Abstract:
A multi-operand decimal adder is described that performs addition on multiple binary coded decimal (BCD) operands. The multi-operand decimal adder uses binary carry-save adders to produce intermediate sums and carries, and outputs a decimal result based on the intermediate sums and carries. In various configurations, the multi-operand decimal adder may perform speculative or non-speculative binary carry-save addition. The multioperand decimal adders achieve a reasonable critical path. As a result, the decimal adders and the techniques described herein may be especially suited for numerically intensive commercial applications, such as spreadsheet or financial applications where large amounts of decimal data typically need to be processed quickly.

Description:
[0001]     This application claims the benefit of U.S. Provisional Application Ser. No. 60/612,421, filed Sep. 23, 2004, the entire content of which is incorporated herein by reference. 
     
    
     TECHNICAL FIELD  
       [0002]     The invention relates to computing devices and, more particularly, to techniques for performing arithmetic operations within computing devices.  
       BACKGROUND  
       [0003]     Although most people use decimal arithmetic when performing manual calculations, computers typically only support binary arithmetic in hardware. This is primarily due to the fact that modem computers typically only represent two logic values: zero and one. While it is possible to use these two logic values to represent decimal numbers, doing so is wasteful in terms of storage space and often computationally less efficient. For example, in binary, four bits can represent sixteen values; while in binary coded decimal (BCD), four bits only represent ten values. Since most computer systems do not provide hardware support for decimal arithmetic, numbers are typically inputted in decimal, converted from decimal to binary, processed using binary arithmetic and then converted back to decimal for output.  
         [0004]     In spite of the current dominance of hardware support for binary arithmetic, there are several motivations that encourage the provision of support for decimal arithmetic. First, applications that deal with financial and other real-world data often introduce errors since many common decimal numbers cannot be represented exactly in binary. For example, the decimal number “0.1” is a repeating fraction when represented in binary. Second, people typically think about computations in decimal, even when using computers that operate only on binary representations, and therefore may experience what is perceived as incorrect behavior when processing decimal values. Third, converting between binary and decimal numbers is computationally intensive. For example, conversion on modem processors may take thousands of processing cycles.  
         [0005]     In an effort to alleviate some of the problems that occur when a computer only supports binary arithmetic in hardware, several software packages and programming language extensions for decimal arithmetic have been developed. Although some of these packages are successful in eliminating binary-to-decimal and decimal-to-binary conversion errors, the packages are often hundreds to thousands of times slower than binary operations implemented in hardware.  
         [0006]     One conventional method of performing decimal addition is to add two decimal operands to produce a BCD sum. If the sum is greater than nine, a correction value of six is added to the sum. Adding six skips the invalid bit patterns and yields the correct BCD sum digit and a carry-out of the digit. For example, to compute 7+7=14 in BCD, the two sevens are added to produce 1110 binary. Since this sum is greater than 9, a correction of 6 (0110 binary) is added, which yields a carry-out of 1 and a sum of 0100. The carry-out of 1 yields the next decimal digit of the sum. For example, the produced sum, 0001 0100, is the BCD representation of 14.  
         [0007]     Adding two BCD operands in this manner is a computationally slow process. One technique used to improve the performance of two operand BCD addition is to speculate that the sum for each pair of digits will be greater than nine. This is done by speculatively adding six along with the original operands. The advantage of this approach is that the correct carries between digits are generated during the first addition. The digits of the resulting sums need to be corrected by subtracting six if the carryout for that digit is zero. The final correction is subtracted using only digit-wide subtractions because all sums are guaranteed to be at least six.  
       SUMMARY  
       [0008]     In general, the invention is directed to various techniques for high-speed, multiple-operand (multioperand) decimal addition and, more particularly, multioperand decimal addition using binary carry-save addition (CSA). For example, three techniques using speculative binary to binary coded decimal (BCD) correction are described. A non-speculative technique is also described.  
         [0009]     In one embodiment, a method comprises receiving at least three decimal operands with a decimal adder of a processing unit, and performing binary carry-save addition with the decimal multioperand adder to produce a set of sum bits and a set of carry bits. The method further comprises outputting a decimal result based on the set of sum bits and the set of carry bits.  
         [0010]     In another embodiment, a processing unit comprises a multioperand decimal adder that receives at least three decimal operands and performs binary carry-save addition and correction on the operands to produce a decimal result.  
         [0011]     In another embodiment, a method comprises (a) receiving at least three decimal operands with a decimal multioperand adder of a processing unit, and (b) adding a digit of a first one of the operands with a digit of a second one of the operands with a first binary carry-save adder to produce an intermediate sum and an intermediate carry. The method further comprises (c) adding a digit of a third one of the operands with a correction value in parallel with step (b) to produce a corrected version of the third one of the operands, and (d) selectively adding either the third one of the operands or the corrected version to the intermediate sum of step (b) with a second binary carry-save adder based on the intermediate carry from step (b).  
         [0012]     In another embodiment, a multioperand decimal adder comprises a plurality of inputs to receive a plurality of decimal operands, and a first binary carry-save adder that adds a digit of a first one of the operands with a digit of a second one of the operands to produce a first intermediate sum and a first intermediate carry. The multioperand decimal adder further comprises correction logic in parallel the first binary carry-save adder to produce a corrected version of a third operand, a multiplexer to selectively output either the third one of the operands or the corrected version of the third operand based on the first intermediate carry; and a second binary carry-save adder to add the first intermediate sum and the first intermediate carry with the output of the multiplexer.  
