Patent Publication Number: US-6223192-B1

Title: Bipartite look-up table with output values having minimized absolute error

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is a continuation-in-part of U.S. Pat. Application No. 09/015,436, filed on Jan. 29, 1998, now abandoned. 
     This application claims the benefit of U.S. Provisional Application No. 60/063,600, filed Oct. 23, 1997, titled “Method And Apparatus For Reciprocal And Reciprocal Square Root,” by Norbert Juffa and Stuart F. Oberman. 
    
    
     BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates to the field of floating-point arithmetic, and, more specifically, to a method for generating look-up table entries for evaluation of mathematical functions. 
     2. Description of the Related Art 
     Floating-point instructions are used within microprocessors to perform high-precision mathematical operations for a variety of numerically-intensive applications. Floating-point arithmetic is particularly important within applications that perform the rendering of three-dimensional graphical images. Accordingly, as graphics processing techniques grow more sophisticated, a corresponding increase in floating-point performance is required. 
     Graphics processing operations within computer systems are typically performed in a series of steps referred to collectively as the graphics pipeline. Broadly speaking, the graphics pipeline 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 in model space coordinates. Through a number of steps in the front end of the pipeline, these vertices are assembled into graphical primitives (such as triangles) which are converted into screen space coordinates. One distinguishing feature of these front-end operations (which include view transformation, clipping, and perspective division) is that they are primarily performed using floating-point numbers. The back end of the pipeline, on the other hand, is typically integer-intensive and involves the rasterization (drawing on a display device) of geometric primitives produced by the front end of the pipeline. 
     High-end graphics systems typically include graphics accelerators coupled to the microprocessor via the system bus. These graphics accelerators include dedicated hardware specifically designed for efficiently performing operations of the graphics pipeline. Most consumer-level graphics cards, however, only accelerate the rasterization stages of the graphics pipeline. In these systems, the microprocessor is responsible for performing the floating-point calculations in the initial stages of the graphics pipeline. The microprocessor then conveys the graphics primitives produced from these calculations to the graphics card for rasterizing. For such systems, it is clear that increased microprocessor floating-point performance may result in increased graphics processing capability. 
     One manner in which floating-point performance may be increased 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 within floating-point code sequences, please refer to “Design Issues in Division and Other Floating-Point Operations”, by Stuart F. Oberman and Michael J. Flynn, published in  IEEE Transactions on Computers , Vol. 46, No. 2, February 1997, pp. 154-161). With regard to the front-end stages of the graphics pipeline, division (or, equivalently, the reciprocal operation) is particularly critical during the perspective correction operation. A low-latency divide operation may thus prevent a potential bottleneck and result in increased graphics processing performance. 
     One means of increasing performance of the divide operation is through the use of dedicated floating-point division hardware. Because floating-point hardware is relatively large as compared to comparable fixed-point 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 existing floating-point element (such as a multiplier) to implement division based on iterative techniques like the Goldschmidt or Newton-Raphson algorithms. 
     Iterative algorithms for division require a starting approximation for the reciprocal of the divisor. A predetermined equation is then evaluated using 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 in the division operation is to increase 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 floating-point operations such as the reciprocal function are typically obtained through the use of a look-up table. A look-up table is a read-only memory (ROM) which stores a predetermined output value for each of a number of regions within a given input range. For floating-point functions such as the division operation, the look-up table is located within the microprocessor&#39;s floating-point unit. An input range for a floating-point function is typically bounded by a single binade of floating point values (a “binade” refers to a range of numbers between consecutive powers of 2). Input ranges for other floating-point functions, however, may span more than one binade. 
     Because a single output value is assigned for each region within a function&#39;s input range, some amount of error is inherently introduced into the result provided by the table look-up operation. One means of reducing this error is to increase the number of entries in the look-up table. This limits the error in any given entry by decreasing the range of input arguments. Often times, however, the number of entries required to achieve a satisfactory degree of accuracy in this manner is prohibitively large. Large tables have the unfortunate properties of occupying too much space and slowing down the table look-up (large tables take longer to index into than relatively smaller tables). 
     In order to decrease table size while still maintaining accuracy, “bipartite” look-up tables are utilized. Bipartite look-up tables actually include two separate tables: a base value table and a difference value table. The base table includes function output values (or “nodes”) for various regions of the input range. The values in the difference table are then used to calculate function output values located between nodes in the base table. This calculation may be performed by linear interpolation or various other techniques. Depending on the slope of the function for which the bipartite look-up table is being constructed, table storage requirements may be dramatically reduced while maintaining a high level of accuracy. If the function changes slowly, for example, the number of bits required for difference table entries is much less than the number of bits in the base table entries. This allows the bipartite table to be implemented with fewer bits than a comparable naïve table (one which does not employ interpolation). 
     Prior art bipartite look-up tables provide output values having a minimal amount of maximum relative error over a given input 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”. Wobbling precision refers to the fact that a difference in the least significant bit of an input 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 binade. A look-up table constructed in 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. 
     SUMMARY OF THE INVENTION 
     The problems outlined above are in large part solved by a method for generating entries 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) within a predetermined input 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 comprises partitioning the input range of the function into intervals, subintervals, and sub-subintervals. This first includes dividing the predetermined input range into a predetermined number (I) of intervals. Next, the I intervals are each divided into J subintervals, resulting in I*J subintervals for the input range. Finally, each of the I*J subintervals is divided into K sub-subintervals, for a total of I*J*K sub-subintervals over the input range. 
     The method next includes generating K difference table entries for each interval in the predetermined input range. Each of the K difference table entries for a given interval corresponds to a particular group of sub-subintervals within the given interval. In one embodiment, this particular group of sub-subintervals includes one sub-subinterval per subinterval of the given interval. Additionally, each sub-subinterval in the particular group has the same relative position within its respective subinterval. For example, one of the K difference table entries for the given interval may correspond to a first group of sub-subintervals wherein each sub-subinterval is the last sub-subinterval within its respective subinterval. 
     In order to generate a first difference table entry for a selected interval (M), a group of sub-subintervals (N) within interval M is selected to correspond to the first entry. The calculation of the first entry then begins with a current subinterval (P), which is bounded by input values A and B. A midpoint X 1  is calculated for current subinterval P such that f(A)−f(X 1 ))=f(X 1 )−f(B). (By calculating the midpoint in this way, maximum possible absolute error is minimized for all input values within the sub-subinterval). Next, a midpoint X 2  is computed in a similar fashion for a predetermined reference sub-subinterval within current subinterval P. (The reference sub-subinterval refers to the sub-subinterval within each subinterval that corresponds to the base table entry). A difference value, f(X 1 )−f(X 2 ), is then computed for current subinterval P. 
     In this manner, a difference value is computed for each sub-subinterval in group N. A running total is maintained of each of these difference values. The final total is then divided by J, the number of subintervals in the selected intervals, in order to generate the difference value average for interval M, sub-subinterval group N. In one embodiment, the difference value average is converted into an integer 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 remaining difference table entries for the selected interval, each remaining group of sub-subintervals is selected in turn, and a corresponding difference value average is computed. In this manner, the additional K−1 difference table entries may be generated for interval M. Difference table entries for any additional intervals in the predetermined input range are calculated in a similar manner. 
     The method next includes generating J base table entries for each interval in the predetermined input range. Each of the J base table entries for a given interval corresponds to a particular subinterval within the given interval. For example, one of the J base table entries for the given interval may correspond to the first subinterval of the given interval. 
     In a similar manner to the difference table computations, a particular interval (M) of the predetermined input range is selected for which to compute the J base table entries. Next, a subinterval of interval M is chosen as a currently selected subinterval P. Typically, the first subinterval is initially chosen as subinterval P. 
     The method then includes calculating an initial base value, B, where B=f(X 2 ). (As stated above, X 2  is the midpoint of the reference sub-subinterval of subinterval P of interval M). Subsequently, a difference value, D, is computed, where D=f(X 3 ). (X 3  is the midpoint of the sub-subinterval within subinterval P which is furthest from the reference sub-subinterval. For example, if the reference sub-subinterval is the last sub-subinterval in subinterval P, X 3  is computed for the first sub-subinterval in P). 
     The actual maximum midpoint difference for subinterval P is given by D−B. A reference is then made to the previously computed difference table entry for the sub-subinterval (or, more appropriately, the sub-subinterval group) within interval M which corresponds to the sub-subinterval for which D is computed. Since this value is computed by difference averaging as described 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 error for subinterval 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 in order to evenly distribute the error over the entire subinterval). The final base value is calculated by adding 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 integer for storage to the base table portion of the bipartite look-up table. The steps described above are repeated for the remaining subintervals in the selected interval, as well as for the subintervals of the remaining intervals of the predetermined input range. 
