of signals 512 as far path select signal 352 Turning now to Figs 10A-B, examples of addition accurately predicted by selection sub-block 510A are shown Smce sub-block 510A only predicts for addition, selection of sum+1 is used for roundmg purposes only. Fig. 10A depicts an addition example 550A in which sum is selected Roundmg is not performed smce G(L+S,) is not true Conversely, Fig 10B depicts an addition example 550B m which sum+1 is selected. Because G and S, are set, the result is closer to 1 01011 than to 1 01010. Accordmgly, sum+1 (1 01011) is selected. Turning now to Figs 10C-10D, examples of addition accurately predicted by selection sub-block 510B are shown. Smce sub-block 510B only predicts for addition, selection of sum+1 is used for roundmg purposes only. The examples shown in Figs 10C-D are similar to those shown m Figs. 10A-B except that overflow conditions are present in examples 550C-D shown m Figs. 10C-D Accordingly, the equation for selectmg sum+1 is slightly different than for selection sub-block 510A. Fig. 10C depicts an addition example 550C m which sum is selected. Conversely, Fig 10D depicts an addition example 550D m which sum+1 is selected, effectively roundmg up the result (after a 1-bit right shift to correct for overflow) Selection sub-block 510B selects sum+1 accordmg to the equation L(N+G+S,) Turning now to Figs. 10E-F, examples of addition accurately predicted by selection sub-block 510C are shown. Smce sub-block 510C is used to predict selection for subtraction operations which have properly normalized results , selection of sum+1 is performed to coπect for one's complement subtraction and for roundmg purposes. As shown m example 550E, sum is mdicated by select signal 512C smce the guard and sticky bits are set before the subtract (ensuring that the result of the subtraction is closer to sum than sum+1). Conversely, m example 550F, the guard and sticky bits are both zero Accordmgly, a one-bit addition to the LSB is needed; therefore, sum+1 is selected. Generally speakmg, selection sub-block 510C selects sum+1 according to the equation G'+LS,', where G' and S,' represent the complements of the G and S, bits. Turning now to Figs. 10G-H, examples of addition accurately predicted by selection sub-block 510D are shown Smce sub-block 510D is used to predict selection for subtract operations which require a 1-bit left shift of the result, selection of sum+1 is performed for both one's complement coπection and roundmg. In example 550G, sum is chosen as the result since both the guard and round bits are set before the subtract (ensuring that the result of the subtraction is closer to sum than sum+1) For this particular example, a zero is shifted into the LSB when the result is normalized. (In other examples, a one may be shifted m). In example 550H, both the guard and round bits are zero, which causes the result of the subtraction to be closer to sum+1 than sum. Accordmgly, sum+1 is selected A zero is shifted m at the LSB. Generally speakmg, selection sub- block 510D selects sum+1 according to the equation G'(R'+S'), while the shift value is generated accordmg to the equation GR'+G'RS It is noted that other embodiments of selection unit 350 are also possible For example, m selection sub-blocks 510C and 510D, the guard and round bit inputs may be inverted if the sticky bit is set, resulting m different roundmg equations. Various other modifications to the selection logic are possible as well Turning now to Fig. 11, a block diagram of multiplexer-shift umt 360 is depicted in one embodiment of far data path 230. As shown, multiplexer-shift unit 360 is coupled to receive adder outputs 342A-B and shift value 514. A concatenation unit 610 receives outputs 342 and shift value 514, and conveys shifted multiplexer outputs 604A-D to multiplexer 600 Multiplexer 600 receives signals 352 (far path select signal), 534 (S MSB ), and 536 (C MSB ) as control mputs In response to these control signals, multiplexer 600 selects one of signals 342 or 604 as far path mantissa result 612 The exponent portion of far path result 232 is conveyed by exponent adjustment unit 370, which adjusts the oπgmal larger exponent value, m one embodiment, by the amount of normalization (or coπection for overflow) required by the result As shown, multiplexer 600 mcludes three groups of mputs, denoted as A, B, and C Inputs A0 and Al are adder outputs 342, representing sum and sum+1 Inputs B0 and Bl (signals 604A-B), on the other hand, represent adder outputs 342 adjusted for overflow (a '0' is routed as the MSB by concatenation unit 610) Finally, inputs CO and Cl represent adder outputs 342 after a one-bit left shift Concatenation unit 610 utilizes the shift value conveyed from selection sub-block 510D to append as the LSB of the conveyed outputs 604C-D In one embodiment, signals 534 and 536 are usable to determine whether adder output 342A is normalized properly (mput group A), has an overflow condition (mput group B), or requires a one-bit left shift (mput group C) Far path select signal 352 is then usable to determine which mput withm the selected mput group is to be conveyed as far path mantissa result 612 Turning now to Fig 12, a block diagram of one embodiment of close data path 240 is depicted As descnbed above, close data path 240 is configured to perform effective subtraction operations for operands havmg an absolute exponent difference of 0 or 1 Subtraction operations with operands havmg other absolute exponent difference values (and all addition operations) are handled as descnbed above m far data path 230 As shown, close data path 240 receives a vaπety of mputs from mput unit 210 Close data path 240 mcludes an exponent prediction unit 704, which receives the two least significant exponent bits of exponents E A and E B In one embodiment, exponent prediction unit 704 generates a prediction 706 regardmg the relationship of the full values of E A and E B As shown m Table 1, prediction 706 may be one of four values 0 (predicting E A =E B ), +1 (predicting E A =E B +1), -1 (predictmg E B =E A +1), and X (predicting d > 1, meaning the result of close path 240 is invalid) It is noted that m other embodiments, different values for prediction 706 are possible Table 1 Because exponent prediction unit 704 only operates on the two least significant bits, the prediction may often be incoπect, due to differences in the upper order bits not considered by umt 704 For this reason, in one embodiment, the actual exponent difference is computed m far data path 230 and utilized as a final select signal to determine whether far path 230 or close path 240 mcludes the coπect result value. Data path 240 further includes a shift-swap unit 710, which is coupled to receive an exponent prediction from unit 704, as well as mantissa values M A and M B from mput unit 210. Shift-swap unit 710, m response to receiving the mput mantissa values, generates shifted mantissa values 712A-B, which are conveyed to an adder unit 720. Unit 710 additionally generates a guard bit 714 which is conveyed to selection unit 730 Adder unit 720 is configured to generate a plurality of outputs (722A-B), representmg sum and sum+1, respectively. Adder unit 720 also conveys a plurality of signals to selection unit 730 as will be descnbed below. Selection unit 730, in response to receivmg an exponent prediction from unit 704 and a plurality of control signals from adder unit 720 and shift-swap unit 710, generates a close path select signal 732, conveyed to a multiplexer-mverter unit 740. Signal 732 is usable to select either adder output 722A or 722B to be conveyed as close path prelimmary result 742 Result 742 is conveyed to a left shift unit 750, which also receives a shift value from selection unit 730 and a predicted shift amount 772. Left shift unit 750 is configured to shift close path prelimmary result 742 left by a number of bits mdicated by shift amount 772. In one embodiment, the shift value conveyed by selection unit 730 is shifted m at the LSB. The output of left shift unit 750 is the mantissa portion of close path result 242. The exponent portion of close path result 242 is generated by an exponent adjustment unit 780, which receives the largest mput exponent value 309 from far data path 230. Unit 780 is configured to adjust exponent 309 by predicted shift amount 772 to produce the final close path exponent. As will be descnbed below, the value of this exponent portion may be off by one in some cases due to the nature of the prediction mechanism. In one embodiment, this possible eπor is checked and coπected if needed m the final multiplexer stage. Predicted shift amount 772 is the output of a shift prediction unit 752. Unit 752, m one embodiment, is coupled to receive three sets of mputs at prediction units 754A-C. Prediction unit 754A is coupled to receive an unshifted version of mantissa value M A , and a negated version of M B which is πght-shifted by one bit (this represents a prediction that operand 204A has an exponent value one greater than the exponent value of operand 204B). Prediction unit 754B is coupled to receive unshifted, non-negated versions of M A and M B , representmg a prediction that the exponent values of both operands are equal. Finally, prediction unit 754C is coupled to receive an unshifted version of mantissa value M B and a negated version of M A which is nght-shifted by one bit (representmg a prediction that operand 204B has an exponent value one greater than the exponent value of operand 204A) The predictions of units 754A-C are concuπently conveyed to a shift prediction multiplexer 760, which receives an exponent prediction from unit 704 as a control signal. The output of shift prediction multiplexer 760 is conveyed to a pπonty encoder 770, which generates predicted shift amount 772 Turning now to Fig 13, a block diagram of one embodiment of shift-swap unit 710 is shown. As shown, shift-swap unit 710 is coupled to receive exponent prediction value 706 from exponent prediction unit 704, as well as mantissa values M A and M B from input unit 210. Exponent prediction value 706 is conveyed to a pair of operand multiplexers 802A-B, as well as a guard bit generation unit 804 Operand multiplexer 802A is coupled to receive unshifted versions of M A and M B , while operand multiplexer 802B receives an unshifted version of M B and versions of M A and M B which are nght shifted by one bit These right shifted values are generated by a parr of right shift units 806. (In one embodiment, the shift units 806 simply route the bits of the mput values one place nghtward, appendmg a "0" as the MSB) If exponent prediction value 706 mdicates that E A =E B , operand multiplexer 802A selects M A to be conveyed as shift output 712A and operand multiplexer 802B selects M B to be conveyed as shift output 712B The output of guard bit generation unit 804, G bit 714, is not used (in one embodiment) m the equal exponent case. If exponent prediction 706 mdicates that E A =E B +1, operand multiplexer 802A selects M A to be conveyed as shift output 712A, and operand multiplexer 802B selects a one-bit-nght-shifted version of M B to be conveyed as shift output 712B. Additionally, the bit shifted out of M B is conveyed as guard bit 714 If exponent prediction 706 mdicates that E B =E A +1, operand multiplexer 802A selects M B to be conveyed as a shift output 712A, while operand multiplexer 802B selects a one-bit-nght-shifted version of M A to be conveyed as shift output 712B Additionally, the bit shifted out of M A is conveyed as guard bit 714 (If exponent prediction value 706 predicts the exponents are not valid close path values, the output of shift-swap unit 710 is undefined m one embodiment since the far path result is selected m such a case) Smce, m the embodiment shown, shift-swap unit 710 ensures that operand 712A is larger than operand 712B, the exponent difference for subsequent operations withm close data path 240 is either 0 or 1 (-1 is no longer applicable). Accordmgly, logic unit 810 is configured to receive exponent prediction value 706 and generate a coπesponding exponent equality signal 812. As will be descnbed below, exponent equality signal is utilized in selection unit 730 m order to generate close path select signal 732. Because m the embodiment shown, close path 240 handles only subtraction operations, the output of multiplexer 802B, 712B, is inverted (one's complemented) before conveyance to adder unit 720. Turning now to Fig. 14, a block diagram of one embodiment of adder unit 720 is depicted. As shown, adder unit 720 mcludes a parr of adders units, 900A-B. Adder unit 900A receives shift outputs/adder mputs 712A-B and carry m signal 904A, and generates an adder output 722A. Similarly, adder unit 900B receives shift outputs/adder mputs 712A-B and carry m signal 904B, and generates adder output 722B. Adder unit 720 generates outputs coπespondmg to sum and sum+1 by havmg cany in signal 904 A at a logical zero and carry m signal 904B at a logical one. As will be descnbed below, selection unit 730 generates a signal which selects either adder output 722A or 722B based upon a number of mput signals Adder unit 720 conveys a number of signals to selection unit 730 which are used m this calculation These signals mclude sign bits A s and B s of operands 204, most significant bits A MSB and B MSB of operands 204, carry m signal 906 to MSB adder block 902B, and least significant bit S LSB of result 722A. As with adders 400 descnbed with reference to Fig. 8 above, adders 900A-B may be implemented as a single adder producmg sum and sum+1 Turning now to Fig 15, a block diagram of one embodiment of selection unit 730 is depicted. As shown, selection unit 730 receives a number of inputs m the embodiment shown, mcludmg least significant bit S LSB (L) from adder unit 720, guard bit (G) 714 from shift-swap unit 710, most significant bit B MSB , C MSB 906, and exponent equality signal 812, mdicatmg whether exponents E A and E B are equal or differ by one. Selection unit 730 mcludes a selection logic block 950, which mcludes a plurality of selection sub-blocks 950A-D. Each sub-block 950A-D generates a coπespondmg select signal 952. Selection sub-block 950D also generates a shift value 954, which is conveyed to left shift unit 750. Select signals 952A-D are conveyed to a close path result multiplexer 960, which also receives a plurality of control signals These control signals mclude exponent equality signal 812, an MSB value 956, and a sign value 958 In one embodiment, MSB value 956 and sign value 958 are generated by a prediction select unit 962. As shown, prediction select unit 962 mcludes two multiplexers 970A-B Multiplexer 970A is coupled to receive B MSB , and also has another mput hardwired to receive a logic high signal. The output of multiplexer 970A, C s 957, is selected by C MSB 906 C s 957 is inverted by inverter 972 and conveyed as sign value 958, representmg the sign of the output of adder unit 720 Multiplexer 970B, on the other hand, is configured to receive inverted and non-inverted versions of B MSB . C MSB also provides selection for multiplexer 970B. The output of 970B is conveyed to multiplexer 960 as the MSB of the output of adder unit 720 Because close data path 240 performs subtraction operations for a limited set of operands (E d] f < 1), only a small number of cases must be considered in order to perform prediction of selection values. In the embodiment shown, there are four cases (coπespondmg to four predicted select values 952) covered by selection logic 950. Selection sub-block 950A coπesponds to the case in which the operand exponents are equal (E A =E B ) and the subtraction result is positive (M A >M B ). For this particular case, smce there is no boπow from the guard bit position, the output of selection sub-block 950A (952A) always mdicates a predicted selection of adder output 722B (sum+1). Selection sub-block 950B coπesponds to the case m which the operand exponents are equal (E A =E B ) and the subtraction result is negative (M A <M B ). Smce this case results m a negative number, the output of selection sub-block 950B (952B) always mdicates a predicted selection of adder output 722A (sum). (As will be descnbed below, this value is later inverted to return it to sign-magnitude form). Selection sub-block 950C coπesponds to the case m which the exponent values differ by one (E A =E B +1) and adder output 722A (sum) is not normalized (S MSB =0). It is noted that, m the embodiment shown, at this stage in the pipelme, the possible exponent difference is either 0 or 1 smce the operands are swapped (if needed) in shift-swap unit 710. Thus, while an exponent difference of -1 may exist for operands entering close data path 240, the inputs to selection logic block 950 have an exponent difference of either 0 or 1. Selection sub-block 950C generates a predicted selection value (952C) equal to the complement of guard bit 714 If the guard bit is zero, there is no boπow from the LSB, and adder output 722B (sum+1) is mdicated by selection value 952C. Furthermore, shift value 954 is zero Conversely, if the guard bit is one, there is a boπow from the LSB. This effectively cancels out the need for coπection of one's complement subtraction, accordmgly, adder output 722 A (sum) is selected (and guard bit 714 is conveyed as shift value 954). Lastly, selection sub-block 950D coπesponds to the case in which the exponent values differ by one (E A =E B +1) and adder output 722 A (sum) is normalized (S MSB =1). Selection sub-block 950D generates a predicted selection value (952D) which is indicative of (sum+1) according to the equation L+G', where G' represents the complement of guard bit 714 (If G=0, there is no boπow from the LSB and sum+1 is selected. If L=0 and G=l, there is a boπow, so sum is selected. If L=l and G=l, there is a boπow, but roundmg occurs, so sum+1 is selected) It is noted that m one embodiment, selection logic 730 mcludes a separate zero detect unit which is configured to recognize the case when the result of the close path subtraction is zero (E A =E B and M A =M B ). A separate zero detect unit may be utilized because in floating point representations such as IEEE standard 754, zero values are treated in a special fashion A zero detect unit is not pictured m Fig. 15 for simplicity and claπty Select signals 952A-D are conveyed to close path result multiplexer 960 The control signals also received by multiplexer 960 are usable to convey one of select signals 952 as close path select signal 732. As descnbed above, these control signals for multiplexer 960 mclude, m one embodiment, exponent equality value 812, MSB value 956, and sign value 958. Exponent equality signal 812 is usable to determine whether close path select signal is one of signals 952A-B (equal exponents) or 952C-D (unequal exponents). If exponent equality signal 812 is indicative of equal exponents, sign value 958 is usable to determine whether adder output 722A is positive or negative Accordmgly, either signal 952A or 952B may be selected. Alternately, if exponent equality signal 812 is indicative of unequal exponents, MSB value 956 may be utilized to determine whether adder output 722A is properly normalized, allowing for selection of either signal 952C or 952D. Although sign and MSB values are generated by adder unit 720 and are mcluded m adder output 722 A, MSB value 956 and sign value 958 are generated m parallel by selection unit 730. This allows close path select signal to be determined more quickly and speed operation of close data path 240 In order to perform this parallel generation, B MSB and C MSB are conveyed from adder unit 900A. (It is noted that for the embodiment of close data path 240 depicted m Fig. 15, A MSB =1, A s =l, and B s =l This allows the logic of prediction unit 962 to be simplified). MSB value 956 is generated by multiplexer 970B usmg C MSB 906, which is the carry m signal to the MSB of adder output 722A. Because it is known that A MSB =1, S MSB is thus equal to B MSB ' if C MSB =0, and B MSB if C MSB = 1 • MSB value 956 may thus be quickly generated and conveyed to multiplexer 960. Sign value 958 is generated by multiplexer 970A and inverter 972. Because A MSB =1 for close data path 240, a carry out of the MSB of adder output 722A (refeπed to m Fig. 15 as C s ) is dependent upon C MSB 906. If C MSB 906 is 0, C s 957 is equal to B MSB ; otherwise, C s 957 is 1. With A s =l and B s =0, the sum of the sign bit of adder output 722A is thus equal to the inverted value of C s 957. The output of inverter 972 is conveyed to multiplexer 960 as sign value 958. Other embodiments of prediction selection unit 962 are also contemplated. For mstance, C MSB signal 957 may be directly conveyed from adder unit 900A instead of bemg generated by prediction selection umt 960 Vanous other embodiments of unit 960 are also possible. Turning now to Fig. 16A, an example 1000A of subtraction withm close data path 240 is shown accordmg to one embodiment of the mvention. Example 1000A is representative of the close path case predicted by selection sub-block 950A, m which E A =E B and M A >M B . Because guard bit 714 is zero m this case, no boπowing is performed and the coπection for one's complement addition is always needed. (This can be seen in the difference between actual result 1002A and computed result 1002B, which coπesponds to adder output 722A). As a result, adder output 722B, or sum+1, is mdicated by select signal 952A Turning now to Fig. 16B, an example 1000B of subtraction withm close data path 240 is shown accordmg to one embodiment of the mvention. Example 1000B is representative of the close path case predicted by selection sub-block 950B, m which E A =E B and M B >M A . As with example 1000A, guard bit 714 is zero m this case, so bonowmg is not performed Because M B is larger than M A , however, the subtraction result is negative It is noted that acmal result 1004 A is the bit-inverted (one's complement) of computed result 1004B, which coπesponds to adder output 722A. Accordingly, acmal result 1004A may be computed by selecting adder output 722A for this case, inverting the resultant mantissa, and setting the sign bit of the result to indicate a negative number. This relationship may be seen from the following formulas: S = A + B'; (4) S = A + l 's comp(B), (5) S' = l 's comp(A + l 's comp(B)), (6) S' = 2 N - (A+2 N -B-1) - 1, (7) S' = B - A. (8) Turning now to Fig. 16C, an example 1000C of subtraction withm close data path 240 is shown accordmg to one embodiment of the mvention. Example 1000C is representative of the close path case predicted by selection sub-block 950C, m which E A =E B +1 and S MSB =0. As shown m Fig. 15, adder output 722B (sum+1) is mdicated by select signal 952C accordmg to the equation G'. As can be seen m example 1000C, the fact that G=0 results m no boπowing, and actual result 1006A is equal to computed result 1006B plus one Accordmgly, adder output 722B (sum+1) is selected. Turning now to Fig. 16D, an example 1000D of subtraction withm close path 240 is shown for the case predicted by selection sub-block 950C m which G=l. In this case, there is a boπow from the LSB smce guard bit 714 is set. Accordmgly, select signal 952C is indicative of adder output 722A (sum). This can be seen from the fact that actual subtraction result 1008 A is equal to computed subtraction result 1008B. Turning now to Fig. 16E, an example 1000E of subtraction withm close path 240 is shown for the case predicted by selection sub-block 950D m which L=0 and G=l. Example 1000E is representative of the close path case predicted by selection sub-block 950D, m which E A =E B +1 and S MSB =1. As shown m Fig. 15, adder output 722B (sum+1) is mdicated by select signal 952D accordmg to the equation L+G'. In example 1000E, a borrow is performed, cancelmg out the need for the one's complement coπection. Furthermore, no roundmg is performed smce L=0. Accordmgly, adder output 722A (sum) is selected by select signal 952D. This can be seen from the fact that actual subtraction result 1010A m Fig. 16E is equal to computed subtraction result 1010B Turning now to Fig. 16F, an example 1000F of subtraction withm close path 240 is shown for the case predicted by selection sub-block 950D m which L=l and G=0. In contrast to example 1000E, no boπow is performed m example 1000F, necessitatmg a one's complement coπection of +1. Accordmgly, adder output 722B (sum+1) is selected by select signal 952D This can be seen from the fact that actual subtraction result 1010A m Fig. 16E is equal to computed subtraction result 1010B plus one. Turning now to Fig. 16G, an example 1000G of subtraction withm close path 240 is shown for the case predicted by selection sub-block 950D in which L=l and G=l As with example 1000E, a boπow is performed from the LSB, cancelling the need for a one's complement coπection of +1. Because both the LSB and guard bit are set in the result, however, the subtraction result is rounded up, accordmg to an embodiment in which results are rounded to the nearest number (an even number m the case of a tie). Accordmgly, even though actual subtraction result 1014A and computed subtraction result 1014B are equal, adder output 722B is selected, effectively roundmg the difference value to the nearest number (which is chosen to be the even number smce the computed subtraction result 1014B is halfway between two representable numbers) Turning now to Fig. 17, a block diagram of one embodiment of multiplexer-mverter unit 740 is shown. Unit 740 is configured to select one of adder outputs 722 as close path preliminary result 742 Result 7412 is then conveyed to left shifter 750, descnbed below with reference to Fig. 18. Multiplexer-mverter unit mcludes an AND gate 1106, a bit XOR block 1110, and a close path result multiplexer 1100. Bit XOR block 1110 is coupled to receive adder output 722A, as well as XOR enable signal 1108 from AND gate 1106. XOR enable signal 1108 is asserted for the case (descnbed above with reference to Fig 16B) m which E A =E B and M B >M A Bit XOR block 1110, m one embodiment, mcludes a two-input XOR gate for each bit m adder output 722A. One mput of each XOR gate is a coπespondmg bit of output 722A; the other bit is XOR enable signal 1108. If signal 1108 is de-asserted, then, XOR block output 1104 is identical to adder output 722A. If signal 1108 is asserted, however, XOR block output 1104 is equal to the one's complement of adder output 722A. Signal 1108 is only enabled for the case m which the result of the close path subtraction is negative. In addition to receivmg XOR block output 1104, close path result multiplexer 1100 also receives adder output 722B. Close path select signal 732, calculated m selection unit 730 as descnbed above, is usable to select either output 1104 or 722B to be conveyed as close path preliminary result 742. Result 742 is then conveyed to left shift umt 750, descnbed next with reference to Fig. 18. By selectmg sum or sum+1 as prelimmary result 742, multiplexer-mverter unit 740 is configured to quickly perform the IEEE round-to-nearest operation. By generatmg more than one close path result and selectmg from between the results (accordmg to vanous roundmg equations), a result 742 is generated for forwardmg to a normalization umt (left shifter). The value conveyed to the normalization unit of Fig. 18 is such that shifted output value is coπectly rounded to the nearest number. This roundmg apparatus advantageously eliminates the need to perform an add operation (subsequent to the add operation of adder unit 720) m order to perform roundmg. Additionally, recomplementation is also achieved quickly smce adder output 722A need only be inverted rather than havmg to perform a two's complement invert and add Turning to Fig. 18, a block diagram of one embodiment of left shifter unit 750 is shown As depicted, left shift unit 750 mcludes a left shift register 1200 and a shift control unit 1210. Shift control unit 1210 receives predicted shift amount 772 from shift prediction unit 752 and shift value 954 from selection logic 950C. In response to these inputs, shift control unit 1210 controls the number of bits the value in register 1200 is shifted leftward. Shift control unit 1210 additionally controls what bit is shifted m at the LSB of register 1200 with each left shift The result after shiftmg is conveyed as close path result 242. For close path subtraction operations, preliminary result 742 is either normalized or requires one or more bits of left shift for normalization Furthermore, smce the loss of precision due to operand alignment is at most one bit, only one value need be generated to shift in at the LSB. This value (shift value 954 m the embodiment shown) is shifted m at the LSB for the first left shift (if needed). If more than a one bit left shift is required, zeroes are subsequently shifted in at the LSB. The output of register 1200 is conveyed as close path result 242 Turning now to Fig 19, a block diagram of one embodiment of result multiplexer unit 250 is shown As depicted, result multiplexer unit 250 includes a final result shift control unit 1310, a 1-bit left shift unit 1312, a exponent coπection adder 1313, and a parr of final multiplexers 1320 Fmal multiplexer 1320A selects to the exponent portion of result value 252, while final multiplexer 1320B selects the coπespondmg mantissa portion Fmal multiplexer 1320A receives the exponent portions of both far path result 232 and close path result 242 Additionally, multiplexer 1320A receives the output of adder 1313, equal to the close path exponent plus one As will be described below, m some cases predicted shift amount 772 is one less than the shift value needed to normalize the mantissa portion of close path 242 If this is the case, the close path exponent is one less than its true value Accordmgly, in addition the far and close path exponent values, the output of adder 1313 is also conveyed to multiplexer 1320A Similarly, multiplexer 1320B receives far and close mantissa portions, along with a coπected close path mantissa value generated by shift unit 1312 The coπected close path mantissa value is generated for the case m which the mantissa of close path result 242 is not properly normalized Guard bit 714 is shifted mto the LSB m such a case Shift control unit 1310 utilized exponent difference select 313 and close path MSB 1314 m order to generate final select signals 1322A-B As described above, the actual exponent difference (calculated m far path 230) mdicates whether far path result 232 or close path result 242 is to be selected Exponent difference select 313 is thus used (along with signal 1314) to select one of the mputs to each of multiplexers 1320 If signal 313 mdicates that the exponent difference is greater than one, far path result 232 exponent and mantissa portions are selected as result value 252 On the other hand, if the absolute exponent difference is mdicated to be 0 or 1, close path MSB 1314 selects whether the calculated or coπected versions of close path result 242 are conveyed as result value 252 As descnbed above, predicted shift amount 772 is generated by a shift prediction unit 752 In one embodiment of close path 240, shift prediction unit 752 mcludes three leadmg 0/1 prediction units 754 Prediction unit 754A is for the case m which E A =E B +1, unit 754B is for the case m which E A =E B , and unit 754C is for the case m which E B =E A +1 As will be descnbed below, units 754A and 754C may be configured to provide improved speed and reduced space requirements Turning now to Fig 20, a block diagram of a prior art leadmg 0/1 prediction unit 1400 is depicted Prediction unit 1400 is configured to receive two operands and generate an mdication of the location of the leading 0 (or 1) m the result value As will be described below, the prediction generated by unit 1400 is