Patent Publication Number: US-7219117-B2

Title: Methods and systems for computing floating-point intervals

Description:
RELATED APPLICATION 
   U.S. patent application Ser. No. 10/320,450, filed on even date herewith in the name of Guy L. Steele Jr. and entitled “METHODS AND SYSTEMS FOR COMPUTING THE QUOTIENT OF FLOATING-POINT INTERVALS,” assigned to the assignee of the present application, is hereby incorporated by reference. 
   TECHNICAL FIELD 
   The present invention relates to the field of floating-point numbers. More particularly, the present invention, in various specific embodiments, involves methods and systems directed to computing floating-point intervals. 
   BACKGROUND 
   Digital electronic devices, such as digital computers, calculators and other devices, perform arithmetic calculations on values in integer, or “fixed point,” format, in fractional, or “floating point” format, or both. Institute of Electrical and Electronic Engineers (IEEE) Standard 754, (hereinafter “IEEE Std. 754” or “the Standard”) published in 1985 and adopted by the American National Standards Institute (ANSI), defines several standard formats for expressing values in floating point format and a number of aspects regarding behavior of computation in connection therewith. In accordance with IEEE Std. 754, a representation in floating point format comprises a plurality of binary digits, or “bits,” having the structure
 
se msb  . . . e lsb f msb  . . . f lsb  
 
where bit “s” is a sign bit indicating whether the entire value is positive or negative, bits “e msb  . . . e lsb ” comprise an exponent field that represents the exponent “e” in unsigned binary biased format, and bits “f msb  . . . f lsb ” comprise a fraction field that represents the fractional portion “f” in unsigned binary format (“msb” represents “most significant bit” and “lsb” represents “least significant bit”). The Standard defines two general formats. A “single” format comprises thirty-two bits while a “double” format comprises sixty-four bits. In the single format, there is one sign bit “s,” eight bits “e 7  . . . e 0 ” comprising the exponent field and twenty-three bits “f 22  . . . f 0 ” comprising the fraction field. In the double format, there is one sign bit “s,” eleven bits “e 10  . . . e 0 ” comprising the exponent field and fifty-two bits “f 51  . . . f 0 ” comprising the fraction field.
 
   As indicated above, the exponent field of the floating point representation e msb  . . . e lsb  represents the exponent “E” in biased format. The biased format provides a mechanism by which the sign of the exponent is implicitly indicated. In particular, the bits “e msb  . . . e lsb ” represent a binary encoded value “e” such that “e=E+bias.” This allows the exponent E to extend from −126 to +127, in the eight-bit “single” format, and from −1022 to +1023 in the eleven-bit “double” format, and provides for relatively easy manipulation of the exponents in multiplication and division operations, in which the exponents are added and subtracted, respectively. 
   IEEE Std. 754 provides for several different formats with both the single and double formats which are generally based on the bit patterns of the bits “e msb  . . . e lsb ” comprising the exponent field and the bits “f msb  . . . f lsb ” comprising the fraction field. If a number is represented such that all of the bits “e msb  . . . e lsh ” of the exponent field are binary ones (i.e., if the bits represent a binary-encoded value of “255” in the single format or “2047” in the double format) and all of the bits f msb  . . . f lsb  of the fraction field are binary zeros, then the value of the number is positive or negative infinity, depending on the value of the sign bit “s.” In particular, the value “v” is v=(−1) s ∞, where “∞” represents the value “infinity.” On the other hand, if all of the bits “e msb  . . . e lsb ” of the exponent field are binary ones and if the bits “f msb  . . . f lsb ” of the fraction field are not all zeros, then the value that is represented is deemed “not a number,” which is abbreviated in the Standard by “NaN.” 
   If a number has an exponent field in which the bits “e msb  . . . e lSb ” are neither all binary ones nor all binary zeros (i.e., if the bits represent a binary-encoded value between 1 and 254 in the single format or between 1 and 2046 in the double format), the number is said to be a “normalized” format. For a number in the normalized format, the value represented by the number is v=(−1) s  2 e-bias (1.|f msb  . . . f lsb ), where “|” represents a concatenation operation. Effectively, in the normalized format, there is an implicit most significant digit having the value “one,” so that the twenty-three digits in the fraction field of the single format, or the fifty-two digits in the fraction field of the double format, will effectively represent a value having twenty-four digits or fifty-three digits of precision, respectively, where the value is less than two, but not less than one. 
   On the other hand, if a number has an exponent field in which the bits “e msb  . . . e lsb ” are all binary zeros, representing the binary-encoded value of “zero,” and a fraction field in which the bits f msb  . . . f lsb  are not all zero, the number is said to be a “de-normalized” format. For a number in the de-normalized format, the value represented by the number is v=(−1) s 2 e-bias+1 (0.|f msb  . . . f lsb ) It will be appreciated that the range of values of numbers that can be expressed in the de-normalized format is disjointed from the range of values of numbers that can be expressed in the normalized format, for both the single and double formats. Finally, if a number has an exponent field in which the bits “e msb  . . . e lsb ” are all binary zeros, representing the binary-encoded value of “zero,” and a fraction field in which the bits f msb  . . . f lsb  are all zero, the number has the value “zero”. It will be appreciated that the value “zero” may be positive zero or negative zero, depending on the value of the sign bit. 
   The discipline of interval arithmetic represents a range of values as a pair of numbers. For example, the interval [a,b] may represent the set of numbers x such that x is not less than a and b is not less than x:
 