     In one embodiment, the output values of the bipartite look-up table are simply the sum of selected base and difference table entries. If these entries are calculated as described above, the resultant output values of the table will have a minimal amount of possible absolute error. Additionally, this minimized absolute error is achieved within a bipartite table configuration, which allows reduced storage requirements as compared to a naive table of similar accuracy. Furthermore, in an embodiment in 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 increases the speed of the table look-up operation, in contrast to prior art systems which often require lengthy multiply-add or multiply instructions as part of the interpolation process. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     Other objects and advantages of the invention will become apparent upon reading the following detailed description and upon reference to the accompanying drawings in which: 
     FIG. 1 is a block diagram of a microprocessor which configured according to one embodiment of the present invention; 
     FIG. 2 is a graph depicting a portion of a function f(x) which is partitioned for use with a prior art naive look-up table; 
     FIG. 3 is a prior art naive look-up table usable in conjunction with the function partitioned according to FIG. 1; 
     FIG. 4 is a graph depicting a portion of a function f(x) which is partitioned for use with a prior art bipartite look-up table; 
     FIG. 5 is a prior art bipartite look-up table usable in conjunction with the function partitioned according to FIG. 4; 
     FIG. 6 is a graph depicting a portion of a function f(x) which is partitioned for use with a bipartite look-up table according to one embodiment of the present invention; 
     FIG. 7 is a bipartite look-up table usable in conjunction with the function partitioned according to FIG. 6; 
     FIG. 8 depicts one format for an input value to a bipartite look-up in accordance with one embodiment of the present invention; 
     FIG. 9A illustrates a look-up table input value according to the format of FIG. 8 in one embodiment of the present invention; 
     FIG. 9B depicts the mantissa portion of a look-up table input value for the reciprocal function; 
     FIG. 9C depicts a base table index for a bipartite look-up table for the reciprocal function, according to one embodiment of the present invention; 
     FIG. 9D depicts a difference table index for a bipartite look-up table for the reciprocal function, according to one embodiment of the present invention; 
     FIG. 10A depicts the mantissa portion of a look-up table input value for the reciprocal square root function; 
     FIG. 10B depicts a base table index for a bipartite look-up table for the reciprocal square root function, according to one embodiment of the present invention; 
     FIG. 10C depicts a difference table index for a bipartite look-up table for the reciprocal square root function, according to one embodiment of the present invention; 
     FIG. 11 is a bipartite look-up table for the reciprocal and reciprocal square root functions according to one embodiment of the present invention; 
     FIG. 12 is one embodiment of an address control unit within the bipartite look-up table of FIG. 11; 
     FIG. 13A is a graph depicting a prior art midpoint calculation for a bipartite look-up table; 
     FIG. 13B is a graph depicting a midpoint calculation for a bipartite look-up table according to one embodiment of the present invention; 
     FIG. 14A 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. 14B is a graph depicting difference value averaging 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; 
     FIG. 15A-B are graphs comparing table output values for a portion of a function f(x) to computed midpoint values for the function portion; 
     FIG. 15C-D are graphs comparing table outputs with adjusted base values for a portion of a function f(x) to computed midpoint values for the function portion; 
     FIG. 16 is a flowchart depicting a method for computation of base table entries for a bipartite look-up table according to one embodiment of the present invention; and 
     FIG. 17 is a block diagram of a computer system according to one embodiment of the present invention. 
     While the invention is susceptible to various modifications and alternative forms, specific embodiments thereof are shown by way of example in 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 intention is to cover all modifications, equivalents and alternatives falling within the spirit and scope of the present invention as defined by the appended claims. 
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     Turning now to FIG. 1, a block diagram of one embodiment of a microprocessor  10  is shown. As depicted, microprocessor  10  includes a predecode logic block  12  coupled to an instruction cache  14  and a predecode cache  15 . Caches  14  and  15  also include an instruction TLB  16 . A cache controller  18  is coupled to predecode block  12 , instruction cache  14 , and predecode cache  15 . Controller  18  is additionally coupled to a bus interface unit  24 , a level-one data cache  26  (which includes a data TLB  28 ), and an L 2  cache  40 . Microprocessor  10  further includes a decode unit  20 , which receives instructions from instruction cache  14  and predecode data from cache  15 . This information is forwarded to execution engine  30  in accordance with input received from a branch logic unit  22 . 
     Execution engine  30  includes a scheduler buffer  32  coupled to receive input from decode unit  20 . Scheduler buffer  32  is coupled to convey decoded instructions to a plurality of execution units  36 A-E in accordance with input received from an instruction control unit  34 . Execution units  36 A-E include a load unit  36 A, a store unit  36 B, an integer X unit  36 C, an integer Y unit  36 D, and a floating point unit  36 E. Load unit  36 A receives input from data cache  26 , while store unit  36 B interfaces with data cache  26  via a store queue  38 . Blocks referred to herein with a reference number followed by a letter will be collectively referred to by the reference number alone. For example, execution units  36 A-E will be collectively referred to as execution units  36 . 
     Generally speaking, floating point unit  36 E within microprocessor  10  includes one or more bipartite look-up tables usable to generate approximate output values of given mathematical functions. As will be described in greater detail below, these bipartite look-up tables are generated such that absolute error is minimized for table output values. In this manner, floating point unit  36 E may achieve an efficient implementation of such operations as the reciprocal and reciprocal square root functions, thereby increasing the performance of applications such as three-dimensional graphics rendering. 
     In one embodiment, instruction cache  14  is organized as sectors, with each sector including two 32-byte cache lines. The two cache lines of a sector share a common tag but have separate state bits that track the status of the line. Accordingly, 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 in instruction cache  14 , with the required cache line being supplied by external memory via bus interface unit  24 . The cache line within 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 line is marked as invalid. The required cache line is supplied by external memory, but, unlike the sector replacement case, the cache line within the sector that was not requested remains in the same state. In alternate embodiments, other organizations for instruction cache  14  may be utilized, as well as various replacement policies. 
     Microprocessor  10  performs prefetching only in the case of sector replacements in one embodiment. During 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 line is in the second half of the sector, no prefetching is performed. It is noted that other prefetching methodologies may be employed in different embodiments of microprocessor  10 . 
     When cache lines of instruction data are retrieved from external memory by bus interface unit  24 , this data is conveyed to predecode logic block  12 . In one embodiment, the instructions processed by microprocessor  10  and stored in cache  14  are variable-length (e.g., the x86 instruction set). Because decode of variable-length instructions is particularly complex, predecode logic  12  is configured to provide additional information to be stored in instruction cache  14  to aid during decode. In one embodiment, predecode logic  12  generates predecode bits for each byte in instruction cache  14  which indicate the number of bytes to the start of the next variable-length instruction. These predecode bits are stored in predecode cache  15  and are passed to decode unit  20  when instruction bytes are requested from cache  14 . 
     Instruction cache  14  is implemented as a 32 Kbyte, two-way set associative, writeback cache in one embodiment of microprocessor  10 . The cache line size is 32 bytes in this embodiment. Cache  14  also includes a TLB  16 , which includes 64 entries used to translate linear addresses to physical addresses. Many other variations of instruction cache  14  and TLB  16  are possible in other embodiments. 
     Instruction fetch addresses are supplied by cache controller  18  to instruction cache  14 . In one embodiment, up to 16 bytes per clock cycle may be fetched from cache  14 . The fetched information is placed into an instruction buffer that feeds into decode unit  20 . In one embodiment of microprocessor  10 , fetching may occur along a single execution stream with seven outstanding branches taken. 
     In one embodiment, the instruction fetch logic within cache controller  18  is capable of retrieving any  16  contiguous instruction bytes within a 32-byte boundary of cache  14 . There is no additional penalty when the 16 bytes cross a cache line boundary. Instructions are loaded into the instruction buffer as the current instructions are consumed by decode unit  20 . (Predecode data from cache  15  is also loaded into the instruction buffer as well). Other configurations of cache controller  18  are possible in other embodiments. 
     Decode logic  20  is configured to decode multiple instructions per processor clock cycle. In one embodiment, decode unit  20  accepts instruction and predecode bytes from the instruction buffer (in x86 format), locates actual instruction boundaries, and generates corresponding “RISC ops”. RISC ops are fixed-format internal instructions, most of which are executable by microprocessor  10  in a single clock cycle. RISC ops are combined to form every function of the x86 instruction set in one embodiment of microprocessor  10 . 
     Microprocessor  10  uses a combination of decoders to convert x86 instructions into RISC ops. The hardware includes 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 instructions (moves, shifts, branches, etc.) into 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  20  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 combination 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 in which four RISC ops cannot be generated, decode unit  20  places RISC NOP operations in the remaining slots of the grouping. These groupings of RISC ops (and possible NOPs) are then conveyed to scheduler buffer  32 . 
     It is noted that in another embodiment, an instruction format other than x86 may be stored in instruction cache  14  and subsequently decoded by decode unit  20 . 
     Instruction control logic  34  contains the logic necessary to manage out-of-order execution of instructions stored in scheduler buffer  32 . Instruction control logic  34  also manages data forwarding, 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 time, equating to a maximum of 12 x86 instructions. When possible, instruction 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 in one embodiment. 
     In one embodiment, microprocessor  10  includes five execution units ( 36 A-E). Store unit  36 A and load unit  36 B are two-staged pipelined designs. Store unit  36 A performs data memory and register writes which are available for loading after one clock cycle. Load unit  36 B performs memory reads. The data from these reads is available after two clock cycles. Load and store units are possible in other embodiments with varying latencies. 
     Execution unit  36 C (Integer X unit) is a fixed point 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  36 D (Integer Y unit) is a fixed point execution unit which is configured to operate on the basic word and double word ALU operations (ADD, AND, CMP, etc.). 
     Execution units  36 C and  36 D are also configured to accelerate performance of software written using multimedia instructions. Applications that can take advantage of multimedia instructions include graphics, video and audio compression and decompression, speech recognition, and telephony. Units  36 C-D are configured to execute multimedia instructions in 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 processing). In one embodiment, unit  36 C-D uses registers which are mapped on to the stack of floating point unit  36 E. 