accurate to withm one bit position The operation of prediction unit 1400 is described in order to provide a contrast to an improved leadmg 1 prediction unit described below with reference to Fig 26 As shown, prediction unit 1400 mcludes a parr of operand input registers 1404A-B Operand register 1404A receives operand A, storing bits A' MSB to A' LSB Operand register 1404B receives a bit-inverted version of operand A, storing bits B' MSB to B' LSB The contents of register 1404A are denoted as A' (even though A', = A,) for purposes of consistency since the inverted contents of register 1404B are denoted as B' Prediction unit 1400 further includes a TGZ logic stage 1408, which mcludes TGZ generation units 1410A-1410Z (The TGZ generation unit which is coupled to A' LSB and B' LSB is denoted as "1410Z" simply to show that this unit is the final sub-block with logic stage 1408 The number of TGZ generation units 1410 withm logic stage 1408 coπesponds to the length of operands A and B) Each TGZ generation unit 1410 receives a pair of coπespondmg bits from operands A and B and produces, m turn, outputs T, G, and Z on a coπespondmg TGZ bus 1412 TGZ generation unit 1410A, for example, produces T, G, and Z outputs on TGZ bus 1412A Prediction unit 1400 further mcludes a leadmg 0/1 detection logic block 1418, which mcludes a plurality of sub- blocks 1420A-1420Z. Logic block 1418 typically includes either n or n+1 sub-blocks, where n is the number of bits m each of operands 1404 Each sub-block 1420 receives three TGZ bus 1412 mputs. Withm prediction unit 1400, a given logic sub-block 1420 has a coπesponding TGZ generation unit 1410. TGZ generation unit 1410B, for example, coπesponds to logic sub-block 1420B. Generally speakmg, then, a given logic sub-block 1420 receives TGZ bus values from its coπespondmg TGZ generation unit, from the TGZ generation unit coπespondmg to the next most significant sub-block 1420, and from the TGZ generation unit coπespondmg to the next least significant sub-block 1420. (As shown, logic sub-block 1420B receives TGZ bus 1412B from unit 1410B, TGZ bus 1412A from unit 1410A, and TGZ bus 1412C from unit 1410C. Unit 1410C is not pictured in Fig. 20). The first and last sub-blocks 1420 receive predefined TGZ values m one embodiment in order to handle the boundary cases. Each logic sub-block 1420 generates a prediction bit value 1430. Each value 1430 is usable to mdicate the presence of leadmg 0 or 1 bits m its coπespondmg bit position Collectively, values 1430A-Z make up leadmg 0/1 detection bus 1428. As will be descnbed below, prediction unit 1400 may be optimized to reduce space requirements and increase performance. Such an unproved prediction unit is descnbed below with reference to Fig. 26 This prediction unit is particularly useful for speedmg leadmg 1 predictions performed m close path 240 of add/subtract pipelme 220. Turning now to Fig. 21, a logic diagram of pnor art TGZ generation unit 1410 is depicted. Umt 1410 shown m Fig. 21 is representative of units 1410A-Z shown in Fig. 20. As shown, unit 1410 mcludes logic gates 1502A, 1502B, and 1502C, each of which receives mputs A', and B'„ where I mdicates a coπespondmg bit position withm A and B. In one embodiment, logic gate 1502A is an AND gate which generates an asserted value G, when both A', and B', are both true. Logic gate 1502B is an exclusive-OR gate which generates an asserted T, value if one of A', and B', is true. Finally, logic gate 1502C is a NOR gate which generates an asserted Z, value if A', and B', are both zero. The values G„ T„ and Z, make up TGZ bus 1412 for bit position l. For the configuration of logic gates shown m Fig. 21, one (and only one) of signals T, G, and Z is asserted for each bit position m the result of A'+B' Thus, for a given set of operands, the output of logic stage 1408 may be represented by a suing of T's, G's, and Z's. It is known that a leadmg 1 may be predicted by matchmg the strmg T * GZ\ where the "*" may be read as "0 or more occuπences of. Conversely, a leadmg 0 may be predicted by matchmg the strmg T'ZG * . As stated above, predictions generated by usmg these strings may be subject to a 1-bit coπection. Turning now to Figs. 22A-C, examples of leading 0/1 prediction usmg T-G-Z strings are shown. Fig. 22A depicts an example 1600A of leadmg 1 prediction for the case of A-B, where A=10110b and B=10010b As shown, the actual leadmg 1 position is found in the third most significant bit position of the subtraction result This operation is performed in hardware as A'+B', where A' is equal to A and B' is the inverted version of B For this set of mput operands, the resultmg T-G-Z strmg is shown as TTGTT This stnng stops matchmg the regular expression T'GZ * in the fourth most significant bit position The leadmg 1 is thus mdicated as bemg in the last bit position which matches the target string (the third most significant bit), which happens for this case to be the coπect prediction Turning now to Fig. 22B, another example of leadmg 1 prediction is shown Example 1600B depicts the case of A-B, where A=10110b and B=10011b For these operands, the actual leadmg 1 position is in the fourth most significant bit. When the subtraction is performed m hardware as A'+B', the resultmg T-G-Z strmg is TTGTZ As with example 1600A, this strmg stops matchmg m the third most significant bit. This results m a leading 1 prediction which is off by one bit position In one embodiment, final result multiplexer 250 may be configured to coπect this one-bit position eπor as described above Turning now to Fig. 22C, an example of leadmg 0 prediction is shown Example 1600C depicts the case of A-B, where A=10010b and B=l 1001b. For this set of operands, the leadmg 0 is found m the third most significant bit position. When this subtraction is performed m hardware as A'+B', the resultmg T-G-Z strmg is TZTGZ. This string stops matchmg the target strmg T * ZG * after the second bit position. This results m a leadmg 0 prediction which is off by one bit position Turning now to Fig. 23, a logic diagram is shown for leadmg 0/1 detection sub-block 1420 (representative of sub-blocks 1420A-Z m Fig. 20) As shown, sub-block 1420 mcludes logic gate 1702A-C, 1704A-C, 1706, 1708, and 1710 An asserted prediction bit value 1430 mdicates that either a leadmg 0 or leadmg 1 is present m this bit position. In one embodiment, when a leadmg 1 value is predicted, the output of one of AND gates 1702 is asserted. Each of AND gates 1702 receives values from the cuπent bit position, the previous bit position, and the next bit position. An assertion of one of gates 1702 mdicates that the T-G-Z strmg produced by logic stage 1408 stops matchmg the target suing T'GZ * m the next bit position. Each logic sub-block 1420 mcludes these gates 1702 in order to coπespond to each of the possible ways a strmg match may end. It is noted that only one of the outputs of AND gates 1702 may be asserted at a given time. An assertion of one of the outputs of gates 1702 causes the output of gate 1706, leadmg 1 prediction 1707, to also be asserted. Conversely, AND gates 1704A-C coπespond to leadmg 0 detection m one embodiment. Each of these gates also receives TGZ values from the cuπent bit position, the previous bit position, and the next bit position. An assertion of one of gates 1704 indicates that the T-G-Z strmg produced by logic stage 1408 stops matchmg the target strmg T'ZG * in the next bit position. Each of sub-blocks 1420 mcludes three gates m order to coπespond to each of the possible ways a string match may end. It is noted that only one of the outputs of AND gates 1704 may be asserted at a given time An assertion of any of the outputs of gates 1704 causes the output of OR gate 1708, leadmg 0 prediction 1709, to also be asserted OR gate 1710 asserts signal 1430 if either of signals 1707 or 1709 is asserted The most significant position withm result bus 1430A-Z which is asserted indicates the position of the leadmg 0 or 1. The configuration of sub-block 1420 is typically used when both leadmg 0 and 1 determination is to be performed As such, this configuration is used m prediction unit 754B Prediction unit 754B coπesponds to the indeterminate case in which E A =E B , and it is not known whether the subtraction operation A-B will produce a positive or negative result (leading 1 and leadmg 0 determination, respectively). As will be shown with reference to Fig. 24, prediction unit 1400 may be configured differently if more information is known regardmg operands A and B. Turning now to Fig 24, a logic diagram of a prior art prediction unit sub-block 1800 is shown Sub- block 1800 is another embodiment of logic sub-block 1420 shown in Fig 20 Sub-block 1800 is usable for operands with the restnction A>B Sub-block 1800 receives T and Z values for each bit position m the sum of A'+B' The T and Z values are coupled to inverters 1802A and 1802B, respectively The outputs of inverters 1802, T, and Z, , are coupled to an AND gate 1810, which conveys result bus 1820 as an output. Sub-block 1800 illustrates an improved method for generating leadmg 1 prediction when A>B (Leadmg 0 prediction is not relevant since the result of subtraction is positive for A>B). The configuration of sub-block 1800 is accomplished noting that the leading 1 target string T * GZ * stops matchmg when the cuπent bit position is not a T and the next bit position is not a Z A prediction unit which mcludes sub-block 1800 for each bit may omit logic for generatmg G on a bit-by-bit basis, smce this signal is not utilized m order to generate result bus 1820. Although logic sub-block 1800 provides improved performance over logic sub-block 1420, the operation of a prediction unit may be further improved for the case of E A =E B +1, which is particularly important for the operation of close data path 240. Turning now to Fig. 25, an illustration 1900 is shown depictmg the deπvation of an unproved prediction unit 754A C for close data path 240 As described above, operands m close data path 240 have an exponent difference E dlff of either 0, +1, or -1 Prediction unit 754B handles the E d]ff =0 case, while units 754 and 754C handle the +1 and -1 cases, respectively The example shown in illustration 1900 coπesponds to the case m which E A =E B +1 (unit 754A), although it is equally applicable to the case m which E B =E A +1 (unit 754C) with a few minor modifications. Illustration 1900 depicts operands A and B after operand B (the smaller operand m this case) is aligned with operand A. Because operand A is the larger operand, the MSB of A is a 1. Furthermore, smce it is predicted that E A =E B +1, the MSB of B (after alignment) is a 0. Accordmgly, the MSB of B' (the inverted version of B) is a 1. This combmation of bits in the MSB results in a G value for the T-G-Z string coπespondmg to the result of A'+B'. The T-G-Z value of the subsequent bits m the result of A'+B' is not known. It may be ascertained however, that the next bit position which equals Z mdicates that the target suing T * ZG * stopped matchmg in the previous bit position A prediction unit 754 which utilizes this detection technique is descnbed with reference to Fig. 26 Turning now to Fig. 26, a block diagram of one embodiment of prediction unit 754A C is shown. As described above, unit 754A/C is optimized for the case m which E A =E B +1 (or E B =E A +1). Accordmgly, the prediction unit shown in Fig. 26 is mdicated as coπespondmg to unit 754A or 754C as shown in Fig 12. Unit 754A/C includes input registers 2000A-B Input register 2000A receives operand A, stormg bits A' MSB through A' LSB , while input register 2000B receives a bit-inverted version of operand B, stonng bits B' MSB through B' LSB Prediction unit 754A/C further includes a plurality of OR gates 2002A-Z, each coupled to receive a pair of input values from input registers 2000 The outputs of OR gates 2002 are conveyed to output register 2010. The collective output of register 2010 (prediction bit values 2011A-Z) forms prediction stnng 2012. In one embodiment, prediction bit value 2011Z is hardwired to a logic high value m order to produce a default leadmg 1 value. The prediction stnng 2012 generated by unit 754 A C is conveyed to shift prediction multiplexer 760 Multiplexer 760 receives prediction strings from each of prediction units 754, and is configured to choose a prediction stnng based on exponent prediction value 706 For example, if exponent prediction value 706 indicates that E A =E B , the prediction string conveyed by prediction unit 754B is selected by multiplexer 760. This string is then conveyed to priority encoder 770, which converts the string into predicted shift amount 772. As described above, given the restriction that E A =E B +1, the contents of output register 2010 may be performed by using a single OR gate for each bit position. As shown in Fig. 25, the first T-G-Z value of the result A'+B' is a G. (This results from A having an MSB of 1 and the inverted version of B, B', also having an MSB of 1). Given a starting string value of G, the result stops matching the target string of T * GZ * when Z is encountered in a bit position. Therefore, when the first Z value is detected at a particular bit position i, the prediction bit value 2011 for bit position i+1 (where i+1 is one bit more significant than position i) should indicate that a leading one value is present. Such a configuration is shown in Fig. 26. Prediction bit value 2011A is asserted if either the second most significant bit of A' or the most significant bit of B' is set. (It is noted that the bit values conveyed to OR gates 2002 from operand B' have a 1-bit relative bit position to those bit values conveyed from operand A'. This routing effectively performs the functionality of aligning A' and B'. In another embodiment, B' may be shifted prior to conveyance to register 2000B. In such a case, the bit values routed to a particular gate 2002 would have common relative bit positions within input registers 2000). If either of these bits is set the second T-G-Z value in the result string is either G or T, but not Z. Accordingly, the strings stops matching in the second most significant bit position. This coπesponds to a leading one being present in the most significant bit position. Hence, prediction bit value 2011 A is asserted. The remaining prediction bit values 2011 are formed similarly. The final prediction bit value 2011Z is hardwired to a logical one (as a default in case none of the other bits are set). It is noted that although many bit values within prediction string 2012 may be asserted, typically only the most significant asserted position is utilized in determining the leading 1 position. Prediction unit 754A/C achieves an optimal implementation of leading 1 prediction for the case in which E A -E B =±1. This case is particularly useful in close data path 240. Prediction unit 754A/C represents a considerable space savings relative to designs such as that shown in Fig. 24. For Fig. 24, each bit position includes an XOR gate (to generate T s ), a NOR gate (to generate Z_), two inverters, and a final AND gate. Prediction unit 754A C includes just a single OR gate for each bit position. Furthermore, each value within prediction string 2010 is generated using bit values from only a single bit position (two bits) in the input operands. This is in contrast to prior art designs in which prediction values are generated using bit values from at least two positions (for a total of four input bits). Such a prediction unit may provide considerable space savings (up to 75% relative to prior art designs). The speed of such a prediction unit is also coπespondingly increased due to fewer gate delays. As described above, the use of far data path 230 and close data path 240 provides an efficient implementation of add/subtract pipeline 220 by eliminating operations not needed for each path. The versatility of add/subtract pipeline 220 may also be increased by expanding the pipeline to handle additional operations. Figs. 27-30 describe an embodiment of far data path 230 which is configured to perform floating point-to- integer conversions. Similarly, Figs. 31-99 describe an embodiment of close data path 240 which is configured to perform integer-to-floating point conversions. As will be shown below, this additional functionality may be achieved with only a minimal number of hardware changes. Turning now to Fig. 27 A, a floating point number 2100 is shown (in single-precision IEEE format) along with its coπesponding integer equivalent, integer number 2102. As shown, number 2100 is equal to 1.00111010011110100001101 x 2 16 . (The exponent field in number 2100 includes a bias value of +128). Integer number 2102 represents the integer equivalent of floating point number 2102, assuming a 32-bit integer format (with one bit designated as the sign bit). Accordingly, to convert floating point number 2100 to its integer equivalent, the floating point mantissa is shifted such that the most significant bit of the mantissa (in one embodiment, a leading "1" bit) ends up in the bit position representing the floating point exponent (16) in the integer format. As shown, depending on the value of the floating point exponent, not all bits of the floating point mantissa portion may be included in the integer representation. Turning now to Fig. 27B, a floating point number 2200 is shown along with coπesponding integer representation, integer number 2202. As shown, number 2200 is equal to -1.1 x 2 30 , with an implied leading "1" bit. Because the true exponent of floating point number 2200 (30) is greater than the number of mantissa bits (23+hidden 1), integer number 2202 includes all mantissa bits of the original number. Turning now to Fig. 28, a block diagram of one embodiment of far data path 2300 is shown. Far data path 2300 is similar to far data path 230 described above with reference to Fig. 6; however, far data path 2300 is modified in order to perform floating point-to-integer (f2i) conversions. The components of far data path 2300 are numbered similarly to the components of far data path 230 in order to denote similar functionality. Exponent difference unit 2310A receives exponent values E B and E A as in far data path 230. Exponent difference unit 2310B, however, receives the output of a multiplexer 2302 and exponent value E B , where E B coπesponds to the floating point value which is to be converted to integer format. Multiplexer 2302 receives an exponent value E A and a maximum integer exponent constant, and selects between these two values based on an f2i signal 2304. In one embodiment, signal 2304 is generated from the opcode of an float-to-integer conversion instruction. In the case of standard far path addition/subtraction, f2i signal 2304 is inactive, and E A is conveyed to exponent difference unit 2310B. If signal 2304 is active, however, this indicates that a floating point-to- integer conversion is being performed on the floating point number represented by E B and M B . In this case, multiplexer 2302 conveys the maximum integer exponent constant to exponent difference unit 2310B. The maximum integer exponent is indicative of the exponent of largest possible floating point value which may be converted to an integer (without clamping) by far data path 2300. If far data path 2300 is configured to handle the 32-bit signed integer format shown in Figs. 27A-B, the value 31 is used as the maximum integer exponent constant. In one embodiment, far data path 2300 may be configured to convert floating point numbers to different size integer formats. In such a case, a plurality of maximum exponent values may be multiplexed (selected by a size select signal) to provide the second input to multiplexer 2302. For standard addition subtraction in far data path 2300, exponent difference units 2310A-B operate as described above. For f2i conversions, however, only the shift amount 2312B generated by unit 2310B is utilized. As will described below, shift amount 2312A is effectively discarded since the "A" operand is set to zero in one embodiment of the f2i instruction. Shift amount 2312B, on the other hand, represents the amount that M B has to be shifted in order to provide the proper integer representation. For a floating point input of 1.0 x 2 30 , shift amount 2312B would be computed as 31-30=1. To allow far data path 2300 to accommodate f2ι conversions, the entire data path is configured to handle max(m, n) bits, where m is the number of bits m mantissa values M Λ and M B , and n is the number of bits in the target mteger format. In other words, far data path 2300 is wide enough to handle the largest possible data type for its defined operations. In order to perform f2ι conversion for 32-bit mtegers, then, nght shift units 314 are 32 bits wide. Shift units 314A-B receive mantissa values M A and M B , respectively, each of which is left aligned Shift outputs 2316A-B are then conveyed to multiplexer-mverter unit 2330 Multiplexer-mverter unit 2330 receives shift outputs 2316, along with M A , M B , and an operand which is set to zero. (It is also noted that m another embodiment, mantissa value M A may itself be set to zero before conveyance to far data path 2300). Unit 2330, in response to receivmg f2ι signal 2304, is configured to convey the zero operand as adder mput 2332A and the shifted version of M B as adder input 2332B. By setting add/subtract mdication 202 to specify addition for the f2ι conversion function, adder output 2342A is equal to adder input 2332B (M B ) Selection unit 2350 is thus configured to select adder output 2342A (sum) to perform the f2ι operation. Adder unit 2340, as descnbed above, produces sum and sum+1 outputs in response to the adder mputs. For f2ι conversions, however, smce one operand is zero, adder output 2342A is equal to adder mput 2332B Accordmgly, selection unit 2350, in response to receivmg f2ι signal 2232, selects adder output 2342A (sum) withm multiplexer-shift unit 2360. A multiplexer 2306 coupled between exponent adjust unit 2370 and multiplexer-shift unit 2360 is configured to provide the proper upper order bits for one embodiment of far path result 232. For standard far path operation (add and subtract operations), 24 bits (m one embodiment) of mantissa value are conveyed as the 24 least significant bits of result 232. Sign and exponent portions are conveyed as the upper order bits. Hence, when f2ι signal 2304 is inactive, the output of exponent adjust unit 2370 and a sign bit (not shown) is conveyed as the upper order bits of far path result 232. On the other hand, when signal 2304 is active, the upper order bits of adder output 2342A are conveyed as the upper order bits of far path result 232. For one embodiment of f2ι conversions, far path result 232 includes one sign bit followed by 31 integer bits. As will be descnbed below, floating point values above or below the maximum/minimum mteger values are clamped to predetermmed values. In one embodiment of a 32-bit representation, these maximum and minimum integer values are 2 31 -1 and -2 31 , respectively. Turning now to Fig. 29, a block diagram of one embodiment of multiplexer-inverter unit 2330 is depicted. Unit 2330 is modified slightly from multiplexer-mverter unit 330 descnbed above with reference to Fig. 7 in order to handle floatmg pomt-to-mteger conversions. As shown, multiplexer-mverter unit 2330 includes control unit 2431, mput multiplexers 2434A-B, and inverter 2436. Input multiplexer 2434A receives three mputs: M A , M B , and an zero operand set to zero, while mput multiplexer 2434B receives the outputs 2316A-B of shift units 2314. Multiplexer 2434B receives another version of shift output 2316B as described below During standard operation of far data path 2300, two 24-bit floatmg pomt mantissas are added by adder unit 2340. In order to accommodate 32-bit mteger values, however, adder unit 2340 (and other elements of data path 2300) are 32 bits wide. Accordingly, the 24-bit M A and M B values are routed to the least significant 24 bits of the adder (with the upper order bits padded with zeroes) in order to perform addition and subtraction. For the case in which E A >E B , control unit 2431 generates select signals 2433 such that multiplexer 2434A selects M A and multiplexer 2434B selects the 24-bit version of M B (shift output 316B) Conversely, for the case m which E B >E A , select signals 2433 are generated such that multiplexer 2434A selects M B and multiplexer 2434B selects the 24-bit version of M A (shift output 2316A) In one embodiment, far data path 2300 performs the f2ι function by addmg zero to an appropnately shifted version of operand B, usmg the sum as the mteger result If f2ι signal 2304 is active, control unit 2431 generates select signals 2433A-B so that the zero operand is selected by multiplexer 2434A as adder mput 2332A and that the 32-bit version of shift output 2316B is selected by multiplexer 2434B For the f2ι instruction function, inverter 2436 is inactive m one embodiment Hence, the output of multiplexer 2434B is conveyed as adder mput 2332B For floating pomt-to-integer conversions, the exponent value of the floatmg pomt number may often exceed the maximum representable mteger value In one embodiment, if an overflow (or underflow) occurs, the converted mteger may be clamped at the maximum (or minimum) representable value to provide a usable result for subsequent operations An example of result clampmg for the f2ι mstruction is descnbed below with reference to Fig 30 Turning now to Fig 30, a block diagram of one embodiment of result multiplexer unit 2500 is depicted Umt 2500 is similar to multiplexer unit 250 depicted m Fig 19, with additional hardware added to perform clampmg of f2ι conversion results As shown, result multiplexer unit 2500 mcludes comparators 2504A-B, a shift control unit 2510, a left shift unit 2512, and a final multiplexer 2520 Like final multiplexer 1320, multiplexer 2520 is configured to select result value 252 from a plurality of mputs accordmg to a final select signal 2522 generated by shift control unit 2510 Control unit 2510 generates select signal 2522 from exponent difference select 2313, comparator outputs 2504A-B, and the most significant bit of close path result 242 (denoted m Fig 30 as numeral 2514) Exponent difference signal 2313 is indicative of either far path result 232 or close path result 242, with an additional mdication of whether far path result 232 is an f2ι result If signal 2313 does mdicate that far path result is an f2ι result, comparator outputs 2506 mdicate whether the f2ι result should be clamped Comparator 2504A mdicates an overflow if E B (the ongmal floating pomt exponent of operand B) is greater than or equal to 31 , smce the maximum positive mteger for the embodiment shown is 2 31 -1 Similarly, comparator 2504B mdicates an underflow if E B is greater than 31 or E B =31 and M B is greater than 1 0 If exponent difference select signal 2313 is mdicative of close path result 242, either result 242 or its one-bit left shifted version (the output of shifter 2512) is chosen, dependmg on the whether result 242 is properly normalized As described above, far data path 2300 is similar to far data path 230, but with the additional f2ι functionality Because minimal hardware is needed to handle this extra instruction, the versatility of data path 2300 is mcreased with relativity little overhead This provides an effective implementation of f2ι conversion instructions through re-use of existmg hardware Similarly, mteger- to-floating pomt conversion (ι2f) may also be performed withm add/subtract pipeline 220 One embodiment of pipelme 220 is descnbed below with reference to Figs 31-35 in which ι2f conversions are performed in close data path 240 Turning now to Fig 31 A, a 32-bit integer number 2550 is shown along with its coπespondmg IEEE single-precision equivalent 2552 The quantity represented by both numbers is 1 1 x 2 30 Because the number of significant bits (2) m number 2550 is less than the number of mantissa bits m number 2552, no precision is lost. It is noted that m the embodiment shown, the mantissa portion of floatmg pomt number 2552 has a hidden 1 bit. Turning now to Fig 3 IB. a 32-bit integer number 2560 is shown along with its coπespondmg single- precision IEEE floatmg point equivalent 2562 Unlike mteger 2550, mteger 2560 includes more significant bits than are available m the mantissa portion of floatmg pomt number 2562 Accordmgly, these extra bits are lost m the conversion process. It is noted that if the target floatmg pomt format mcludes a larger number of bits than are m the source mteger format, no precision is lost during integer-to-float conversions. Turning now to Fig. 32, a block diagram of one embodiment of close data path 2600 is depicted Close data path 2600 has a similar structure to that of close data path 240 descnbed above with reference to Fig 12, but data path 2600 is additionally configured to perform ι2f conversions The differences m functionality between data path 240 and data path 2600 are descnbed below Other embodiments are possible in which the leadmg 1 bit is explicit In one embodiment, ι2f conversions are performed by setting operand A to zero. Accordmgly, multiplexer 2601 receives both mantissa value M A and an operand set to zero An ι2f signal 2602 is utilized to select one of these mput values to be conveyed as the output of multiplexer 2601. If ι2f signal 2602 is inactive, mantissa value M A is conveyed to both shift-swap unit 2610 and prediction 2654B, m which case close data path 2600 operates identically to close data path 240. If ι2f signal 2602 is active, however, the zero operand is conveyed to both units 2610 and 2654B. Shift-swap unit 2610, m response to receivmg ι2f signal 2602, selects 0 and M B to be conveyed as adder mputs 2620. In one embodiment, close data path 2600 is only configured to perform subtraction. In such an embodiment, a positive mteger mput to close data path 2600 produces a negative result from adder unit 2620 (smce the mteger is effectively subtracted from zero). In this case, as with close data path 240, the "sum" output of adder 2620 may be inverted m order to produce the coπect result. Conversely, a negative integer mput (in 2's complement form) to close data path 2600 produces a positive result from adder unit 2620. As will be described below, the 2's complement mteger mput is negated m shift-swap unit 2610 by takmg the l's complement This results m an adder mput havmg a magnitude which is one less than the ongmal negative number Accordmgly, the coπect output of adder unit 2620 is obtained by selectmg the "sum+1" output, which coπects for the one's complement addition Restatmg, selection unit 2630 selects the output of adder unit 2620 based on the sign of operand B if ι2f signal 2602 is active If an ι2f instruction is bemg performed, adder output 2622A (sum) is chosen (and subsequently inverted) if the sign of operand B is 0 (indicating a positive number) On the other hand, adder output 2622B (sum+1) is chosen if the sign of operand B is 1 (mdicatmg a negative number) Multiplexer- inverter unit 2640, m response to receivmg close path select signal 2632, conveys the selected adder output 2622 as close path preliminary result 2642 Close path preliminary result 2642 is then normalized m left shift unit 2650 accordmg to predicted shift amount 2672. If ι2f signal 2602 is active, prediction unit 2654B receives a zero operand and a negated version of M B as mputs. The prediction string generated by unit 2654B is then selected by shift prediction multiplexer 2660 m response to signal 2602 Prionty encoder 2670 then generates a predicted shift amount 2672 which is usable to left-align close path prelimmary result withm left shift unit 2650. In one embodiment, left shift unit 2650 is an n+1 bit shifter, where n is the width of close data path 2600 (32 bits in one embodiment). The shifter is configured to be n+1 bits in order to account for the one bit position prediction eπor which may occur using the T-G-Z methodology for leading 0/1 detection. All n+1 bits may thus be conveyed to final multiplexer unit 2500. If the most significant bit is set (indicating proper normalization), the most significant n bits of the n+1 bits conveyed to unit 250 are selected as the mantissa portion of result value 252. Conversely, if the most significant bit is not set, the least significant n bits of the n+1 bits conveyed to unit 2500 are selected as the mantissa portion of result value 252. The exponent portion of close path result 242 is calculated by an exponent adjustment unit 2680 using either exponent large input 309 or the maximum exponent value for the given integer representation. For the 32-bit integer format described above, the maximum exponent value is 31 in one embodiment. This coπesponds to the largest exponent possible for an integer value within the given format. The operation of adjustment unit 2680 is described below with reference to Fig. 35. Turning now to Fig. 33, a block diagram of one embodiment of shift-swap unit 2610 is depicted. Shift-swap unit 2610 is similar to unit 710 described above with reference to Fig. 13. Unit 2610 is additionally configured, however, to select the proper operands for the i2f operation. As shown, unit 2610 is coupled to receive i2f signal 2602. In response to signal 2602 being asserted, input multiplexers 2702A is configured to output the zero operand (conveyed as the output of multiplexer 2601) as adder input 2612 A, while input multiplexer 2702B is configured to output operand M B . Operand M B is then negated by inverter 2708 and conveyed as adder input 2612B. Turning now to Fig. 34, a block diagram of one embodiment of multiplexer-inverter unit 2640 is depicted. Unit 2640 is similar in structure to unit 740 described above with reference to Fig. 17. Unit 2640 is additionally configured to provide proper selection for i2f conversions in addition to standard close path subtraction. As shown, unit 2640 is coupled to receive adder outputs 2622A-B. For standard close path subtraction, close path select signal 2632 selects of one of the adder inputs to be conveyed as close path preliminary result 2642. Adder input 2622A may be inverted before selection by multiplexer 2800 for the case in which E A =E B and the output of adder unit 2620 is negative. The selection process for i2f conversion is similar. In one embodiment, selection unit 2630 generates close path select signal according to the sign of the integer input number is i2f signal 2602 is active. If the i2f input is a positive number, close path select signal 2632 is generated to be indicative of adder output 2622A (sum). Because a positive i2f input in close path 2600 produces a negative output from adder 2620 in one embodiment, proper recomplementation is provided by inverting adder output 2622A in XOR block 2810. This produces a result of the coπect magnitude which may be conveyed as close preliminary result 2642. If, on the other hand, the i2f input is a negative number (expressed in two's complement form), selection of adder output 2622B by select signal 2632 produces a result of the coπect magnitude. Sign bit logic (not shown) is also included in close data path 2600 to ensure that the target floating point nur. .