         [0013]     In another embodiment, a method comprises (a) receiving at least four decimal operands with a decimal adder of a processing unit, and (b) adding a digit of a first one of the operands with a digit of a second one of the operands with a first binary carry-save adder to produce an intermediate sum S 1  and an intermediate carry C 1 . The method further comprises (c) adding the intermediate sum S 1  and the intermediate carry C 1  with a digit of a third one of the operands with a second binary carry-save adder to produce an intermediate sum S 2  and an intermediate carry C 2 , (d) adding a digit of a fourth one of the operands with a correction value in parallel with steps (b) and (d) to produce a corrected version of the fourth one of the operands, and (e) selectively adding either the fourth one of the operands or the corrected version to the intermediate sum S 2  and the intermediate carry C 2  with a third binary carry-save adder based on the intermediate carry C 2 .  
         [0014]     In another embodiment, a multioperand decimal adder comprises a plurality of inputs to receive a plurality of decimal operands, a first binary carry-save adder that adds a digit of a first one of the operands with a digit of a second one of the operands to produce a first intermediate sum S 1  and a first intermediate carry C 1 , and a second binary carry-save adder that adds the intermediate sum S 1  and the first intermediate carry C 1  with a digit of a third one of the operands to produce a second intermediate sum S 2  and a second intermediate carry C 2 . The multioperand decimal adder further comprises correction logic in parallel the first binary carry-save adder and the second binary carry-save adder that outputs a corrected version of a fourth operand, a multiplexer to selectively output either the fourth one of the operands or the corrected version of the fourth operand based on the second intermediate carry C 2 , and a third binary carry-save adder to add the output from the multiplexer with the second intermediate sum S 2  and the second intermediate carry C 2  based on a logical value of the second intermediate carry C 2 .  
         [0015]     In another embodiment, a method comprises (a) receiving a plurality of decimal operands with a decimal adder of a processing unit, and (b) adding a first subset of the operands with a first binary speculative adder to produce an intermediate sum S 0  and an intermediate carry C 0 . The method further comprises (c) adding a second subset of the operands with a second binary speculative adder in parallel with step (b) to produce an intermediate sum S 1  and an intermediate carry C 1 , and (d) outputting a decimal result based on the intermediate sums S 0  and S 1  and the intermediate carries C 0  and C 1 .  
         [0016]     In another embodiment, a multioperand decimal adder comprises a plurality of inputs to receive a plurality of decimal operands, and a first speculative adder to add a first subset of the operands to produce a first inter intermediate sum S 1 . The multioperand decimal adder further comprises a second speculative adder in parallel with the first speculative adder to add a second subset of the operands to produce a second intermediate sum S 2 , and at least one decimal carry propagate adder to output the decimal result as a function of the intermediate sums S 1  and S 2 .  
         [0017]     In another embodiment, a method comprises receiving a plurality of decimal operands with a decimal adder of a processing unit, wherein the decimal adder includes a carry-save adder tree having a plurality of carry-save adders. The method further comprises adding the operands using a carry-save adder tree to produce an intermediate binary sum and a plurality of carries, and producing a decimal result based on the intermediate sum and a correction value selected based on the intermediate carries.  
         [0018]     In another embodiment, a multioperand decimal adder comprises a plurality of inputs to receive a plurality of decimal operands, and a carry-save adder tree having a plurality of carry-save adders that perform non-speculative binary addition to produce an intermediate binary sum and a plurality of carries. The multioperand decimal adder further comprises at least one decimal carry propagate adder to output the decimal result as a function of the intermediate binary sum and the plurality of carries.  
         [0019]     The technique may provide one or more advantages. For example, the multioperand decimal adders described herein may be used in a processing unit to provide fast and efficient decimal addition or subtraction of multiple operands. In addition, the multioperand decimal adders may be utilized in the implementation of fast decimal multipliers and dividers. The multioperand decimal adder may also be useful in spreadsheet or other applications where large amounts of decimal data typically need to be processed quickly. Moreover, embodiments of the described multioperand decimal adders achieve a reasonable critical path delay. As a result, the decimal adders and the techniques described herein may be especially suited for numerically intensive commercial applications.  
         [0020]     The details of one or more embodiments of the invention are set fourth in the accompanying drawings and the description below. Other features, objects, and advantages of the invention will be apparent from the description and drawings, and from the claims. 
     
    
     BRIEF DESCRIPTION OF DRAWINGS  
       [0021]      FIG. 1  is a block diagram illustrating a processing unit that performs decimal arithmetic in accordance with the invention.  
         [0022]      FIG. 2  is a block diagram illustrating a single-digit multioperand decimal adder, which uses single correction speculation according to an embodiment of the invention.  
         [0023]      FIG. 3  is a flowchart illustrating exemplary operation of a multioperand decimal adder that performs decimal arithmetic using single correction speculation in accordance with the invention.  
         [0024]      FIG. 4  is a block diagram illustrating a single-digit multioperand adder, which uses double correction speculation according to an embodiment of the invention.  
         [0025]      FIG. 5  is a flowchart illustrating exemplary operation of a multioperand decimal adder that performs decimal arithmetic using double correction speculation according to an embodiment of the invention.  
         [0026]      FIG. 6  is a block diagram illustrating parallel single-digit multioperand decimal adders according to an embodiment of the invention.  
         [0027]      FIG. 7  is a flowchart illustrating exemplary operation of parallel multioperand decimal adders according to an embodiment of the invention.  
         [0028]      FIG. 8  is a block diagram illustrating word-wide binary coded decimal addition with speculative adders according to an embodiment of the invention.  