     Execution unit  36 E contains an IEEE 734-compatible floating point unit designed to accelerate the performance of software which utilizes the x86 instruction set. Floating point software is typically written 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. Floating point unit includes 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 in as few as two clock cycles. 
     Branch resolution unit  35  is separate from branch prediction logic  22  in that it resolves conditional branches such as JCC and LOOP after the branch condition has been evaluated. Branch resolution unit  35  allows efficient speculative execution, enabling microprocessor  10  to execute instructions beyond conditional branches before knowing whether the branch prediction was correct. As described above, microprocessor  10  is configured to handle up to seven outstanding branches in one embodiment. 
     Branch prediction logic  22 , coupled to decode unit  20 , is configured to increase the accuracy with which conditional branches are predicted in microprocessor  10 . Ten to twenty percent of the instructions in typical applications include conditional branches. Branch prediction logic  22  is configured to handle this type of program behavior and its negative effects on instruction execution, such as stalls due to delayed instruction fetching. In one embodiment, branch prediction logic  22  includes 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 algorithm using 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 in 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 within branch logic  22  supplies the first 16 bytes at that address directly to the instruction buffer (if a hit occurs in the branch target cache). In one embodiment, this branch prediction logic achieves branch prediction rates of over 95%. 
     Branch logic  22  also includes special circuitry designed to optimize the CALL and RET instructions. This circuitry allows the address of the next instruction following the CALL instruction in memory to be pushed onto a return address stack. When microprocessor  10  encounters a RET instruction, branch logic  22  pops this address from the return stack and begins fetching. 
     Like instruction cache  14 , L 1  data cache  26  is also organized as two-way set associative 32 Kbyte storage. In one embodiment, data TLB  28  includes 128 entries used to translate linear to physical addresses. Like instruction cache  14 , L 1  data cache  26  is also sectored. Data cache  26  implements a MESI (modified-exclusive-shared-invalid) protocol to track cache line status, although other variations are also possible. In order to maximize cache hit rates, microprocessor  10  also includes on-chip L 2  cache  40  within the memory sub-system. 
     Turning now to FIG. 2, a graph  50  of a function f(x) is depicted which corresponds to a prior art look-up table described below with reference to FIG.  3 . Graph  50  includes a portion  80  of function f(x), with output values 82A-E plotted on a vertical axis  60  against corresponding input values on a horizontal axis  70 . 
     As will be described below, a look-up table for function f(x) is designed by dividing a predetermined input range into one or more various sub-regions. A single value is generated for each of the one or more sub-regions, and then stored into the look-up table. When an input value is presented to the look-up table, an index is formed which corresponds to one of the sub-regions of the input range. This index is then usable to select one of the predetermined output values. 
     In FIG. 2, input range portion  64  corresponds to portion  80  of function f(x). As shown, input range  64  is divided into a plurality of intervals  72 . Interval  72 A, for example, corresponds to input values located between points  71 A and  71 B on the horizontal axis. Interval  72 B corresponds to input values located between points  71 B and  71 C, etc. It is noted that while only four intervals are shown in graph  50 , many intervals are typically computed for a given function. Only four are shown in FIG. 2 for simplicity. 
     As mentioned, each interval  72  has a corresponding range of output values. Interval  72 A, for example, includes a range of output values spanning between points  82 A and  82 B. In order to construct a look-up table for function f(x), a single output value is selected for interval  72 A which has a value between points  82 A and  82 B. The method of selecting this output value varies 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. 3, 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 m , where m is some integer value. For example, if table portion  108  includes 64 entries, six highorder 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  corresponds to one of these intervals as shown in FIG.  3 . For example, table entry  110 A corresponds to interval  32 A, table entry  10 B corresponds to interval  32 B, 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 error. Since a table entry  110  is provided for each interval  32 , an increase in the number of intervals leads to a corresponding increase in table size. (Table size is equal to P*2 index  bits, where P is the number of bits per table entry, and 2 index  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 correspondence 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. 4, a graph  120  is depicted of a portion  150  of function f(x). The partitioning of function portion  150  corresponds to a prior art look-up table described below with reference to FIG.  5 . Graph  120  includes a portion  150  of function f(x), with output values  152 A-E plotted on a vertical axis  130  against corresponding input values on a horizontal axis  140 . 
     FIG. 4 illustrates a different input range partitioning for function f(x) than is shown in FIG.  2 . This partitioning allows an interpolation scheme to be implemented for the look-up table described below with reference to FIG.  5 . The input range of function f(x) is, as above, divided into intervals. Intervals  142 A and  142 B are shown in FIG. 4, although a given function may have any number of intervals depending upon the particular embodiment. Each interval  142  is then divided into subintervals. Interval  142 A, for example, is divided into subintervals  144 A-D, while interval  142 B is divided into subintervals  146 A-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  142 A is located between points  152 A and  152 E. Which subinterval  144  the base value for interval  142 A 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  142 A. 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 subintervals  144 . The output value chosen to represent interval  142 A is then used as the corresponding 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  142 A. 
     Turning now to FIG. 5, 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 index  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  32 A-D of FIG.  2 . 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  32 A-D partitioned as in FIG. 4, however, intervals  32 A-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 required to represent the difference values of difference table  224 . Note that the above example is only for a single interval of a given function. In typical embodiments of look-up tables, function input ranges are divided into a large number of input sub-regions, and table size savings is applicable over each of these sub-regions. 
     Turning now to FIG. 6, a graph  250  of a function f(x) is depicted which corresponds to a look-up table according to one embodiment of the present invention. This look-up table is described below with reference to FIG.  7 . Graph  250  includes a portion  280  of function f(x), with output values  282 A-Q plotted on a vertical axis  260  against corresponding input values x on a horizontal axis  270 . 
     FIG. 6 depicts yet another partitioning of the range of inputs for function f(x). This partitioning allows an interpolation scheme to be implemented for the look-up table of FIG. 7 which allows further reduction in table storage from that offered by the configuration of table  200  in FIG.  5 . The input range of function f(x) is, as above, divided into intervals. Only one interval,  272 A, is shown in FIG. 6 for simplicity, although a given function may have any number of intervals, depending upon the embodiment. As shown, interval  272 A is divided into a plurality of subintervals  274 A-D. Additionally, each subinterval  274  is divided into a plurality of sub-subintervals. Subinterval  274 A is divided into sub-subintervals  276 A-D, subinterval  274 B is divided into sub-subintervals  277 A-D, etc. 
     With the partitioning shown in FIG. 6, a bipartite look-up table  300  may be constructed which is similar to table  200  shown in FIG.  5 . Table  300  is described in detail below with reference to FIG.  7 . Like table  200 , table  300  includes a base table portion and a difference table portion. The entries of these tables, however, correspond to regions of the input range of function f(x) in a slightly different manner than the entries of table  200 . The base table portion of table  300  includes an entry for each subinterval in the input range. Each base table entry includes a single output value to represent its corresponding subinterval. The base table entry for subinterval  274 A, for example, is an output value between those represented by points  282 A and  282 E. Instead of including a separate difference table entry for each sub-subinterval in each subinterval, however, table  300  has a number of difference table entries for each interval equal to the number of sub-subintervals per subinterval. Each of these entries represents an averaging of difference values for a particular group of sub-subintervals within the interval. 
     Consider the partitioning shown in FIG.  6 . An output value is determined for each subinterval  274 , and each sub-subinterval  276 - 279 . As will be described below, in one embodiment of the present invention, the output value for each subinterval and sub-subinterval is chosen such that maximum possible absolute error is minimized for each input region. The base table entries are computed by using the assigned output value for each of subintervals  274 . A separate entry is entered for each of regions  274 A-D. Then, difference values are computed for each sub-subinterval which are equal to the difference between the output value for the sub-subinterval and the output value assigned for the subinterval. Then, the difference values are averaged for sub-subintervals having common relative positions within the subintervals. These values are then used as the difference table entries. 
     For example, difference values are computed for each of sub-subintervals  276 - 279  and their respective subintervals. Then difference values for sub-subintervals  276 A,  277 A,  278 A, and  279 A are averaged to form the first difference entry for interval  272 . Difference values for sub-subintervals  276 B,  277 B,  278 B, and  279 B 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. 7, 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.  6 ), 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. 6, 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. 7, each of table entries  322 A-D corresponds to a different subinterval  274  within interval  272 A. 
     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  272 A. Entry  326 A corresponds to sub-subintervals  276 A,  277 A,  278 A, and  279 A, 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 corresponding 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.  7 ). 
     In the configuration of table  100 , the  16  input regions of FIG. 6 correspond to intervals. Each of the  16  intervals has a corresponding entry of P bits in table portion  108 . Thus, the partitioning of FIG. 6 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. 6 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 corresponding 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. 7, function portion  280  includes an interval  272 A divided into four subintervals  274 . Each subinterval  274  is divided into sub-subintervals, for a total of 16 input regions. Each subinterval has a corresponding 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  272 A, 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. 8, 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 =(−1) S ·2 expo   ·mant,   (1) 
       