-,er has the same sign as the input integer number. Turning now to Fig. 35, a block diagram of one embodiment of exponent adjustment unit 2680 is depicted. As shown, unit 2680 includes an exponent multiplexer 2902, an inverter 2904, a shift count adjustment multiplexer 29030, a half adder 2910, and a full adder 2920 Exponent adjustment unit 2680 is configured to subtract the predicted shift amount from an initial exponent m order to generate the exponent portion of close path result 242 In the case of standard close path subtraction (non-ι2f operations), a coπection factor is added back mto the exponent to account for the difference in width between the mteger and floating pomt formats This function is descnbed in greater detail below Consider an embodiment of close data path 2600 which is configured to handle a 32-bit mteger format and a floating pomt format with a 24-bit mantissa portion For standard close path subtraction, large exponent 309 is calculated withm far data path 230 and conveyed to multiplexer 2902 Concuπently, predicted shift amount 2672 is calculated by shift prediction unit 2652 and conveyed to inverter 2904 The negated shift amount and large exponent 309 may then be added usmg half adder 2910 and full adder 2920 This adder configuration allows a coπection constant conveyed from multiplexer 2930 to be added m as the second operand at bit 3 of full adder 2920 For standard close path operation, this constant is 1 (which is equivalent addmg the value 2 3 =8 as a third operand to exponent adjustment calculation) The exponent adjustment calculation for standard close path subtraction becomes adjusted_exponent_value = expo_large - (shιft_count - 8) (9), adjusted_exponent_value = expo_large - shιft_count + 8 (10) This coπection constant is used smce standard close path subtractions are over-shifted by 8 bits by left shift unit 2650 Because shift prediction unit 2652 is configured to generate predicted shift amounts for both mteger and floatmg pomt values withm data path 2600, the shift amounts are based on left-aligning both sets of values with the larger format, which m this embodiment is the 32-bit mteger format Stated another way, normalizing the floating pomt values produced by close path subtraction only requires the MSB of the subtraction result to be left aligned with a 24-bit field In order to accommodate 32-bit mtegers, however, all close path results are left-aligned with a 32-bit field Accordmgly, the predicted shift amount minus 8 is subtracted from large exponent 309 m order to produce the adjusted exponent The carry in to bit 0 of full adder 2920 is set in order to compensate for the one's complement addition of shift amount 2672 For ι2f conversions, the exponent adjustment calculation is similar to that performed for standard close path subtraction If ι2f signal 2602 is active, however, the output of multiplexer 2902 is 31 and the coπection constant conveyed from multiplexer 2930 is 0 Consider an ι2f conversion m which the most significant bit of the adder output is located m bit 28 out of bits [31 0] The floatmg pomt number resultmg from this mteger is 1 xxx x 2 28 The floatmg pomt exponent may thus be calculated by subtracting the shift amount (3) from the predetermmed maximum mteger exponent (31) without usmg a coπection constant Although exponent adjustment unit 2680 is shown m Fig 35 as bemg implemented with half adder 2910 and full adder 2920, vanous other adder configurations are also possible to produce the exponent portion of close path result 242 As with the inclusion of floatmg pomt-to-mteger conversion capability m far data path 2300, the expansion of close data path 2600 to handle mteger-to-floatmg point conversion also provides extra versatility to add/subtract pipelme 220 The additional functionality is mcluded withm data path 2600 with a minimum number of changes. Accordmgly, ι2f conversion capability is achieved with an efficient hardware implementation. The embodiments shown above depict a smgle add subtract pipelme 220 withm each of execution units 136C and 136D. These embodiments allow concuπent execution of floatmg pomt add and subtract mstructions, advantageously mcreasmgly floatmg point performance. By configuring pipelmes 220 to handle integer-to- float and float-to-mteger conversions as descnbed above, execution units 136C-D may concuπently perform these operations as well. Performance may further be mcreased by configunng each of execution units 136C-D to include a plurality of add/subtract pipelmes 220. As will be descnbed below, this allows each of execution units 136C-D to perform vector operations (the ability the concuπently perform the same anthmetic/logical operations on more than one set of operands). This configuration also allows a number of other operations to be efficiently implemented by pipelmes 220 at a small additional hardware cost. These mstructions are particularly useful for the types of operations typically performed by units 136C-D Turning now to Fig. 36, a block diagram of one embodiment of execution unit 136C/D is depicted. As shown, execution umt 136C/D is coupled to receive operands 204A-D and an mstruction indication 3002, and includes mput unit 3010 and add/subtract pipelmes 220A-B. Each of pipelmes 220 mcludes a far and close data path which is configured to operate as descnbed above. The outputs of each pipelme 220 is selected by one of result multiplexers 250. The outputs of multiplexers 250 are conveyed as result values 3008A-B for storage in output register 3006. Instruction mdication 3002 specifies which operation is performed concuπently m each pipelme 220. For example, if mdication 3002 specifies an add operation, both pipelmes 220 concuπently execute an add operation on operands 204. Pipelme 220 A may add operands 204 A and 204C, for mstance, while pipelme 220B adds operands 204B and 204D. This operation is descnbed in greater detail below. In one embodiment, indication 3002 may specify any of the mstructions descnbed below with reference to Figs. 37-49. Additional operand instruction information specifies the mput values by referencmg one or more storage locations (registers, memory, etc.). As descnbed above, add, subtract, float-to-mteger, and rnteger-to-float conversion mstruction may be performed m add/subtract pipelme 220 usmg far data path 230 and close data path 240 Vectored versions of these mstructions for one embodiment of pipelme 220 are descnbed below with reference to Figs. 37-42. The configuration of Fig. 36 with a plurality of pipelmes 220 may additionally be expanded to handle a number of other vectored mstructions such as reverse subtract, accumulate, compares, and extreme value mstructions Specific embodiments of such mstructions are descnbed with reference to Figs. 43-49. (Other embodiments of these mstructions are also possible). Turning now to Fig. 37A, the format of a vectored floating pomt add mstruction ("PFADD") 3100 is shown accordmg to one embodiment of microprocessor 100. As depicted, PFADD mstruction 3100 mcludes an opcode value 3101 and two operand fields, first operand field 3102 A and second operand field 3102B. The value specified by first operand field 3102 A is shown as bemg "mmregl", which, m one embodiment, maps to one of the registers on the stack of floatmg pomt execution unit 136E. In another embodiment, mmregl specifies a storage location withm execution unit 136C or 136D or a location m mam memory. The value specified by second operand field 3102B is shown in one embodiment as either bemg another of the floatmg pomt stack registers or a memory location ("mmreg2/mem64") Similarly, mmreg2 may also specify a register withm execution umt 136C or 136D m another embodiment As used m the embodiment shown m Fig 36, operand fields 3102 A-B each specify a pair of floatmg pomt values havmg a sign value, an exponent value, and a mantissa portion Turning now to Fig 37B, pseudocode 3104 illustrating operation of PFADD mstruction 3100 is given As shown, upon execution of PFADD mstruction 3100, a first vector portion (such as mput value 204A in Fig 36) of the value specified by first operand field 3102 A is added to a first vector portion (e g , 204C) of the mput value specified by second operand field 3102B As descnbed above, this sum is computed withm far path 230A of pipelme 220A In the embodiment shown, this sum is then wntten back to the upper portion of operand 3102A (mmregl [63 32]) In another embodiment of the mstruction, a destmation storage location may be specified which is different than either of the source operands PFADD mstruction 3100 also specifies that a second vector portion of the mput value specified by first operand field 3102A (e g , 204B) is added to a second vector portion (e g , 204D) of the mput value specified by second operand field 3102B This sum is computed m far data path 230B of add/subtract pipelme 220B This sum is then wntten, m one embodiment, to the lower portion of the location specified by operand 3102 A (mmregl [31 0]), although an alternate destmation location may be specified m another embodiment In one embodiment, the two add operations specified by mstruction 3100 are performed concuπently to improve performance Turning now to Fig 38 A, the format of a floatmg-pomt vectored subtract mstruction ("PFSUB") 3110 is shown accordmg to one embodiment of microprocessor 100 The format of PFSUB mstruction 3110 is similar to that descnbed above for PFADD mstruction 3100 As depicted, PFSUB mstruction 3110 mcludes an opcode value 3111 and two operands, first operand field 3112A and second operand field 3112B The value specified by first operand field 3112A is shown as bemg "mmregl", which, m one embodiment, maps to one of the registers on the stack of floatmg point execution unit 136E In another embodiment, mmregl specifies a register or storage location withm execution unit 136C D The value specified by second operand field 3112B is shown, m one embodiment, as either bemg another of the floatmg point stack registers or a memory location ("mmreg2/mem64") Similarly, mmreg2 may also specify a register within execution unit 136C/D in another embodiment As with PFADD mstruction 3100, the values specified by operand fields 3112A-B for PFSUB instruction 3110 each specify a pair of floatmg pomt numbers each havmg a sign value, an exponent value, and a mantissa portion Turning now to Fig 38B, pseudocode 3114 illustrating operation of PFSUB mstruction 3110 is given As shown, upon execution of PFSUB instruction 3110, a first vector portion (such as mput value 204C shown m Fig 36) of the mput value specified by second operand field 3112B is subtracted from a first vector portion of the value (e g , value 204 A) specified by first operand field 3112A As descnbed above, this difference may be computed m either far path 230A or close path 240A of pipelme 220A dependmg on the exponent difference value between the operands In the embodiment shown, this difference value is wntten back to the upper portion of the value specified by first operand field 3112A (mmregl [63 32]), although an alternate destmation may be specified m other embodiments PFSUB instruction 3110 also specifies that a second vector portion (such as value 204D) of the value specified by second operand field 3112B be subtracted from a second vector portion (e g , 204B) of the mput value specified by first operand field 3112 A. This difference is wntten to the lower portion of operand 3112B (mmregl [31:0]) m one embodiment, but may be wntten to another location m other embodiments. In a configuration such as that shown m Fig. 36, both difference calculations are performed concuπently m respective add/subtract pipelmes 220 to improve performance. Turning now to Fig. 39A, the format of a vectored floatmg pomt-to-mteger conversion mstruction ("PF2ID") 3120 is shown accordmg to one embodunent of microprocessor 100 The format of PF2ID mstruction 3120 is similar to those described above As depicted, PF2ID instruction 3120 mcludes an opcode value 3121 and two operand fields, first operand field 3122A and second operand field 3122B The value specified by first operand field 3122 A is shown as bemg "mmregl", which, in one embodiment, maps to one of the registers on the stack of floatmg pomt execution unit 136E In another embodiment, mmregl specifies a register or storage location withm one of execution units 136C-D As will be descnbed below, mmregl specifies a destination location for the result of mstruction 3120. The value specified by second operand field 3122B is shown as either bemg another of the floatmg point stack registers or a memory location ("mmreg2/mem64"). (Operand field 3122B may also specify a register or storage location withm one of execution units 136C-D). Operand field 3122B specifies a pair of floatmg pomt numbers havmg a sign value, an exponent value, and a mantissa portion. It is noted that mstruction 3120 produces a parr of 32-bit signed mteger values m the embodiment shown. A floatmg pomt-to-mteger mstruction which produces a parr of 16-bit signed mtegers is descnbed below with reference to Figs. 40A-C. Turning now to Fig. 39B, pseudocode 3124 for PF2ID instruction 3120 is given. In the embodiment descnbed by pseudocode 3124, PF2ID mstruction 3120 operates separately on the first and second floatmg pomt numbers specified by second operand field 3122B. If the first floatmg pomt number specified by operand 3122B is outside the allowable conversion range, the coπespondmg output value is clamped at either the maximum or mmimum value. If the first floatmg pomt mput value is withm the allowable mput range, a float- to-integer conversion is performed m far data path 220A as descnbed above. In one embodiment, the resultmg integer is wntten to the upper portion of the storage location specified by operand field 3122 A. This storage location may map to a floatmg pomt register withm execution unit 136E, or may alternately be located withm execution unit 136C/D or m mam memory. Pseudocode 3124 also specifies a similar conversion process for the second floatmg pomt mput value specified by operand field 3122B. This floatmg pomt number is converted to a signed 32-bit mteger and wntten to the upper half of the storage location specified by operand field 3122 A m one embodiment. If microprocessor 100 is configured to mclude a plurality of add/subtract pipelmes 220, the second f2ι conversion may be performed m add/subtract pipelme 220B concuπently with the first conversion to improve performance. Turning now to Fig. 39C, a table 3128 is given illustrating the mteger output values resulting from vanous floatmg pomt mput values. It is noted that the f2ι conversion process truncates floating pomt numbers, such that the source operand is rounded toward zero m this embodiment Turning now to Figs. 40A-C, the format and operation of another floatmg pomt-to-mteger ("PF2rW") mstruction 3130 is shown. PF2IW mstruction 3130 mcludes an opcode 3131 and a pair of operands fields 3132A-B. Fig. 40-B gives pseudocode 3134 which describes the operation of PF2IW instruction 3130. Instruction 3130 operates in a similar fashion to instruction 3120 except that the target integers are signed 16-bit integers rather than signed 32-bit integers. The maximum and minimum values for instruction 3130 reflect this change. The f2i conversions are performed in far data paths 230A-B in the configuration of execution unit 136C/D shown in Fig. 36. Table 3138 shown in Fig. 40C illustrates the output values of instruction 3130 for various ranges of input values. Turning now to Fig. 41 A, the format of an integer-to-floating point ("PI2FD") instruction 3140 is given. Instruction 3140 includes an opcode value 3141 and a pair of operand fields 3142 A-B. In the embodiment shown, instruction 3140 is usable to convert a pair of signed 32-bit integers (specified by operand field 3142B) to a pair of coπesponding floating point numbers (specified by operand field 3142A). In other embodiments, instruction 3140 may be used to convert floating point numbers of other sizes. Turning now to Fig. 41B, pseudocode 3144 illustrating operation of instruction 3140 is given. As shown, instruction 3140 performs integer-to-float conversions on each of the values specified by operand field 3142B. Using the execution unit 136C/D shown in Fig. 36, each of the conversions may be performed concuπently within close data paths 240A-B of add/subtract pipelines 220A-B. Turning now to Figs. 42A-B, the format and operation of another integer-to-floating point ("PI2FW") instruction 3150 is shown. As depicted, instruction 3150 includes an opcode value 3151, and a pair of operand fields 3152A-B. In the embodiment shown, the source values are a pair of floating point numbers specified by operand field 3152B. Pseudocode 3154 given in Fig. 42B illustrates the operation of instruction 3150. Instruction 3150 operates similarly to PI2FD instruction 3140 described above with reference to Figs. 41A-B, but instruction 3150 converts a pair of 16-bit signed integers to coπesponding floating point values. In one embodiment, these floating point output values are written to respective portions of the storage location specified by operand field 3152A. Execution unit 136C/D shown in Fig. 36 is configured to handle vectored add, subtract, f2i, and i2f instructions as described above. As will be shown below, pipelines 220A-B may be enhanced to handle additional vectored instructions as well. These instructions include, but are not limited to, additional arithmetic instructions, comparison instructions, and extreme value (min/max) instructions. These instructions may be realized within pipelines 220 with relatively little additional hardware, yielding an efficient implementation. Specific embodiments of such instructions are described below with reference to Figs. 43-49, although other instruction formats are possible in other embodiments. Turning now to Fig. 43 A, the format of a floating point accumulate instruction ("PFACC") 3160 is shown according to one embodiment of the invention. As depicted, PFACC instruction 3160 includes an opcode value 3161 and two operand fields, first operand field 3162A and first operand field 3162B. First operand field 3162 A ("mmregl") specifies a first pair of floating point input values in one embodiment. Operand field 3162 A may specify a location which maps to one of the registers on the stack of floating point execution unit 136E. In another embodiment, operand field 3162 A specifies a register or storage location within execution unit 136C/D. Second operand field 3162B ("mmreg2") specifies a second pair of floating point input values. These input values may be located on the floating point stack of unit 136E or within a storage location in execution unit 136C/D. Turning now to Fig. 43B, pseudocode 3164 illustrating operation of instruction 3160 is shown. Accumulate instruction 3160 is slightly different than other floating point vector operations described above (such as PFADD instruction 3100 and PFSUB instruction 3110). In the embodiments described above, instructions 3100 and 3110 operate on coπesponding parts of two different register values to produce an output value. For example, PFADD instruction 3100 forms a first portion of a vector output value by adding a first vector portion of a first input register to a first vector portion of a second input register. In contrast, PFACC instruction 3160 adds the component values of each floating point input register separately. As shown in Fig. 43B, the first portion of the vector output value produced by instruction 3160 is equal to the sum of the pair of floating point input values within the storage location specified by first operand field 3162A. This addition operation is performed within far data path 230 A of add/subtract pipeline 220 A. The second portion of the vector output value for instruction 3160 is produced similarly within far data path 230B of add/subtract pipeline 220B. Because PFACC instruction 3160 operates on vectored components of a single input storage location, this instruction is particularly advantageous in matrix multiply operations. Matrix multiply operations may be effectuated by performing vector multiply operations, then summing the resulting values to obtain a sum of products. It is noted that PFACC instruction 3160 provides an advantageous means for summing the result of these vector multiply operations, particularly if these results reside in a single vector register. Because matrix multiply operations are quite prevalent in 3-D graphics operations, the use of instruction 3160 may significantly increase the graphics processing capabilities (particularly with regard to front-end geometry processing) of a system which includes microprocessor 100. Turning now to Fig. 44A, the format of a floating-point vectored reverse subtract instruction ("PFSUBR") 3170 is shown according to one embodiment of microprocessor 100. The format of PFSUBR instruction 3170 is similar to that described above for PFSUB instruction 3110. As depicted, PFSUBR instruction 3110 includes an opcode value 3171 and two operands, first operand field 3172 A and second operand field 3172B. In a similar fashion to operands for instructions described above, the floating point input values specified by operand fields 3172 A-B may map to the stack of floating point unit 136E in one embodiment. These values may additionally be located within a register or storage location within execution unit 136C/D. It is noted that in the embodiment shown, the only difference between PFSUBR instruction 3170 and PFSUB instruction 3110 is the "direction" of the subtraction. In PFSUB instruction 3110, portions of the values specified by operand field 3112B are subtracted from coπesponding portions of the values specified by operand field 3112 A. Conversely, in PFSUBR instruction 3170, portions of the values specified by operand field 3172 A are subtracted from the coπesponding portions of the values specified by operand field 3172B. Turning now to Fig. 44B, pseudocode 3174 illustrating operation of PFSUBR instruction 3170 is given. As shown, upon execution of PFSUBR instruction 3170, a first vector portion (such as input value 204A) of the value specified by first operand field 3172A is subtracted from a first vector portion (e.g., 204C) of the value specified by second operand field 3172B. This subtraction operation may either be performed within far data path 230A or close data path 240A depending upon the exponent difference value of the operands. In the embodiment shown, this difference value is written back to the upper portion of operand 3172A (mmregl[63:32]). In other embodiments, the difference value may be wntten back to a different destination storage location. Concuπently, a second vector portion of the value specified by first operand field 302A is subtracted from a second vector portion of the value specified by second operand field 302B. This difference is wntten, m one embodiment, to the lower portion of the location specified by operand 302 A (mmregl [31.0]). In the configuration of execution unit 136C/D shown in Fig. 36, this second reverse subtract operation is performed either m far data path 230B or close data path 230B of add/subtract pipelme 220B The vectored floatmg pomt mstructions descnbed above are particularly useful m the geometry processmg stages of a 3-D graphics pipelme Another class of functions commonly utilized m graphics processmg are extreme value functions As used herem, "extreme value functions" are those functions which return as a result either a maximum or minimum value selected among a plurality of values. In typical multimedia systems, a mmimum value or a maximum value is obtamed through the execution of several sequentially executed mstructions. For example, a compare mstruction may first be executed to determine the relative magnitudes of a parr of operand values, and subsequently a conditional branch mstruction may be executed to determine whether a move operation must be performed to move the extreme value to a destmation register or other storage location. These sequences of commands commonly occur m multimedia applications, such as m clipping algonthms for graphics rendermg systems. Smce extreme value functions are implemented through the execution of multiple mstructions, however, a relatively large amount of processmg time may be consumed by such operations. Graphics processmg efficiency may be advantageously mcreased by dedicated extreme value mstructions as descnbed below with reference to Figs. 45-46. Turning now to Fig. 45 A, the format of a floatmg pomt maximum value mstruction ("PFMAX") 3180 is shown accordmg to one embodiment of the mvention. As depicted, PFMAX mstruction 3180 mcludes an opcode value 3181 and two operands, first operand field 3182 A and first operand field 3182B. The value specified by first operand field 3182A is shown as bemg "mmregl", which, m one embodiment, is one of the registers on the stack of floatmg point execution unit 136E. As with operands descnbed above for other instructions, the storage locations specified by operand field 3182A may be located m alternate locations such as execution unit 136C/D. Similarly, the values specified by second operand field 3182B, mmreg2, may also specify the floatmg pomt stack registers, a memory location, or a register withm umt 136C/D In another embodiment, second operand field 3182B specifies an immediate value. Turning now to Fig. 45B, pseudocode illustrating operation of PFMAX mstruction 3180 is given. As shown, upon execution of PFMAX mstruction 3180, a companson of a first vector portion (such as value 204A) of the value specified by first operand field 3182A and a first vector portion of the value specified by second operand 3182B (e.g., 204C) is performed. Concuπently, a companson of a second vector portion (such as value 204B) of the value specified by first operand field 3182A and a second vector portion of the value specified by second operand field 3182B (e.g., 204D) is also performed. If the first vector portion of the value specified by first operand field 3182A is found to be greater than the first vector portion of the value specified by second operand field 3182B, the value of the first vector portion of the value specified by first operand field 3182A is conveyed as a first portion of a result of mstruction 3180. Otherwise, the value of the first vector portion of value specified by second operand field 3182B is conveyed as the first vector portion of the result of instruction 3180. The second vector portion of the result of the PFMAX mstruction is calculated m a similar fashion usmg the second vector portions of the values specified by operands fields 3182 A-B Turning now to Fig. 45C, a table 3188 is shown which depicts the output of mstruction 3180 for vanous mputs. Table 3188 mcludes cases m which operands 3182 are set to zero or m unsupported formats Turning now to Figs. 46A-C, the format and operation of a vectored floatmg pomt ("PFMIN") mstruction 3190 is shown. As depicted, mstruction 3190 mcludes an opcode value 3191, and a pair of operands fields 3192A-B Operation of PFMIN mstruction 3190 is similar to that of PFMAX mstructton 3180, although instruction 3190 performs a mmimum value function mstead of a maximum value function The operation of instruction 3190 is given by pseudocode 3194 in Fig 45B Fig 45C mcludes a table 3198 which illustrates outputs of PFMIN mstruction 3190 for vanous mput values, mcludmg zero values and unsupported formats As descnbed above, vectored extreme value functions such as PFMAX mstruction 3180 and PFMIN instruction 3190 are particularly useful for performing certain graphics processmg functions such as clipping. Because the operands m extreme value functions are compared m order to produce a result value, vectored comparison mstructions may also be realized withm an execution unit 136C/D which is configured to perform extreme value mstructions 3180 and 3190. Three such companson instructions are descnbed below with reference to Figs. 47-49. Turning now to Fig 47 A, the format of a floatmg point equality compare mstruction ("PFCMPEQ") 3200 is shown accordmg to one embodiment of microprocessor 100. As depicted, PFCMPEQ mstruction 3200 mcludes an opcode value 3201 and two operands, first operand field 3202A and first operand field 3202B. The value specified by first operand field 3202A is shown as bemg "mmregl", which, m