         [0029]      FIG. 9  is a block diagram illustrating exemplary operation of a single-digit non-speculative multioperand decimal adder that performs decimal arithmetic according to an embodiment of the invention.  
         [0030]      FIG. 10  is a flowchart illustrating exemplary operation of a non-speculative multioperand decimal adder that performs decimal arithmetic according to an embodiment of the invention.  
         [0031]      FIGS. 11-14  are graphs showing delay and area of multioperand decimal adders in accordance with embodiments of the invention. 
     
    
     DETAILED DESCRIPTION  
       [0032]      FIG. 1  is a block diagram illustrating an example embodiment of a processing unit  10  that performs multioperand decimal arithmetic in accordance with one or more of the techniques described herein. In particular,  FIG. 1  illustrates a portion of processing unit  10  that includes a multioperand decimal adder  20  for performing decimal arithmetic. Multioperand decimal adder  20  receives binary coded decimal (BCD) inputs  12  labeled “A 0 -A(N−1).” Each BCD operand, e.g., A 0 , of BCD operands  12  comprises a BCD digit. Multioperand decimal adder  20  sums BCD operands  12  using binary carry-save addition to produce a BCD sum  14 , “Z.” For example, let BCD operands  12  comprise exactly three double-digit inputs, wherein A 0 =19 (decimal)=0001 1001 (BCD), A 1 =28=0010 1000 and A 2 =86=1000 0110. Multioperand decimal adder would produce a BCD sum Z of 0001 0011 0011, which is 133 in BCD format.  
         [0033]     In various embodiments described herein, multioperand decimal adder  20  one of four techniques described herein for performing fast decimal addition on BCD operands  12 . As further described below, three of the techniques speculate BCD correction values and use chaining to correct intermediate results. The first technique speculates over one addition. The second technique speculates over two additions. The third technique employs multiple instances of the second technique in parallel and then merges the results. The fourth technique uses a binary carry-save adder tree and produces a binary sum. Combinational logic is then used to correct the sum and determine the carry into the next digit.  
         [0034]     Multioperand decimal adder  20  uses binary carry-save addition (CSA). In different embodiments, multioperand decimal adder  20  may speculate binary to BCD correction values, in which case a speculation correction is required to produce BCD sum  14 . In other embodiments, multioperand decimal adder  20  may comprise one or more non-speculative decimal adders.  
         [0035]     Processing unit  10  may be a microprocessor or coprocessor for use within a laptop computer, general-purpose computer or high-end computing system. Alternatively, processing unit  10  may be a microcontroller, application-specific integrated circuit (AS 1 C) or other component. Moreover, processing unit  10  may be implemented as a single integrated circuit in which adder  20  constitutes only a portion of the implemented functionality. Alternatively, adder  20  may be implemented in one or more stand-alone integrated circuits. Further, components of processing unit  10  and adder  20  may be implemented as discrete combinatorial logic, logic arrays, microcode, firmware or combinations thereof.  
         [0036]      FIG. 2  is a block diagram illustrating a single-digit multioperand decimal adder  21  (herein, “adder  21 ”) that uses speculative binary to BCD correction according to one embodiment of the invention. In the illustrated embodiment, adder  21  includes binary carry-save adders  22 A,  22 B,  22 C,  22 D and  22 E (collectively, “CSAs  22 ”). Adder  21  also includes speculative binary to BCD correction logic  24 A,  24 B and  24 C (collectively, “logic  24 ”) and multiplexers  26 A,  26 B,  26 C and  26 D (collectively, “multiplexers  26 ”). Lastly, adder  21  includes carry propagate adders  28 A and  28 B (collectively, “CPAs  28 ”) and final correction logic  29 .  
         [0037]     In this description, a digit referenced with brackets (e.g. A 1 [ 3 ]) denotes a single bit of that digit. With respect to  FIG. 2 , CSAs  22  each output a sum S digit and a carry C digit. Each carry digit is a 4-bit quantity having bit positions [4:1].  
         [0038]     Adder  21  applies single correction speculation. With single correction speculation, BCD digits from the first two input operands, A 0  and A 1 , are added using binary carry-save addition by CSA  22 A to produce a 4-bit sum digit, S 1 , and a 4-bit carry digit, C 1 , such that S 1 +C 1 =A 0 +A 1 . When performing word-wide decimal multioperand addition, i.e. adding operands having more than one BCD digit, bit position C[ 0 ] is set to the carry-out from the previous, less-significant, carry digit and bit position C[ 4 ] is passed to the least significant bit of the next more significant carry digit.  
         [0039]     If the most significant bit of the first carry digit, C 1 [ 4 ], is equal to one, then a carry-out of the current digit has occurred and the sum of the first two input operands is at least sixteen and a correction value of six needs to be added. To keep the addition of the correction value off the critical delay path, the correction value for the sum of A 0  and A 1  is added in advance to the BCD digit of the next input operand, i.e., operand A 2 , by logic  24 A. C 1 [ 4 ] is used by multiplexer  26 A to selected the next value to be added by CSA  22 B, i.e., either operand A 2  or operand A 2 +6. When C 1 [ 4 ] equals zero, i.e., no carry occurs, multiplexer  26 A selects operand A 2 . When C 1 [ 4 ] equals one, indicating a carry out has occurred, multiplexer  26 A selects operand A 2 +6.  