[ a,b]={x|a&lt;=x&lt;=b} 
 
   Any numerical function f of one numerical argument is then extended to accept an interval as an argument by considering the set
 
 F={f ( x )| a&lt;=x&lt;=b} 
 
and then defining
 
 f ([ a,b ])=[ p,q ]where  p =inf  F  and  q =sup  F  
 
where “inf F” (also called the greatest lower bound of F) is the largest number that is not greater than any number in the set F, and “sup F” (also called the least upper bound of F) is the smallest number that is not less than any number in the set F. Thus, the result is the smallest possible interval that contains every possible result of applying f to some number in the argument interval.
 
   Similarly, any numerical function g of two numerical arguments is extended to accept intervals as arguments by considering the set
 
 G={g ( x,y )| a&lt;=x=b  and  c&lt;=y&lt;=d} 
 
and then defining
 
 g  ([ a,b], [c,d] )=[ p,q ] where  p =inf  G  and  q =sup  G.  
 
   The result is the smallest possible interval that contains every possible result of applying g to two numbers such that the first number lies in the first argument interval and the second number lies in the second argument interval. It may be difficult in some cases to ascertain this result set precisely because of mathematical difficulty or limits on computational resources. Therefore, it may be acceptable to compute an approximation, [p′,q′], to the true interval result such that p′&lt;=p and q&lt;=q′, so that the approximate result interval completely contains the true result interval. 
   For certain very well behaved functions f and g, it is relatively easy to specify the true interval result in terms of the endpoints of the argument interval(s) without the need to refer to applications of f or g to all possible numerical values in the specified intervals. 
   For example, if f is “−,” the negation operation,
 
−[ a,b]=[−b,−a] 
 
and if g is the binary addition operation “+,”
 
[ a,b]+[c,d]=[a+c,b+d].  
 
   Similarly, if g is the binary subtraction operation “−,”
 
[ a,b]−[c,d]=[a−c,b−d].  
 
   If g is the binary multiplication operation “*,”
 
[ a,b]*[c,d ]=[min( a*c,a*d,b*c,b*d ), max( a*c,a*d,b*c,b*d )]
 
where “min” is a function that returns a result equal to the smallest of its arguments and “max” is a function that returns a result equal to the largest of its arguments.
 
   And if g is the binary division operation “/,”
 
[ a,b]/[c,d ]=[min( a/c, a/d, b/c, b/d ), max( a/c, a/d, b/c, b/d )],
 
provided that either c&gt;0 or d&lt;0 (so that the divisor interval does not contain the value 0).
 