     
     where s represents the value sign bit  402 , expo represents the true exponent value of the floating point number (as opposed to the biased exponent value found in portion  404 ), and mant represents the value of mantissa portion  406  (including the leading one bit). 
     An important floating-point 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 pipeline. The reciprocal function may be generally expressed as follows:                  1   x     =     1         (     -   1     )     s     ·     2   expo     ·   mant         ,   or           (   2   )                   1   x     =       1       (     -   1     )     s       ·     1     2   expo       ·     1   mant         ,           (   3   )                         
     which simplifies to                1   x     =           (     -   1     )     s     ·     2     -   expo       ·     1   mant                     or             (4a)                 1   x     =         (     -   1     )     s     ·     2       -   1     -   expo       ·       2   mant                .               (4b)                         
     Since the reciprocal of mant is clearly the difficult part of the operation, it is advantageous to implement an approximation to this value using table look-up. Since table input values (e.g., input value  302 ) are normalized, mant is restricted to 
     
       
         2 N   ≦mant &lt;2 N+1   (5) 
       
     
     for some fixed N. In order to compute the reciprocal of all floating-point numbers, then, it suffices to compute 1/mant over the primary range [2 N ,2 N+1 ), and map all other inputs to that range by appropriate exponent manipulation (which may be performed in parallel with the table look-up). 
     Another common graphics operation is the reciprocal square root operation (x −½ ), used in distance and normalization calculations. Defining sqrt(−x)=−sqrt(x) in order to handle negative inputs, this function may be expressed as follows:                  1     x       =     1           (     -   1     )     s     ·     2   expo     ·   mant           ,   or           (   6   )                   1     x       =       1         (     -   1     )     s         ·     1       2   expo         ·     1     mant           ,           (   7   )                         
     which simplifies to                1     x       =         (     -   1     )     s     ·     2     -     (     expo   2     )         ·       1     mant       .               (   8   )                         
     Because having the exponent of 2 be a whole number in equation (8) is desirable, the reciprocal square root function may be written as two separate equations, depending upon whether expo is odd or even. These equations are as follows:                  1     x       =           (     -   1     )     s     ·     2     (     -     expo   2       )       ·     1     mant              (     expo                 even     )         ,   and           (   9   )                 1     x       =       (     -   1     )          s   ·     2     (     -       expo   -   1     2       )       ·     1       2   ·   mant                  (     expo                 odd     )     .               (   10   )                         
     As with the reciprocal function, the difficult part of the reciprocal square root function is the computation of 1/sqrt(mant) or 1/sqrt(2*mant). Again, this is implemented as a table look-up function. From equations (9) and (10), it can be seen that in one embodiment of a look-up table for the reciprocal square root function, the look-up table inputs may span two consecutive binades in order to handle both odd and even exponents. For true exponent values that are even, then, the input range is [2 N , 2 N+1 ), with odd true exponent values occupying the next binade, [2 N+1 , 2 N+2 ). 
     It is noted that the order of the binades 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 contain entries for the binade defined by [2,4), while the upper order addresses include entries for the binade [1,2). Alternatively, the least significant bit of the biased exponent value may be inverted so that binade [1,2) entries are in the lower half of the look-up table. 
     For any binary floating-point format (such as format  400 ), a table look-up mechanism 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 input value. The look-up table includes P bits for each entry, for a total size (in a naïve implementation) 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 appropriately combined with the table output to generate the output value (be it a reciprocal or a reciprocal square root). Note that since the numeric value of each mantissa bit is fixed for a given binade, extracting high-order bits automatically ensures equidistant nodes over the binade, such that interpolation may be performed easily. 
     As described above, the table look-up mechanism for the reciprocal square root has input values ranging over two consecutive binades. If it is desired to have equidistant nodes across both binades, IDX high-order bits may extracted from mantissa value  406  for the lower binade, with IDX+1 bits extracted from value  406  for the upper binade (this is done since the numeric value of each fractional bit in the upper binade is twice that of the same bit in 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 required table accuracy allows table size to be reduced to 2*P*2 IDX =P*2 IDX+1  bits by always extracting IDX leading fractional mantissa bits for each binade. This results in reducing the distance between the nodes in the upper binade. For the reciprocal square root function (1/sqrt(x)), the slope decreases rapidly for increasing x, which offsets table quantization error in the upper binade. Thus, nodes in a given binade (either upper or lower) are equidistant, but the distance between nodes varies in adjacent binades by a factor of two. 
     In one embodiment, performing table look-up for the reciprocal square root function may be accomplished by making one table for each of the two binades and multiplexing their output based upon the least significant bit of the value of exponent portion  404 . In another embodiment, a single table may be implemented. This single table is addressed such that the IDX leading fractional bits of mantissa value  406  constitute bits &lt;(IDX−1):0&gt; of the address, with the least significant bit of exponent value  404  bit &lt;IDX&gt; of the table address. Such a table is discussed in greater detail below. 
     Turning now to FIG. 9A, a look-up table input value  420  according to format  400  is depicted. Input value  420  includes a sign bit (IS)  422 , an exponent value (IEXPO)  424 , and a mantissa value (IMANT)  426 . In the embodiment shown, input value  420  is normalized, and mantissa value  426  does not include the leading one bit. Accordingly mantissa value  426  is shown as having N−1 bits (mantissa value  426  would be shown as having N bits in an embodiment in which the leading one bit is stored explicitly). The most significant bit in mantissa value  426  is represented in FIG. 9A as IMANT&lt;N−2&gt;, while the least significant bit is shown as IMANT&lt;0&gt;. 
     Turning now to FIG. 9B, an exploded view of mantissa value  426  is shown according to one embodiment of the present invention. In one embodiment, the bits of mantissa value  426  may be grouped according to the scheme shown in FIG. 9B in order to index into base and difference table portions of a look-up table for the reciprocal function. Other bit grouping are possible in alternate embodiments of the present invention. 
     The first group of bits is XHR  430 , which is HR consecutive bits from IMANT&lt;N−2&gt; to IMANT&lt;N−2−HR&gt;. Similarly, the second group of bits is XMR  432 , which includes MR consecutive bits from position IMANT&lt;N−2−HR&gt; to IMANT&lt;N− 1 −HR−MR&gt;, while the third group of bits, XLR  434 , includes LR consecutive bits from IMANT&lt;N−2−HR−MR&gt; to IMANT&lt;N−2−HR−MR−LR&gt;. As will be described below, XHR  430  is used to specify the interval in the input range which includes the input value. Likewise, XMR  432  identifies the subinterval, and XLR the sub-subinterval 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. 9B 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. 9C, 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. 9D, 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. 10A, mantissa value  426  is shown with different groupings of bits. Mantissa value  426  is partitioned in this manner when input value  420  corresponds to a second function, the reciprocal square root. The base and difference indices generated from the bit groupings of FIG. 10A 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. 9B, 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. 10A 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. 10B, a reciprocal square root base table index  470  is depicted. Similarly, FIG. 10C depicts a reciprocal square root difference table index  480 . Both indices  470  and  480  are formed from the bit groups shown in FIG. 10A, and usable in a similar manner to indices  440  and  450  shown in FIGS. 8C and 8D. 
     Turning now to FIG. 11, 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. 9A) 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. 11, 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  522 A and  522 B. Table  520  further includes base output values for the reciprocal function over a single binade in entries within base table region  522 C. 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  532 A and  532 B), and over a single binade for the reciprocal function (within entries in region  532 C). 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. 11, 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. 12, 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&gt;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 subintervals/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 in magnitude that occurs in the primary input interval). 
     For example, for the reciprocal function, c=1 since the maximum slope in interval [1,2) is 1 (at x=1). Similarly, for the reciprocal square root function, c=0, since the maximum slope in [1,4) is 0.5 (at x=1). Generally speaking, a function with a relatively high slope requires more bits in the difference entry to represent change from a corresponding 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 maintaining a high degree of accuracy. 
     Adder  540  is configured to be an R-bit adder, where R is sufficient to represent the maximum value in 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 rounding constant  542 , such that the least significant bits of the addends are aligned. This add operation results in an output value  550  being produced. In one embodiment, the use of optional rounding constant  542  results in a number of least significant bits being discarded from output value  550 . 
     In the embodiment shown in FIG. 11, adder  540  does not generate a carry out signal (a carry out signifies that output value  550  exceeds 2 R ). Since all the entries of tables  520  and  530  have been determined before table  500  is to be used (during operation of a microprocessor in one embodiment), it may be determined if any of the possible combinations of base/difference entries (plus the rounding constant) result in an output value  550  which necessitates providing a carry out signal. 
     As shown, result  560  for the two functions of table  500  includes 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  in writing output mantissa portion  566 . With regard to the value of output sign bit portion  562 , the value of input 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 input exponent portion  422  of input value  420 , and is calculated differently for the reciprocal function than it is for the reciprocal square root function. 
     In one embodiment, the true input exponent, TIEXPO, is related to the value of field  424  in input value  420 , IEXPO. Similarly, the true output exponent, TOEXPO, is related to the value to be written to field  564 , OEXPO. The value written to OEXPO is dependent upon the particular function being evaluated. 
     For the reciprocal function, the value written 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 in which IEXPO=TIEXPO+BIAS and OEXPO=TOEXPO+BIAS, it follows that OEXPO=2*BIAS−1−EXPO[+CR]. 
     For the reciprocal square root function, OEXPO is computed such that TOEXPO=(−1−(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+½))[+CR]. For the common case in which IEXPO=TIEXPO+BIAS and OEXPO=TOEXPO+BIAS, OEXPO=((3*BIAS −1−IEXPO)&gt;&gt;1)[+CR]. 
     Turning now to FIG. 12, a block diagram of address control  510  within multi-function look-up table  500  is depicted according to one embodiment of the present invention. Address control unit  510  receives input value  420  and function select signal  502  and generates base table index  512  and difference table index  514 . 
     Input value  420  includes sign bit field  422  having a value IS, exponent field  424  having a value IEXPO (the biased exponent value), and mantissa field  426  having a value IMANT. As shown, mantissa field  426  includes three bit groups ( 573 ,  574 , and  575 ) usable to form indices  512  and  514 . Because input 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, since group  573  is usable to specify an interval for both functions within 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 having 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 combined as shown in FIGS. 8C-D (and  9 B and  9 C) in order to form portions of indices  512  and  514 . 
     The most significant bits of indices  512  and  514  are used for function selection. In the embodiment shown in FIG. 12, the most significant 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 index  512  and difference table index  514  access entries within table regions  522 A-B and  532 A-B (the reciprocal square root entries). Conversely, when signal  502  is low, indices  512  and  514  access entries within table regions  522 C and  532 C (the reciprocal entries). The second most significant bit of indices  512 / 514  is used (if applicable) to select one of the two binades for the reciprocal square root entries. That is, these bits select between table regions  522 A and  522 B in base table  520 , and between table regions  532 A and  532 B in difference table  530 . Furthermore, these second-most-significant bits are only set (in the embodiment shown) if function select  502  is high and the LSB of the true exponent value is set (meaning the true exponent is odd and the biased exponent,  511 , is even). Thus, these bits are not set if function select  502  is low, indicating the reciprocal function. 
     The equations for index  512  in the embodiment shown in FIG. 11 may be summarized as follows: 
     
       
           BADDR&lt;XH+XM+ 1&gt;=!(Signal  502 ),  (11) 
       
     
     
       
           BADDR&lt;XH+XM&gt;=!IEXPO&lt; 0&gt;&amp;&amp;( 502 ),  (12) 
       
     
     
       
           BADDR&lt;XH+XM− 1 :XM&gt;=IMANT&lt;N− 2 :N −1 −XH &gt;,  (13) 
       
     
     
       
           BADDR&lt;XM −1:0 &gt;=IMANT&lt;N −2 −XH:N −1 −XH−XM&gt;.   (14) 
       
     
     Similarly, the equations for index  514  are as follows: 
     
       
           DADDR&lt;XH+XL +1&gt;=!(Signal  502 ),  (15) 
       
     
     
       
           DADDR&lt;XH+XL&gt;=IEXPO &lt;0&gt;&amp;&amp;( 502 ),  (16) 
       
     
     
       
           DADDR&lt;XH+XL −1 :XL&gt;=IMANT&lt;N −2 :N −1 −XH&gt;,   (17) 
       
     
     
       
           DADDR&lt;XL −1:0 &gt;=IMANT&lt;N −2 −XH−XM:N −1 −XH−XM−XR&gt;.   (18) 
       
     
     Other equations are possible in other embodiments. 
     Turning now to FIG. 13A, a graph  578  of an input region  580  is shown according to a prior art method for calculating a midpoint value. Input region  580  is bounded by input values A and B, located at points  582  and  584 , respectively, on the horizontal axis of graph  578 . Point A corresponds to an output value (for the reciprocal function) denoted by point  581  on the vertical axis of graph  578 . Point B, likewise, corresponds to an output value denoted by point  583 . 
     As shown in FIG. 13A, a midpoint X 1  is calculated for input region  580  by determining the input value halfway in between A and B. This input value X 1  is located at point  586 , and corresponds to an output value denoted by point  585  on the vertical axis. In prior art systems, the output value corresponding to point  585  is chosen to represent all values in input region  580 . An output value calculated in this manner has the effect of minimizing maximum relative error over a given input region. Although this midpoint calculation method is shown in FIG. 13A for the reciprocal function, this method is applicable to any function. 
     Turning now to FIG. 13B, a graph  590  of input region  580  is shown according to a method for calculating a midpoint value according to the present invention. As in FIG. 13A, input region  580  is bounded by input values A and B located at points  582  and  584 , respectively. Input value A corresponds to an output value denoted by point  581 , while input value B corresponds to an output value at point  583 . As depicted in FIG. 13B, both of these output values correspond to the reciprocal function. 
     Unlike the midpoint calculation in FIG. 13A, the midpoint calculation in FIG. 13B produces an output value for input region  580  which minimizes absolute error. The midpoint calculation is FIG. 13A is independent of the particular function, since the midpoint (X 1 ) is simply calculated to be halfway between the input values (A and B) which bound region  580 . Midpoint X 2 , on the other hand, is calculated such that the corresponding output value, denoted by point  587 , is halfway between the output values ( 581  and  583 ) corresponding to the input region boundaries. That is, the difference between  581  and  587  is equal to the difference between  587  and  583 . The calculation of X 2  (denoted by point  588 ) is function-specific. For the reciprocal function, X 2  is calculated as follows:                    1   A     -     1   X2       =       1   M2     -     1   B         ,   or           (   19   )                 A   ·   X2   ·     B        (         1     A   ·       -     1   X2       =       1   X2     -     1   B         )         ,           (   20   )                         
     which simplifies to 
     
       
           X 2 ·B−A−B=A·B−A·X 2  (21). 
       