         [0040]     A similar advanced correction process continues for operands A 3  and A 4 , which are added by CSAs  22 C and  22 D with appropriate correction values selected by multiplexers  26 B and  26 C. For each level of adder  21 , the most significant bit of the carry digit C in the previous level is examined. For example, if C 2 [ 4 ] is 1, then a carry-out of that level has occurred and A 3 +6 is selected by multiplexer  26 B for addition to S 2  and C 2  by CSA  22 C to produce S 3 +C 3 =S 2 +C 2 +(A 3 +6). Otherwise, if C 2 [ 4 ] is 0, no correction is needed, and A 3  is added to S 2  and C 2  to produce S 3 +C 3 =S 2 +C 2 +A 3 .  
         [0041]     Next, a speculation correction value, SC, is added to S 4  and C 4  based on C 4 [ 4 ] by CSA  22 E. In particular, when C 4 [ 4 ]=0, no correction value is needed, i.e., SC=0 If, however, C 4 [ 4 ]=1, a correction value of SC=6 is used.  
         [0042]     A 1-digit carry-propagate addition is then performed by CPA  28 A to compress the sum and carry digitis to obtain an intermediate BCD digit Z′ =S 5 +C 5 . Last, the final sum is corrected back to a valid BCD digit, Z, by final correction logic  29  and CPA  28 B, and a digit carry, C 0 , is produced.  
         [0043]     Table 1 illustrates a final correction value F output by final correction logic  29  based on C 5 [4], Z′[ 4 ], and Z′. Correction value F ensures the final digit, Z, is a valid BCD digit in the range of 0 to 9. Since each addition has been corrected by adding six whenever there is a carry-out of the current digit position, the final correction, F, is either 0, 6, or 12, based on the values of C 5 [ 4 ], Z′[ 4 ], and Z′, as shown in Table 1.  
                                                         TABLE 1                                   C5[4]   Z′[4]   Z′   F                                        0   0   &lt;10   0x0           0   0   &gt;=10   0x6           0   1   &lt;10   0x6           0   1   &gt;=10   0xc           1   0   &lt;10   0x6           1   0   &gt;=10   0xc           1   1   &lt;10   0xc           1   1   &gt;=10   N/A                      
 
         [0044]     In the illustrated embodiment of  FIG. 2 , adder  21  receives five single-digit operands and quickly produces a single BCD sum digit Z and a digit carry C 0 . However, more or less levels may similarly be used to implement a single correction speculation multioperand decimal adder capable of receiving more or less than five operands.  
         [0045]     Moreover, as illustrated below in reference to  FIG. 8 , adder  21  may comprise multiple 1-digit single correction speculation adders and a word-wide decimal carry-propagate adder (CPA) to add multiple digit BCD operands. In word-wide decimal multioperand addition, Z represents the correct BCD digit having the same significance as the input operands and C 0  would be passed along to be included in the calculation of the next most significant digit of the total sum.  
         [0046]      FIG. 3  is a flowchart illustrating exemplary operation of a multioperand decimal adder, such as adder  21  of  FIG. 2 , that performs decimal arithmetic using single correction speculation in accordance with the techniques described herein. Initially, adder  21  receives at least three operands ( 32 ). Next, adder  21  sums the first operand, or a single BCD digit from the first operand if the first operand has more than one BCD digit, A 0 , with the second operand, A 1 , using binary carry-save addition ( 36 ). This produces a 4-bit sum S 1 , and a 4-bit carry C 1 . If the most significant bit of the carry is 0, then S 1 , C 1  and A 2  are summed using a carry-save adder to produce S 2  and C 2  ( 38 A). If instead the most significant bit of carry is 1, then S 1 , C 1 , and (A 2 +6) are summed using a carry-save adder to produce S 2  and C 2  ( 38 B).  
         [0047]     A similar process continues for N−2 iterations (2≦i&lt;N), until all input operands: A 0 , A 1 , . . . A(N−1), are added with appropriate correction values. For each iteration, the most significant bit of the carry digit in the previous iteration, C(i−1)[ 4 ], is examined ( 37 ). If C(i−1)[ 4 ] is one, then a carry-out of the current digit has occurred and A(i)+6 is added to S(i−1) and C(i−1) using carry-save addition ( 38 B) to produce S(i)+C(i)=S(i−1)+C(i−1)+(A(i)+6). Otherwise, no correction is needed and A(i) is added to S(i−1) and C(i−1) using carry-save addition ( 38 A) to produce S(i)+C(i)=S(i−1)+C(i−1)+A(i).  
         [0048]     Once the last operand A(N−1) has been added ( 39 ), a final speculation correction is required. Specifically, a speculation correction value SC is added to S(N−1) and C(N−1) based on C(N−1)[ 4 ] to produce SN and CN ( 40 ). Then, a 1-digit carry-propagate addition is performed to compress the sum, SN, and carry, CN, to obtain Z′ =SN+CN using 4-bit carry-propagate addition ( 42 ). Last, the final sum Z′ is corrected back to a valid BCD digit, Z, and a digit carry, C 0 , is also produced ( 44 ). For illustrative purposes, the following pseudo code also illustrates the single correction speculation algorithm performed by adder  21 :  
                                                           S1 + C1 = A0 + A1 + 0;   // using carry save addition       for (i = 2; i &lt;N; i++)   // for each of the remaining operands              if (C(i−1)[4]==1)   S(i) + C(i) = S(i−1) + C(i−1) + (A(i) + 6);         else   S(i) + C(i) = S(i−1) + C(i−1) + (A(i) + 0);            SN + CN = S(N−1) + C(N−1) +   // speculation correction       SC;       Z′ = SN +CN;   // 4-bit carry-propagate addition       {CO, Z} = Z′ + F;   // F is the final correction from Table 1                  
 
         [0049]      FIG. 4  is a block diagram illustrating an exemplary multioperand decimal adder  50  (herein, “adder  50 ”) that uses double correction speculation according to an embodiment of the invention. As illustrated, adder  50  includes carry-save adders  52 A,  52 B,  52 C,  52 D and  52 E (collectively, “CSAs  52 ”). Adder  50  also includes speculative binary to BCD correction logic  54 A and  54 B (collectively, “logic  54 ”) and multiplexers  56 A,  56 B and  56 C (collectively, “multiplexers  56 ”). Lastly, adder  50  includes carry propagate adders  58 A and  58 B (collectively, “CPAs  58 ”) and final correction logic  59 .  