one embodiment, is one of the registers on the stack of floatmg pomt execution unit 136E. First operand field 3202A may also specify a register or storage location withm execution unit 136C/D. The value specified by second operand field 3202B, "mmreg2", is shown as either bemg another of the floatmg pomt stack registers or a memory location. In another embodiment, second operand field 3202B specifies an immediate value or a register/storage location withm unit 136C/D. Turning now to Fig. 47B, pseudocode 3204 illustrating operation of PFCMPEQ mstruction 3200 is given. As shown, upon execution of PFCMPEQ instruction 3200, a companson of a first vector portion (such as value 204A) of the value specified by first operand field 3202 A and a first vector portion of the value second operand 3202B (e.g., 204C) is performed. Concuπently, a companson of a second vector portion (e g., 204B) of the value specified by first operand field 3202A and a second vector portion of the value specified by second operand field 3202B (204D) is also performed. If the first vector portion of the value specified by first operand field 3202 A is found to be equal to the first vector portion of the value specified by second operand field 3202B, a first mask constant is conveyed as a first portion of a result of mstruction 3200. In the embodiment shown, this first mask constant is all l 's (FFFF FFFFh), but may be different m other embodiments. Otherwise, a second mask constant (0000_0000h m one embodiment) is conveyed as the first vector portion of the result of mstruction 3200. Similarly, if the second vector portion of the value specified by first operand field 3202A is found to be equal to the second vector portion of the value specified by second operand field 302B, the first mask constant is conveyed as a second portion of a result of mstruction 3200 Otherwise, the second vector portion of the result of mstructton 3200 is conveyed as the second mask constant Fig 47C is a table which shows the output of mstruction 3200 given various mputs, mcludmg cases m which operands 3202 are zero or m unsupported formats The result (both the first and second vector portions) of instruction 3200 is subsequently wntten to the storage location specified by operand field 3202A In another embodiment of mstructton 3200, the result value may be stored to mmreg2, a memory location, or a third register specified by an additional operand It is noted that m other embodiments of operands 3202, these values may mclude additional vector values beyond the two vector values shown m Fig 47A Turning now to Figs 48A-C, the format and operation of a vectored floating pomt greater than compare operation ("PFCMPGT") instruction 3210 is shown As depicted, mstruction 3210 mcludes an opcode value 3211, and a pair of operand fields 3212A-B Instruction 3210 is performed in a similar fashion to mstruction 3200, although a greater than compaπson test is performed instead of an equality test The operation of PFCMPGT mstructton 3210 is given by pseudocode listmg 3214 m Fig 48B Fig 48C mcludes a table 3218 which gives outputs for vanous mput values of mstruction 3210 Turning now to Figs 49A-C, the format and operation of a vectored floatmg pomt greater than or equal compare operation ("PFCMPGE") mstruction 3220 is shown As depicted, mstruction 3220 mcludes an opcode value 3221, and a pair of operand fields 3222A-B Instruction 3220 is performed m a similar fashion to mstructions 3200 and 3210, although mstruction 3220 effectuates a greater than or equal to companson test The operation of PFCMPGE mstruction 3220 is given by pseudocode listmg 3224 in Fig 49B Fig 49C mcludes a table 3228 which gives outputs for vanous mput values of mstructton 3220 Turning now to Fig 50, a block diagram of another embodiment of execution unit 136C/D is shown Like the embodiment shown m Fig 36, execution unit 136C/D mcludes a pair of add/subtract pipelmes 220A-B with respective far and close data paths for performing add, subtract, £2ι, and ι2f mstructions as descnbed above The embodiment of execution unit 136C/D shown in Fig 50, however, additionally mcludes an mput umt 3310 and an output unit 3320 which allow implementation of a number of other mstructions, particularly those descnbed above with reference to Figs 37-49 As depicted, execution unit 136C/D is coupled to receive inputs mto a parr of mput registers 3304A-B In one embodiment, each register 3304 is configured to store a first vector value and a second vector value For example, input register 3304A is configured to store first vector portion 204A and second vector portion 204B Similarly, mput register 3304B is configured to store first vector portion 204C and second vector portion 204D As descnbed above, these registers may mclude either mteger or floatmg pomt values dependmg upon the type of operation bemg performed The type of operation to be performed by execution unit 136C/D is conveyed by mstruction mdication 3302 Instruction mdication 3302 may specify any number of operations, mcludmg those descnbed above (add/subtract, accumulate, f2ι, ι2f, extreme value, compare) For the embodiment of execution unit 136C/D shown m Fig 50, all of the instructions descnbed above are performed In alternate embodiments, a unit 136C/D may only execute a subset of these mstructions In still other embodiments, execution unit 136C/D may also execute additional mstructions to those descnbed above (a vectored floatmg pomt mstructton which performs a less than companson test, for example) In response to receiving mstruction mdication 3302, mput unit 3310 is configured to route the appropriate combmation of operand values 204 to add subtract pipelmes 220A-B via operand buses 3012A-D Each data path withm each of pipelines 220A-B receives an "A" operand value and a "B" operand value, even if one or more of these values is not utilized withm a particular data path. For example, an f2ι mstruction is performed m the far data path 230A of pipelme 220A m one embodiment Accordmgly, the values conveyed to close data path 230B m pipelme 220A are not utilized for that particular mstruction Furthermore, different portions of the A and B operands may be conveyed to data paths 230 and 240 As descnbed above , m one embodiment, far data paths 230A-B receive full exponent values, while close data paths 240A-B receive only the two least significant bits of each exponent for performing leadmg 0/1 prediction. With appropnate routmg by mput unit 3310, a number of similar anthmetic mstructions may be performed withm execution unit 136C/D with minimal additional overhead. Table 2 given below shows the routing of operands for vanous values of mstruction mdication 3302. It is noted that mstruction mdication 3302 may mdicate an effective operation (e g., effective addition or subtraction) rather than an explicit operation denoted by an opcode. Table 2 With operands 204 appropnately routed to pipelmes 220, far data paths 230A-B and close data paths 240A-B operate substantially as descnbed above. Far data paths 230A-B perform effective addition, as well as effective subtraction for operands with E dlfl l. Conversely, close data paths 240A-B perform effective subtraction on operands with E^l. Each pipelme 220 selects its coπespondmg far path result 232 or close path result 242 to be conveyed as result value 252. Pipelme 220A generates result value 252A, while pipelme 220B generates result value 252B. Result values 252A-B are conveyed to output unit 3320 and utilized as descnbed below to generate output values 3008A-B. In addition to receivmg result values 252A-B, output unit 3320 is coupled to receive a maximum mteger value 3321, a rnimmum mteger value 3322, first and second mask constants 3324A-B, and operands 204A-D (A„ A 0 , B,, and B 0 ). Output unit 3320 mcludes clampmg comparators 3030A-D, extreme value comparator 3340, output selection logic 3350, and output multiplexer 3360 Output multiplexer 3360 is configured to convey output values 3008A-B to output register 3006. The values conveyed to the mput of output multiplexer 3360 represent the possible outputs for all of the mstructions descnbed above with reference to Figs 37-49 Result values 252A-B convey output values for add, subtract, f2ι, ι2f, and accumulate mstructions Maximum mteger value 3321 and mmimum mteger value 3322 are used for clampmg f2ι mstruction results if needed Operand values 204A-D are used to generate the output of the extreme value (mm max) mstructions First and second mask constants 3324A-B are used as outputs of the companson mstructions such as the equality compare, greater than compare, and greater than or equal to compare mstructions descnbed above With the outputs for each of the mstructions descnbed above conveyed to output multiplexer 3360, output selection logic 3350 may be used to select the appropnate multiplexer 3360 mputs to be conveyed as output values 3308A-B It is noted that because of the vector nature of the mput and output registers of execution unit 136C/D, output multiplexer 3360 accordmgly selects a pair of output values Accordmgly, multiplexer 3360 is shown in Fig 50 as havmg sub-portion 3360A (configured to convey output 3308A) and sub-portion 3360B (configured to convey output 3308B) Output selection logic 3350 generates a pair of coπespondmg select signals, 3352A-B, to control each of these multiplexer sub-portions Output selection logic receives mstruction mdication 3302, the outputs of clampmg comparators 3030A-D, and the output of extreme value comparator 3340 If mstruction mdication 3302 specifies that an aπthmetic mstruction is bemg performed, result values 252A-B are conveyed as output values 3008A-B to output register 3006 If a floating pomt-to-mteger mstruction is specified by mdication 3302, result values 252A and 252B (calculated m far data paths 230A-B, respectively) are conveyed as output values 3008A-B unless one or both values exceed maximum mteger value 3321 or minimum mteger value 3322 Overflow and underflow conditions are detected by clampmg comparators 3330A-D and conveyed to output selection logic 3350 In one embodiment, the maximum and mmimum mteger values are conveyed as output values 3008 in place of the values which caused the overflow/underflow condition The f2ι mstruction specified by mdication 3302 may generate mtegers of a vanety of sizes as descnbed above If an mteger-to- floatmg pomt mstruction is specified by mstruction mdication 3302, result values 252A and 252B (calculated in close data paths 240A-B, respectively) are conveyed as output values 3008A-B It is noted that m the embodiment shown, the dynamic range of the floatmg point format exceeds the maximum and minimum mteger values, so overflow/underflow detection logic is not used for the ι2f mstruction The ι2f mstruction may specify conversion of mtegers of a vanety of sizes as descnbed above If an extreme value mstruction is mdicated by mstruction mdication 3302, extreme value comparator 3350 generates a plurality of outputs usable to determme the maximum and mmimum values from each mput pair For example, if mstruction mdication 3302 specifies a maximum value mstruction, comparator 3350 tests whether operand 204A is greater than operand 204C If operand 204A is greater, it is conveyed as output value 30O8A Otherwise, operand 204C is conveyed The outputs generated by comparator 3350 are also usable to implement the companson mstructions descnbed above If a companson mstruction is specified by mdication 3302, comparator outputs 3350 determme whether first or second mask constant 3324 is conveyed for each output value 3008 It is noted that different mask constants may be generated for each portion of output register 3006 dependmg upon the particular mput values m question The embodiments of execution units 136C/D shown above provide an efficient means for performing floatmg pomt anthmetic operations such as add and subtract The improved selection logic implemented m one embodiment of close path 240 results m an add subtract pipelme 220 with only one full add and one full shift in each of data paths 230 and 240 Still further, data paths 230 and 240 may additionally be configured to perform floating pomt-to-mteger and integer-to-floating pomt conversions with little additional hardware Such a capability is particularly important for an embodiment of execution unit 136C/D which handles both mteger and floating pomt data (which may or may not be vectored) By mcludmg a plurality of add/subtract pipelmes in execution units 136C and D, vectored floatmg pomt mstructions may be performed This capability is advantageous m applications such as geometry processmg for graphics primitives, m which identical operations are performed repetitively on large sets of data By configuring each of units 136C-D with a pair of add/subtract pipelmes 220, up to four vectored floating pomt operations may be performed concuπently m microprocessor 100 By proper mput multiplexing of mput operands, execution unit 136C/D may be expanded to handle additional anthmetic operations such as reverse subtract and accumulate functions Finally, proper output multiplexing allows execution unit 136C/D to accommodate additional mstruction such as extreme value and compaπson mstructions Turning now to Fig 51, a block diagram of one embodiment of a computer system 3400 mcludmg microprocessor 100 coupled to a variety of system components through a bus bndge 3402 is shown Other embodiments are possible and contemplated In the depicted system, a mam memory 3404 is coupled to bus bndge 3402 through a memory bus 3406, and a graphics controller 3408 is coupled to bus bndge 3402 through an AGP bus 3410 Finally, a plurality of PCI devices 3412A-3412B are coupled to bus bndge 3402 through a PCI bus 3414 A secondary bus bndge 3416 may further be provided to accommodate an electncal mterface to one or more EISA or ISA devices 3418 through an EISA/ISA bus 3420 Microprocessor 100 is coupled to bus bndge 3402 through a CPU bus 3424 Bus bndge 3402 provides an mterface between microprocessor 100, mam memory 3404, graphics controller 3408, and devices attached to PCI bus 3414 When an operation is received from one of the devices connected to bus bndge 3402, bus bridge 3402 identifies the target of the operation (e g a particular device or, m the case of PCI bus 3414, that the target is on PCI bus 3414) Bus bndge 3402 routes the operation to the targeted device Bus bndge 3402 generally translates an operation from the protocol used by the source device or bus to the protocol used by the target device or bus In addition to providmg an mterface to an ISA/EISA bus for PCI bus 3414, secondary bus bndge 3416 may ftirther incorporate additional functionality, as desued For example, m one embodiment, secondary bus bndge 3416 mcludes a master PCI arbiter (not shown) for arbitrating ownership of PCI bus 3414 An input/output controller (not shown), either external from or integrated with secondary bus bndge 3416, may also be mcluded withm computer system 3400 to provide operational support for a keyboard and mouse 3422 and for vanous senal and parallel ports, as desired An external cache unit (not shown) may further be coupled to CPU bus 3424 between microprocessor 100 and bus bndge 3402 m other embodiments Alternatively, the external cache may be coupled to bus bridge 3402 and cache control logic for the external cache may be mtegrated mto bus bndge 3402 Mam memory 3404 is a memory m which application programs are stored and from which microprocessor 100 pnmanly executes A suitable mam memory 3404 compπses DRAM (Dynamic Random Access Memory), and preferably a plurality of banks of SDRAM (Synchronous DRAM) PCI devices 3412A-3412B are illustrative of a vanety of penpheral devices such as, for example, network mterface cards, video accelerators, audio cards, hard or floppy disk dnves or drive controllers, SCSI (Small Computer Systems Interface) adapters and telephony cards Similarly, ISA device 3418 is illustrative of vanous types of penpheral devices, such as a modem, a sound card, and a variety of data acquisition cards such as GPIB or field bus mterface cards. Graphics controller 3408 is provided to control the rendering of text and images on a display 3426 Graphics controller 3408 may embody a typical graphics accelerator generally known m the art to render three- dimensional data structures which can be effectively shifted mto and from mam memory 3404. Graphics controller 3408 may therefore be a master of AGP bus 3410 m that it can request and receive access to a target mterface withm bus bndge 3402 to thereby obtam access to mam memory 3404 A dedicated graphics bus accommodates rapid retneval of data from mam memory 3404 For certain operations, graphics controller 3408 may further be configured to generate PCI protocol transactions on AGP bus 3410. The AGP mterface of bus bndge 3402 may thus mclude functionality to support both AGP protocol transactions as well as PCI protocol target and initiator transactions. Display 3426 is any electronic display upon which an image or text can be presented. A suitable display 3426 mcludes a cathode ray tube ("CRT"), a liquid crystal display ("LCD"), etc. It is noted that, while the AGP, PCI, and ISA or EISA buses have been used as examples m the above descnption, any bus architectures may be substituted as desired It is further noted that computer system 3400 may be a multiprocessing computer system mcludmg additional microprocessors (e g. microprocessor 100a shown as an optional component of computer system 3400) Microprocessor 100a may be similar to microprocessor 100. More particularly, microprocessor 100a may be an identical copy of microprocessor 100 Microprocessor 100a may share CPU bus 3424 with microprocessor 100 (as shown in Fig 51) or may be connected to bus bndge 3402 via an mdependent bus. Numerous vanations and modifications will become apparent to those skilled m the art once the above disclosure is fully appreciated. It is mtended that the following claims be interpreted to embrace all such vanations and modifications. Multi-function Bipartite Look-up Table Turning now to Fig. 52, a block diagram of one embodiment of a microprocessor 10 is shown. As depicted, microprocessor 10 mcludes a predecode logic block 12 coupled to an mstruction cache 14 and a predecode cache 15. Caches 14 and 15 also mclude an mstruction TLB 16 A cache controller 18 is coupled to predecode block 12, mstruction cache 14, and predecode cache 15. Controller 18 is additionally coupled to a bus mterface unit 24, a level-one data cache 26 (which mcludes a data TLB 28), and an L2 cache 40 Microprocessor 10 further mcludes a decode unit 20, which receives mstructions from mstruction cache 14 and predecode data from cache 15 This information is forwarded to execution engme 30 m accordance with mput received from a branch logic unit 22 Execution engme 30 includes a scheduler buffer 32 coupled to receive mput from decode unit 20 Scheduler buffer 32 is coupled to convey decoded mstructions to a plurality of execution units 36A-E m accordance with mput received from an mstruction control unit 34 Execution units 36A-E mclude a load unit 36A, a store unit 36B, an mteger X unit 36C, an mteger Y unit 36D, and a floating pomt unit 36E Load unit 36A receives mput from data cache 26, while store unit 36B mterfaces with data cache 26 via a store queue 38 Blocks refeπed to herem with a reference number followed by a letter will be collectively refeπed to by the reference number alone For example, execution units 36A-E will be collectively refeπed to as execution units 36 Generally speakmg, floating pomt unit 36E withm microprocessor 10 mcludes one or more bipartite look-up tables usable to generate approximate output values of given mathematical functions As will be descnbed m greater detail below, these bipartite look-up tables are generated such that absolute eπor is minimized for table output values In this manner, floating pomt unit 36E may achieve an efficient implementation of such operations as the reciprocal and reciprocal square root functions, thereby mcreasmg the performance of applications such as three-dimensional graphics rendering In addition, floating pomt unit 36E withm microprocessor 10 mcludes a multi-function look-up table usable to generate approximate output values of a plurality of given mathematical functions As will be descnbed m greater detail below, this multi-function look-up table is configured such that an efficient implementation of the look-up function is achieved for more than one mathematical function In this manner, floating pomt unit 36E may mcrease the performance of such operations as the reciprocal and reciprocal square root functions, thereby enhancmg three-dimensional graphics rendering capabilities of microprocessor 10 In one embodiment, mstruction cache 14 is organized as sectors, with each sector mcludmg two 32- byte cache lmes The two cache lmes of a sector share a common tag but have separate state bits that track the stams of the Ime Accordmgly, two forms of cache misses (and associated cache fills) may take place sector replacement and cache line replacement In the case of sector replacement, the miss is due to a tag mismatch m mstruction cache 14, with the required cache Ime bemg supplied by external memory via bus mterface unit 24 The cache Ime withm the sector that is not needed is then marked invalid In the case of a cache Ime replacement, the tag matches the requested address, but the Ime is marked as invalid The requued cache Ime is supplied by external memory, but, unlike the sector replacement case, the cache Ime within the sector that was not requested remams m the same state In alternate embodiments, other organizations for mstruction cache 14 may be utilized, as well as vanous replacement policies Microprocessor 10 performs prefetchmg only m the case of sector replacements m one embodiment During sector replacement, the required cache Ime is filled If this reqmred cache Ime is m the first half of the sector, the other cache Ime m the sector is prefetched If this required cache Ime is m the second half of the sector, no prefetchmg is performed It is noted that other prefetchmg methodologies may be employed m different embodiments of microprocessor 10 When cache lmes of instruction data are retrieved from external memory by bus mterface unit 24, this data is conveyed to predecode logic block 12 In one embodiment, the mstructions processed by microprocessor 10 and stored in cache 14 are vanable-lengfh (e g., the x86 mstruction set) Because decode of vaπable-length mstructions is particularly complex, predecode logic 12 is configured to provide additional information to be stored m mstruction cache 14 to aid during decode. In one embodiment, predecode logic 12 generates predecode bits for each byte m mstruction cache 14 which mdicate the number of bytes to the start of the next vanable-length mstruction. These predecode bits are stored m predecode cache 15 and are passed to decode unit 20 when mstruction bytes are requested from cache 14 Instruction cache 14 is implemented as a 32Kbyte, two-way set associative, wπteback cache m one embodiment of microprocessor 10. The cache Ime size is 32 bytes m this embodiment Cache 14 also mcludes a TLB 16, which mcludes 64 entnes used to translate lmear addresses to physical addresses Many other vanations of mstruction cache 14 and TLB 16 are possible m other embodiments Instruction fetch addresses are supplied by cache controller 18 to mstruction cache 14. In one embodiment, up to 16 bytes per clock cycle may be fetched from cache 14 The fetched information is placed mto an mstruction buffer that feeds mto decode unit 20. In one embodiment of microprocessor 10, fetchmg may occur along a smgle execution stream with seven outstandmg branches taken. In one embodiment, the mstruction fetch logic withm cache controller 18 is capable of retrieving any 16 contiguous mstruction bytes withm a 32-byte boundary of cache 14. There is no additional penalty when the 16 bytes cross a cache Ime boundary. Instructions are loaded mto the mstruction buffer as the cuπent mstructions are consumed by decode unit 20. (Predecode data from cache 15 is also loaded mto the mstruction buffer as well). Other configurations of cache controller 18 are possible m other embodiments. Decode logic 20 is configured to decode multiple mstructions per processor clock cycle. In one embodiment, decode unit 20 accepts mstruction and predecode bytes from the mstruction buffer (m x86 format), locates actual mstruction boundanes, and generates coπespondmg "RISC ops". RISC ops are fixed- format internal mstructions, most of which are executable by microprocessor 10 m a smgle clock cycle. RISC ops are combmed to form every function of the x86 mstruction set in one embodiment of microprocessor 10. Microprocessor 10 uses a combmation of decoders to convert x86 mstructions mto RISC ops The hardware mcludes three sets of decoders' two parallel short decoders, one long decoder, and one vectoring decoder. The parallel short decoders translate the most commonly-used x86 mstructions (moves, shifts, branches, etc.) mto zero, one, or two RISC ops each. The short decoders only operate on x86 mstructions that are up to seven bytes long. In addition, they are configured to decode up to two x86 mstructions per clock cycle. The commonly-used x86 mstructions which are greater than seven bytes long, as well as those seπu- commonly-used mstructions are up to seven bytes long, are handled by the long decoder The long decoder in decode unit 20 only performs one decode per clock cycle, and generates up to four RISC ops. All other translations (complex mstructions, mterrupts, etc.) are handled by a combmation of the vector decoder and RISC op sequences fetched from an on-chip ROM. For complex operations, the vector decoder logic provides the first set of RISC ops and an initial address to a sequence of further RISC ops. The RISC ops fetched from the on-chip ROM are of the same type that are generated by the hardware decoders. In one embodiment, decode unit 20 generates a group of four RISC ops each clock cycle. For clock cycles m which four RISC ops cannot be generated, decode unit 20 places RISC NOP operations m the remammg slots of the groupmg These groupmgs of RISC ops (and possible NOPs) are then conveyed to scheduler buffer 32 It is noted that m another embodiment, an mstruction format other than x86 may be stored m mstruction cache 14 and subsequently decoded by decode unit 20 Instruction control logic 34 contains the logic necessary to manage out-of-order execution of mstructions stored m scheduler buffer 32 Instruction control logic 34 also manages data forwardmg, register renaming, simultaneous issue and retirement of RISC ops, and speculative execution In one embodiment, scheduler buffer 32 holds up to 24 RISC ops at one tune, equating to a maximum of 12 x86 mstructions When possible, mstruction control logic 34 may simultaneously issue (from buffer 32) a RISC op to any available one of execution units 36 In total, control logic 34 may issue up to six and retire up to four RISC ops per clock cycle m one embodiment In one embodiment, microprocessor 10 mcludes five execution units (36A-E) Store unit 36A and load umt 36B are two-staged pipelined designs Store unit 36A performs data memory and register wntes which are available for loadmg after one clock cycle Load unit 36B performs memory reads The data from these reads is available after two clock cycles Load and store units are possible m other embodiments with varying latencies Execution unit 36C (Integer X unit) is a fixed pomt execution unit which is configured to operate on all ALU operations, as well as multiplies, divides (both signed and unsigned), shifts, and rotates In contrast, execution unit 36D (Integer Y unit) is a fixed pomt execution unit which is configured to operate on the basic word and double word ALU operations (ADD, AND, CMP, etc ) Execution units 36C and 36D are also configured to accelerate performance of software wntten usmg multimedia mstructions Applications that can take advantage of multimedia mstructions mclude graphics, video and audio compression and decompression, speech recognition, and telephony Units 36C-D are configured to execute multimedia mstructions m a smgle clock cycle in one embodiment Many of these instructions are designed to perform the same operation of multiple sets of data at once (vector processmg) In one embodiment, unit 36C-D uses registers which are mapped on to the stack of floatmg pomt unit 36E Execution unit 36E contams an IEEE 754-compatible floatmg pomt unit designed to accelerate the performance of software which utilizes the x86 mstruction set Floating pomt software is typically wntten to manipulate numbers that are either very large or small, requue a great deal of precision, or result from complex mathematical operations such as transcendentals Floatmg pomt unit mcludes an adder unit, a multiplier unit, and a divide/square root unit In one embodiment, these low-latency units are configured to execute floating pomt mstructions in as few as two clock cycles Branch resolution unit 35 is separate from branch prediction logic 22 m that it resolves conditional branches such as JCC and LOOP after the branch condition has been evaluated Branch resolution umt 35 allows efficient speculative execution, enablmg microprocessor 10 to execute instructions beyond conditional branches before knowmg whether the branch prediction was coπect As descnbed above, microprocessor 10 is configured to handle up to seven outstanding branches m one embodiment Branch prediction logic 22, coupled to decode unit 20, is configured to mcrease the accuracy with which conditional branches are predicted m microprocessor 10 Ten to twenty percent of the mstructions m typical applications mclude conditional branches Branch prediction logic 22 is configured to handle this type of program behavior and its negative effects on mstruction execution, such as stalls due to delayed mstruction fetchmg. In one embodiment, branch prediction logic 22 mcludes an 8192-entry branch history table, a 16- entry by 16 byte branch target cache, and a 16-entry return address stack. Branch prediction logic 22 implements a two-level adaptive history algoπthm usmg the branch history table. This table stores executed branch information, predicts individual branches, and predicts behavior of groups of branches. In one embodiment, the branch history table does not store predicted target addresses m order to save space These addresses are instead calculated on-the-fly during the decode stage. To avoid a clock cycle penalty for a cache fetch when a branch is predicted taken, a branch target cache withm branch logic 22 supplies the first 16 bytes at that address duectly to the mstruction buffer (if a hit occurs m the branch target cache). In one embodiment, this branch prediction logic achieves branch prediction rates of over 95%. Branch logic 22 also mcludes special circuitry designed to optimize the CALL and RET mstructions. This circuitry allows the address of the next mstruction following the CALL mstruction m memory to be pushed onto a return address stack When microprocessor 10 encounters a RET mstruction, branch logic 22 pops this address from the return stack and begms fetchmg Like mstruction cache 14, LI data cache 26 is also organized as two-way set associative 32Kbyte storage. In one embodiment, data TLB 28 mcludes 128 entnes used to translate lmear to physical addresses Like mstruction cache 14, LI data cache 26 is also sectored. Data cache 26 implements a MESI (modified- exclusive-shared-mvahd) protocol to track cache Ime status, although other vanations are also possible. In order to maximize cache hit rates, microprocessor 10 also mcludes on-chip L2 cache 40 withm the memory