   The theory of interval arithmetic is sometimes used as the basis of a computational discipline within digital computers. In particular, sometimes the endpoints “a” and “b” of an interval are represented as floating-point numbers, and (for example) sometimes these floating-point numbers are represented according to IEEE Std. 754 for Binary Floating-Point Arithmetic. 
   SUMMARY OF THE INVENTION 
   Consistent with the present invention, methods and systems for computing floating-point intervals are provided that avoid problems associated with prior methods and systems for computing floating-point intervals. 
   In one aspect, a method for computing an output interval comprises producing a first product resulting from a conditional multiplication using a first operand, a second operand, and a third operand, the operands respectively including a first input interval lower-point, a second input interval lower-point, and a second input interval upper-point, producing a second product resulting from the conditional multiplication using the first the operand, the third operand, the operands respectively including a first input interval upper-point, the second input interval lower-point, and the second input interval upper-point, producing a third product resulting from the conditional multiplication using the first operand, the second operand, and the third operand, the operands respectively including the first input interval lower-point, the second input interval upper-point, and the second input interval lower-point, producing a fourth product resulting from the conditional multiplication using the first operand, the second operand, and the third operand, the operands respectively including the first input interval upper-point, the second input interval upper-point, and the second input interval lower-point, and producing the output interval including an output interval lower-point and an output interval upper-point, the output interval lower-point being the minimum of the first product and the third product, and the output interval upper-point being the maximum of the second product and the fourth product. 
   In another aspect, a system for computing an output interval comprises a component for producing a first product resulting from a conditional multiplication using a first operand, a second operand, and a third operand, the operands respectively including a first input interval lower-point, a second input interval lower-point, and a second input interval upper-point, a component for producing a second product resulting from the conditional multiplication using the first operand, the second operand, and the third operand, the operands respectively including a first input interval upper-point, the second input interval lower-point, and the second input interval upper-point, a component for producing a third product resulting from the conditional multiplication using the first operand, the second operand, and the third operand, the operands respectively including the first input interval lower-point, the second input interval upper-point, and the second input interval lower-point, a component for producing a fourth product resulting from the conditional multiplication using the first operand, the second operand, and the third operand, the operands respectively including the first input interval upper-point, the second input interval upper-point, and the second input interval lower-point, and a component for producing the output interval including an output interval lower-point and an output interval upper-point, the output interval lower-point being the minimum of the first product and the third product, and the output interval upper-point being the maximum of the second product and the fourth product. 
   In yet another aspect, a computer-readable medium on which is stored a set of instructions for computing an output interval, which when executed perform stages comprising producing a first product resulting from a conditional multiplication using a first operand, a second operand, and a third operand, the operands respectively including a first input interval lower-point, a second input interval lower-point, and a second input interval upper-point, producing a second product resulting from the conditional multiplication using the first operand, the second operand, and the third operand, the operands respectively including a first input interval upper-point, the second input interval lower-point, and the second input interval upper-point, producing a third product resulting from the conditional multiplication using the first operand, the second operand, and the third operand, the operands respectively including the first input interval lower-point, the second input interval upper-point, and the second input interval lower-point, producing a fourth product resulting from the conditional multiplication using the first operand, the second operand, and the third operand, the operands respectively including the first input interval upper-point, the second input interval upper-point, and the second input interval lower-point, and producing the output interval including an output interval lower-point and an output interval upper-point, the output interval lower-point being the minimum of the first product and the third product, and the output interval upper-point being the maximum of the second product and the fourth product. 
   Both the foregoing general description and the following detailed description are exemplary and are intended to provide further explanation of the invention as claimed. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The accompanying drawings provide a further understanding of the invention and, together with the detailed description, explain the principles of the invention. In the drawings: 
       FIG. 1  is a functional block diagram of a system for computing an output interval consistent with an embodiment of the present invention; 
       FIG. 2  is a flow chart of an exemplary method for computing an output interval consistent with an embodiment of the present invention; 
       FIG. 3  is a flow chart of an exemplary method for producing at least one of the first product, the second product, the third product, and the fourth product consistent with an embodiment of the present invention; 
       FIG. 4  illustrates a first table of special cases involving the product of zero or infinity consistent with an embodiment of the present invention; 
       FIG. 5  illustrates a second table of special cases involving the product of zero or infinity consistent with an embodiment of the present invention; 
       FIG. 6  illustrates a third table of special cases involving the product of zero or infinity consistent with an embodiment of the present invention; and 
       FIG. 7  illustrates a fourth table of special cases involving the product of zero or infinity consistent with an embodiment of the present invention. 
   