         [0050]     With double correction speculation, BCD digits from the first two input operands, A 0  and A 1 , are added using binary carry-save addition by CSA  52 A to produce a 4-bit sum digit, S 1 , and a 4-bit carry digit, C 1 , such that S 1 +C 1 =A 0 +A 1 . Then, the next input operand, A 2 , is added with S 1  and C 1  by CSA  52 B such that S 2 +C 2 =S 1 +C 1 +A 2 .  
         [0051]     If the most significant bit of the first carry digit, C 1 [ 4 ], is equal to one, then a carry-out of the CSA  52 A has occurred and a correction value of six needs to be added. To keep the addition of the correction value off the critical delay path, the correction value is added in advance to the digit of input operand A 3  by logic  54 A. Moreover, by adding the correction to A 3  instead of A 2 , as with adder  21 , multiplexer  56 A is also kept off the critical path as the selection process occurs in parallel with the addition operation of CSA  52 B. The same process continues for A 4 , which is added by CSA  52 D to S 3  and C 3  with an appropriate correction selected by multiplexer  56 B.  
         [0052]     At the end of the process, multiplexer  56 C selects a speculation correction value, SC, which is added to S 4  and C 4  based on C 4 [ 4 ] and C 3 [ 4 ] by CSA  52 E. Next, a carry-propagate addition is performed by CPA  58 A to compress the sum and carry digits to obtain Z′=S5sum Z′ is corrected back to a valid BCD digit, Z and a digit carry C 0  by final correction logic  59  and CPA  58 B.  
         [0053]     As with adder  21  of  FIG. 2 , adder  50  speculates that the addition of A 0  with A 1  does not need to be corrected. Adder  50  also speculates that the addition of A 2  with S 1  and C 1  does not need to be corrected. Using double correction speculation, logic  54  uses C(i− 2 )[ 4 ] to select whether A(i) or A(i)+6 is added to S(i−1) and C(i−1). Compared with adder  21  of  FIG. 2 , which uses single correction speculation, this removes the multiplexers  56  that select between A(i) and A(i)+6 from the critical path, since the correction for A(i+1) is selected while the carry-save addition of A(i) or A(i)+6 with S(i−1) and C(i−1) is being performed. It also removes the logic to produce A 2 +6, since A 2  is always added without a correction value. However, determination of the speculation correction value, SC, is slightly more complex with adder  50  than adder  21  because double correction speculation requires two speculative additions whereas single correction speculation only requires one. Final correction logic  59  selects a value of 0, 6, or 12 for speculation correction SC based on C(N−2)[4] and C(N−1)[4] as shown in Table 2.  
                       TABLE 2                       C(N − 2)[4]   C(N − 1)[4]   SC                   0   0   0000 (+0)       0   1   0110 (+6)       1   0   0110 (+6)       1   1    1100 (+12)                  
 
         [0054]     For ease of illustration, adder  50  of  FIG. 4  is shown to receive five operands A 0 -A 4 . In this embodiment, adder  50  utilizes five 4-bit carry-save adders  52 , two 4-bit 2:1 multiplexers  56 A adn  56 B, two combinational logic blocks  54  to find A(i)+6, one 4-bit 4:1 multiplexer  56 C to produce SC, two 4-bit carry-propagate adders  58 , and one 4-level combinational logic block  59  to produce the final correction, F. In this configuration, the critical delay path of adder consists of five carry-save additions, one 4-bit 4:1 multiplexer delay, two 4-bit carry propagate additions, and 4-levels of logic to implement Table 2. Compared to the single correction speculation Adder  21  in  FIG. 2 , the double correction speculation adder  50  removes three 4-bit 2:1 multiplexers from the critical delay path.  
         [0055]     In accordance with other embodiments of the invention, the techniques shown in  FIG. 4  may be utilized to implement a double correction speculation decimal adder capable of receiving and adding more or less than five decimal operands. Further, a processing unit may comprise multiple 1-digit double correction speculation adders  50  and a word-wide decimal carry-propagate adder (CPA) for operation on multiple digit BCD operands.  
         [0056]      FIG. 5  is a flowchart illustrating exemplary operation of a multioperand decimal adder, such as adder  50 , that performs decimal arithmetic using double correction speculation in accordance with an embodiment of the invention.  
         [0057]     First, adder  50  receives at least three operands A 0 -A(N−1) ( 62 ). For each digit, operand A 0  is summed with the second operand A 1  using binary carry-save addition ( 65 ). This produces a 4-bit sum, S 1 , and a 4-bit carry, C 1 . Next the third operand, A 2 , is summed with the second operand, S 1  and C 1  using binary carry-save addition, producing S 2  and C 2  ( 65 ). If the most significant bit of the carry from the addition of A 0  and A 1  is 0, then S 2 , C 2  and A 3  are summed using a carry-save adder to produce S 3  and C 3 . ( 68 A). If instead the most significant bit of the carry is 1, then S 2 , C 2 , (A 3 +6) are summed using a carry-save adder to produce S 3  and C 3 . ( 68 B).  