subsystem. Turning now to Fig. 53, a graph 50 of a function f(x) is depicted which coπesponds to a pnor art lookup table descnbed below with reference to Fig. 54 Graph 50 mcludes a portion 80 of function f(x), with output values 82A-E plotted on a vertical axis 60 agamst coπespondmg mput values on a horizontal axis 70 As will be descnbed below, a look-up table for function f(x) is designed by dividing a predetermmed mput range mto one or more vanous sub-regions. A smgle value is generated for each of the one or more sub- regions, and then stored mto the look-up table When an mput value is presented to the look-up table, an mdex is formed which coπesponds to one of the sub-regions of the mput range. This mdex is then usable to select one of the predetermmed output values In Fig. 53, mput range portion 64 coπesponds to portion 80 of function f(x). As shown, mput range 64 is divided mto a plurality of mtervals 72. Interval 72A, for example, coπesponds to mput values located between pomts 71A and 71B on the horizontal axis. Interval 72B coπesponds to mput values located between pomts 71B and 71C, etc. It is noted that while only four mtervals are shown m graph 50, many mtervals are typically computed for a given function. Only four are shown m Fig. 53 for simplicity. As mentioned, each mterval 72 has a coπespondmg range of output values. Interval 72A, for example, mcludes a range of output values spannmg between pomts 82 A and 82B. In order to construct a look-up table for function f(x), a smgle output value is selected for mterval 72A which has a value between pomts 82A and 82B. The method of selecting this output value vanes between look-up tables. The method used for selecting output values for various input sub-regions in one embodiment of the present invention is described in detail below. Turning now to Fig. 54, a block diagram of a prior art look-up table 100 is depicted. Look-up table 100 is configured to receive an input value 102 and generate an output value 112. Input value 102 is conveyed to an address control unit 104, which in turn generates an index 106 to a table portion 108. Table portion 108 includes a plurality of table entries 110. Index 106 selects one of table entries 110 to be conveyed as output value 112. The implementation of look-up table 100 is advantageous for several reasons. First, index 106 is readily generated from input value 102. Typically, input value 102 is represented in binary format as a floating point number having a sign bit, a mantissa portion, and an exponent. Index 106, then, is formed by selecting a sufficient number of high-order mantissa bits to table portion 108, which usually includes a number of entries 2 , where m is some integer value. For example, if table portion 108 includes 64 entries, six high-order bits from the mantissa portion of input value 102 are usable as index 106. Another advantage of look-up table 100 is that output value 112 is usable as a output value of function f(x) without the additional step of interpolation (which is used in other look-up tables described below). No interpolation is needed because input range portion 24 (and any additional range of input values) is divided into intervals for which a single output value is assigned. Each table entry 110 coπesponds to one of these intervals as shown in Fig. 54. For example, table entry 110A coπesponds to interval 32A, table entry HOB coπesponds to interval 32B, etc. With this configuration, in order to increase the accuracy of output value 112, the number of intervals 32 are increased. This decreases the range of input values in each interval, and hence, the maximum possible eπor. Since a table entry 110 is provided for each interval 32, an increase in the number of intervals leads to a coπesponding increase in table size. (Table size is equal to p*2 ,ndc bits, where P is the number of bits per table entry, and 2 ,ndtx is the number of table entries.) For many functions, in order to achieve the desired degree of accuracy, the input range is divided into a large number of intervals. Since there is a one-to-one coπespondence between the number of intervals 32 and the number of table entries 110, achieving the desired degree of accuracy for many functions may lead to a prohibitively large look-up table. Turning now to Fig. 55, a graph 120 is depicted of a portion 150 of function f(x). The partitioning of function portion 150 coπesponds to a prior art look-up table described below with reference to Fig. 56. Graph 120 includes a portion 150 of function f(x), with output values 152A-E plotted on a vertical axis 130 against coπesponding input values on a horizontal axis 140. Fig. 55 illustrates a different input range partitioning for function f(x) than is shown in Fig. 53. This partitioning allows an interpolation scheme to be implemented for the look-up table described below with reference to Fig. 56. The input range of function f(x) is, as above, divided into intervals. Intervals 142A and 142B are shown in Fig. 55, although a given function may have any number of intervals depending upon the particular embodiment. Each interval 142 is then divided into subrntervals. Interval 142 A, for example, is divided into subrntervals 144A-D, while interval 142B is divided into subrntervals 146A-D. With the input range of function f(x) partitioned as shown, a bipartite table look-up may thus be constructed which includes separate base and difference portions. The base portion of the bipartite look-up table includes an output value for each interval 142. The output value is located somewhere within the range of output values for the interval. For example, the output value selected for interval 142A is located between points 152A and 152E. Which subinterval 144 the base value for interval 142A is located in depends upon the particular embodiment. The difference portion of the bipartite look-up table includes an output value difference for each subinterval. This output value difference may then be used (along with the base value for the interval) to compute an output of the bipartite look-up table. Typically, the output value difference is either added to the base value or subtracted from the base value in order to generate the final output. For example, consider this method as applied to interval 142. First, an output value is chosen to represent each subinterval 144. Then, an output value is chosen for the entire interval 142A. In one embodiment, the chosen output value for interval 142 A may be identical to one of the output values chosen to represent one of subrntervals 144. The output value chosen to represent interval 142 A is then used as the coπesponding base portion value. The differences between this base portion value and the values chosen to represent each of subintervals 144 are used as the difference portion entries for interval 142A. Turning now to Fig. 56, a block diagram of a prior art look-up table 200 is depicted. Look-up table 200 is configured to receive an input value 202 and generate an output value 232. Input value 202 is conveyed to an address control unit 210, which in turn generates a base table index 212 and a difference table index 214. Base table index 212 is conveyed to a base table 220, while difference table index 214 is conveyed to a difference table 224. Base table 220 includes a plurality of table entries 222. Base table index 212 selects one of entries 222 to be conveyed to an output unit 230 as a base table value 223. Similarly, difference table 224 includes a plurality of entries 226. Difference table index 214 selects one of entries 226 to be conveyed to output unit 230 as a difference table value 227. Output unit 230 then generates output value 232 in response to receiving base table value 223 and difference table value 227. The indexing scheme of look-up table 200 is only slightly more complicated than that of look-up table 100. Similar to index 106, base table index 212 is formed by a number of high-order mantissa bits in the binary representation of input value 202. Like table portion 108, base table 220 includes an entry 222 for each interval 142 in the predetermined input range of function f(x). Typically there are 2 mieκ entries, where index is the number of bits in base table index 212. The bits of index 212 plus an additional number of bits are used to form index 214. If the number of subintervals per interval, s, is a power of two, this number of additional bits is equal to log 2 s. In general, the number of additional bits is sufficient to specify all subintervals per interval s. This implementation may result in a savings of table storage for table 200 with respect to table 100. Consider intervals 32A-D of Fig. 53. In table 100, entries in table portion 108 each include P bits. Thus, the storage requirement for these four intervals is 4*P bits in a scheme in which no interpolation is utilized. With the intervals 32A-D partitioned as in Fig. 55, however, intervals 32A-D become a single interval having four subintervals. The storage requirements for this partitioning would be a single base table entry 222 of P bits (for the one interval) and four difference table entries 226 (one per subinterval) of Q bits each. For this example, then, the total storage requirement for this bipartite scheme is P + 4*Q bits, where Q is the number of bits in each difference entry. If Q is sufficiently smaller than P, the bipartite implementation of table 200 results in a reduced storage requirement vis-a-vis table 100. This condition is typically satisfied when function f(x) changes slowly, such that few bits are requned to represent the difference values of difference table 224 Note that the above example is only for a single mterval of a given function In typical embodiments of look-up tables, function mput ranges are divided mto a large number of input sub-regions, and table size savmgs is applicable over each of these sub-regions Turning now to Fig 57, a graph 250 of a function f(x) is depicted which coπesponds to a look-up table accordmg to one embodiment of the present mvention This look-up table is descnbed below with reference to Fig 58 Graph 250 mcludes a portion 280 of function f(x), with output values 282A-Q plotted on a vertical axis 260 agamst coπespondmg mput values x on a horizontal axis 270 Fig 57 depicts yet another partitioning of the range of mputs for function f(x) This partitioning allows an interpolation scheme to be implemented for the look-up table of Fig 58 which allows further reduction m table storage from that offered by the configuration of table 200 m Fig 56 The mput range of function f(x) is, as above, divided mto mtervals Only one mterval, 272A, is shown m Fig 57 for simplicity, although a given function may have any number of mtervals, dependmg upon the embodiment As shown, mterval 272A is divided mto a plurality of subintervals 274A-D Additionally, each submterval 274 is divided mto a plurality of sub-submtervals Submterval 274A is divided mto sub-submtervals 276A-D, submterval 274B is divided mto sub-submtervals 277 A-D, etc With the partitioning shown m Fig 57, a bipartite look-up table 300 may be constructed which is similar to table 200 shown m Fig 56 Table 300 is descnbed m detail below with reference to Fig 58 Like table 200, table 300 mcludes a base table portion and a difference table portion The entnes of these tables, however, coπespond to regions of the mput range of function f(x) m a slightly different manner than the entnes of table 200 The base table portion of table 300 mcludes an entry for each submterval m the mput range Each base table entry mcludes a smgle output value to represent its coπespondmg submterval The base table entry for submterval 274A, for example, is an output value between those represented by pomts 282A and 282E Instead of mcludmg a separate difference table entry for each sub-submterval m each submterval, however, table 300 has a number of difference table entnes for each mterval equal to the number of sub-submtervals per submterval Each of these entnes represents an averaging of difference values for a particular group of sub- submtervals withm the mterval Consider the partitioning shown in Fig 57 An output value is determined for each submterval 274, and each sub-submterval 276-279 As will be descnbed below, m one embodiment of the present mvention, the output value for each submterval and sub-submterval is chosen such that maximum possible absolute eπor is mmimized for each mput region The base table entnes are computed by usmg the assigned output value for each of submtervals 274 A separate entry is entered for each of regions 274A-D Then, difference values are computed for each sub-submterval which are equal to the difference between the output value for the sub- submterval and the output value assigned for the submterval Then, the difference values are averaged for sub- submtervals havmg common relative positions withm the submtervals These values are then used as the difference table entnes For example, difference values are computed for each of sub-subintervals 276-279 and then respective submtervals Then difference values for sub-submtervals 276A, 277A, 278A, and 279A are averaged to form the first difference entry for mterval 272 Difference values for sub-submtervals 276B, 277B, 278B, and 279B are averaged to form the second difference entry, etc. This results in a number of difference entries per interval equal to the number of sub-subintervals per interval. Like table 200, the base and difference table values may be combined to form a final output value. While the configuration of table 300 may result in a reduced table size, a slight increase in the number of bits in each table may be needed in order to achieve the same result accuracy as table 200. Turning now to Fig. 58, a block diagram of look-up table 300 is depicted according to one embodiment of the present invention. Look-up table 300 is configured to receive an input value 302 and generate an output value 332. Input value 302 is conveyed to an address control unit 310, which in turn generates a base table index 312 and a difference table index 314. Base table index 312 is conveyed to a base table 320, while difference table index 314 is conveyed to a difference table 324. Base table 320 includes a plurality of table entries 322. Base table index 312 selects one of entries 322 to be conveyed to an output unit 330 as a base table value 323. Similarly, difference table 324 includes a plurality of entries 326. Difference table index 314 selects one of entries 326 to be conveyed to output unit 230 as difference table value 327. Output unit 330 then generates output value 332 in response to receiving base table value 323 and difference table value 327. The indexing scheme of look-up table 300 is slightly different than that used to address table 200. In one embodiment, three groups of bits from a binary representation of input value 302 are used to generate indices 312 and 314. The first group includes a number of high-order mantissa bits sufficient to uniquely specify each interval of the input range of function f(x). For example, the first group includes four bits if the input range of function f(x) is divided into 16 intervals. Similarly, the second bit group from the binary representation of input value 302 has a number of bits sufficient to uniquely specify each subinterval included within a given interval. For example, if each interval includes four subintervals (such as is shown in Fig. 57), the second bit group includes two bits. Finally, the third bit group includes a number of bits sufficient to uniquely identify each group of sub-subintervals within a given interval. In this context, a group of sub- subintervals includes one sub-subinterval/subinterval, with each sub-subinterval in the group having the same relative position within its respective subinterval. The third bit group thus includes a number sufficient to specify the number of sub-subintervals in each subinterval. For the partitioning shown in Fig. 57, two bits are needed in the third bit group in order to specify each group of sub-subintervals. This addressing scheme is described in greater detail below. Because base table 320 includes an entry for each subinterval in the input range of function f(x), base table index 312 includes the first and second bit groups described above from the binary representation of input value 302. Base table index 312 is thus able to select one of entries 322, since the first bit group effectively selects an input interval, and the second bit group selects a subinterval within the chosen interval. As shown in Fig. 58, each of table entries 322A-D coπesponds to a different subinterval 274 within interval 272A. Difference table 324 includes a set of entries for each interval equal to the number of sub-subintervals per subinterval. As shown, difference table 324 includes four entries 326 for interval 272A. Entry 326A corresponds to sub-subintervals 276A, 277A, 278A, and 279A, and includes an average of the actual difference values of each of these sub-subintervals. Difference table index 314 thus includes the first and third bit groups described above from the binary representation of input value 302. The first bit group within index 314 effectively selects an interval within the input range of function f(x), while the third bit group selects a relative position of a sub-subinterval within its coπesponding subinterval. The configuration of table 300 may result in a savings in table storage size with respect to tables 100 and 200. Consider the partitioning of function portion 280 shown in graph 250. Function portion 280 is divided into 16 equal input regions (called "sub-subintervals" with reference to Fig. 58). In the configuration of table 100, the 16 input regions of Fig. 57 coπespond to intervals. Each of the 16 intervals has a coπesponding entry of P bits in table portion 108. Thus, the partitioning of Fig. 57 results in a table size of 16*P bits for the configuration of table 100. By contrast, in the configuration of table 200, the 16 input regions in Fig. 57 would represent intervals divided into subintervals. In one embodiment, the 16 input regions are divided into four intervals of four subintervals each. Each interval has a coπesponding entry of P bits in base table 220, while each of the 16 subintervals has a difference entry of Q bits in difference table 224. For this partitioning, then, the table storage size of table 200 is 4*P + 16*Q bits. The configuration of table 200 thus represents a storage savings over table 100 if function f(x) changes slowly enough (Q is greater for functions with steeper slopes, since larger changes are to be represented). The configuration of table 300 represents even greater potential storage savings with respect to tables 100 and 200. As shown in Fig. 58, function portion 280 includes an interval 272A divided into four subintervals 274. Each subinterval 274 is divided into sub-subintervals, for a total of 16 input regions. Each subinterval has a coπesponding entry of P' bits in base table 320 (P' is potentially slightly larger than P in order to achieve the same degree of accuracy). For interval 272A, difference table 224 has four entries of Q' bits each (Q' is potentially slightly larger than Q since averaging is used to compute the difference values). The total table storage requirement for table 300 is thus 4*P' + 4*Q' bits. Depending on the slope of function f(x), this represents a potential savings over both tables 100 and 200. The configuration of table 300 is well-suited for large, high-precision tables. Turning now to Fig. 59, a format 400 for input values used in one embodiment of the invention is illustrated. Generally speaking, look-up tables according to the present invention are compatible with any binary floating-point format. Format 400 (the IEEE single-precision floating-point format) is one such format, and is used below in order to illustrate various aspects of one embodiment of the invention. Format 400 includes a sign bit 402, an 8-bit exponent portion 404, and a 23-bit mantissa portion 406. The value of sign bit 402 indicates whether the number is positive or negative, while the value of exponent portion 404 includes a value which is a function of the "true" exponent. (One common example is a bias value added to the true exponent such that all exponent 404 values are greater than or equal to zero). Mantissa portion 406 includes a 23-bit fractional quantity. If all table inputs are normalized, values represented in format 400 implicitly include a leading "1" bit. A value represented by format 400 may thus be expressed as x = (-\) s - 2 exp ° - mant , (1) where s represents the value sign bit 402, expo represents the true exponent value of the floating pomt number (as opposed to the biased exponent value found m portion 404), and mant represents the value of mantissa portion 406 (including the leadmg one bit). An important floatmg-pomt operation, particularly for 3-D graphics applications, is the reciprocal function (1/x), which is commonly used during the perspective division step of the graphics pipelme. The reciprocal function may be generally expressed as follows- (2) x (-\) s ■ 2 expo - mant 1 1 1 1 x (-1) 2 p mant which simplifies to - = (-l) , - 2- β *° — — or (4a) x: mant 1 2 - = (-l) s • 2- 1 - etpo . (4b) x mant Smce the reciprocal of mant is clearly the difficult part of the operation, it is advantageous to implement an approximation to this value usmg table look-up Smce table mput values (e g , mput value 302) are normalized, mant is restricted to 2 N ≤ mant < 2 N+ (5) for some fixed N. In order to compute the reciprocal of all floatmg-pomt numbers, then, it suffices to compute l/mant over the primary range [2 N ,2 N+1 ), and map all other mputs to that range by appropnate exponent manipulation (which may be performed in parallel with the table look-up). Another common graphics operation is the reciprocal square root operation (x " 2 ), used m distance and normalization calculations. Defmmg sqrt(-x) = -sqrt(x) m order to handle negative mputs, this function may be expressed as follows- which simplifies to l -f^l l -7= = (-l) s 2 2 ) = (8) /J -jmant Because havmg the exponent of 2 be a whole number m equation (8) is desnable, the reciprocal square root function may be wntten as two separate equations, dependmg upon whether expo is odd or even These equations are as follows l f-^1 l -= - (-\γ 2 2 ) • , (expo even) (9), and : jmant (expo odd) (10). As with the reciprocal function, the difficult part of the reciprocal square root function is the computation of l/sqτt(mant) or l/sqrt(2*mαn Agam, this is implemented as a table look-up function From equations (9) and (10), it can be seen that m one embodiment of a look-up table for the reciprocal square root function, the look-up table mputs may span two consecutive bmades m order to handle both odd and even exponents For true exponent values that are even, then, the mput range is [2 N ,2 N+1 ), with odd true exponent values occupymg the next bmade, [2 N+I ,2 N+2 ) It is noted that the order of the bmades may be reversed for a look-up table that receives biased exponent values with a format that has an odd bias value Thus, the lower half of a look-up table for the reciprocal square root function may contam entnes for the bmade defined by [2,4), while the upper order addresses mclude entnes for the bmade [1,2) Alternatively, the least significant bit of the biased exponent value may be inverted so that bmade [1,2) entnes are m the lower half of the look-up table For any binary floatmg-pomt format (such as format 400), a table look-up mechamsm may be constructed for the reciprocal and reciprocal square root functions by extracting some number IDX of high- order bits of mantissa portion 406 of the mput value The look-up table includes P bits for each entry, for a total size (in a naive unplementation) of P*2 IDX bits The computation of the output sign bit and the output exponent portion are typically computed separately from the table look-up operation and are appropπately combmed with the table output to generate the output value (be it a reciprocal or a reciprocal square root) Note that smce the numenc value of each mantissa bit is fixed for a given bmade, extracting high-order bits automatically ensures equidistant nodes over the bmade, such that interpolation may be performed easily As descnbed above, the table look-up mechanism for the reciprocal square root has mput values rangmg over two consecutive bmades If it is desired to have equidistant nodes across both bmades, IDX high- order bits may extracted from mantissa value 406 for the lower bmade, with IDX+1 bits extracted from value 406 for the upper binade (this is done smce the numenc value of each fractional bit m the upper bmade is twice that of the same bit m the lower binade) In this implementation, the reciprocal square root function has a storage size of p*2 IDX +p*2 IDX+1 =3*P*2 IDX bits In one embodiment, the reqmred table accuracy allows table size to be reduced to 2*P*2 IDX =P*2 IDX+1 bits by always extracting IDX leadmg fractional mantissa bits for each bmade This results m reducmg the distance between the nodes m the upper binade For the reciprocal square root function (l/sqrt(x)), the slope decreases rapidly for mcreasmg x, which offsets table quantization eπor m the upper bmade Thus, nodes m a given bmade (either upper or lower) are equidistant, but the distance between nodes vanes m adjacent bmades by a factor of two In one embodiment, performing table look-up for the reciprocal square root function may be accomplished by makmg one table for each of the two bmades and multiplexing then output based upon the least significant bit of the value of exponent portion 404 In another embodiment, a smgle table may be implemented This smgle table is addressed such that the IDX leadmg fractional bits of mantissa value 406 constitute bits <(IDX-1) 0> of the address, with the least significant bit of exponent value 404 bit <IDX> of the table address Such a table is discussed m greater detail below Turning now to Fig 60A, a look-up table mput value 420 accordmg to format 400 is depicted Input value 420 mcludes a sign bit (IS) 422, an exponent value (IEXPO) 424, and a mantissa value (IMANT) 426 In the embodiment shown, mput value 420 is normalized, and mantissa value 426 does not mclude the leadmg one bit Accordmgly mantissa value 426 is shown as havmg N-1 bits (mantissa value 426 would be shown as havmg N bits m an embodiment m which the leadmg one bit is stored explicitly) The most significant bit m mantissa value 426 is represented m Fig 60A as IMANT<N-2>, while the least significant bit is shown as IMANT<0> Turning now to Fig 60B, an exploded view of mantissa value 426 is shown accordmg to one embodiment of the present mvention In one embodiment, the bits of mantissa value 426 may be grouped accordmg to the scheme shown m Fig 60B m order to mdex mto base and difference table portions of a look-up table for the reciprocal function Other bit groupmg are possible m alternate embodiments of the present mvention The first group of bits is XHR 430, which is HR consecutive bits from IMANT<N-2> to IMANT<N-1- HR> Similarly, the second group of bits is XMR 432, which mcludes MR consecutive bits from position IMANT<N-2-HR> to IMANT<N-1-HR-MR>, while the thud group of bits, XLR 434, mcludes LR consecutive bits from IMANT<N-2-HR-MR> to IMANT<N-1-HR-MR-LR> As will be descnbed below, XHR 430 is used to specify the mterval m the mput range which mcludes the mput value Likewise, XMR 432 identifies the submterval, and XLR the sub-submterval group In one embodiment, the input value range for the reciprocal function for which look-up values are computed is divided into a plurality of intervals, each having a plurality of subintervals that are each divided into a plurality of sub-subintervals. Accordingly, XHR 430, XMR 432, and XLR 434 may each be as short as one bit in length (although the representation in Fig. 60B shows that each bit group includes at least two bits). Because each of these quantities occupies at least one bit in mantissa value 426, none of bit groups 430, 432, and 434 may be more than N-3 bits in length. Turning now to Fig. 60C, a reciprocal base table index 440 is shown. As depicted, index 440 is composed of bit group XHR 430 concatenated with bit group XMR 432. As will be described below, index 440 is usable to select a base entry in a bipartite look-up table according to one embodiment of the present invention. In one embodiment, XHR 430 includes sufficient bits to specify each interval in the input range, while XMR 432 includes sufficient bits to specify each subinterval within a given interval. Accordingly, index 440 is usable to address a base table portion which includes an entry for each subinterval of each interval. Turning now to Fig. 60D, a reciprocal difference table index 450 is shown. As depicted, index 450 is composed of bit group XHR 430 concatenated with bit group XLR 434. As will be described below, index 450 is usable to select a difference entry in a bipartite look-up table according to one embodiment of the present invention. As described above, XHR 430 includes sufficient bits to specify each interval in the input range, while XLR 432 includes sufficient bits to specify a group of sub-subintervals within a given interval. (As stated above, each group of sub-subintervals includes one sub-subinterval per subinterval, each sub-subinterval having the same relative position within its respective subinterval). Accordingly, index 450 is usable to address a difference table portion which includes an entry for each sub-subinterval group of each interval. Turning now to Fig. 61A, mantissa value 426 is shown with different groupings of bits. Mantissa value 426 is partitioned in this manner when input value 420 coπesponds to a second function, the reciprocal square root. The base and difference indices generated from the bit groupings of Fig. 61 A are usable to obtain base and difference values for the reciprocal square root function within a bipartite look-up table according to one embodiment of the present invention. Like the groupings of Fig. 60B, mantissa value 426 includes a first bit group XHS 460 which includes HS bits. This first group is followed by a second bit group XMS 462, having MS bits, and a third bit group XLS 464, with LS bits. In one embodiment, groups 460, 462, and 464 have the same length restrictions as groups 430, 432, and 434. Fig. 61A is illustrative of the fact that the indices for each function in a multi- function bipartite look-up table do not have to be identical. Instead, the indices may be adjusted according to how the individual input ranges for the different functions are partitioned. For example, in one embodiment, a bipartite look-up table may include base and difference values for a first and second function. If greater accuracy is required for the second function in comparison to the first function, the input range of the second function may be partitioned differently than that of the first (the second function input range may be divided into more intervals, subintervals, etc.). Accordingly, this leads to more bits in the base and difference table indices for the second function. As will be shown below, however, it is often advantageous for the base and difference table indices to be identical in length (HR=HS, MR=MS, and LR=LS). Turning now to Fig. 61B, a reciprocal square root base table index 470 is depicted. Similarly, Fig. 61C depicts a reciprocal square root difference table index 480. Both indices 470 and 480 are formed from the bit groups shown in Fig. 61 A, and usable in a similar manner to indices 440 and 450 shown in Figs. 8C and 8D. Turning now to Fig. 62, a block diagram of a multi-function bipartite look-up table 