   DETAILED DESCRIPTION 
   Reference will now be made to various embodiments according to this invention, examples of which are shown in the accompanying drawings and will be obvious from the description of the invention. In the drawings, the same reference numbers represent the same or similar elements in the different drawings whenever possible. 
   A floating-point number format may not represent all possible real numbers; therefore, it may not be possible to represent all possible intervals in interval arithmetic. For example, the mathematical values ⅓ and ⅔ may not be represented exactly as a binary floating-point number, so it may not be possible to represent the interval [⅓, ⅔] using binary floating-point numbers. (Such an interval would be the mathematical result of dividing the interval [1, 2] by the interval [3, 3], for example, and those two example intervals can be represented exactly using floating-point numbers.) Therefore, such a computational discipline may rely on approximations. If the result of the division ⅓ is “rounded down” to a floating-point number t that is less than ⅓ in value, and the result of the division ⅔ is “rounded up” to a floating-point number u that is greater than ⅔in value, then a digital computer may deliver the interval [t,u] as the result of dividing [1, 2] by [3, 3]. The interval [t, u] will contain the interval [⅓, ⅔], so this result may be considered acceptable for many computational purposes. In practice, t may be chosen to be the largest possible floating-point number that is not greater than ⅓ and u may be chosen to be the smallest possible floating-point number that is not less than ⅔, so that [t, u] may be as good an approximation as possible to the true mathematical interval result [⅓, ⅔]. 
   IEEE Std. 754 provides for rounding the results of arithmetic operations either up or down on request. While IEEE Std. 754 calls for specifying the rounding through “rounding mode” information stored in a separate register, the direction of rounding may be specified as part of the instruction operation code. 
   For computational purposes, the formulas for four binary arithmetic operations may be implemented as follows:
 
[ a,b]+[c,d ]=[down( a+c ), up( b+d )]
 
[ a,b]−[c,d ]=[down( a−c ), up( b−d )]
 
[ a,b]*[c,d ]=[min(down( a*c ), down( a*d ), down( b*c ), down( b*d )), max(up( a*c ), up( a*d ), up( b*c ), up( b*d ))]
 
[ a,b]/[c,d ]=[min(down( a/c ), down( a/d ), down( b/c ), down( b/d )), max(up( a/c ), up( a/d ), up( b/c ), up( b/d ))]
 
where “down” indicates that the result r of an arithmetic operation is to be rounded down to the largest representable floating-point value that is not larger than r, and “up” indicates that the result r of an arithmetic operation is to be rounded up to the smallest representable floating-point value that is not less than r.
 
   In the above computational scheme, while addition or subtraction of intervals is only approximately twice as expensive as addition or subtraction of single floating-point numbers, multiplication or division, using the formulas shown above, is more than eight times as expensive. This is because eight multiplications or divisions may be needed in the general case, in addition to the min and max operations. 
   A case analysis may be performed on the signs of the endpoints of the interval arguments. If each endpoint of an interval is replaced by its sign, + or −, then there are three possibilities: [−,−],[−,+], and [+,+]. For a binary operation such as multiplication, there are then three possibilities for each argument, or nine in all. It is then possible to prove for each of the nine cases that certain of the operands to the min operation may not be the smallest, and that certain operands to the max operation may not be the largest. As a result, a different formula can be used for each of the nine cases: 
   
     
       
         
             
             
             
           
             
                 
             
             
               [a, b] 
               [c, d] 
               Result 
             
             
                 
             
           
          
             
               [+, +] 
               [+, +] 
               [down(a*c), up(b*d)] 
             