         [0058]     A similar process continues for N−3 iterations (3≦i&lt;N), until all input operands: A 0 , A 1 , . . . A(N−1), are added with appropriate correction values. For each iteration i, the most significant bit of the carry digit C(i−2)[ 4 ] is examined ( 67 ). If C(i−2)[ 4 ] is a logical one, then a carry-out has occurred and A(i)+6 is added to S(i−1) and C(i−1) using carry-save addition ( 68 B) to produce S(i)+C(i)=S(i−1)+C(i−1)+(A(i)+6). Otherwise, no correction is needed and A(i) is added to S(i−1) and C(i−1) using carry-save-addition ( 68 A) to produce S(i)+C(i)=S(i−1)+C(i−1)+A(i).  
         [0059]     Once the process has completed for the final iteration, such that there are no more operands ( 69 ), a speculation correction is required. The speculation correction value is added to S(N−1) and C(N−1) based on C(N−1)[ 4 ] and C(N−2)[ 4 ] using Table 2 ( 70 ). Then, carry-propagate addition is performed to compress the sum, SN, and carry, CN, to obtain Z′ =SN+CN using 4-bit carry-propagate addition ( 72 ). Last, the final sum is corrected back to a valid BCD digit, Z, and a digit carry, CO, is also produced ( 74 ). In word-wide decimal multioperand addition, Z represents the correct BCD digit having the same significance as the input operands and CO would be passed along to be included in the calculation of the next most significant digit of the total sum.  
         [0060]     For illustrative purposes, pseudocode for the double speculative correction process in accordance with the invention is included below:  
                                                           S1 + C1 = A0 + A1 + 0;   // using carry save addition       S2 + C2 = S1 + C1 + A2;   // using carry save addition       for (i = 3; i &lt;N; i++)   // for each of the remaining           operands             if (C(i−2)[4]==1)   S(i) + C(i) = S(i−1) + C(i−1) + (A(i) + 6);        else   S(i) + C(i) = S(i−1) + C(i−1) + (A(i) + 0);            SN + CN = S(N−1) + C(N−1) + SC;   // special correction from Table 2       Z′ = SN +CN;   // 4-bit carry-propagate addition       {CO, Z} = Z′ + F;   // F is the final correction from           Table 1                  
 
         [0061]      FIG. 6  is a block diagram illustrating an exemplary parallel multioperand decimal adder  80  that incorporates multiple speculative adders  81  A and  81  B (collectively, speculative adders  81 ). Parallel adder  80  also includes carry-save adders  82 A,  82 B and  82 C (collectively, “CSAs  82 ”) and carry-propagate adders  86 A and  86 B (collectively, “CPAs  86 ”) Parallel adder  80  further comprises a multiplexer  84  and final correction logic  88 .  
         [0062]     Speculative adders  81  operate in parallel to produce partial sums S 0  and S 1  and partial carries C 0  and C 1 . Either single or double correction speculation adders can be used as adders  81 .  
         [0063]     In this example, adder  80  receives six operands A 0 -A 5 , and speculative adders  81  each sum three of the operands as described above until the point where the speculation corrections are added (e.g., step  70  in  FIG. 5 ). Then, the outputs of adders  81  (S 0 , S 1 , C 0  and C 1 ) are merged by CSA  82 A and CSA  82 B to produce S 3  and C 3 . Next, CSA  82 C adds a speculative correction SC selected by multiplexer  84 . CPA  86 A then compresses the S 4  and C 4  into a sum Z′ and a carryout Z′[ 4 ]. Finally, logic  88  produces a final correction F in accordance with Table 1, which is added by CPA  86 B to produce a BCD digit sum, Z and a carry-out, CO.  
         [0064]     In this embodiment, parallel adder  80  comprises exactly two speculative adders  81  operating in parallel. In other embodiments parallel adder  80  may have additional speculative adders operating in parallel, but the benefits of using more adders in parallel are offset by the costs of merging the results. As illustrated in  FIG. 6 , merging the results of parallel adders  81  can be accomplished using three carry-save additions. With P parallel adders, there are 3×┌log 2 (P)┐ extra carry-save additions on the critical path.  
         [0065]     Embodiments of a 1-digit, N-operand Parallel Correction Speculation Adder composed of P parallel adders requires N 4-bit carry-save adders, (N−3) 4-bit 2:1 multiplexers, (N−3) 2-level combinational logic blocks to find A(i)+6, (2P) 4-bit carry-propagate adders, P 4-level combinational logic blocks to select F for each parallel adder, and 3(P−1) carry-save adders plus (P−1) 4-level correction logic blocks to merge results. Its critical path is ┌N/P┐ carry-save additions, one 4-bit 2:1 multiplexer delay, two 4-bit carry-propagate additions, and  4  levels of logic to implement speculative correction. Merging the results requires 3×┌log 2 (P)┐ carry-save additions and 4×┌log 2 (P)┐ levels of logic.  
         [0066]     For ease of illustration, adder  80  is shown as capable of summing single digit operands. However, adder  20 C may also comprise additional parallel adders and a word-wide decimal carry-propagate adder (CPA) and be capable of adding multiple digit BCD operands. Word-wide decimal multioperand addition is described in greater detail in the description of  FIG. 8 .  