     
     Solving for X 2  gives        X2   =         2   ·   A   ·   B       A   +   B       .                     
     Calculating X 2  for the reciprocal square root function gives        X2   =         4   ·   A   ·   B       A   +     2          A   ·   B         +   B       .                     
     Calculation of midpoint X 2  in this manner ensures that maximum absolute error is minimized by selecting f(X 2 ) as the output value for input region  580 . This is true because the absolute error at both A and B is identical with f(X 2 ) selected as the output value for region  580 . 
     Another method of calculating error, “ulp” (unit in last place) error, is currently favored by the scientific community. Generally speaking, ulp error is scaled absolute error 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. 14A, 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. 14B, which is a graph  640  of a portion  642  of function f(x). Method  600  is described generally in relation to FIG. 14A, while FIG. 14B 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 narrowly-partitioned input range generally leads to reduced output error. FIG. 14B illustrates a single interval  650  of the input range of f(x). Interval  650  is partitioned into four subintervals,  652 A-D, each of which is further partitioned into four sub-subintervals. Subinterval  652 A, for example, includes sub-subintervals  654 A,  654 B,  654 C, and  654 D. 
     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.  14 B. 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  654 C,  655 C,  656 C, and  657 C, 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. 14B, 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 (referred to in FIG. 14A as “Group N”) are selected for which to calculate a difference entry. Typically, the groups are selected sequentially. For example, in FIG. 14B, group  0  (consisting of sub-subintervals  654 A,  655 A,  656 A, and  657 A) 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  610 A, a subinterval is selected in which to begin computation of the current difference table entry being calculated. The current subinterval is referred to as “P” within FIG.  14 A. Subintervals are also typically selected in sequential order. For example, in calculating table entries for groups  0 - 3  in FIG. 14B, computations first begin in subinterval  652 A, then subinterval  652 B, etc. 
     In sub-step  610 B, the midpoint (X 1 ) and corresponding output value (R=f(X 1 )) are computed for the sub-subinterval of group N located within current subinterval P. For example, if the current subinterval P is  652 A and the current group N is group  0 , the midpoint and corresponding output value are computed for sub-subinterval  654 A. In one embodiment, midpoint X 1  is calculated as shown in FIG.  13 B. That is, the midpoint X 1  is calculated such that f(X 1 ) is halfway between the maximum and minimum output values for the sub-subinterval for which the midpoint is being calculated. The midpoints ( 660 A- 660 P) are shown in FIG. 14B for each sub-subinterval within interval  650 . 
     Next, in sub-step  610 C, a midpoint(X 2 ) and corresponding output value (R2=f(X 2 )) are calculated for a reference sub-subinterval within current subinterval P. This reference sub-subinterval is the sub-subinterval within current subinterval P for which the base value is ultimately calculated (as is described below with reference to FIG.  15 A). In one embodiment, the reference sub-subinterval is the last sub-subinterval within a given subinterval. In FIG. 14B, for example, the reference sub-subintervals are those in group  3 . 
     In sub-step  610 D, the difference between the midpoint output values (R1−R2) is added to the current 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.  14 B. Note that the difference value for the reference sub-subinterval in each subinterval is zero. 
     In step  612 , a determination is made whether current subinterval P is the last (J-1th) subinterval in interval M. If P is not the last subinterval in interval M, processing returns to step  610 . In sub-step  610 A, the next subinterval (sequential to that previously processed) is selected as subinterval P. Computations are made in sub-steps  610 B-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  610 D. 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 current subinterval P is the last subinterval in interval M, method  600  continues with step  620 . In step  620 , the current 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  corresponds to the sub-subintervals in group  0 . This entry is calculated by the average of difference values represented by lines  662 A,  662 D,  662 G, and  662 J in FIG.  14 B. 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. 14B, this approximate function value for each subinterval corresponds to the midpoint of the reference sub-subinterval within the subinterval. For example, the approximate function value for subinterval  652 A in FIG. 14B is the function value at midpoint  660 D of sub-subinterval  654 D. An approximate function value for another sub-subinterval within subinterval  652 A may then be calculated by adding the function value at midpoint  660 D 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  652 A-B). Conversely, the differences (and result of the addition) are too large in the last subintervals in interval  650  (subintervals  652 C-D). Furthermore, within a given subinterval, error varies according to the sub-subinterval position due to difference value averaging. Difference value error 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 error is zero. In the first sub-subinterval within the subinterval, however, this error is at its maximum. In one embodiment, it is desirable to compute base table entries for a given subinterval such that maximum error is distributed evenly throughout the subinterval. Graphs illustrating the result of this process are depicted in FIGS. 14A-D, with an actual method for this computation described with reference to FIG.  16 . 
     Turning now to FIG. 15A, a graph  700  is shown of a portion of function f(x) (denoted by reference numeral  642 ) from FIG.  14 B. Only subinterval  652 A is shown in FIG.  15 A. As in FIG. 14B, subinterval  652 A includes four sub-subintervals ( 654 A-D), each having a corresponding midpoint  660 . Graph  700  further includes a line segment  702 , which illustrates the actual look-up table outputs  704  for each sub-subinterval  654  of subinterval  652 A. 
     These actual look-up table outputs are equal to the base entry plus the corresponding difference table entry. As described above, for the first subintervals (such as  652 A) in subinterval  650 , the result of the base-difference addition is smaller than computed midpoints for the sub-subintervals in the subinterval. This can be seen in FIG. 15A, as actual look-up table output  704 A is less than computed midpoint  660 A. Furthermore, for the embodiment shown in FIG. 15A, the sub-subinterval with the maximum error within subinterval  652 A is sub-subinterval  654 A. The difference between computed midpoint  660 A and actual look-up table output  704 A is shown as maximum error value  706 . Actual look-up table outputs  704 B and  704 C in sub-subintervals  654 B and  654 C are also less than their respective computed midpoints, but not by as large a margin as in sub-subinterval  654 A. Sub-subinterval  654 D, however, is used as the reference sub-subinterval, and as a result, actual look-up table output  704 D is equal to computed midpoint  660 D. 
     Turning now to FIG. 15B, a graph  710  is shown of a portion of function f(x) (denoted by reference numeral  642 ) from FIG.  14 B. Only subinterval  652 D is shown in FIG.  15 B. As in FIG. 14B, subinterval  652 D includes four sub-subintervals ( 657 A-D), each having a corresponding midpoint  660 . Graph  710  further includes a line segment  712 , which depicts the actual look-up table outputs  714  for each sub-subinterval  657  of subinterval  652 D. 
     As in FIG. 15A, these actual look-up table outputs are equal to the base entry plus the corresponding difference table entry. As described above, for the last subintervals (such as  652 D) in subinterval  650 , the result of the base/difference addition is larger than computed midpoints for the sub-subintervals in the subinterval. This can be seen in FIG. 15B, as actual look-up table output  714 A is greater than computed midpoint  660 M. For the embodiment shown in FIG. 15B, the sub-subinterval with the maximum error is within subinterval  652 D is sub-subinterval  657 A. This difference between computed midpoint  660 M and actual look-up table output  714 A is shown as maximum error value  716 . Actual look-up table outputs  714 B and  714 C in sub-subintervals  657 B and  657 C are also greater than their respective computed midpoints, but not by as large a margin as in sub-subinterval  657 A. Sub-subinterval  657 D, however, is used as the reference sub-subinterval, and as a result, actual look-up table output  714 D is equal to computed midpoint  660 P. 
     In one embodiment, the base value for a subinterval may be adjusted (from the function output value at the midpoint of the reference sub-subinterval) in order to more evenly distribute the maximum error value. Although adjusting the base values increases error within the reference sub-subinterval, overall error is evenly distributed across all sub-subintervals in a subinterval. This ensures that error is minimized within a subinterval no matter which sub-subinterval bounds the input value. 
     Turning now to FIG. 15C, a graph  720  is depicted which illustrates portion  642  of function f(x) corresponding to subinterval  652 A. 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 error in sub-subinterval  654 A is equal to the error in sub-subinterval  654 D. The error in sub-subinterval  654 A is given by the difference between computed midpoint  660 A of sub-subinterval  654 A and adjusted look-up table output value  722 A. This difference is denoted by −Δf(x)  726 A in FIG.  15 C. The error in sub-subinterval  654 D is given by the difference between adjusted look-up table output value  722 D and computed midpoint  660 D of subinterval  654 D. This difference is denoted by Δf(x)  726 B. Thus, the error in sub-subinterval  654 A and the error in sub-subinterval  654 D are equal in magnitude, but opposite in sign. 
     Turning now to FIG. 15D, a graph  730  is depicted which illustrates portion  642  of function f(x) corresponding to subinterval  652 D. 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. 15C, which is positive, the offset value in FIG. 15D is negative. With this offset value, the values which make up line segment  734  are adjusted such that the error in sub-subinterval  657 A is equal to the error in sub-subinterval  657 D. The error in sub-subinterval  657 A is given by the difference between adjusted look-up table output value  732 A and computed midpoint  660 M. This difference is denoted by Δf(x)  736 A in FIG.  15 D. Similarly, the error in sub-subinterval  657 D is given by the difference between computed midpoint  660 P of subinterval  657 D and adjusted look-up table output value  732 D. This difference is denoted by −Δf(x)  736 B. Thus, the error in sub-subinterval  657 A and the error in sub-subinterval  657 D 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.  16 . 
     Turning now to FIG. 16, 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.  11 . Method  800  may be performed in conjunction with method  600  of FIG. 14A, or with other methods employed for computation of difference table entries. As needed, method  800  is also described with reference to FIGS. 14A-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. 16, 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 (referred 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. 16, 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 . Currently 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 currently selected subinterval P. In one embodiment, B corresponds to the function value at the midpoint (X 2 ) of a predetermined reference sub-subinterval, where the midpoint is calculated as described with reference to FIG.  13 B. (The midpoint of the reference sub-subinterval for subinterval P is denoted as X 2  in order to be consistent with the terminology of FIG.  14 A). The initial base value is thus given by the equation B=f(X 2 ). In one embodiment of look-up table  500  (such as in FIGS.  13 B and  14 A-D), the reference sub-subinterval (Q) is the last, or (K−1)th, sub-subinterval in each subinterval, where each subinterval includes sub-subintervals 0 to K−1. 
     Next, in step  812 , a function value (D) is computed which corresponds to the midpoint (X 3 ) 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. 15A, sub-subinterval  654 D is reference sub-subinterval Q, while sub-subinterval  654 A is sub-subinterval R. The function value D is thus given by the equation D=f(X 3 ), where X 3  is the midpoint of sub-subinterval R calculated as described above with reference to FIG. 13B in one embodiment. 
     In step  820 , the difference, (referred to as “actual difference” in FIG.  16 ), 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 referred to as the averaged difference value (“avg. diff.”). 
     In step  824 , the maximum error that results from using averaged difference values is calculated. This is performed by setting max error =actual diff.−avg. diff. As shown in FIGS. 14C and 14D, the maximum error from the averaged difference table values occurs in the first sub-subinterval in the subinterval (e.g., sub-subintervals  654 A and  657 A). In fact, the max error computed in step  824  of method  800  is equal to max error values  706  and  716  in FIGS. 14C and 14D. 
     In order to distribute the maximum error of step  824  throughout subinterval P, an adjust value is computed as a fraction of max error in step  826 . In order to evenly distribute the error throughout the subinterval, the adjust value is computed as half the maximum error 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−1  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 subinterval P is the last subinterval in interval M. If more subintervals exist, method  800  continues with step  808 . In step  808 , a next subinterval within interval M is selected, and the succeeding steps are usable to calculate the base value for the newly-selected subinterval. On the other hand, if P is the last subinterval in interval M, method  800  concludes with step  836 . 
     Methods for calculation of difference and base table entries are described 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 invention. Note that the #define&#39;s for HIGH, MID, and LOW effectively partition the input range of these functions into four intervals, four subintervals/interval, and four sub-subintervals/subinterval. 
     