             
               [+, +] 
               [−, +] 
               [down(b*c), up(b*d)] 
             
             
               [+, +] 
               [−, −] 
               [down(b*c), up(a*d)] 
             
             
               [−, +] 
               [+, +] 
               [down(a*d), up(b*d)] 
             
             
               [−, +] 
               [−, +] 
               [min(down(a*d), down(b*c)), max(up(a*c), up(b*d))] 
             
             
               [−, +] 
               [−, −] 
               [down(b*c), up(a*c)] 
             
             
               [−, −] 
               [+, +] 
               [down(a*d), up(b*c)] 
             
             
               [−, −] 
               [−, +] 
               [down(a*d), up(a*c)] 
             
             
               [−, −] 
               [−, −] 
               [down(b*d), up(a*c)] 
             
             
                 
             
          
         
       
     
   
   The above case analysis may be expressed in software as a set of nested if-then-else statements that cause one of nine formulas to be chosen and executed. This has the advantage that at most four multiplications need to be executed rather than eight, and in many cases only two multiplications may be needed. But the complexity of the case analysis, and that one of the nine formulas has a greater implementation cost than the other eight, may discourage hardware implementations of this strategy. Also, with the case analysis possibly including multiple conditional branches, implementation may be time-consuming in software, especially on contemporary pipelined microprocessors that rely on branch prediction for good performance. In current RISC microprocessor designs it is not conventional to allow a single instruction to accept four distinct floating-point operands or to generate more than one floating-point result. Most instructions accept only two floating-point operands and perhaps a limited number of instructions accept three floating-point operands at the most. And it is rare to allow a single instruction to deliver a result to more than one floating-point register. (These restrictions may stem from the costs associated with providing a register file with multiple read ports and write ports.) 
   IEEE Std. 754 provides representations for positive and negative zero and for positive and negative infinity. When these values are used as endpoints of an interval, they may be interpreted in more than one way, according to whether the signs of zeros are regarded as significant. For example, positive zero times positive infinity may be regarded as potentially producing any real value as its result or as producing only a value with a positive sign. In the first case, rounding the result of positive zero times positive infinity down should produce minus infinity, and rounding the result up should produce positive infinity. In the second case, rounding the result of positive zero times positive infinity down should produce positive zero, and rounding the result up should produce positive infinity. 
   IEEE Std. 754 specifies that when a zero is multiplied by an infinity, the result should be a NaN (Not A Number) value, which is not appropriate for use as the endpoint of an interval. Therefore, implementations of interval arithmetic may handle the case of zero times infinity through special-case checks in software. A multiplication circuit that rounds the result of multiplying a zero times an infinity up or down to produce a result appropriate to interval arithmetic when the signs of zeros are regarded as significant is disclosed and described U.S. patent application Ser. No. 10/035,580, entitled “FLOATING-POINT MULTIPLIER WITH EMBEDDED STATUS INFORMATION”, which is hereby incorporated by reference. 
   Consistent with the general principles of the present invention, a system for computing an output interval comprises a component for producing a first product resulting from a conditional multiplication using a first operand, a second operand, and a third operand, the operands respectively including a first input interval lower-point, a second input interval lower-point, and a second input interval upper-point, a component for producing a second product resulting from the conditional multiplication using the first operand, the second operand, and the third operand, the operands respectively including a first input interval upper-point, the second input interval lower-point, and the second input interval upper-point, a component for producing a third product resulting from the conditional multiplication using the first operand, the second operand, and the third operand, the operands respectively including the first input interval lower-point, the second input interval upper-point, and the second input interval lower-point, a component for producing a fourth product resulting from the conditional multiplication using the first operand, the second operand, and the third operand, the operands respectively including the first input interval upper-point, the second input interval upper-point, and the second input interval lower-point, and a component for producing the output interval including an output interval lower-point and an output interval upper-point, the output interval lower-point being the minimum of the first product and the third product, and the output interval upper-point being the maximum of the second product and the fourth product. 
   The component for producing the first product, the component for producing the second product, the component for producing the third product, the component for producing the fourth product, and the component for producing the output interval may comprise elements of, be disposed within, or may otherwise be utilized by or embodied within a mobile phone, a personal computer, a hand-held computing device, a multiprocessor system, microprocessor-based or programmable consumer electronic device, a minicomputer, a mainframe computer, a personal digital assistant (PDA), a facsimile machine, a telephone, a pager, a portable computer, or any other device that may receive, transmit, and/or process information. The above list of devices are exemplary and the component for producing the first product, the component for producing the second product, the component for producing the third product, the component for producing the fourth product, and the component for producing the output interval may comprise elements of, be disposed within, or may otherwise be utilized by or embodied within many other devices or system without departing from the scope and spirit of the invention. 