         [0067]      FIG. 7  is a flowchart illustrating exemplary operation of a parallel multioperand decimal adder according to an embodiment of the invention. First, the multioperand decimal adder receives at least four operands ( 92 ). Next, the input operand digits are divided in P 4-bit groups, where P equals the number of speculative adders being used in parallel ( 96 ). Speculative addition, e.g., single or double speculative addition, is then used to produce P partial sums and partial carries, corresponding to one partial sum and partial carry for each speculative adder ( 97 ). The sums and carries from each speculative adder are then merged using carry-save addition ( 98 ). After the sums and carries are merged, another correction is needed to correct for the carry-outs for which are currently unaccounted. Then, speculation correction steps are taken, and a CPA compresses the sum and carry produced from step  98 , followed by the addition of the addition of a final correction factor F in accordance with Table 1 ( 99 ).  
         [0068]      FIG. 8  is a block diagram illustrating a portion of an exemplary word-wide decimal adder  100  that uses 1-digit (4-bit) speculative multioperand adders  104 A,  104 B and  104 C, (collectively, “adders  104 ”) and a word-wide decimal carry-propagate adder  106 . Word-wide decimal carry-propagate adder  106  includes correction logic  108 , a 2-bit counter  112  and a carry-propagate adder  110  for each of adders  104 .  
         [0069]     Each of multioperand speculative adders  104  may be any of the 1-digit (4-bit) multioperand speculative adders described above in reference to  FIGS. 2, 4  and  6 . Each of multioperand speculative adders  104  adds different digits of the input operands  102 . The most significant bit of each carry digit, C(i)[ 4 ], becomes the least significant bit of the next more significant carry digit, which is known an inter-digit carry.  FIG. 8  shows inter-digit carries  105 A,  105 B and  105 C. For example, adder  104 A outputs inter-digit carries  105 A to adder  104 B.  FIG. 8  illustrates a portion of word-wide multioperand decimal adder  100  that is composed of M 1-digit speculative decimal multioperand adders  104  operating in parallel and feeding carries from one digit to the next more significant digit, where M equals the number of digits in each of operands  102 . The 4-bit sum and 1-bit carry digits produced by the 1-digit multioperand adders  104  are fed into a word-wide decimal carry-propagate adder  106  to obtain the final result, digits  114 A,  114 B and  114 C.  
         [0070]     For example, logic  108 A receives a sum, Z and a carry-out, CO from adder  104 A. Logic  108 A then determines if Z is greater than nine, which would require a binary to BCD correction of +6. Next, CPA  110 A adds Z with the correction from logic  108 A to produce a digit sum  114 A and a 1-bit carry if required. 2-bit counter  112 A receives a 1-bit carry from each of the C 0  and CPA  110 A, which are passed to CPA  110 B for determining the digit sum  114 B. This process is repeated within word-wide decimal carry-propagate adder  106  for each sum, Z and carry-out, CO produced by adders  104 .  
         [0071]      FIG. 9  is a block diagram illustrating one embodiment of a non-speculative multioperand decimal adder  130  according to an embodiment of the invention. Specifically,  FIG. 9  shows a single-digit, non-speculative multioperand decimal adder  130  that uses a binary CSA tree  135  for m ultioperand decimal addition according to an embodiment of the invention. Adder  130  includes carry-save adders  132 A,  132 B and  132 C (collectively, “CSAs  132 ”). Adder  130  also includes carry-propagate adders  138 A and  138 B, (collectively, “CPAs  138 ”) and carry correction and generation logic  139 .  
         [0072]     Non-speculative adder  130  sums BCD input operands A 0 , A 1 , A 2 , A 3  and A 4  in a binary carry-save tree that includes CSAs  132  that passes carries, C(i)[ 4 ] along each level from lower significant digits to more significant digits. In the example of  FIG. 9 , five BCD operand digits A 0 -A 4  are added using binary carry-save tree  135  and a 4-bit CPA  138 A. The result is a 5-bit binary sum, z′, produced by CPA  138 A and three intermediate carry-outs (C 1 [ 4 ], C 2 [ 4 ], and C 3 [ 4 ]).  
         [0073]     The sum and carry-outs from the carry-save adder tree  135  are fed into correction logic  139 , which includes combinatorial logic to produce a decimal sum correction and additional carry-outs, if needed. Specifically, a correction value of six needs to be added for each carry-out generated in the binary CSA tree  135 . Thus, correction logic  139  produces a correction value G that is a multiple of six. Thus, the correction digit is always even and a 3-bit CPA is used. The correction digit, G, and the lower four bits of the binary sum, Z′[3:0], are passed through 1-digit CPA  138 B to produce the correct BCD sum, Z. Correction logic  139  ensures that the final sum digit, Z, is a valid BCD digit and produces the additional carries, C-OUT, for the next most significant digit. For example, adding 8+2+6+5+3=24 gives C 1 [ 4 ]=0, C 2 [ 4 ]=0, C 3 [ 4 ]=1, and Z′=01000. Carry and correction generation logic  139  produces a correction of G=6×2 (mod  16 )=12=1100, and C-OUT=01. Since 1000+1100=1 0100, the proper BCD representation of  24  is produced.  
         [0074]     The exemplary non-speculative adder  130  shown in  FIG. 10  can sum digits for up to five operands A 0 -A 5 . However, other embodiments of non-speculative adders in accordance with the invention may be implemented to sum more or less operands. A 1-digit, N-operand non-speculative adder requires (N−2) 4-bit carry-save adders, one 4-bit carry-propagate adder, one five-level combinational logic block to generate the carry-out and correction digits (for up to sixteen input operands), and one three-bit carry-propagate adder to add the correction digit to the binary sum. Its critical delay path consists of roughly └log 3/2 (N−1)┘ carry-save additions, one 4-bit carry-propagate addition, one 5-level logic block, and one 3-bit carry-propagate addition. Unlike the correction speculation adders shown in  FIGS. 2, 4  and  6 , which use an array of binary carry-save adders and have a linear delay, non-speculative adders use a tree of binary carry-save adders and have logarithmic delay.  