       
         
           
               
             
               
                   
               
             
            
               
                 #define HIGH   2 
               
               
                 #define MID   2 
               
               
                 #define LOW   2 
               
               
                 #define OUT   16 
               
               
                 #define OUTP   16 
               
               
                 #define OUTQ   (OUTP−(HIGH+MID)+1) 
               
               
                 #define RECIPENTRIES (1L &lt;&lt; (HIGH+MID)) 
               
               
                 #define ROOTENTRIES (2L &lt;&lt; (HIGH+MID)) 
               
               
                 #define BIAS 127L    /* exponent bias for single precision format */ 
               
               
                 #define POW2(x) (1L &lt;&lt; (x)) /* helper function */ 
               
               
                 typedef union { 
               
               
                  float   f; 
               
               
                  unsigned long i; 
               
               
                 } SINGLE; 
               
               
                 #define SIGN_SINGLE(var) ((((var).i)&amp;0x80000000L)?1L:0L)   /* sign bit */ 
               
               
                 #define EXPO_SINGLE(var) ((((var).i)&gt;&gt;23L)&amp;0xFFL   /* 8 bit exponent */ 
               
               
                 #define MANT_SINGLE(var) (((var).i)&amp;0x7FFFFFL)   /* 23 bit mantissa */ 
               
               
                 #define SETSIGN_SINGLE(var,sign) \ 
               
               
                 (((var).i)=((sign)&amp;1)?(((var).i)|0x80000000L):(((var).i)&amp;0x7FFFFFFFL)) 
               
               
                 #define SETEXPO_SINGLE(var,expo) \ 
               
               
                 (((var).i)=(((var).i)&amp;0x807FFFFFL)|(((expo)&amp;0xFFL)&lt;&lt;23)) 
               
               
                 #define SETMANT_SINGLE(var,mant) \ 
               
               
                 (((var).i)=(((var).i)&amp;0xFF800000L)|(((mant)&amp;0x7FFFFFL))) 
               
               
                 extern unsigned long rom_p[ ]; 
               
               
                 extern unsigned long rom_q[ ]; 
               
               
                 #define TRUE 1 
               
               
                 #define FALSE 0 
               
               
                 #define HIGHMID (HIGH+MID) 
               
               
                 #define HIGHLOW (HIGH+LOW) 
               
               
                 #define ALL (HIGH+MID+LOW) 
               
               
                 #define POW2(x) (1L &lt;&lt;(x)) 
               
               
                 #define CONCAT(a,b,c) ((0X7FL &lt;&lt;23)|\ 
               
            
           
           
               
               
            
               
                   
                 (((a) &amp; (POW2(HIGH) − 1)) &lt;&lt; (23 − (HIGH)))|\ 
               
               
                   
                 (((b) &amp; (POW2(MID) − 1)) &lt;&lt; (23 − (HIGHMID)))|\ 
               
               
                   
                 (((c) &amp; (POW2(LOW) − 1)) &lt;&lt; (23 − (ALL)))) 
               
            
           
           
               
            
               
                 #define CONCAT2(e,a,b,c) (((e) &lt;&lt; 23)|\ 
               
            
           
           
               
               
            
               
                   
                 (((a) &amp; (POW2(HIGH) − 1)) &lt;&lt; (23 − (HIGH)))|\ 
               
               
                   
                 (((b) &amp; (POW2(MID) − 1)) &lt;&lt; (23 − (HIGHMID)))|\ 
               
               
                   
                 (((c) &amp; (POW2(LOW) − 1)) &lt;&lt; (23 − (ALL)))) 
               
            
           
           
               
            
               
                 void make_recip_bipartite_table (unsigned long *tablep, unsigned long *tableq) 
               
               
                 { 
               
               
                  unsigned long xh, xm, x1, indexp, indexq, maxq, minq, maxp, minp; 
               
               
                  SINGLE   temp1, temp2; 
               
               
                  double   midpoint1, midpoint2; 
               
               
                  double   reset, sumdiff, result1, result2, adjust; 
               
               
                  printf (“\nCreating lookup tables . . .\n”); 
               
               
                  for (xh = 0; xh &lt; POW2(HIGH); xh++) { 
               
               
                   for (x1 = 0; x1 &lt;POW2(LOW); xl++) { 
               
            
           
           
               
               
            
               
                   
                 indexq = (xh &lt;&lt; LOW)|xl; 
               
               
                   
                 sumdiff = 0.0; 
               
               
                   
                 for (xm = 0; xm &lt; POW2(MID); xm++) { 
               
               
                   
                  temp1.i = CONCAT (xh, xm, xl); 
               
               
                   
                  temp2.i = (temp1.i (POW2(23 − ALL) − 1)) + 1; 
               
               
                   
                  midpoint1 = (2.0 * temp1.f * temp2.f) / (temp1.f + temp2.f); 
               
               
                   
                  temp1.i = CONCAT (xh, xm, POW2(LOW)−1); 
               
               
                   
                  temp2.i = (temp1.i | (POW2(23 − ALL) − 1)) + 1; 
               
               
                   
                  midpoint2 = (2.0 * temp1.f * temp2.f) / (temp1.f + temp2.f); 
               
               
                   
                  sumdiff = sumdiff + ((1.0 / midpoint1) − (1.0 / midpoint2)); 
               
               
                   
                 } 
               
               
                   
                 result = 1.0/((double)(POW2(MID))) * sumdiff; 
               
               
                   
                 tableq [indexq]= (unsigned long)(POW2(OUTP+1) * result + 0.5); 
               
            
           
           
               
            
               
                  } 
               
               
                 } 
               
               
                 for (xh = 0; xh &lt; POW2(HIGH); xh++) { 
               
               
                  for (xm = xm &lt; POW2(MID); xm++) { 
               
            
           
           
               
               
            
               
                   
                 indexp = (xh &lt;&lt; (MID)) | xm; 
               
               
                   
                 temp1.i = CONCAT (xh, xm, 0); 
               
               
                   
                 temp2.i = (temp1.i | (POW2(23 − ALL) − 1)) + 1; 
               
               
                   
                 midpoint1 = (2.0 * temp1.f * temp2.f) / (temp1.f + temp2.f); 
               
               
                   
                 result1 = 1.0 / midpoint1; 
               
               
                   
                 temp1.i = CONCAT (xh, xm, POW2(LOW) − 1); 
               
               
                   
                 temp2.i = (temp1.i (POW2(23 − ALL) − 1)) + 1; 
               
               
                   
                 midpoint2 = (2.0 * temp1.f * temp2.f) / (temp1.f + temp2.f); 
               
               
                   
                 result2 = 1.0 / midpoint2; 
               
               
                   
                 adjust = 0.5 * ((result1 − result2) − (1.0/POW2(OUTP+1)) * tableq[xh &lt;&lt; LOW]); 
               
               
                   
                 tablep [indexp] = (unsigned long)(POW2(OUTP+1) * (result2 + adjust) + 0.5); 
               
               
                   
                 tablep [indexp]−= (1L &lt;&lt; OUTP); /* subtract out integer bit */ 
               
            
           
           
               
            
               
                   } 
               
               
                  } 
               
               
                 } 
               
               
                 void make_recipsqrt_bipartite_table (unsigned long *tablep, 
               
            
           
           
               
               
            
               
                   
                 unsigned long *tableq) 
               
            
           
           
               
            
               
                 { 
               
               
                  unsigned long xh, xm, xl, indexp, indexq, maxq, minq, start, end, 
               
            
           
           
               
               
            
               
                   
                 maxp, minp, expo; 
               
            
           
           
               
            
               
                  SINGLE   temp1, temp2; 
               
               
                  double   midpoint1, midpoint2; 
               
               
                  double   result, adjust, sumdif, result1, result2; 
               
               
                  printf(“\nCreating lookup tables . . . \n”); 
               
               
                  for (expo = 0x7F; expo &lt;= 0x80; expo++) { 
               
               
                   for (xh = 0; xh &lt;POW2(HIGH); xh++) { 
               
               
                   for (xl = 0; xl &lt;POW2(LOW); xl++) { 
               
            
           
           
               
               
            
               
                   
                 indexq = ((expo &amp; 1) &lt;&lt; (HIGHLOW)) (xh &lt;&lt; LOW) | xl; 
               
               
                   
                 sumdiff = 0.0; 
               
               
                   
                 for (xm = 0; xm &lt;POW2(MID); xm++) { 
               
               
                   
                  temp1.i = CONCAT2 (expo, xh, xm, xl); 
               