   Moreover, embodiments of the invention may be practiced in electrical circuits comprising discrete electronic elements, packaged or integrated electronic chips containing logic gates, circuits utilizing microprocessors, or on a single chip containing electronic elements or microprocessors. Furthermore, embodiments of the invention may be provided using other technologies capable of performing logical operations such as, for example, AND, OR, and NOT, including but not limited to mechanical, optical, fluidic, and quantum technologies. In addition, the invention may be practiced within a general purpose computer, may be implemented in software either as part of a runtime library routine or software generated by an otherwise conventional compiler, or in any other circuits or systems. 
   As herein embodied and illustrated in  FIG. 1 , the system for computing the output interval may comprise an output interval computing system  100 . The following description of the system for computing an output interval is exemplary and many other configurations may be employed. System  100  may include a first operand register  105  for receiving a first operand, a second operand register  110  for receiving a second operand, and a third operand register  115  for receiving a third operand. Operand registers  105 ,  110 , and  115  receive and store respective operands and may comprise, for example, a set of registers in a conventional manner. 
   In addition to operand registers  105 ,  110 , and  115 , system  100  may include a multi-bit multiplexer  120  configured to receive the output of second operand register  110 , third operand register  115 , and a sign bit  125  of first operand register  105 . Once input is received, multiplexer  120  is configured to place on a multiplexer output  130 , either the output of second operand register  110  or the output of third operand register  115 , based upon sign bit  125 . For example, if sign bit  125  comprises a digital “0”, multiplexer  120  may place the output of second operand register  110  on multiplexer output  130 . If sign bit  125  comprises a digital “1”, however, multiplexer  120  may place the output of third operand register  115  on multiplexer output  130 . 
   System  100  also includes a multiplier core  135  configured to receive multiplexer output  130 , the output of first operand register  105 , and a rounding mode indicator  145 . Multiplier core  135  generates a product of multiplexer output  130  and the output of first operand register  105  as a product result  140 . Multiplier core  135  may generate this product in accordance with IEEE Std. 754. Furthermore, product result  140  may be rounded up or rounded down depending upon the status of rounding mode indicator  145 . 
   Although not illustrated in  FIG. 1 , system  100  may include other elements configured to handle special cases involving the product of zero or infinity in a manner illustrated in tables  405 ,  505 ,  605 , and  705  of  FIGS. 4–7 , respectively. For example, if the output of first operand register  105  or multiplexer output  130  comprises −∞, +∞, +0, or −0, product result  140  may comprise a value as indicated in  FIGS. 4–7 . The elements of system  100  configured to handle special cases involving the product of zero or infinity, for example, may comprise a decision logic table. The decision logic table may be implemented using many different circuit elements including, but not limited to, programmable logic arrays, ASIC circuits, general memory registers, other addressable memory storage devices, or combinations thereof. The aforementioned elements are exemplary and many other elements or configurations may be employed. Moreover, special cases involving the product of zero or infinity may be processes using computer software including, for example, a runtime library routine or software generated by a compiler. 
     FIG. 4  illustrates the special case where the product result is rounded down with the signs of zeros treated as significant.  FIG. 5  illustrates the special case where the product result is rounded up with the signs of zeros treated as significant.  FIG. 6  illustrates the special case where the product result is rounded down with the signs of zeros not treated as significant.  FIG. 7  illustrates the special case where the product result is rounded up with the signs of zeros not treated as significant. As shown in table  405  of  FIG. 4 , for example, if one of two operands to be multiplied comprises +inf (plus infinity or +∞) as shown in column  410 , and the other operand to be multiplied comprises −0 as shown in row  415 , the product result may comprise −inf (minus infinity or −∞). Similarly, as shown in table  705  of  FIG. 7 , for example, if one of two operands to be multiplied comprises a negative number (−p) as shown in column  710 , and the other operand to be multiplied comprises −0 as shown in row  415 , the product result may comprise +0. Tables  505  and  605 , of  FIGS. 5 and 6  respectively, may be utilized in a similar manner. 
     FIG. 1  illustrates an embodiment that uses a 32-bit representation for floating-point numbers. This 32-bit representation is exemplary and other representations may be employed including, but not limited to, a 64-bit representation. 
     FIG. 2  is a flow chart setting forth the general stages involved in an exemplary method for computing an output interval consistent with an embodiment of the present invention. The implementation of the stages of exemplary method  200  in accordance with an exemplary embodiment of the present invention will be described in detail below. 
   The multiplication of an interval may be described as follows:
 