         [0075]     A word-wide BCD non-speculative adder may use decimal carry-lookahead logic. The addition is can be done using a variation of direct decimal addition, in which each 1-digit adder takes a sum and carry digit and produces digit propagate and generate signals. The digit propagate and generate signals are then sent to carry-lookahead logic, which is used to compute digit carries in parallel. Finally, the digit carries and additional carry-lookahead logic within each digit are used to quickly produce the sum digits. The word-wide adder is less complex than the word-wide decimal carry-propagate adder  106  in  FIG. 8 , since only a 1-bit carry into each digit is necessary.  
         [0076]      FIG. 10  is a flowchart illustrating exemplary operation of a non-speculative multioperand decimal adder, such as adder  130  of  FIG. 9 , that performs decimal arithmetic according to an embodiment of the invention. First, adder  130  receives operands A 0 -AN ( 142 ). Adder  140  then sums the input operands in a binary carry-save tree, passing carries generated along the way to the more significant digits ( 146 ). Adder  140  then produces a binary sum from the results of the binary carry-save tree using carry-propagate addition ( 148 ). This sum and carry-outs from the carry-propagate addition are fed into combinational correction logic, which produces a decimal correction and additional carry-outs, if needed ( 150 ). After the decimal correction value is produced, the sum produced in step  148  is added to the decimal correction using carry-propagate addition to produce a final resultant decimal digit ( 152 ).  
         [0000]     Simulation Results  
         [0077]     The multioperand decimal adders described herein have been modeled in Verilog and simulated extensively. From the Verilog models, multioperand decimal adders were synthesized using a 0.18-micron CMOS standard cell library. When performing synthesis, the designs were optimized for area. Both 4-bit (1-digit) and 32-bit (8-digit) multioperand decimal adders were constructed for: (1) single correction speculation, (2) double correction speculation, (3) parallel correction speculation using two double correction speculation adders, (4) parallel correction speculation using four double correction speculation adders, and (5) non-speculative addition. Each 32-bit decimal multioperand adder was constructed from eight 1-digit multioperand decimal adders, followed by a word-wide decimal carry-lookahead adder, as described above. Each 32-bit binary multioperand adder was constructed using a linear array or tree of carry-save adders, followed by a word-wide binary carry-lookahead adder.  
         [0078]     For comparison, binary multioperand carry-save adders were built to evaluate the additional cost of performing multioperand decimal addition. One set of binary multioperand adders was designed to be similar to the Correction Speculation Adders and contains a linear array of binary carry-save adders. The other set was designed to be similar to the Non-Speculative Adders and uses a tree of binary carry-save adders. Both types of binary multioperand adders use the same word-wide carry-propagate adder. In the word-wide carry-propagate adder, two levels of carry-lookahead logic are implemented. The first level produces group generate and propagate signals for 4-bit blocks. The second level uses the group generate and propagate signals to obtain the carries into each 4-bit block.  
         [0079]     Overall, seven different adder types for each of 4, 6, 8, 10, 12, and 16 operands were constructed for both 4-bit and 32-bit operands for a total of 94 different multioperand adders. The delay and area for the constructed 4-bit multioperand adders are shown in  FIGS. 11 and 12 , respectively. The delay and area for the constructed 32-bit adders are shown in  FIGS. 13 and 14 , respectively. Similar conclusions can be reached using either the 4-bit or the 32-bit multioperand adder results. The 32-bit multioperand adder results, which show the overall area and delay due to processing multiple digits and performing word-wide carry-lookahead addition, are discussed throughout the rest of this section.  
         [0080]     In reference to  FIG. 13 , for all decimal multioperand adders that speculate corrections and also the binary array multioperand adders, the delay increases linearly with the number of input operands. The difference in delay between double and single correction speculation adders grows with more input operands because the multiplexer delays to select A(i) or A(i)+6 are hidden in the double correction speculation adders. The parallel correction speculation adders have a larger overhead (from result merging) than the other adders. This is seen in longer relative delays for fewer input operands. As the number of input operands increases, the difference in delay is less, because the Parallel Correction Speculation Adders operate on two or four sets of operands in parallel.  
         [0081]     The non-speculative adders have lower delays than all of the other decimal multioperand adders. One advantage of the non-speculative adders is that the delay grows logarithmically, rather than linearly, since operands are added using a tree of binary carry-save adders. Their logarithmic delay may be particularly useful when a large number of input operands are added. Although the area for the 32-bit non-speculative decimal adder, single correction speculation decimal adder, and double correction speculation adder are similar, the double correction speculation adders have the lowest average area. This makes the double correction speculation adders desirable when area is more important than delay. The parallel correction speculation adders require more area than the other adders due to the overhead of merging the results.  
         [0082]     The areas and delays for binary adders are shown for comparison. The cost of performing multioperand decimal addition versus multioperand binary addition is calculated by comparing the non-speculative adders, which have the smallest delay and small overall area, to the binary tree adders. The non-speculative adders have 1.44 to 2.34 times more delay and 1.61 to 2.03 times more area than the binary tree adders.  
         [0083]     Various embodiments of the invention have been described. These and other embodiments are within the scope of the following claims.