               
                   
                  temp2.i = (temp1.i (POW2(23 − ALL) − 1)) +1; 
               
               
                   
                  midpoint1 = (4.0 * temp1.f* temp2.f)/ 
               
            
           
           
               
            
               
                 ((sqrt(temp1.f)+sqrt(temp2.f))* (sqrt(temp 1.f)+sqrt(temp2.f))); 
               
            
           
           
               
               
            
               
                   
                  temp1.i = CONCAT2 (expo, xh, xm, POW2(LOW)−1); 
               
               
                   
                  temp2.i = (temp1.i (POW2(23 − ALL) − 1)) + 1; 
               
               
                   
                  midpoint2 = (4.0 * temp1.f* temp2.f) / 
               
            
           
           
               
            
               
                 ((sqrt(temp 1.f)+sqrt(temp2.f))*(sqrt(temp 1.f)+sqrt(temp2.f))); 
               
            
           
           
               
               
            
               
                   
                  sumdiff = sumdiff + ((1.0 / sqrt(midpoint1)) − (1.0 / sqrt(midpoint2))); 
               
               
                   
                 } 
               
               
                   
                 result = 1.0/((double)(POW2(MID))) * sumdiff; 
               
               
                   
                 tableq [indexq] = (unsigned long)(POW2(OUTP+1) * result + 0.5); 
               
            
           
           
               
            
               
                   } 
               
               
                  } 
               
               
                  for (xh = 0; xh &lt; POW2(HIGH); xh++) { 
               
               
                   for (xm = 0; xm &lt; POW2(MID); xm++) { 
               
            
           
           
               
               
            
               
                   
                 indexp = ((expo &amp; 1) &lt;&lt; (HIGHMID)) | (xh &lt;&lt; (MID)) | xm; 
               
               
                   
                 temp1.i = CONCAT2 (expo, xh, xm, 0); 
               
               
                   
                 temp2.i = (temp1.i (POW2(23 − ALL) − 1)) + 1; 
               
               
                   
                 midpoint1 = (4.0 * temp1.f* temp2.f)/ 
               
            
           
           
               
            
               
                 ((sqrt(temp 1.f)+sqrt(temp2.f))*(sqrt(temp 1.f)+sqrt(temp2.f))); 
               
            
           
           
               
               
            
               
                   
                 result1 = 1.0 / sqrt(midpoint1); 
               
               
                   
                 temp1.i = CONCAT2 (expo, xh, xm,POW2(LOW) − 1); 
               
               
                   
                 temp2.i = (temp1.i (POW2(23 − ALL) − 1)) + 1; 
               
               
                   
                 midpoint2 = (4.0 * temp1.f * temp2.f) / 
               
            
           
           
               
            
               
                 ((sqrt(temp1.f)+sqrt(temp2.f))*(sqrt(temp 1.f)+sqrt(temp2.f))); 
               
            
           
           
               
               
            
               
                   
                 result2 = 1.0 / sqrt(midpoint2); 
               
               
                   
                 adjust = 0.5 * ((result1 − result2) − (1.0/POW2(OUTP+1)) * tableq[((expo &amp; 1) &lt;&lt; 
               
            
           
           
               
            
               
                 (HIGH+LOW)) | (xh &lt;&lt; LOW)]); 
               
            
           
           
               
               
            
               
                   
                 tablep [indexp] = (unsigned long)(POW2(OUTP+1) * (result2 + adjust) + 0.5); 
               
               
                   
                 tablep [indexp] −= (1L &lt;&lt; OUTP); /* subtract out integer bit */ 
               
            
           
           
               
            
               
                   } 
               
               
                   } 
               
               
                  } 
               
               
                 } 
               
               
                 void recip_approx_bipartite ( 
               
               
                  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, indexq, indexp, p, q; 
               
               
                  /* handle zero separately */ 
               
               
                  if ((arg—&gt;i &amp; 0x7F800000L) == 0) { 
               
               
                  approx—&gt;i = (arg—&gt;i &amp; 0x80000000L) | 0x7F7FFFFFL; 
               
               
                  return; 
               
               
                  } 
               
               
                  /* unpack arg */ 
               
               
                  expo = (arg—&gt;i &gt;&gt; 23) &amp; 0xFF; 
               
               
                  sign = (arg—&gt;i &gt;&gt; 31) &amp; 1; 
               
               
                  mant = (arg—&gt;i &amp; 0x7FFFFFL); 
               
               
                  /* do table lookup on tables P and Q */ 
               
               
                  indexp = (mant &gt;&gt; (23 − (high + mid))); 
               
               
                  indexq = ((mant &gt;&gt; (23 − (high))) &lt;&lt; low) 
               
            
           
           
               
               
            
               
                   
                 ((mant &gt;&gt; (23 − (high+mid+low))) &amp; (POW2(low) − 1)); 
               
            
           
           
               
            
               
                  p = tablep [indexp]; 
               
               
                  q = tableq [indexq]; 
               
               
                  /* generate result in single precision format */ 
               
               
                  approx—&gt;i = ((2*BIAS + ˜expo) &lt;&lt; 23L) + 
               
            
           
           
               
               
            
               
                   
                 (((p + q)) &lt;&lt; (23L − out)); 
               
            
           
           
               
            
               
                  /* check for underflow */ 
               
               
                  if((((approx—&gt;i &gt;&gt; 23) &amp; 0xFFL) == 0x00L) ∥ 
               
               
                   (((approx—&gt;i &gt;&gt; 23) &amp; 0xFFL) == 0xFFL)) { 
               
               
                  approx—&gt;i = 0L; 
               
               
                  } 
               
               
                  /* mask sign bit because exponent above may have overflowed into sign bit */ 
               
               
                  approx—&gt;i = (approx—&gt;i &amp; 0x7FFFFFFFL) (sign &lt;&lt; 31L); 
               
               
                 } 
               
               
                 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, indexq, indexp, p, q; 
               
               
                  long expo; 
               
               
                  /* Handle zero separately. Returns maximum normal */ 
               
               
                  if ((arg—&gt;i &amp; 0x7F800000L) == 0L) { 
               
               
                  approx—&gt;i = 0x7F7FFFFFL | (arg—&gt;i &amp; 0x80000000L); 
               
               
                  return; 
               
               
                  } 
               
               
                  expo = (arg—&gt;i &gt;&gt; 23) &amp; 0xFFL; 
               
               
                  sign = (arg—&gt;i &gt;&gt; 31) &amp; 1; 
               
               
                  mant = (arg—&gt;i &amp; 0x7FFFFFL); 
               
               
                  indexp = ((expo &amp; 1) &lt;&lt; (high + mid)) | (mant &gt;&gt; (23 − (high + mid))); 
               
               
                  indexq = ((expo &amp; 1) &lt;&lt; (high + low)) | ((mant &gt;&gt; (23 − (high))) &lt;&lt; low) | 
               
            
           
           
               
               
            
               
                   
                 ((mant &gt;&gt; (23 − (high + mid + low))) &amp; (POW2(low) −1)); 
               
            
           
           
               
            
               
                  p = tablea [indexp]; 
               
               
                  q = tableb [indexq]; 
               
               
                  approx—&gt;i = (((3*BIAS + ˜expo) &gt;&gt; 1) &lt;&lt; 23) + 
               
            
           
           
               
               
            
               
                   
                 (((p + q)) &lt;&lt; (23 − out)); 
               
            
           
           
               
            
               
                  approx—&gt;i |= sign &lt;&lt; 31; 
               
               
                 } 
               
               
                   
               
            
           
         
       
     
     To further clarify calculation of base and difference table entries in the embodiment corresponding 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 entries are calculated similarly. The input range (1.0 inclusive to 2.0 exclusive) for this example is divided into four intervals, four subintervals/interval, and four sub-subintervals/subinterval. The table values are only shown for the first interval (1.0 inclusive to 1.25 exclusive) for simplicity. 
     The difference table for this example receives a four bit index (two bits for the interval, two bits for the sub-subinterval group). The base table also receives a four bit index (two bits for the interval, two bits for the subinterval). The base table includes 16 bits, while the difference table includes 13 bits for this embodiment. 
     
       
         
           
               
               
               
               
               
               
             
               
                 TABLE 1 
               
               
                   
               
               
                   
                   
                 Sub- 
                   
                   
                   
               
               
                 Int. 
                 Sub int. 
                 Sub. 
                 A 
                 B 
                 A (Binary) 
               
               
                   
               
             
            
               
                 0 
                 0 
                 0 
                 1.0 
                 1.015625 
                 1.00 00 00 . . . 
               
               
                 0 
                 0 
                 1 
                 1.015625 
                 1.03125 
                 1.00 00 01 . . . 
               
               
                 0 
                 0 
                 2 
                 1.03125 
                 1.046875 
                 1.00 00 10 . . . 
               
               
                 0 
                 0 
                 3 
                 1.046875 
                 1.0625 
                 1.00 00 11 . . . 
               
               
                 0 
                 1 
                 0 
                 1.0625 
                 1.078125 
                 1.00 01 00 . . . 
               
               
                 0 
                 1 
                 1 
                 1.078125 
                 1.093125 
                 1.00 01 01 . . . 
               
               
                 0 
                 1 
                 2 
                 1.093125 
                 1.109375 
                 1.00 01 10 . . . 
               
               
                 0 
                 1 
                 3 
                 1.109375 
                 1.125 
                 1.00 01 11 . . . 
               
               
                 0 
                 2 
                 0 
                 1.125 
                 1.140625 
                 1.00 10 00 . . . 
               
               
                 0 
                 2 
                 1 
                 1.140625 
                 1.15625 
                 1.00 10 01 . . . 
               
               
                 0 
                 2 
                 2 
                 1.15625 
                 1.171875 
                 1.00 10 10 . . . 
               
               
                 0 
                 2 
                 3 
                 1.171875 
                 1.1875 
                 1.00 10 11 . . . 
               
               
                 0 
                 3 
                 0 
                 1.1875 
                 1.203125 
                 1.00 11 00 . . . 
               
               
                 0 
                 3 
                 1 
                 1.203125 
                 1.21875 
                 1.00 11 01 . . . 
               
               
                 0 
                 3 
                 2 
                 1.21875 
                 1.234375 
                 1.00 11 10 . . . 
               
               
                 0 
                 3 
                 3 
                 1.234375 
                 1.25 
                 1.00 11 11 . . . 
               