500 is shown according to one embodiment of the present invention. Look-up table 500 receives input value 420 (depicted above in Fig. 60A) and a function select signal 502, and generates an output value 550 as a result of the table look-up operation. Input value 420 and function select signal 502 are conveyed to an address control unit 510, which in turn generates a base table index 512 and a difference table index 514. Base table index 512 is conveyed to base table 520, which, in one embodiment, includes base output values for both the reciprocal function and the reciprocal square root function. Similarly, difference table index 514 is conveyed to difference table 530. Difference table 530 may also, in one embodiment, include difference output values for both the reciprocal and reciprocal square root functions. In the embodiment shown in Fig. 62, base table 520 includes output base values for the reciprocal square root function over an input range of two binades. These base values are stored within locations in base table regions 522A and 522B. Table 520 further includes base output values for the reciprocal function over a single binade in entries within base table region 522C. Each region 522 includes a number of entries equal to the number of intervals in the allowable input range times the number of subintervals/interval. Difference table 530, on the other hand, is configured similarly to base table 520, only it includes output difference values for the two functions. Like table 520, table 530 includes difference values over two binades for the reciprocal square root function (within entries in difference table regions 532A and 532B), and over a single binade for the reciprocal function (within entries in region 532C). Each of regions 532 includes a number of entries equal to the number of intervals in the input range times the number of sub- subintervals/subinterval. Ultimately, base table index 512 and difference table index 514 select entries from base table 520 and difference table 530, respectively. The output of base table 520, base table output 524, is conveyed to an adder 540, which also receives difference table output 534, selected from difference table 530 by difference table index 514. Adder 540 also receives an optional rounding constant 542 as a third addend. If rounding is not needed, constant 542 is zero. Adder 540 adds quantities 524, 534, and 542, generating output value 550. As described above, an efficient indexing implementation may be achieved by partitioning the input range identically for each function provided by look-up table 500. This allows the entries for both functions within tables 520 and 530 to each be addressed by a single index, even though each table includes values for two functions. In the embodiment shown in Fig. 62, the input range for the two functions (reciprocal and reciprocal square root) are partitioned such that a single index is generated per table portion. As will be shown in Fig. 63, the number of index bits is equal to the number of bits necessary to select a table region 522/532, plus the number of bits needed to select an entry within the chosen table region (the number of entries in each storage region for tables 520 and 530 is described above). In one embodiment, each of the entries in base table 520 is P bits (P > 1). Each entry in difference table 530 is Q bits, where Q is less than P. As described above, the ratio of P to Q depends upon the slope of the function being represented. In general, where I is the number of intervals in a predetermined input range and J is the number of subrntervals/interval, Q is related to P by the relationship Q=P-(I+J)+c, where c is a constant which depends upon the slope of the function (specifically the largest slope m magnitude that occurs m the primary mput mterval). For example, for the reciprocal function, c=l smce the maximum slope m mterval [1,2) is 1 (at x=l) Similarly, for the reciprocal square root function, c=0, smce the maximum slope m [1,4) is 0 5 (at x=l) Generally speakmg, a function with a relatively high slope requues more bits m the difference entry to represent change from a coπespondmg base value In one embodiment, for example, both the reciprocal and reciprocal square root functions have slopes which allow Q to be less than 0.5 *P, while still mamtammg a high degree of accuracy. Adder 540 is configured to be an R-bit adder, where R is sufficient to represent the maximum value m base table 520 (R may be equal to P in one embodiment) Adder 540 is configured to add table outputs 524 and 534, plus optional roundmg constant 542, such that the least significant bits of the addends are aligned. This add operation results m an output value 550 bemg produced In one embodiment, the use of optional roundmg constant 542 results m a number of least significant bits bemg discarded from output value 550 In the embodiment shown m Fig. 62, adder 540 does not generate a carry out signal (a carry out signifies that output value 550 exceeds 2 R ). Smce all the entnes of tables 520 and 530 have been determined before table 500 is to be used (during operation of a microprocessor m one embodiment), it may be determined if any of the possible combinations of base/difference entnes (plus the roundmg constant) result m an output value 550 which necessitates providmg a carry out signal. As shown, result 560 for the two functions of table 500 mcludes an output sign bit portion 562, an output exponent portion 564, and an output mantissa portion 566. Output value 550 is usable as mantissa portion 566, although some bits may be discarded from output value 550 m wnting output mantissa portion 566. With regard to the value of output sign bit portion 562, the value of mput sign portion 422 is usable as the value of portion 562 for both the reciprocal and reciprocal square root functions The value of output exponent portion 564 is generated from the value of mput exponent portion 422 of mput value 420, and is calculated differently for the reciprocal function than it is for the reciprocal square root function In one embodiment, the true mput exponent, TIEXPO, is related to the value of field 424 m mput value 420, IEXPO Similarly, the true output exponent, TOEXPO, is related to the value to be wntten to field 564, OEXPO. The value wntten to OEXPO is dependent upon the particular function bemg evaluated. For the reciprocal function, the value wntten to OEXPO is computed such that TOEXPO— 1- TIEXPO[+CR], where [+CR] is part of the equation if carry out generation is applicable. For the common case m which IEXPO-TIEXPO+BIAS and OEXPO=TOEXPO+BIAS, it follows that OEXPO=2*BIAS-l- EXPO[+CR]. For the reciprocal square root function, OEXPO is computed such that TOEXPO=(-l- (TIEXPO/2))[+CR] if TIEXPO is greater than or equal to zero. Conversely, if TIEXPO is less than zero, OEXPO is computed such that TOEXPO=(-(TIEXPO+l/2))[+CR]. For the common case m which IEXPO=TIEXPO+BIAS and OEXPO=TOEXPO+BIAS, OEXPO=((3*BIAS-l-IEXPO)»l)[+CR] Turning now to Fig 63, a block diagram of address control 510 withm multi-function look-up table 500 is depicted accordmg to one embodiment of the present mvention Address control unit 510 receives mput value 420 and function select signal 502 and generates base table mdex 512 and difference table mdex 514 Input value 420 mcludes sign bit field 422 havmg a value IS, exponent field 424 havmg a value IEXPO (the biased exponent value), and mantissa field 426 havmg a value IMANT As shown, mantissa field 426 mcludes three bit groups (573, 574, and 575) usable to form mdices 512 and 514 Because mput value 420 is used to select base/difference values for both the reciprocal and reciprocal square root functions, these bit groups are equivalent to the bit groups of Figs 8B and 9A More specifically, group 573 is equivalent to groups 430 and 460, respectively, smce group 573 is usable to specify an mterval for both functions withm table 500 Similarly, group 574 is equivalent to groups 432/462, while group 575 is equivalent to groups 434/464 Bit group 573 is shown as havmg XH bits, where XH=HR=HS Similarly, bit group has XM bits (XM=MR=MS), while bit group 575 has XL bits (XL=LR=LS) Bit groups 573-575 are combmed as shown m Figs 8C-D (and 9B and 9C) m order to form portions of mdices 512 and 514 The most sigmficant bits of mdices 512 and 514 are used for function selection In the embodiment shown m Fig 63, the most sigmficant bit is low when function select signal 502 is high (as signal 502 is conveyed through an inverter 570) Thus, when signal 502 is high, base table mdex 512 and difference table mdex 514 access entnes withm table regions 522A-B and 532A-B (the reciprocal square root entnes) Conversely, when signal 502 is low, mdices 512 and 514 access entnes withm table regions 522C and 532C (the reciprocal entnes) The second most sigmficant bit of mdices 512/514 is used (if applicable) to select one of the two bmades for the reciprocal square root entnes That is, these bits select between table regions 522A and 522B m base table 520, and between table regions 532A and 532B m difference table 530 Furthermore, these second-most-significant bits are only set (m the embodiment shown) if function select 502 is high and the LSB of the true exponent value is set (meanmg the true exponent is odd and the biased exponent, 511, is even) Thus, these bits are not set if function select 502 is low, mdicatmg the reciprocal function The equations for mdex 512 m the embodiment shown in Fig 62 may be summarized as follows BADDR<XH+XM+1> ='(Sιgnal 502), (11) BADDR<XH+XM> ='IEXPO<0>&&(502), (12) BADDR<XH+XM- 1 XM> =IMANT<N-2 N- 1 -XH>, (13) BADDR<XM-1 0> =IMANT<N-2-XH N-1-XH-XM> (14) Similarly, the equations for mdex 514 are as follows DADDR<XH+XL+1> ='(Sιgnal 502), (15) DADDR<XH+XL> =IEXPO<0>&&( 502), ( 16) DADDR<XH+XL-1 XL> =IMANT<N-2 N-1-XH>, (17) D ADDR<XL- 1 0> =IMANT<N-2-XH-XM N- 1 -XH-XM-XR> (18) Other equations are possible m other embodiments Turning now to Fig 64A, a graph 578 of an mput region 580 is shown accordmg to a pnor art method for calculating a midpomt value Input region 580 is bounded by mput values A and B, located at pomts 582 and 584, respectively, on the horizontal axis of graph 578 Pomt A coπesponds to an output value (for the reciprocal function) denoted by pomt 581 on the vertical axis of graph 578 Pomt B, likewise, coπesponds to an output value denoted by pomt 583 As shown m Fig 64A, a midpomt XI is calculated for mput region 580 by determining the mput value halfway m between A and B This mput value XI is located at pomt 586, and coπesponds to an output value denoted by pomt 585 on the vertical axis In prior art systems, the output value coπespondmg to point 585 is chosen to represent all values m mput region 580 An output value calculated in this manner has the effect of minimizing maximum relative eπor over a given mput region Although this midpomt calculation method is shown m Fig 64A for the reciprocal function, this method is applicable to any function Turning now to Fig. 64B, a graph 590 of mput region 580 is shown accordmg to a method for calculating a midpomt value accordmg to the present mvention As m Fig 64A, mput region 580 is bounded by mput values A and B located at pomts 582 and 584, respectively Input value A coπesponds to an output value denoted by pomt 581, while mput value B coπesponds to an output value at pomt 583 As depicted m Fig 64B, both of these output values coπespond to the reciprocal function Unlike the midpomt calculation m Fig 64A, the midpomt calculation m Fig 64B produces an output value for mput region 580 which mmimizes absolute eπor The midpomt calculation is Fig. 64A is mdependent of the particular function, smce the midpomt (XI) is simply calculated to be halfway between the mput values (A and B) which bound region 580 Midpomt X2, on the other hand, is calculated such that the coπespondmg output value, denoted by pomt 587, is halfway between the output values (581 and 583) coπespondmg to the mput region boundanes That is, the difference between 581 and 587 is equal to the difference between 587 and 583 The calculation of X2 (denoted by pomt 588) is function-specific For the reciprocal function, X2 is calculated as follows j_ 1 A ~ X2 ri (19) ' or 1 1 A - X2 - B (20), \ A X2 ~ X2 B which simplifies to X2 - B - A - B = A - B - A - X2 (21). Solvmg for X2 gives 2 - A - B X2 = A + B Calculating X2 for the reciprocal square root function gives A - A - B X2 = A + 2JA - B + B ' Calculation of midpoint X2 in this manner ensures that maximum absolute eπor is minimized by selecting f(X2) as the output value for input region 580. This is true because the absolute eπor at both A and B is identical with f(X2) selected as the output value for region 580. Another method of calculating eπor, "ulp" (unit in last place) eπor, is cuπently favored by the scientific community. Generally speaking, ulp eπor is scaled absolute eπor where the scale factor changes with a) precision of the floating point number and b) the binade of a particular number. For example, for IEEE single-precision floating point format, 1 ulp for a number in binade [1,2) is 2 "23 . The ulp method of midpoint calculation is utilized below in a method for computation of base and difference table values in one embodiment of the present invention. Turning now to Fig. 65A, a flowchart of a method 600 for calculations of difference table entries is depicted according to one embodiment of the present invention. Method 600 is described with further reference to Fig. 65B, which is a graph 640 of a portion 642 of function f(x). Method 600 is described generally in relation to Fig. 65 A, while Fig. 65B illustrates a particular instance of the use of method 600. Method 600 first includes a step 602, in which the input range of f(x) is partitioned into I intervals, J subintervals/interval, and K sub-subintervals/subinterval. The partitioning choice directly affects the accuracy of the look-up table, as a more naπowly-partitioned input range generally leads to reduced output eπor. Fig. 65B illustrates a single interval 650 of the input range of f(x). Interval 650 is partitioned into four subintervals, 652A-D, each of which is further partitioned into four sub-subintervals. Subinterval 652A, for example, includes sub-subintervals 654A, 654B, 654C, and 654D. These partitions affect the input regions for which difference table entries are generated. In one embodiment, a difference table entry is generated for each group of sub-subintervals in a subinterval of an input range. As described above, each sub-subinterval group includes one sub-subinterval subinterval within a given interval, with each sub-subinterval in the group having the same relative position within its respective subinterval. For example, if an interval includes eight subintervals of eight sub-subintervals each, a difference table according to one embodiment of the present invention would include eight entries for the interval. Consider Fig. 65B. Interval 650 is shown as having four subintervals 652 of four sub-subintervals each. Each sub-subinterval within a given subinterval belongs to one of four groups. Each group has a number of entries equal to the number of subintervals/interval, and each member of a particular group has the same relative position within its respective subinterval. Group 2, for instance, includes sub-subintervals 654C, 655C, 656C, and 657C, all of which are the third sub-subinterval within their respective subintervals. As will be described below, a difference table entry is computed for each group within a given interval. In step 604, a particular interval M is selected for which to calculate K difference table entries. In Fig. 65B, interval M is interval 650. Method 600 is usable to calculate difference table entries for a single interval; however, the method may be applied repeatedly to calculate entries for each interval in an input range. Next, in step 606, a group of K sub-subintervals (refeπed to in Fig. 65 A as "Group N") are selected for which to calculate a difference entry. Typically, the groups are selected sequentially. For example, in Fig. 65B, group 0 (consisting of sub-subintervals 654A, 655A, 656A, and 657A) would typically be selected first. In step 608, a counter variable, SUM, is reset. As will be described, this variable is used to compute an average of the difference values in each group. SUM is reset each time a new group of sub-subintervals is processed. Step 610 includes several sub-steps which make up a single iteration in a loop for calculating a single difference entry. In sub-step 610A, a subinterval is selected in which to begin computation of the cuπent difference table entry being calculated. The cuπent subinterval is refeπed to as "P" within Fig. 65A. Subintervals are also typically selected in sequential order. For example, in calculating table entries for groups 0-3 in Fig. 65B, computations first begin in subinterval 652A, then subinterval 652B, etc. In sub-step 610B, the midpoint (XI) and coπesponding output value (R=f(Xl)) are computed for the sub-subinterval of group N located within cuπent subinterval P. For example, if the cuπent subinterval P is 652A and the current group N is group 0, the midpoint and coπesponding output value are computed for sub- subinterval 654A. In one embodiment, midpoint XI is calculated as shown in Fig. 64B. That is, the midpoint XI is calculated such that f(Xl) is halfway between the maximum and minimum output values for the sub- subinterval for which the midpoint is being calculated. The midpoints (660A-660P) are shown in Fig. 65B for each sub-subinterval within interval 650. Next, in sub-step 610C, a midpoint(X2) and coπesponding output value (R2=f(X2)) are calculated for a reference sub-subinterval within cuπent subinterval P. This reference sub-subinterval is the sub-subinterval within cuπent subinterval P for which the base value is ultimately calculated (as is described below with reference to Fig. 66A). In one embodiment, the reference sub-subinterval is the last sub-subinterval within a given subinterval. In Fig. 65B, for example, the reference sub-subintervals are those in group 3. In sub-step 610D, the difference between the midpoint output values (R1-R2) is added to the cuπent value of SUM. This effectively keeps a running total of the difference values for the group being calculated. The difference values for each sub-subinterval are represented by vertical lines 662 in Fig. 65B. Note that the difference value for the reference sub-subinterval in each subinterval is zero. In step 612, a determination is made whether cuπent subinterval P is the last (J-lth) subinterval in interval M. If P is not the last subinterval in interval M, processing returns to step 610. In sub-step 610A, the next subinterval (sequential to that previously processed) is selected as subinterval P. Computations are made in sub-steps 610B-C of the midpoint and midpoint output values for the group N sub-subinterval and reference sub-subinterval within the newly-selected subinterval P. The new R1-R2 computation is performed and added to the SUM variable in sub-step 610D. This processing continues until all subintervals in interval M have been traversed. For example, step 610 is executed four times to calculate a difference table entry for group 0 sub- subintervals in interval 650. When step 612 is performed and cuπent subinterval P is the last subinterval in interval M, method 600 continues with step 620. In step 620, the cuπent value of SUM is divided by the number of times step 610 was performed (which is equal to the number of subintervals/intervals, or J). This operation produces a value AVG, which is indicative of the average of the difference values for a particular group. Entry 0 of the difference table for interval 650 coπesponds to the sub-subintervals in group 0. This entry is calculated by the average of difference values represented by lines 662A, 662D, 662G, and 662J in Fig. 65B. Note that the difference entries for group 3 in this embodiment are zero since group 3 includes the reference sub-subintervals. In step 622, the floating-point value AVG is converted to an integer format for storage in difference table 530. This may be performed, in one embodiment, by multiplying AVG by 2 P+1 , where P is the number of bits in base table 520, and the additional bit accounts for the implicit leading one bit. A rounding constant may also be added to the product of AVG*2 P+1 in one embodiment. In step 624, the integer computed in step 622 may be stored to the difference table entry for interval M, sub-subinterval group N. Typically, all the entries for an entire table are computed during design of a microprocessor which includes table 500. The table values are then encoded as part of a ROM within the microprocessor during manufacture. In step 630, a determination is made whether group N is the last sub-subinterval group in interval M. If group N is not the last group, method 600 continues with step 606, in which the next sub-subinterval group is selected. The SUM variable is reset in step 608, and difference table entry for the newly-selected sub- subinterval group is computed in steps 610, 612, 620, and 622. When group N is the last sub-subinterval group in interval M, method 600 completes with step 632. As stated above, method 600 is usable to calculate difference tables for a single interval. Method 600 may be repeatedly executed to calculate difference table entries for additional intervals of f(x). As described above, the base value in look-up table 500 includes an approximate function value for each subinterval. As shown in Fig. 65B, this approximate function value for each subinterval coπesponds to the midpoint of the reference sub-subinterval within the subinterval. For example, the approximate function value for subinterval 652A in Fig. 65B is the function value at midpoint 660D of sub-subinterval 654D. An approximate function value for another sub-subinterval within subinterval 652A may then be calculated by adding the function value at midpoint 660D with the difference table entry for the appropriate interval sub- subinterval group. Because of the averaging between subintervals used to compute difference table 530 entries, for a given interval (interval 650, for example), the differences (and, therefore, the result of the addition) are too small in the first subintervals in interval 650 (i.e., subintervals 652A-B). Conversely, the differences (and result of the addition) are too large in the last subintervals in interval 650 (subintervals 652C-D). Furthermore, within a given subinterval, eπor varies according to the sub-subinterval position due to difference value averaging. Difference value eπor from averaging refers to the difference between a computed midpoint for a sub- subinterval and the actual table output (a base-difference sum) for the group which includes the sub-subinterval. Within the last sub-subinterval in a subinterval, this eπor is zero. In the first sub-subinterval within the submterval, however, this eπor is at its maximum In one embodiment, it is desuable to compute base table entnes for a given submterval such that maximum eπor is distnbuted evenly throughout the submterval Graphs illustrating the result of this process are depicted m Figs. 14A-D, with an actual method for this computation descnbed with reference to Fig. 67 Turning now to Fig. 66A, a graph 700 is shown of a portion of function f(x) (denoted by reference numeral 642) from Fig. 14B Only submterval 652A is shown m Fig. 66A As m Fig 65B, submterval 652A includes four sub-submtervals (654A-D), each havmg a coπespondmg midpomt 660 Graph 700 further mcludes a Ime segment 702, which illustrates the actual look-up table outputs 704 for each sub-submterval 654 of submterval 652A. These actual look-up table outputs are equal to the base entry plus the coπespondmg difference table entry. As descnbed above, for the first submtervals (such as 652A) in submterval 650, the result of the base- difference addition is smaller than computed midpoints for the sub-submtervals m the submterval This can be seen m Fig. 66A, as actual look-up table output 704A is less than computed midpomt 660A Furthermore, for the embodiment shown m Fig 66A, the sub-submterval with the maximum eπor withm submterval 652A is sub-submterval 654A. The difference between computed midpomt 660A and actual look-up table output 704A is shown as maximum eπor value 706. Actual look-up table outputs 704B and 704C m sub-submtervals 654B and 654C are also less than then respective computed midpoints, but not by as large a margin as m sub- submterval 654A. Sub-submterval 654D, however, is used as the reference sub-submterval, and as a result, actual look-up table output 704D is equal to computed midpomt 660D. Turning now to Fig 66B, a graph 710 is shown of a portion of function f(x) (denoted by reference numeral 642) from Fig. 14B. Only submterval 652D is shown m Fig. 66B. As m Fig. 65B, submterval 652D mcludes four sub-submtervals (657A-D), each havmg a coπespondmg midpomt 660 Graph 710 further mcludes a Ime segment 712, which depicts the actual look-up table outputs 714 for each sub-submterval 657 of submterval 652D. As m Fig. 66A, these actual look-up table outputs are equal to the base entry plus the coπespondmg difference table entry As descnbed above, for the last submtervals (such as 652D) m submterval 650, the result of the base/difference addition is larger than computed midpoints for the sub-submtervals m the submterval This can be seen in Fig. 66B, as actual look-up table output 714A is greater than computed midpomt 660M. For the embodiment shown m Fig 66B, the sub-submterval with the maximum eπor is withm submterval 652D is sub-submterval 657A. This difference between computed midpomt 660M and actual look-up table output 714A is shown as maximum eπor value 716 Actual look-up table outputs 714B and 714C m sub-submtervals 657B and 657C are also greater than their respective computed midpoints, but not by as large a margin as m sub- submterval 657A. Sub-submterval 657D, however, is used as the reference sub-submterval, and as a result, actual look-up table output 714D is equal to computed midpomt 660P. In one embodiment, the base value for a submterval may be adjusted (from the function output value at the midpomt of the reference sub-submterval) m order to more evenly distribute the maximum eπor value Although adjusting the base values mcreases eπor withm the reference sub-submterval, overall eπor is evenly distnbuted across all sub-submtervals m a submterval. This ensures that eπor is nununized withm a submterval no matter which sub-submterval bounds the mput value. Turning now to Fig. 66C, a graph 720 is depicted which illustrates portion 642 of function f(x) coπesponding to subinterval 652A. Graph 720 also includes a line segment 724, which is equivalent to line segment 702 with each table value adjusted by an offset. Values making up line segment 724 are adjusted such that the eπor in sub-subinterval 654 A is equal to the eπor in sub-subinterval 654D. The eπor in sub-subinterval 654A is given by the difference between computed midpoint 660A of sub-subinterval 654A and adjusted lookup table output value 722A. This difference is denoted by -Δf(x) 726A in Fig. 66C. The eπor in sub- subinterval 654D is given by the difference between adjusted look-up table output value 722D and computed midpoint 660D of subinterval 654D. This difference is denoted by Δf(x) 726B. Thus, the eπor in sub- subinterval 654A and the eπor in sub-subinterval 654D are equal in magnitude, but opposite in sign. Turning now to Fig. 66D, a graph 730 is depicted which illustrates portion 642 of function f(x) corresponding to subinterval 652D. Graph 730 also includes a line segment 734, which is equivalent to line segment 712 with each table value adjusted by an offset. Unlike the offset value in Fig. 66C, which is positive, the offset value in Fig. 66D is negative. With this offset value, the values which make up line segment 734 are adjusted such that the eπor in sub-subinterval 657A is equal to the eπor in sub-subinterval 657D. The eπor in sub-subinterval 657A is given by the difference between adjusted look-up table output value 732 A and computed midpoint 660M. This difference is denoted by Δf(x) 736A in Fig. 66D. Similarly, the eπor in sub- subinterval 657D is given by the difference between computed midpoint 660P of subinterval 657D and adjusted look-up table output value 732D. This difference is denoted by -Δf(x) 736B. Thus, the eπor in sub-subinterval 657 A and the eπor in sub-subinterval 657D are equal in magnitude, but opposite in sign. The method by which the adjustments of Figs. 14C and 14D are made is described below with reference to Fig. 67. Turning now to Fig. 67, a flowchart of a method 800 is depicted for computing base table entries for a bipartite look-up table such as look-up table 500 of Fig. 62. Method 800 may be performed in conjunction with method 600 of Fig. 65A, or with other methods employed for computation of difference table entries. As needed, method 800 is also described with reference to Figs. 65A-D. Method 800 first includes a step 802 in which the input range of f(x) is partitioned. Step 802 is identical to step 602 of method 600, since base and difference values are computed according to the same partitioning. Method 800 next includes step 804, in which difference table entries are calculated. This may be performed using method 600 or other alternate methods. In the embodiment shown in Fig. 67, difference entries are computed prior to base values since difference values are referenced during base value computation (as in step 822 described below). Once difference table entries are calculated, computation of base table values begins with step 806, in which an interval (refeπed to as "M") is selected for which to calculate the entries. As with method 600, method 800 is usable to calculate entries for a single interval of a function input range. The steps of method 800 may be repeatedly performed for each interval in an input range. In the embodiment shown in Fig. 67, J base tables (one for each subinterval) are calculated for interval M. In step 810, one of the J subintervals of interval M is selected as a current subinterval P. The first time step 808 is performed during method 800, the first subinterval within interval M is selected as subinterval P. Successive subintervals are selected on successive executions of step 808. Cuπently selected subinterval P is the subinterval for which a base table entry is being calculated. In step 810, an initial base value (B) is computed for cuπently selected subinterval P. In one embodiment, B coπesponds to the function value at the midpoint (X2) of a predetermined reference sub- subinterval, where the midpoint is calculated as described with reference to Fig. 64B. (The midpoint of the reference sub-subinterval for subinterval P is denoted as X2 in order to be consistent with the terminology of Fig. 65A). The initial base value is thus given by the equation B=f(X2). In one embodiment of look-up table 500 (such as in Figs. 64B and 65A-D), the reference sub-subinterval (Q) is the last, or (K-l)th, sub-subinterval in each subinterval, where each subinterval includes sub-subintervals 0 to K-l. Next, in step 812, a function value (D) is computed which coπesponds to the midpoint (X3) of a sub- subinterval (R) within subinterval P which has the greatest difference value from reference sub-subinterval Q. If reference sub-subinterval Q is the last sub-subinterval in subinterval P, then sub-subinterval R is the first, or 0th, sub-subinterval. For example, in Fig. 66A, sub-subinterval 654D is reference sub-subinterval Q, while sub- subinterval 654A is sub-subinterval R. The function value D is thus given by the equation D=f(X3), where X3 is the midpoint of sub-subinterval R calculated as described above with reference to Fig. 64B in one embodiment. In step 820, the difference, (refeπed to as "actual difference" in Fig. 67), is computed between D and B. This is representative of what the maximum difference value would be for subinterval P if difference value averaging were not employed, since sub-subinterval R has the maximum difference value in relation to sub- subinterval Q as described above. Next, in step 822, the difference table entry (computed previously in step 804) is referenced for subinterval P, sub-subinterval R. (In method 600, however, a dedicated difference table entry does not exist solely for subinterval P, sub-subinterval R. Rather, a difference table exists for subinterval P and a group of sub-subintervals N within interval M which includes sub-subinterval R). The difference table entry referenced in step 822 is refeπed to as the averaged difference value ("avg. diff."). In step 824, the maximum eπor that results from using averaged difference values is calculated. This is performed by setting max eπor = actual diff. - avg. diff. As shown in Figs. 14C and 14D, the maximum eπor from the averaged difference table values occurs in the first sub-subinterval in the subinterval (e.g., sub- subintervals 654A and 657A). In fact, the max eπor computed in step 824 of method 800 is equal to max eπor values 706 and 716 in Figs. 14C and 14D. In order to distribute the maximum eπor of step 824 throughout subinterval P, an adjust value is computed as a fraction of max eπor in step 826. In order to evenly distribute the eπor throughout the subinterval, the adjust value is computed as half the maximum eπor value. Then, in step 828, the final base value is computed from the initial base value B by adding the adjust value. In step 830, the final value as computed in step 828 is converted to an integer value. As with the integer conversion of the difference value in step 622 of method 600, the conversion of step 830 may be performed in one embodiment by multiplying the final base value by 2 P+I and adding an optional rounding constant. In alternate embodiments, the integer conversion may be performed differently. In step 832, the converted integer value is ready for storage to the base table entry for interval M, subinterval P. The base table entries may be stored to the table one-by-one, but typically they are all computed then stored to the ROM that includes the look-up table. In step 834, a determination is made of whether submterval P is the last submterval m mterval M. If more submtervals exist, method 800 continues with step 808 In step 808, a next submterval withm mterval M is selected, and the succeedmg steps are usable to calculate the base value for the newly-selected submterval On the other hand, if P is the last submterval m mterval M, method 800 concludes with step 836 Methods for calculation of difference and base table entnes are descnbed in a general manner with reference to Figs 13A and 15, respectively Source code which implements these methods (for the reciprocal and reciprocal square root functions) is shown below for one embodiment of the present mvention Note that the #define's for HIGH, MID, and LOW effectively partition the mput range of these functions mto four mtervals, four subintervals/interval, and four sub-subrntervals/subinterval. #define HIGH 2 #defme MID 2 #define LOW 2 #define OUT 16 #define OUTP 16 #define OUTQ (OUTP-(HIGH+MID)+l) #defme RECIPENTRIES ( 1 L « (HIGH+MID)) #defme ROOTENTRIES (2L « (HIGH+MID)) #defme BIAS 127L /* exponent bias for smgle precision format */ #define POW2(x) (IL « (x)) /* helper function */ typedef union { float f; unsigned long i; } SINGLE; #define SIGN_SINGLE(var) ((((var).