[ a,b]*[c,d]=[e,f] 
 
   In the above equation, a and b may be described as a first input interval lower-point and a first input interval upper-point respectively, c and d may be described as a second input interval lower-point and a second input interval upper-point respectively, and e and f may be described as an output interval lower-point and an output interval upper-point respectively. 
   Exemplary method  200  begins at starting block  205  and proceeds to stage  210  where a first product is produced resulting from a conditional multiplication using a first operand, a second operand, and a third operand. In this case, the operands may respectively include the first input interval lower-point, the second input interval lower-point, and the second input interval upper-point. For example, the conditional multiplication used to produce the first product may be accomplished by the stages of exemplary method  300  as shown in  FIG. 3  and described below. Method  300  is exemplary, and other methods, processes, or procedures may be used, implemented either in hardware or in software for example, without departing from the spirit of the invention. 
     FIG. 3  is a flow chart setting forth the general stages involved in an exemplary method for producing at least one of the first product, the second product, the third product, and the fourth product resulting from the conditional multiplication using the first operand, the second operand, and the third operand. Exemplary method  300  begins at starting block  305  and proceeds to stage  310  where the first operand is received. For example, to produce the first product, the first operand may comprise the first input interval lower-point. 
   Once the first operand is received in stage  310 , exemplary method  300  advances to stage  315  where the second operand is received. For example, to produce the first product, the second operand may comprise the second input interval lower-point. After the second operand is received in stage  315 , exemplary method  300  continues to stage  320  where the third operand is received. For example, to produce the first product, the third operand may comprise the second input interval upper-point. 
   From stage  320  where the third operand is received, exemplary method  300  proceeds to stage  325  where a fourth operand is selected based upon a sign of the first operand. The fourth operand may comprise either of the second operand and the third operand. For example, if sign bit  125  comprises a digital “0”, multiplexer  120  may place the output of second operand register  110  on multiplexer output  130 . If sign bit  125  comprises a digital “1”, however, multiplexer  120  may place the output of third operand register  115  on multiplexer output  130 . 
   Once the fourth operand is selected in stage  325 , exemplary method  300  advances to stage  330  where the first operand and the fourth operand are multiplied. For example multiplier core  135  may receive multiplexer output  130 , the output of first operand register  105 , and the rounding mode indicator  145 . Multiplier core  135  may then generate the product of multiplexer output  130  and the output of first operand register  105  as product result  140 . The product may be generated in accordance with IEEE Std. 754. Product result  140  may be rounded up or rounded down depending upon the status of rounding mode indicator  145 . This rounding may be performed by multiplier core  135  or by other elements as described above with respect to  FIG. 1 . In addition, product result  140  may comprise a special value including the special cases involving the product of zero or infinity as described above with respect to  FIGS. 4–7 . After the first operand and the fourth operand are multiplied in stage  330 , exemplary method  300  ends at stage  335 . 
   Once the first product is produced in stage  210 , exemplary method  200  advances to stage  220  where a second product is produced resulting from the conditional multiplication using the first operand, the second operand, and the third operand. In this case, the operands may respectively include the first input interval Upper-point, the second input interval lower-point, and the second input interval upper-point. For example, the conditional multiplication used to produce the second product may be accomplished by the stages of exemplary method  300  as shown in  FIG. 3  and described above with respect to stage  210 . However, in stage  220 , the first input interval upper-point, the second input interval lower-point, and the second input interval upper-point are respectively received as the first operand, the second operand, and the third operand. 