               
                   
               
            
           
         
       
     
     Table 1 illustrates the partitioning of the first interval of the input range of the reciprocal function. With regard to the binary representation of A, only the six high-order mantissa bits are shown since these are the ones that are used to specify the interval, subinterval, and sub-subinterval group of the input sub-region. Note that the first group of mantissa bits of A corresponds to the interval number, the second group corresponds to the subinterval number, and the third group corresponds to the sub-subinterval group. 
     Table 2 shows the midpoint of each sub-subinterval (computed as in FIG.  13 B), as well as the function evaluation at the midpoint and the difference value with respect to the reference sub-subinterval of the subinterval. (The reference sub-subintervals are those in group  3 ). 
     
       
         
           
               
               
               
               
               
             
               
                 TABLE 2 
               
               
                   
               
               
                 Subint. 
                 Sub-Sub. 
                 Midpoint (M) 
                 f(M) = 1/M 
                 Diff. Value 
               
               
                   
               
             
            
               
                 0 
                 0 
                 1.007751938 
                 .992307692 
                 .04410751672 
               
               
                 0 
                 1 
                 1.023377863 
                 .977156177 
                 .02895600156 
               
               
                 0 
                 2 
                 1.039003759 
                 .962460426 
                 .01426024955 
               
               
                 0 
                 3 
                 1.05462963 
                 .948200175 
                 0 
               
               
                 1 
                 0 
                 1.070255474 
                 .934356352 
                 .03920768144 
               
               
                 1 
                 1 
                 1.085881295 
                 .920910973 
                 .02576230329 
               
               
                 1 
                 2 
                 1.101507092 
                 .907847083 
                 .01269841270 
               
               
                 1 
                 3 
                 1.117132867 
                 .895148670 
                 0 
               
               
                 2 
                 0 
                 1.132758621 
                 .882800609 
                 .03508131058 
               
               
                 2 
                 1 
                 1.148384354 
                 .870788597 
                 .02306929857 
               
               
                 2 
                 2 
                 1.164010067 
                 .859099099 
                 .01137980085 
               
               
                 2 
                 3 
                 1.179635762 
                 .847719298 
                 0 
               
               
                 3 
                 0 
                 1.195261438 
                 .836637047 
                 .03157375602 
               
               
                 3 
                 1 
                 1.210887097 
                 .825840826 
                 .0207775347 
               
               
                 3 
                 2 
                 1.226512739 
                 .815319701 
                 .01025641026 
               
               
                 3 
                 3 
                 1.242138365 
                 .805063291 
                 0 
               
               
                   
               
            
           
         
       
     
     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 3 
               
               
                   
               
               
                 Sub-Sub. 
                 Difference 
                 Integer 
               
               
                 Group 
                 Average 
                 Value (hex) 
               
               
                   
               
             
            
               
                 0 
                 .03749256619 
                 1332 
               
               
                 1 
                 .02464128453 
                 0C9E 
               
               
                 2 
                 .01214871834 
                 0638 
               
               
                 3 
                 0 
                 0000 
               
               
                   
               
            
           
         
       
     
     With regard to the base values for this example, Table 4 below shows midpoints X 2  and X 3 . Midpoint X 2  is the midpoint for the reference sub-subinterval of each subinterval, while X 3  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 4 
               
               
                   
               
               
                   
                   
                 Init. Base 
                   
                   
               
               
                 Subint. 
                 Midpoint X2 
                 Value (1/X2) 
                 Midpoint X3 
                 1/X3 
               
               
                   
               
             
            
               
                 0 
                 1.05462963 
                 .9482001756 
                 1.007751938 
                 .992307692 
               
               
                 1 
                 1.117132867 
                 .8951486698 
                 1.070255474 
                 .934356352 
               
               
                 2 
                 1.179635762 
                 .8477192982 
                 1.132758621 
                 .882800609 
               
               
                 3 
                 1.242138365 
                 .8050632911 
                 1.195261438 
                 .836637047 
               
               
                   
               
            
           
         
       
     
     Next, Table 5 below shows the actual error difference for each subinterval, computed as 1/X 3 −1/X 2 . 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 error for the subinterval. Half of this value is the adjust value. 
     
       
         
           
               
               
               
               
               
             
               
                 TABLE 5 
               
               
                   
               
               
                   
                 Actual Diff. 
                 Average 
                 Maximum 
                 Adjust 
               
               
                 Subint. 
                 (1/X3 − 1/X2) 
                 Diff. 
                 Error 
                 Value 
               
               
                   
               
             
            
               
                 0 
                 .044107516 
                 .03749256619 
                  .00661495 
                  .003307475 
               
               
                 1 
                 .039207682 
                 .03749256619 
                  .001715116 
                  .000857558 
               
               
                 2 
                 .0358081311 
                 .03749256619 
                 −.002411255 
                 −.001205628 
               
               
                 3 
                 .031573756 
                 .03749256619 
                 −.00591881 
                 −.002959405 
               
               
                   
               
            
           
         
       
     
     In Table 6, The adjust value plus the initial base value gives the 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 corresponds to the integer position). 
     
       
         
           
               
               
               
             
               
                 TABLE 6 
               
               
                   
               
               
                   
                 Final Base 
                 Integer 
               
               
                 Subint. 
                 Value 
                 Value (hex) 
               
               
                   
               
             
            
               
                 0 
                 .951507651 
                 E72C 
               
               
                 1 
                 .896006228 
                 CAC1 
               
               
                 2 
                 .846513671 
                 B16A 
               
               
                 3 
                 .802103886 
                 9AAD 
               
               
                   
               
            
           
         
       
     
     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 instruction within the instruction set of the microprocessor. Additional dedicated instructions may be employed in 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 time. 
     In one embodiment, base and difference values calculated as described in FIGS. 13A and 15 result in table output values with minimized absolute error. Advantageously, this minimal absolute error is obtained with a bipartite table configuration, which requires less table storage than a naive table of comparable accuracy. This configuration also allows the interpolation to be achieved with a simple addition. Thus, a costly multiply or multiply-add is not required to generate the final table output, effectively increasing the performance of the table look-up operation. 
     Turning now to FIG. 17, a block diagram of one embodiment of a computer system  900  including microprocessor  10  coupled to a variety of system components through a bus bridge  902  is shown. Other embodiments are possible and contemplated. In the depicted system, a main memory  904  is coupled to bus bridge  902  through a memory bus  906 , and a graphics controller  908  is coupled to bus bridge  902  through an AGP bus  910 . Finally, a plurality of PCI devices  912 A- 912 B are coupled to bus bridge  902  through a PCI bus  914 . A secondary bus bridge  916  may further be provided to accommodate an electrical interface to one or more EISA or ISA devices  918  through an EISA/ISA bus  920 . Microprocessor  10  is coupled to bus bridge  902  through a CPU bus  924 . 
     Bus bridge  902  provides an interface between microprocessor  10 , main memory  904 , graphics controller  908 , and devices attached to PCI bus  914 . When an operation is received from one of the devices connected to bus bridge  902 , bus bridge  902  identifies the target of the operation (e.g. a particular device or, in the case of PCI bus  914 , that the target is on PCI bus  914 ). Bus bridge  902  routes the operation to the targeted device. Bus bridge  902  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 providing an interface to an ISA/EISA bus for PCI bus  914 , secondary bus bridge  916  may further incorporate additional functionality, as desired. For example, in one embodiment, secondary bus bridge  916  includes a master PCI arbiter (not shown) for arbitrating ownership of PCI bus  914 . An input/output controller (not shown), either external from or integrated with secondary bus bridge  916 , may also be included within computer system  900  to provide operational support for a keyboard and mouse  922  and for various serial and parallel ports, as desired. An external cache unit (not shown) may further be coupled to CPU bus  924  between microprocessor  10  and bus bridge  902  in other embodiments. Alternatively, the external cache may be coupled to bus bridge  902  and cache control logic for the external cache may be integrated into bus bridge  902 . 
     Main memory  904  is a memory in which application programs are stored and from which microprocessor  10  primarily executes. A suitable main memory  904  comprises DRAM (Dynamic Random Access Memory), and preferably a plurality of banks of SDRAM (Synchronous DRAM). 
     PCI devices  912 A- 912 B are illustrative of a variety of peripheral devices such as, for example, network interface cards, video accelerators, audio cards, hard or floppy disk drives or drive controllers, SCSI (Small Computer Systems Interface) adapters and telephony cards. Similarly, ISA device  818  is illustrative of various types of peripheral devices, such as a modem, a sound card, and a variety of data acquisition cards such as GPIB or field bus interface cards. 
     Graphics controller  908  is provided to control the rendering of text and images on a display  926 . Graphics controller  908  may embody a typical graphics accelerator generally known in the art to render three-dimensional data structures which can be effectively shifted into and from main memory  904 . Graphics controller  908  may therefore be a master of AGP bus  910  in that it can request and receive access to a target interface within bus bridge  902  to thereby obtain access to main memory  904 . A dedicated graphics bus accommodates rapid retrieval of data from main memory  904 . For certain operations, graphics controller  908  may further be configured to generate PCI protocol transactions on AGP bus  910 . The AGP interface of bus bridge  902  may thus include functionality to support both AGP protocol transactions as well as PCI protocol target and initiator transactions. Display  926  is any electronic display upon which an image or text can be presented. A suitable display  926  includes 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 in the above description, any bus architectures may be substituted as desired. It is further noted that computer system  900  may be a multiprocessing computer system including additional microprocessors (e.g. microprocessor  10   a  shown as an optional component of computer system  900 ). Microprocessor  10   a  may be similar to microprocessor  10 . More particularly, microprocessor  10   a  may be an identical copy of microprocessor  10 . Microprocessor  10   a  may share CPU bus  924  with microprocessor  10  (as shown in FIG. 17) or may be connected to bus bridge  902  via an independent bus. 
     It is noted that while base and difference tables have been described above with reference to the reciprocal and reciprocal square root functions, such tables are generally applicable to any monotonically decreasing function. These tables are also applicable to a function which is monotonically decreasing over the desired input range. 
     In another embodiment, these base and difference tables may be modified to accommodate any monotonically increasing function(such as sqrt(x)), as well as any function which is monotonically increasing over a desired input range. In such an embodiment, the “leftmost” sub-subinterval within an interval becomes the reference point instead of the “rightmost” sub-subinterval, ensuring the values in the difference tables are positive. Alternatively, the “rightmost” sub-subinterval may still be used as the reference point if difference values are considered negative and a subtractor is used to combine base and difference table values. 
     Numerous variations and modifications will become apparent to those skilled in the art once the above disclosure is fully appreciated. It is intended that the following claims be interpreted to embrace all such variations and modifications.