ι)&0x80000000L)?lL:0L) /* sign bit */ #define EXPO_SINGLE(var) ((((var).ι)»23L)&0xFFL) /* 8 bit exponent */ #defme MANT_SINGLE(var) (((var).ι)&0x7FFFFFL) /* 23 bit mantissa */ #define SETSIGN_SINGLE(var,sιgn) \ (((var).ι)=((sιgn)&l)^(((var).ι)|0x80000000L) (((var) ι)&0x7FFFFFFFL)) #define SETEXPO_SINGLE(var,expo) \ (((var) ι)=(((var).ι)&0x807FFFFFL)|(((expo)&0xFFL)«23)) #defme SETMANT_SINGLE(var,mant) \ (((var) ι)=(((var).ι)&0xFF800000L)|(((mant)&0x7FFFFFL))) extern unsigned long rom_p[], extern unsigned long rom_q[], #define TRUE 1 #define FALSE 0 rfdefine HIGHMID (HIGH+MID) #defme HIGHLOW (HIGH+LOW) #define ALL (HIGH+MID+LOW) #defme POW2(x) ( 1 L « (x)) #define CONCAT(a,b,c) ((0x7FL « 23) | \ (((a) & (POW2(HIGH) - 1)) « (23 - (HIGH))) | \ (((b) & (POW2(MID) - 1)) « (23 - (HIGHMID))) | \ (((c) & (POW2(LOW) - 1)) « (23 - (ALL)))) #define CONCAT2(e,a,b,c) (((e) « 23) | \ (((a) & (POW2(HIGH) - 1)) « (23 - (HIGH))) | \ (((b) & (POW2(MID) - 1)) « (23 - (HIGHMID))) | \ (((c) & (POW2(LOW) - 1)) « (23 - (ALL)))) void make_recip_bipartite_table (unsigned long *tablep, unsigned long *tableq) { unsigned long xh, xm, xl, indexp, indexq, maxq, minq, maxp, minp; SINGLE temp 1 , temp2 ; double midpoint 1, midpoint2; double result, sumdiff, resultl, result2, adjust ; printf ("\nCreating lookup tables ...\n"); for (xh = 0; xh < POW2(HIGH); xh++) { for (xl = 0; xl < POW2(LOW); xl++) { indexq = (xh « LOW) | xl; sumdiff = 0.0; for (xm = 0; xm < POW2(MID); xm++) { templ.i = CONCAT (xh, xm, xl); temp2.i = (templ.i | (POW2(23 - ALL) - 1)) + 1; midpointl = (2.0 * templ.f * temp2.f) / (templ.f + temp2.f); templ.i = CONCAT (xh, xm, POW2(LOW)-l); temp2.i = (templ.i | (POW2(23 - ALL) - 1)) + 1; midpoint2 = (2.0 * templ.f * temp2.f) / (templ.f + temp2.f); sumdiff = sumdiff + ((1.0 / midpoint 1 ) - ( 1.0 / midpoint2)); } result = 1.0/((double)(POW2(MID))) * sumdiff; tableq [indexq] = (unsigned long)(POW2(OUTP+ 1 ) * result + 0.5); } } for (xh = 0; xh < POW2(HIGH); xh++) { for (xm = 0; xm < POW2(MID); xm++) { indexp = (xh « (MID)) | xm; tempi .i = CONCAT (xh, xm, 0); temp2.i = (templ.i | (POW2(23 - ALL) - 1)) + 1; midpointl = (2.0 * templ.f * temp2.f) / (templ.f + temp2.f); resultl = 1.0 / midpointl; tempi .i = CONCAT (xh, xm, POW2(LOW) - 1); temp2.i = (templ.i | (POW2(23 - ALL) - 1)) + 1; midpoint2 = (2.0 * templ.f * temp2.f) / (templ .f + temp2.f); resulf2 = 1.0 / midpoint2; adjust = 0.5 * ((resultl - result2) - (1.0/POW2(OUTP+1)) * tableq[xh « LOW]); tablep [indexp] = (unsigned long)(POW2(OUTP+l) * (result2 + adjust) + 0.5); tablep [indexp] -= (IL « OUTP); /* subtract out integer bit */ } void make_recιpsqrt_bιpartιte_table (unsigned long *tablep, unsigned long *tableq) { unsigned long xh, xm, xl, mdexp, indexq, maxq, minq, start, end, maxp, mmp, expo; SINGLE tempi, temp2, double midpoint 1 , mιdpomt2 ; double result, adjust, sumdiff, resultl, result2, pnntf ("VnCreating lookup tables ...\n"); for (expo = 0x7F; expo <= 0x80; expo++) { for (xh = 0; xh < POW2(HIGH); xh++) { for (xl = 0; xl < POW2(LOW); xl++) { indexq = ((expo & 1) « (HIGHLOW)) | (xh « LOW) | xl; sumdiff = 0.0; for (xm = 0; xm < POW2(MID), xm++) { temp l.i = CONCAT2 (expo, xh, xm, xl); temp2.ι = (templ.i | (POW2(23 - ALL) - 1)) + 1; midpomtl = (4.0 * templ.f * temp2.f) / ((sqrt( tempi. f)+sqrt(temp2.f))*(sqrt(templ.f)+sqrt(temp2.f))); templ.i = CONCAT2 (expo, xh, xm, POW2(LOW)-l); temp2.ι = (templ.i | (POW2(23 - ALL) - 1)) + 1; mιdpomt2 = (4.0 * templ.f * temp2.f) / ((sqrt(templ.f)+sqrt(temp2.f))*(sqrt(ternpl.f)+sqrt(temp2.f))); sumdiff = sumdiff + ((1.0 / sqrt(rmdpomtl)) - (1.0 / sqrt(mιdpornt2))); } result = 1.0/((double)(POW2(MID))) * sumdiff; tableq [indexq] = (unsigned long)(POW2(OUTP+l) * result + 0.5); } } for (xh = 0; xh < POW2(HIGH); xh++) { for (xm = 0; xm < POW2(MID); xm++) { mdexp = ((expo & 1) « (HIGHMID)) | (xh « (MID)) | xm; templ.i = CONCAT2 (expo, xh, xm, 0); temp2.ι = (templ.i | (POW2(23 - ALL) - 1)) + 1 ; midpomtl = (4.0 * templ.f * temp2.f) / ((sqrt(templ f)+sqrt(temp2.f))*(sqrt(templ.f)+sqrt(temp2.f))), resultl = 1.0 / sqrt(rmdpomtl), tempi .1 = CONCAT2 (expo, xh, xm, POW2(LOW) - 1); temp2.ι = (templ.i | (POW2(23 - ALL) - 1)) + 1 ; πudpomt2 = (4.0 * templ.f * temp2.f) / ((sqrt(templ.f)+sqrt(temp2.f))*(sqrt(templ .f)+sqrt(temp2.f))); result2 = 1.0 / sqrt(mιdpoint2); adjust = 0.5 * ((resultl - result2) - (1.0/POW2(OUTP+1)) * tableq[((expo & 1) « (HIGH+LOW)) | (xh « LOW)]); tablep [mdexp] = (unsigned long)(POW2(OUTP+l) * (result2 + adjust) + 0.5); tablep [indexp] -= (IL « OUTP); /* subtract out mteger bit */ } } } } void recιp_approx_bιpartιte ( const SINGLE *arg, const unsigned long *tablep, const unsigned long *tableq, unsigned long high, unsigned long mid, unsigned long low, unsigned long out, SINGLE *approx) { unsigned long expo, sign, mant, mdexq, mdexp, p, q, /* handle zero separately */ if ((arg->ι & 0x7F800000L) = 0) { approx->ι = (arg->ι & OxδOOOOOOOL) | 0x7F7FFFFFL, return, } /* unpack arg */ expo = (arg->ι » 23) & OxFF, mant = (arg->ι & 0x7FFFFFL), /* do table lookup on tables P and Q */ mdexp = (mant » (23 - (high + mid))), mdexq = ((mant » (23 - (high))) « low) | ((mant » (23 - (high+mid+low))) & (POW2(low) - 1)), p = tablep [mdexp], q = tableq [mdexq], /* generate result m smgle precision format */ approx->ι = ((2*BIAS + -expo) « 23L) + (((p + q)) « (23L - out)), /* check for underflow */ if ((((approx->ι » 23) & OxFFL) = OxOOL) || (((approx->ι » 23) & OxFFL) = OxFFL)) { approx->ι = 0L, } /* mask sign bit because exponent above may have overflowed mto sign bit */ approx->ι = (approx->ι & 0x7FFFFFFFL) | (sign « 3 IL), } void recipsqrt approx bipartite ( const SINGLE *arg, const unsigned long *tablea, const unsigned long *tableb, unsigned long high, unsigned long mid, unsigned long low, unsigned long out, SINGLE *approx) { unsigned long sign, mant, mdexq, mdexp, p, q, long expo; /* Handle zero separately. Returns maximum normal */ if ((arg->ι & 0x7F800000L) = OL) { approx->ι = 0x7F7FFFFFL | (arg->ι & 0x80000000L), return; } expo = (arg->ι » 23) & OxFFL, sign = (arg->ι » 31 ) & 1 ; mant = (arg->ι & 0x7FFFFFL); mdexp = ((expo & 1) « (high + mid)) | (mant » (23 - (high + mid))), mdexq = ((expo & 1) « (high + low)) | ((mant » (23 - (high))) « low) | ((mant » (23 - (high + mid + low))) & (POW2(low) - 1)), p = tablea [mdexp]; q = tableb [mdexq]; approx->ι = (((3*BIAS + -expo) » 1) « 23) + (((p + q)) « (23 - out)); approx->ι |= sign « 31 , To further clanfy calculation of base and difference table entries m the embodiment coπespondmg to the above source code, sample table portions are given below These table portions are for the reciprocal function only, although the reciprocal square root table entnes are calculated similarly. The mput range (1.0 mclusive to 2.0 exclusive) for this example is divided mto four mtervals, four submtervals/mterval, and four sub-subintervals/subinterval. The table values are only shown for the first mterval (1.0 mclusive to 1.25 exclusive) for simplicity. The difference table for this example receives a four bit mdex (two bits for the mterval, two bits for the sub-submterval group) The base table also receives a four bit mdex (two bits for the mterval, two bits for the submterval). The base table mcludes 16 bits, while the difference table mcludes 13 bits for this embodiment Table 3 Table 1 illustrates the partitioning of the first mterval of the mput range of the reciprocal function. With regard to the bmary representation of A, only the six high-order mantissa bits are shown smce these are the ones that are used to specify the mterval, submterval, and sub-submterval group of the mput sub-region. Note that the first group of mantissa bits of A coπesponds to the mterval number, the second group coπesponds to the submterval number, and the third group coπesponds to the sub-submterval group. Table 2 shows the midpomt of each sub-submterval (computed as m Fig. 54B), as well as the function evaluation at the midpomt and the difference value with respect to the reference sub-submterval of the submterval. (The reference sub-submtervals are those m group 3). Table 4 Table 3 shows the difference value average for each sub-subinterval group. Additionally, Table 3 includes the difference average value in integer form. This integer value is calculated by multiplying the difference average by 2 17 , where 17 is the number of bits in the output value (including the leading one bit). Table 5 With regard to the base values for this example, Table 4 below shows midpoints X2 and X3. Midpoint X2 is the midpoint for the reference sub-subinterval of each subinterval, while X3 is the midpoint of the sub- subinterval within each subinterval that is furthest from the reference sub-subinterval. The table also shows the function values at these midpoints. Table 6 Next, Table 5 below shows the actual eπor difference for each subinterval, computed as 1/X3-1/X2. Table 5 additionally shows the average difference value, which is equal to the previously computed difference value for sub-subinterval group 0. The difference between the actual difference and the average difference is equal to the maximum eπor for the subinterval. Half of this value is the adjust value. Table 7 In Table 6, The adjust value plus the initial base value gives die final base value. This final base value is converted to an 16-bit integer value by multiplying by 2 17 and discarding the most significant 1 bit (which coπesponds to the integer position). Table 8 As stated above, the bipartite table look-up operation is usable to obtain a starting approximation for mathematical functions such as the reciprocal and reciprocal square root implemented within a microprocessor. In one embodiment, the table look-up is initiated by a dedicated mstruction withm the mstruction set of the microprocessor Additional dedicated mstructions may be employed m order to implement the iterative evaluations which use the starting approximation to produce the final result for these functions This, in turn, leads to a faster function evaluation tune In one embodiment, base and difference values calculated as descnbed m Figs 13 A and 15 result m table output values with minimized absolute eπor Advantageously, this mmimal absolute eπor is obtamed with a bipartite table configuration, which requires less table storage than a naive table of comparable accuracy This configuration also allows the mterpolation to be achieved with a simple addition Thus, a costly multiply or multiply-add is not requued to generate the final table output, effectively mcreasmg the performance of the table look-up operation It is noted that while base and difference tables have been descnbed above with reference to the reciprocal and reciprocal square root functions, such tables are generally applicable to any monotomcally decreasing function These tables are also applicable to a function which is monotomcally decreasmg over the desued mput range In another embodiment, these base and difference tables may be modified to accommodate any monotomcally mcreasmg function(such as sqrt(x)), as well as any function which is monotomcally mcreasmg over a desired input range In such an embodiment, the "leftmost" sub-submterval withm an mterval becomes the reference pomt mstead of the "nghtmost" sub-submterval, ensuring the values m the difference tables are positive Alternatively, the "nghtmost" sub-submterval may still be used as the reference pomt if difference values are considered negative and a subtractor is used to combme base and difference table values Numerous vanations and modifications will become apparent to those skilled m the art once the above disclosure is fully appreciated It is mtended that the followmg claims be mteφreted to embrace all such vanations and modifications

Claims: WHAT IS CLAIMED IS: 1 A method for generatmg entries for a bipartite look-up table havmg a base table portion and a difference table portion, wherem said bipartite look-up table is usable to generate output values for a given mathematical function over a predetermined mput range which is divided mto a plurality of mtervals, said method compnsing computmg a first base table entry which coπesponds to a first sub-range withm a first mterval of said predetermmed mput range, wherem said first mterval mcludes a plurality of sub-ranges each divided mto a plurality of sub-sub-ranges, computmg a first difference table entry which coπesponds to a first group of sub-sub-ranges withm said first interval, wherem said first group of sub-sub-ranges mcludes a first sub-sub-range which is located withm said first sub-range, wherem said first base table entry and said first difference table entry are usable to generate a first output value for an input value located withm said first sub-sub-range, wherem said first output value has a mmimized amount of absolute eπor for all mput values withm said first sub-sub-range 2 A bipartite look-up table, compnsmg a base table portion for storing base entnes for a given mathematical function over a predetermmed mput range, wherem said predetermmed mput range is divided mto a plurality of mtervals, wherem said base table portion mcludes a first base table entry corresponding to a first sub-range withm a first mterval of said predetermmed mput range, a difference table portion for storing difference entnes for said given mathematical function over said predetermmed mput range, wherem said difference table portion mcludes a first difference table entry coπespondmg to a first group of sub-sub-ranges withm said first mterval, wherem said first group of sub-subranges includes a first sub-sub-range which is located withm said first sub-range, wherem said bipartite look-up table is configured to generate a first output value m response to receivmg a first input value located withm said first sub-sub-range, wherem said first output value has a mmimized amount of absolute eπor for all mput values withm said first sub-sub-range 3 A method for generatmg entnes for a bipartite look-up table which mcludes a base table and a difference table, wherem said bipartite look-up table is configured to provide an output value for a given mathematical function in response to receivmg a coπespondmg mput value withm a predetermmed mput range, said method compnsmg (a) dividing said predetermmed mput range mto a predetermmed number of equal mtervals mcludmg a first interval, (b) dividing said first mterval mto a predetermmed number of equal submtervals mcludmg a first subinterval, wherem said first submterval mcludes said corresponding mput value, (c) dividing each of said predetermmed number of equal submtervals in said first mterval mto a predetermined number of equal sub-subintervals mcludmg a first sub-submterval withm said first submterval, wherein said first sub-submterval includes said coπespondmg input value, (d) computing a first difference table entry which corresponds to a given group of sub- submtervals withm said first mterval, wherem each of said given group of sub-submtervals is located withm a coπespondmg one of said predetermmed number of equal submtervals withm said first mterval, and wherem each of said given group of sub-submtervals has a common relative position withm said coπespondmg one of said predetermmed number of equal submtervals withm said first mterval, wherem said given group of sub- submtervals mcludes said first sub-submterval, (e) computing a first base table entry which coπesponds to said first submterval withm said first mterval, wherem said first difference table entry and said first base table entry are usable to generate said output value of given mathematical function for said corresponding mput value, and wherem said first difference table entry and said first base table entry are computed such that said output value has a mmimum amount of possible absolute error for all mput values withm said first sub-submterval withm said first submterval of said first mterval 4 The method of claim 3, wherem said computing said first difference table entry mcludes (l) selecting a given submterval withm said first mterval as a currently selected submterval, (n) calculating a first midpomt value for a particular sub-submterval withm said cuπently selected submterval which is mcluded m said given group of sub-submtervals, wherem evaluating said given mathematical function at said first midpomt value produces a first function value which mmimizes absolute eπor for all mput values withm said particular sub-submterval, (in) calculating a second midpomt value for a reference sub-submterval of said currently selected submterval, wherem evaluating said given mathematical function at said second midpomt value produces a second function value which mmimizes absolute error for all input values withm said reference sub-submterval of said cuπently selected submterval, (IV) computing a difference value between said first function value and said second function value, (v) repeating steps (ι)-(ιv) usmg each remammg submterval m said first mterval as said cuπently selected submterval, wherem said repeating mcludes computing a difference total of each said difference value, (vi) determining an difference value average from said difference total, (vn) storing said difference value average as said first difference table entry, wherem said first difference table entry is usable to compute a value of said given mathematical function for an mput value located m any of said given group of sub-submtervals withm said first mterval 5 The method of claim 4 wherem said particular sub-submterval of said cuπently selected submterval for which said first midpomt value is calculated mcludes a first mput range havmg a smallest first input value and a largest first mput value, wherem evaluating said given mathematical function at said smallest first mput value produces a thud function value, and wherem evaluating said given mathematical function at said greatest first mput value produces a fourth function value 6 The method of claim 3, further compnsing dividing each remammg one of said predetermmed number of equal mtervals mto said predetermmed number of equal submtervals, and dividing each of said predetermmed number of equal submtervals withm said each remammg one of said predetermmed number of equal mtervals mto said predetermmed number of equal sub-submtervals 7 A method for generating entnes for a difference table portion of a bipartite look-up table mcludmg difference values of a given mathematical function over a predetermmed mput range, said method compnsmg (a) dividing said predetermmed mput range mto a predetermmed number of equal intervals mcludmg a first mterval, (b) dividing said first mterval mto a predetermmed number of equal submtervals mcludmg a first submterval, (c) dividing each of said predetermmed number of equal submtervals m said first mterval mto a predetermined number of equal sub-submtervals, wherem each of said predetermmed number of equal submtervals m said first interval mcludes a reference sub-submterval, (d) selectmg a first group of sub-submtervals withm said first mterval, wherem each of said first group of sub-submtervals is located m a coπespondmg submterval of said first mterval, and wherem each of said first group of sub-submtervals has a common relative position with said corresponding submterval, (e) calculating a first difference table entry for said first group of sub-submtervals, wherem said first difference table entry is an average of difference values computed between each of said first group of sub- submtervals and said reference sub-submtervals m said coπespondmg submterval, and wherem said first difference table entry is usable to compute a function output value which has mmimal absolute error for all possible mput values withm said first group of sub-submtervals 8 The method of claim 7, wherem said calculating said first difference table entry mcludes (l) selecting a given submterval as a currently selected submterval of said first mterval, (n) calculating a first midpomt value for a particular sub-submterval of said cuπently selected submterval which is mcluded m said first group of sub-submtervals, wherem evaluatmg said given mathematical function at said first midpomt value produces a first function value which mmimizes absolute eπor for all mput values withm said particular sub-submterval of said currently selected submterval, (in) calculating a second midpoint value withm said reference sub-submterval of said cuπently selected submterval, wherem evaluatmg said given mathematical function at said second midpomt value produces a second function value which minimizes absolute eπor for all mput values within said reference sub-submterval of said cuπently selected submterval, (iv) computmg a difference value between said first function value and said second function value, (v) repeatmg steps (ι)-(ιv) usmg each remammg submterval m said first mterval as said cuπently selected submterval, wherein said repeating mcludes summmg each said difference value to produce a difference total, (vi) determining an difference value average from said difference total; (vn) storing said difference value average as said first difference table entry, wherem said first difference entry is usable to compute a value of said given mathematical function for an mput value located of any of said first group of sub-submtervals. 9 The method of claim 8, wherem said given mathematical function is f(x)=l/x or f(x)=l/sqrt(x) 10. A microprocessor, compnsmg an execution unit coupled to receive a first set of mput data values, a second set of mput data values, and an mstruction mdication specifymg an operation to be performed by said execution unit, wherem said execution unit includes: an mput multiplexer unit coupled to receive said first set of mput data values, said second set of input data values, and said mstruction mdication, wherem said mput multiplexer is configured to select a first set of operands and a second set of operands from said first pau of mput data values and said second pau of mput data values m response to said mstruction mdication; a first add/subtract pipelme coupled to receive said first set of operands and said mstruction mdication, wherem said first add subtract pipelme is configured to generate a first result value from said first set of operands by performing an anthmetic operation specified by said mstruction mdication; a second add/subtract pipelme coupled to receive said second set of operands and said mstruction indication, wherem said second add/subtract pipelme is configured to generate a second result value from said second set of operands by performing said anthmetic operation specified by said mstruction mdication, wherem said first result value and said second result value are generated concuπently. 11. The microprocessor of claim 10, wherem said first pair of mput data values mcludes a first data value and a second data value, and wherem said second pair of mput data values mcludes a thud data value and a fourth data value, wherem said mput multiplexer unit is configured to select said first set of operands to mclude said first data value and said thud data value m response to said mstruction mdication specifymg a vectored add operation, and wherem said mput multiplexer unit is further configured to select said second set of operands to include said second data value and said fourth data value m response to said mstruction mdication specifymg said vectored add operation. 12 The microprocessor of claim 11, wherem said first add/subtract pipelme is configured to generate said first result value from a sum of said first data value and said thud data value m response to receivmg said first data value, said thud data value, and said mstruction mdication specifymg said vectored add operation, and wherem said second add subtract pipelme is configured to generate said second result value from a sum of said second data value and said fourth data value m response to receivmg said second data value, said fourth data value, and said mstruction mdication specifying said vectored add operation 13 The microprocessor of claim 12, wherem said mput multiplexer unit is configured to select said first set of operands to mclude said first data value and said second data value in response to said mstruction mdication specifymg a vectored accumulate operation, and wherem said mput multiplexer unit is furthered configured to select said second set of operands to mclude said thud data value and said fourth data value m response to said mstruction mdication specifymg said vectored accumulate operation 14 The microprocessor of claim 11, wherem said mput multiplexer unit is configured to select said first set of operands to mclude said first data value and said thud data value m response to said mstruction indication specifymg either a vectored subtract operation or a vectored reverse subtract operation, and wherem said input multiplexer unit is furthered configured to select said second set of operands to mclude said second data value and said fourth data value m response to said mstruction mdication specifymg either said vectored subtract operation or said vectored reverse subtract operation 15 The microprocessor of claim 10, wherem said first pau of mput data values mcludes a first floating point number, and wherem said second pau of mput data values mcludes a second floating pomt number, wherem said mput multiplexer unit is configured to select said first set of operands to mclude said fust floating pomt number m response to said mstruction mdication specifymg a floating pomt to mteger conversion operation, and wherem said mput multiplexer unit is configured to select said second set of operands to mclude said second floatmg pomt number m response to said mstruction mdication specifymg said floating pomt-to-mteger conversion operation 16 The microprocessor of claim 15, wherem said first add/subtract pipelme is configured to generate said first result value by converting said first floatmg pomt number mto a corresponding first mteger value in response to receivmg said first floatmg pomt number and said mstruction mdication specifying said floating pomt-to-mteger conversion operation, and wherem said second add/subtract pipelme is configured to generate said second result value by converting said second floating pomt number mto a coπespondmg second integer value in response to receivmg said second floatmg pomt number and said mstruction indication specifymg said floating pomt-to-mteger conversion operation 17 The microprocessor of claim 11, wherem said first add/subtract pipelme mcludes a first far data path and a first close data path, and wherem said second add/subtract pipelme mcludes a second far data path and a second close data path, wherein said first far data path and said second far data path are configured to perform effective addition operations on received operands, and wherem said first far data path and said second far data path are further configured to perform effective subtraction operations on pans of received floating pomt operands havmg an absolute exponent difference greater than one 18. A microprocessor, comprising: an execution unit coupled to receive a first set of input data values, a second set of input data values, and an instruction indication specifying an operation to be performed by said execution unit, wherein said execution unit includes: a first add/subtract pipeline coupled to receive a first set of operands and said instruction indication, wherein said first set of operands are selected from said first set of input data values and said second set of input data values, and wherein said first add/subtract pipeline is configured to generate a first result value from said first set of operands according to said instruction indication; a second add/subtract pipeline coupled to receive a second set of operands and said instruction indication, wherein said second set of operands are selected from said first set of input data values and said second set of input data values, and wherein said second add subtract pipeline is configured to generate a second result value from said second set of operands according to said instruction indication; an output multiplexer unit coupled to receive said first result value, said second result value, said instruction indication, and one or more additional input values, wherein said output multiplexer unit is configured to select a first output value and a second output value from said first result value, said second result value, and said one or more additional input values according to said instruction indication; wherein said first result value and said second result value are generated concurrently. 