   After the second product is produced in stage  220 , exemplary method  200  continues to stage  230  where a third product is produced resulting from the conditional multiplication using the first operand, the second operand, and the third operand. In this case, the operands may respectively include the first input interval lower-point, the second input interval upper-point, and the second input interval lower-point. For example, the conditional multiplication used to produce the third product may be accomplished by the stages of exemplary method  300  as shown in  FIG. 3  and described above with respect to stage  210 . However, in stage  230 , the first input interval lower-point, the second input interval upper-point, and the second input interval lower-point are respectively received as the first operand, the second operand, and the third operand. 
   From stage  230  where the third product is produced, exemplary method  200  proceeds to stage  240  where a fourth product is produced resulting from the conditional multiplication using the first operand, the second operand, and the third operand. In this case, the operands respectively include the first input interval upper-point, the second input interval upper-point, and the second input interval lower-point. For example, the conditional multiplication used to produce the fourth product may be accomplished by the stages of exemplary method  300  as shown in  FIG. 3  and described above with respect to stage  210 . However, in stage  240 , the first input interval upper-point, the second input interval upper-point, and the second input interval lower-point are respectively received as the first operand, the second operand, and the third operand. 
   Once the fourth product is produced in stage  240 , exemplary method  200  advances to stage  250  where the output interval is produced comprising the output interval upper-point and the output interval lower-point. The output interval upper-point is the minimum of the first product and the third product, and the output interval lower-point is the maximum of the second product and the fourth product. For example, methods and systems for producing minimums and maximums is disclosed and described U.S. patent application Ser. No. 10/035,746, entitled “CIRCUIT FOR SELECTIVELY PROVIDING MAXIMUMS OR MINIMUMS OF A PAIR OF FLOATING-POINT OPERANDS”, which is hereby incorporated by reference. After the output interval is produced in stage  250 , exemplary method  200  ends at stage  260 . 
   Other formats and bit patterns may be used to represent floating point operand formats without departing from the principles of the present invention. Also, floating point status information contained in the operands could easily be represented by other bit combinations (not shown) without departing from the principles of the present invention. For example, more or fewer bits could be used, a subset or superset of exemplary status bits could be used, or the most significant bits of an operand (or some other subset of bits) could be used to indicate floating point status information, instead of the least significant bits. 
   It will be appreciated that a system in accordance with an embodiment of the invention can be constructed in whole or in part from special purpose hardware or a general purpose computer system, or any combination thereof. Any portion of such a system may be controlled by a suitable program. Any program may in whole or in part comprise part of or be stored on the system in a conventional manner, or it may in whole or in part be provided in to the system over a network or other mechanism for transferring information in a conventional manner. In addition, it will be appreciated that the system may be operated and/or otherwise controlled by means of information provided by an operator using operator input elements (not shown) which may be connected directly to the system or which may transfer the information to the system over a network or other mechanism for transferring information in a conventional manner. 
   The foregoing description has been limited to a specific embodiment of this invention. It will be apparent, however, that various variations and modifications may be made to the invention, with the attainment of some or all of the advantages of the invention. It is the object of the appended claims to cover these and such other variations and modifications as come within the true spirit and scope of the invention. 
   Other embodiments of the invention will be apparent to those skilled in the art from consideration of the specification and practice of the invention disclosed herein. It is intended that the specification and examples be considered as exemplary only, with a true scope and spirit of the invention being indicated by the following claims.