19. The microprocessor of claim 18, wherein said first result value and said second result value are generated by an operation selected from the group consisting of: (i) vectored add operation, (ii) vectored subtract operation, (iii) vectored accumulate operation, (iv) vectored reverse subtract operation, (v) floating point-to- integer conversion operation, and (vi) integer-to-floating point conversion operation. 20. The microprocessor of claim 19, wherein said first result value and said second result value coπespond to one of a plurality of arithmetic operations executable by said first add/subtract pipeline and said second add subtract pipeline, wherein said first add/subtract pipeline includes a first far data path and a first close data path, and wherein said second add subtract pipeline includes a second far data path and a second close data path. 21. The microprocessor of claim 20, wherein said first far data path and said second far data path are configured to perform effective addition operations on received operands, and wherein said first far data path and said second far data path are further configured to perform effective subtraction operations on pairs of received floating point operands having an absolute exponent difference greater than one. 22. The microprocessor of claim 21, wherein said first close data path and said second close data path are configured to perform effective subtraction operations on pairs of received floating point operands having an absolute exponent difference less than or equal to one. 23 A microprocessor, compnsmg: an execution unit coupled to receive a first set of mput data values, a second set of mput data values, and an mstruction mdication specifymg an operation to be performed by said execution unit, wherem said execution unit includes: an mput multiplexer unit coupled to receive said first set of mput data values, said second set of mput data values, and said mstruction mdication, wherem said mput multiplexer is configured to select a first set of operands and a second set of operands from said first pau of mput data values and said second pau of mput data values accordmg to said mstruction mdication, a first add/subtract pipelme coupled to receive said first set of operands and said instruction indication, wherem said first add/subtract pipelme is configured to generate a first result value from said first set of operands accordmg to said mstruction mdication; a second add/subtract pipelme coupled to receive said second set of operands and said mstruction mdication, wherem said second add/subtract pipelme is configured to generate a second result value from said second set of operands accordmg to said mstruction mdication; an output multiplexer unit coupled to receive said first result value, said second result value, said mstruction mdication, and one or more additional mput values, wherem said output multiplexer unit is configured to select a first output value and a second output value from said first result value, said second result value, and said one or more additional mput values accordmg to said mstruction mdication. 24. The microprocessor of claim 23, wherem said mput multiplexer is configured to selectively route data values m said first set of mput data values and said second set of mput data values to said first add/subtract pipelme and said second add/subtract pipelme m response to said mstruction mdication specifymg one of a first plurality of anthmetic operations. 25. A microprocessor, compnsmg: an execution unit coupled to receive a first pau of floating pomt mput values and a first control value mdicative of an operation to be performed on said first pau of floating pomt mput values, wherem said first pan of floating pomt mput values mcludes a first floatmg pomt number and a second floating pomt number, wherem said execution unit mcludes: a far data path coupled to receive said first pau of floating pomt mput values and said first control value; a close data path coupled to concurrently receive said first pau of floatmg pomt mput values and said first control value; wherem an effective addition operation is performed on said first pau of floating pomt mput values m said far data path m response to said first control value mdicatmg said effective addition operation; wherem an effective subtraction operation is performed on said first pau of floating pomt mput values m response to said first control value mdicatmg said effective subtraction operation, wherem said effective subtraction operation is performed m said far data path m response to an absolute exponent difference of said first pair of floating pomt numbers bemg greater than one, and wherem said effective subtraction operation is performed in said close data path in response to said absolute exponent difference of said first pair of floating point numbers being less than or equal to one; and wherein a floating point-to-integer conversion operation is performed on said second floating point number in said far data path in response to said first control value indicating said floating point-to- integer conversion operation. 26. The microprocessor of claim 25, wherein said first floating point number includes a first sign bit, a first exponent value, and a first mantissa value, wherein said second floating point number includes a second sign bit, a second exponent value, and a second mantissa value, and wherein said far data path includes a exponent difference generation unit coupled to receive said first exponent value, said second exponent value, and said first control value, wherein said exponent difference generation unit is configured to generate one or more exponent difference values. 27. The microprocessor of claim 26, wherein said exponent difference generation unit is configured to generate said one or more exponent difference values usable to align said first mantissa value and said second mantissa value in response to said first control signal indicating said effective addition operation or said effective subtraction operation. 28. The microprocessor of claim 27, wherein said exponent difference generation unit is configured to generate a first integer conversion shift count within said one or more exponent difference values in response to said first control signal indicating said floating point-to-integer conversion operation, wherein said first integer conversion shift count is usable to shift said second mantissa value to a bit position within said far data path which coπesponds to said second exponent value. 29. The microprocessor of claim 28, wherein said far data path includes a shift unit coupled to receive said first mantissa value, said second mantissa value, and said one or more exponent difference values, wherein said shift unit is configured to generate a shifted first mantissa value from said first mantissa value and a shifted second mantissa value from said second mantissa value according to said one or more exponent difference values. 30. The microprocessor of claim 29, wherein said shifted second mantissa value is generated from said second mantissa value according to said first integer conversion shift count in response to said first control signal indicating said floating point-to-integer conversion operation, wherein a leading one bit within said shifted second mantissa value is located in bit position having an associated exponent magnitude which coπesponds to said second exponent value. 31. The microprocessor of claim 25, wherein an output of said floating point-to-integer conversion operation is clamped at a maximum representable integer in response to said second floating point number being greater than said maximum representable integer. 32. The microprocessor of claim 25, wherein an output of said floating point-to-integer conversion operation is clamped at a minimum representable integer in response to said second floating point number being less than said minimum representable integer. 33. A microprocessor, comprising: a leading one prediction unit configured to predict a position of a leading one value within a result mantissa value coπesponding to a first floating point subtraction operation performed upon a first floating point number and a second floating point number, wherein said leading one prediction unit is coupled to receive a first operand corresponding to said first floating point number and a second operand coπesponding to said second floating point number, wherein said leading one prediction unit is configured to generate a prediction string including a prediction value for each bit position within said result mantissa value, wherein each prediction value within said prediction string is generated by utilizing values from a single coπesponding bit position within said first operand and said second operand, wherein an indication of said position of said leading one value within said result mantissa value is given by a bit position of a most significant asserted prediction value within said prediction string; wherein said prediction string is generated according to a prediction that said first floating point number includes a first exponent value that is one greater than a second exponent value included in said second floating point number. 34. The microprocessor of claim 33, wherein said prediction string includes a first prediction value coπesponding to a most significant bit position of said prediction string, wherein said first prediction value is generated using only values from a second most sigmficant bit position of said first operand and said second operand. 35. A method for detecting a position of a leading one value in a result mantissa value coπesponding to a first floating point subtract operation performed upon a first floating point number and a second floating point number, said method comprising: receiving a first operand coπesponding to said first floating point number; receiving a second operand coπesponding to said second floating point number; forming a prediction string by generating a prediction value for each bit position within said result mantissa value, wherein each prediction value within said prediction string is generated using values of a single corresponding bit position within said first operand and said second operand; determining a position of said leading one value within said result mantissa value by locating a most significant asserted bit position within said prediction string; wherein said prediction string is generated according to a prediction that a first floating point exponent value included in said first floating point number is one greater than a second floating point exponent value included in said second floating point number. 36 A method for performing effective subtraction for floating pomt mput values havmg an absolute exponent difference less than or equal to one, compnsmg receivmg a first mantissa portion coπespondmg to a first floatmg pomt mput value and an inverted version of a second mantissa portion coπespondmg to a second floating pomt mput value; addmg said first mantissa portion and said inverted version of said second mantissa portion m order to produce a first output value and a second output value, wherem said first output value is equal to said first mantissa portion plus said inverted version of said second mantissa portion, and wherem said second output value is equal to said first output value plus one, generating a fust plurality of prelimmary selection signals mdicative of either said first output value or said second output value, wherem each of said first plurality of prelimmary selection signals is generated accordmg to one of a plurality of input/output prediction values, generating a first set of control signals which mdicate which of said plurality of input/output prediction values actually occurs, wherem said first set of control signals are generated usmg a carry m signal corresponding to a most significant bit position of said first output value; selectmg one of said first plurality of prelimmary selection signals as a final select value by utilizing said first set of control signals, selecting either said first output value or said second output value as a preliminary subtraction result accordmg to said final select value 37. The method of claim 1 , further compnsing- detecting that said first output value is negative; inverting said first output value; selectmg said inverted first output value as said preliminary subtraction result. 38 A microprocessor, compnsmg- a first execution unit coupled to receive a given pau of floating pomt mput values, mcludmg a first close data path configured to perform effective subtraction on said given pau of floating pomt mput values by predictmg said given pau of floating pomt mput values to have an absolute exponent difference less than or equal to one, wherem said first close data path mcludes a first close path selection unit configured to generate a first close path selection signal usable to select either a first close path adder result or a second close path adder result as a first close path prelimmary subtraction result, wherem said first close path adder result is equal to a difference value of said given pau of floatmg pomt mput values, and wherem said second close path adder result is equal to said first close path adder result plus one, wherem said first close path selection unit mcludes: a first plurality of logic units coupled to receive a least significant bit and a guard bit coπespondmg to said fust close path adder result, wherem each of said first plurality of logic units is configured to generate one of a first plurality of close path preliminary select signals, wherem each of said first plurality of close path prelimmary select signals coπesponds to a different set of predictions regardmg said given pau of floatmg pomt mput values and said first close path adder result, a first close path selection multiplexer coupled to receive said first plurality of close path prelimmary select signals, wherem said first close path selection multiplexer is configured to select said first close path selection signal m response to receivmg a first plurality of control signals, wherem said first plurality of control signals mclude a first control signal and a second control signal generated m said first close path selection unit m response to receivmg a carry m signal for a most significant bit position of said first close path adder result, wherem said first control signal is mdicative of a sign value of said first close path adder result, and wherem said second control signal is mdicative of a most sigmficant bit of said first close path adder result. 39 The microprocessor of claim 38, wherem an output of said first close data path is discarded if said absolute exponent difference of said given pau of floating pomt mput values is calculated to be greater than one. 40. The microprocessor of claim 38, wherem selection of said first close path adder result or said second close path adder result effectuates a round-to-nearest-number operation for a result of said effective subtraction of said given pau of floating pomt mput values 41. The microprocessor of claim 38, wherem said first close data path is configured to generate a first close path result m response to receivmg said given pau of floating pomt mput values. 42 A microprocessor, compnsmg: an execution unit coupled to receive a first pan of floating pomt mput values, mcludmg: a close data path configured to perform a first effective subtract operation on said first pair of floating pomt input values if said first pau of floating pomt mput values have an absolute exponent difference less than or equal to one, wherem said close data path includes- a first anthmetic unit configured to generate a first difference value and a second difference value, wherem said first difference value is equal to a difference of mantissa portions of said first pau of floatmg point mput values, and wherem said second difference value is equal to said first difference value plus one; a first multiplexer unit coupled to receive said first output value and said second output value, wherem said first multiplexer unit is configured to select either said first output value or said second output value as a prelunmary subtraction result accordmg to a close path selection signal, a first selection unit configured to generate said close path selection signal from a plurality of prelunmary selection signals, wherem said first selection unit utilizes a carry m signal to a most significant bit position of said first anthmetic unit m order to select one of said plurality of prelunmary selection signals as said close path selection signal 43. The microprocessor of claim 42, wherem selection of either said first output value or said second output value is usable to effectuate a round-to-nearest operation on a result of said first effective subtract operation 44. The microprocessor of claim 42, wherein, if said first difference value is calculated to be negative, said multiplexer unit is configured to convey an inverted version of said first difference value as said preliminary subtraction result. 45. The microprocessor of claim 42, wherein said first selection unit utilizes a least significant bit and a guard bit corresponding to said first output value in order to generate said plurality of preliminary selection signals. 46. The microprocessor of claim 45, wherein said plurality of preliminary selection signals includes a first select signal corresponding to a prediction that exponent values of said first pair of floating point input values are equal and said first output value is negative. 47. A look-up table for determining output values for a first mathematical function and a second mathematical function, said look-up table comprising: a first plurality of storage locations configured to store a first plurality of base values for said fust mathematical function and a second plurality of base values for said second mathematical function; a second plurality of storage locations configured to store a first plurality of difference values for said first mathematical function and a second plurality of difference values for said second mathematical function; an address control unit coupled to receive a first set of input signals indicative of a first input value to said look-up table and whether a first output value corresponding to said first input value is to be generated for said first mathematical function or said second mathematical function, wherein said address control unit is configured to generate a first address value from said first set of input signals and convey said first address value to said first plurality of storage locations and said second plurality of storage locations, and wherein said first plurality of storage locations is configured to output a first base value in response to receiving said first address value, and wherein said second plurality of storage locations is configured to output a first difference value in response to receivmg said first address value; an output unit coupled to receive said first base value from said first plurality of storage locations and said first difference value from said second plurality of storage locations, wherein said output unit is configured to generate said first output value from said first base value and said first difference value. 48. The look-up table of claim 47, wherein said first base value is selected from said first plurality of base values and said first difference value is selected from said first plurality of difference values in response to said first set of control values including an indication that said first output value is to be generated for said first mathematical function. 49. The look-up table of claim 47, wherein said first base value is selected from said second plurality of base values and said first difference value is selected from said second plurality of difference values in response to said first set of control values including an indication that said first output value is to be generated for said second mathematical function. 50. The look-up table of claim 48, wherein each of said first plurality of base values is an output value of said first mathematical function for a coπesponding one of a plurality of first function input regions, wherein each of said plurality of first function input regions is located within a predetermined first input range. 51. The look-up table of claim 50, wherein each of said first plurality of difference values is an output value difference from one of said first plurality of base values, wherein each of said first plurality of difference values is usable with said one of said first plurality of base values to determine output values of said first mathematical function for input values within selected regions of said first input range. 52. The look-up table of claim 51, wherein each of said second plurality of base values is an output value of said second mathematical function for a coπesponding one of a plurality of second function input regions, wherein each of said plurality of second function input regions is located within a predetermined second input range. 53. The look-up table of claim 49, wherein said look-up table is usable to compute values of said first mathematical function for a predetermined first range of input values, and wherein said look-up table is usable to compute values of said second mathematical function for a predetermined second range of input values. 54. The look-up table of claim 53, wherein said first range of input values and said second range of input values are each divided into intervals of continuous input ranges, wherein said first range of input values is divided into a first plurality of intervals, and wherein said second range of input values is divided into a second plurality of intervals. 55. The look-up table of claim 54, wherein said first plurality of intervals and said second plurality of intervals are each divided into subintervals of continuous input ranges, wherein said first plurality of intervals includes a first plurality of subintervals, and wherein said second plurality of intervals includes a second plurality of subintervals. 56. The look-up table of claim 55, wherein said first plurality of subintervals and said second plurality of subintervals are each divided into sub-subintervals of continuous input ranges, wherein said first plurality of subintervals includes a first plurality of sub-subintervals, and wherein said second plurality of subintervals includes a second plurality of sub-subintervals. 57. The look-up table of claim 56, wherein each of said first plurality of difference values is an output value difference of said first mathematical function which coπesponds to a first group of sub-subintervals within a first interval which includes said coπesponding one of said first plurality of subintervals. 58. The look-up table of claim 49, wherein said output unit is configured to add said first base value and said first difference value in order to generate said first output value, and wherein said output unit is coupled to receive a roundmg constant, and wherem said output unit is configured to add said roundmg constant to said first base value and said first difference value m order to generate said first output value. 59 The look-up table of claim 49, wherem said first mathematical function is f(x)=l/x, and said second mathematical function is f(x)=l/sqrt(x) 60. A look-up table for determining output values for a plurality of mathematical functions, said look-up table compnsmg: a first plurality of storage locations configured to store a plurality of base values for each of said plurality of mathematical functions; a second plurality of storage locations configured to store a plurality of difference values for each of said mathematical functions; an address control unit coupled to receive a first set of mput signals mdicative of a first mput value to said look-up table and a selected one of said plurality of mathematical functions, wherem a first output value is to be generated for said selected one of said plurality of mathematical functions from said first mput value, wherem said address control unit is configured to generate a first address value from said first set of mput signals and convey said fust address value to said first plurality of storage locations and said second plurality of storage locations, and wherem said first plurality of storage locations is configured to output a first base value m response to receivmg said first address value, and wherem said second plurality of storage locations is configured to output a first difference value m response to receivmg said first address value; an output unit coupled to receive said first base value from said first plurality of storage locations and said first difference value from said second plurality of storage locations, wherem said output unit is configured to generate said first output value from said first base value and said first difference value. 61. The look-up table of claim 60, wherem first base value is selected from a plurality of base values which coπespond to said selected one of said plurality of mathematical functions, and wherem said first difference value is selected from a plurality of difference values which coπespond to said selected one of said plurality of mathematical functions TITLE: Multifunction Floating Point Addition/Subtraction Pipeline And Bipartite Look-up Table ABSTRACT OF THE DISCLOSURE An optimized multimedia execution unit configured to perform vectored floating pomt and mteger mstructions. In one embodiment, the execution unit mcludes an add/subtract pipelme havmg far and close data paths The far data path is configured to handle effective addition operations, as well as effective subtraction operations for operands havmg an absolute exponent difference greater than one The close data path, conversely, is configured to handle effective subtraction operations for operands havmg an absolute exponent difference less than or equal to one. The close data path mcludes an adder unit configured to generate a first and second output value. The first output value is equal to the first mput operand plus an inverted version of the second mput operand, while the second output value is equal to the first output value plus one. The two output values are conveyed to a multiplexer unit, which selects one of the output values as a prelunmary subtraction result based on a final selection signal received from a selection unit The selection unit generates the final selection signal from a plurality of prelunmary selection signals based on the carry m signal to the most significant bit of the first adder output value Selection of the first or second output value m the close data path effectuates the round-to-nearest operation for the output of the adder. The execution unit may also be configured, m another embodiment, to perform floating pomt-to-mteger and mteger-to-floating pomt conversions. The floating pomt-to-mteger conversions may be efficiently executed m the far data path of the add/subtract pipelme, with the mteger-to- floating pomt mstructions executed m the close data path. The execution unit may also mclude a plurality of add/subtract pipelmes, allowing vectored add, subtract, and integer/floating pomt conversion mstructions to be performed. The execution unit be also expanded to handle additional anthmetic mstructions (such as reverse subtract and accumulate functions) by appropnate mput multiplexing. Finally, functions like extreme value (rmnunum/maxunum) and companson mstructions may also be implemented by proper multiplexing of output results. A method for generatmg entnes for a bipartite look-up table havmg base and difference table portions is also disclosed. In one embodiment, these entnes are usable to form output values for a mathematical function, f(x), m response to receivmg coπespondmg mput values withm a predetermined mput range. The method first compπses partitioning the mput range mto I mtervals, J submtervals/mterval, and K sub-subintervals/subinterval For a given interval M, the method mcludes generatmg K difference table entnes and J base table entnes. Each of the K difference table entnes coπesponds to a particular group of sub-submtervals withm mterval M, each of which has the same relative position withm theu respective submtervals. Each difference table entry is computed by averagmg difference values for the sub-submtervals mcluded m a corresponding group N. Each difference value which makes up this average is equal to f(Xl)-f(X2), where XI is the midpomt of the sub-submterval withm group N, and X2 is the midpomt of a predetermmed reference sub-submterval withm the same submterval as XI. Each of these midpoints is calculated such that maximum absolute eπor is minimized for all possible mput values m the sub-submterval Each of the J base table entnes, on the other hand, corresponds to a submterval within mterval M Each entry is equal to f(X2)+adjust, where X2 is the midpoint of the reference sub-submterval of the submterval corresponding to the base table entry The adjust value is calculated so that eπor introduced by the averagmg of the difference table entnes is evenly distnbuted over the entire submterval. A multi-function look-up table for determining output values for predetermined ranges of a first mathematical function and a second mathematical function. In one embodiment, the multi-function look-up table is a bipartite look-up table including a fust plurality of storage locations and a second plurality of storage locations. The first plurality of storage locations store base values for the fust and second mathematical functions. Each base value is an output value (for either the first or second function) coπesponding to an input region which includes the look-up table input value. The second plurality of storage locations, on the other hand, store difference values for both the first and second mathematical functions. These difference values are used for linear inteφolation in conjunction with a corresponding base value in order to generate a look-up table output value. The multi-function look-up table further includes an address control unit coupled to receive a first input value and a signal which indicates whether an output value is to be generated for the first or second mathematical function. The address control unit then generates a first address value from these signals which is in turn conveyed to the first and second plurality of storage locations. In response to receiving the first address value, the first and second plurality of storage locations are configured to output a fust base value and a first difference value, respectively. The first base value and first difference value are then conveyed to an output unit configured to generate a look-up table output value from the two values.

Inventors: CHERUKURI RAVI KRISHNA, JUFFA NORBERT, OBERMAN STUART F, RAMANI KRISHNAN, ROBERTS MARK E, WEBER FREDERICK D, CHERUKURI, RAVI, KRISHNA, JUFFA, NORBERT, OBERMAN, STUART, F., RAMANI, KRISHNAN, ROBERTS, MARK, E., WEBER, FREDERICK, D.
Applicants: ADVANCED MICRO DEVICES INC, ADVANCED MICRO DEVICES, INC.
Application year: 1998
Main classification: G06F7
IPC codes: G06F7, G06F9, G06F17, H03M7