FPGA in Decimal Arithmetic | Encyclopedia MDPI

FPGA in Decimal Arithmetic: Comparison

Please note this is a comparison between Version 2 by Mário Véstias and Version 3 by Lily Guo.

Decimal operations are executed with slow software-based decimal arithmetic functions. For the fast execution of decimal operations, dedicated hardware units have been proposed and designed in FPGA. Decimal addition and multiplication is found in most decimal-based applications and so its design is very important for fast execution. This entry describes recent solutions for decimal multiplication and addition in FPGA.

decimal multiplication
FPGA
decimal addition

1. Introduction

Financial and commercial applications like accounting, banking, tax calculation, insurance

and currency conversion require a large amount of data computing. Therefore,

they are typically executed in high-performance computing platforms. These applications

run over large databases of numbers which, in many cases, are represented in decimal

format ^[1]. The last revision of the IEEE standard for floating-point arithmetic ^[2] includes

specific definitions and rules for decimal operations and three different formats: decimal

32, decimal 64 and decimal 128, with 7, 16 and 34 coefficient digits.

Most general-purpose processors only have binary arithmetic units. So, the fastest

solution to run decimal operations would be to convert decimal numbers to binary before

being processed and then convert the result to decimal. The problem is that not all

decimal numbers can be represented exactly as binary numbers with a finite number of

bits. So, to avoid errors created from binary calculation that could lead to unwanted result

deviations, arithmetic operations must be done directly over decimal numbers.

Executing decimal operations with binary arithmetic hardware without converting

data to binary requires software algorithms for decimal arithmetic. Software-based decimal

arithmetic is very slow compared to binary arithmetic implemented in hardware. However,

the fast increase of commercial and financial transactions requires fast decimal arithmetic

computing to meet real-time requirements and exact computations. Decimal addition and

multiplication are fundamental arithmetic operations used in many applications. Therefore,

fast decimal multipliers are important to obtain fast decimal-based applications.

FPGAs (Field Programmable Gate Array) are a good alternative for the execution of

decimal arithmetic with dedicated hardware modules.

2. Decimal Addition in FPGA $w + z + 6 \leq 15$

Decimal addition is implemented by direct manipulation of decimal numbers or using a binary adder followed decimal correction.

Some of the first decimal adders

Decimal addition is implemented by direct manipulation of decimal numbers or using

a binary adder followed decimal correction.

Some of the first decimal adders

^[3][4]

were based on 1-digit BCD. Others

^[5] use a direct BCD carry look-ahead adder or a carry-select technique to conditionally sum 6 in binary to do the decimal correction

use a

direct BCD carry look-ahead adder or a carry-select technique to conditionally sum 6 in

binary to do the decimal correction

^[6][7].

Other decimal addition approach considers intermediate representations that reduce the complexity of decimal addition but requires converters from and to BCD of the intermediate representation. For example,

Other decimal addition approach considers intermediate representations that reduce

the complexity of decimal addition but requires converters from and to BCD of the intermediate representation. For example,

^[8][9] consider a redundant BCD representation to achieve carry free operations.

To take advantage of binary arithmetic circuits, some decimal addition solutions consider binary adders followed by number correction

consider a redundant BCD representation to

achieve carry free operations.

To take advantage of binary arithmetic circuits, some decimal addition solutions

consider binary adders followed by number correction

^{[10][11][12][13][14][15][16][17]}.

Considering also binary-based decimal addition, some works use binary to BCD conversion

Considering also binary-based decimal addition, some works use binary to BCD

conversion

^[18]

and BCD to binary conversion

^[19] to use binary adders. The main problem of this solution is the overhead of binary-BCD conversions

to use binary adders. The main problem

of this solution is the overhead of binary-BCD conversions

^[20].

All previous solutions can be implemented in FPGA. However, some works were proposed specifically for decimal addition in FPGA. Decimal adders wer designed for FPGAs with 4-input LUTs

All previous solutions can be implemented in FPGA. However, some works were

proposed specifically for decimal addition in FPGA. Decimal adders wer designed for

FPGAs with 4-input LUTs

^[21]

and with 6-input LUTs

^[16]

. Decimal adders with binary

adders followed by correction stages were also implemented in 6-input LUT FPGAs

^[22][23][24].

Multioperand decimal addition is a particular case of decimal addition where techniques like carry-save addition can be applied efficiently

Multioperand decimal addition is a particular case of decimal addition where techniques

like carry-save addition can be applied efficiently

^[6][25]

. In

^[25] a new BCD adder was proposed to efficiently implement multioperand addition. In

a new BCD adder

was proposed to efficiently implement multioperand addition. In

^[4][7][22] the tree structure was identified as the most efficient approach for multioperand addition in FPGA.

the tree structure

was identified as the most efficient approach for multioperand addition in FPGA.

^[26] a decimal adder was proposed that considers an excess-6 representation to avoid carry propagation of addition. This adder is used in the proposed multipliers to implement the adder tree. It also serves as the base for a novel decimal adder/subtractor necessary for the design of the partial product generators.

a decimal adder was proposed that considers an excess-6 representation to

avoid carry propagation of addition. This adder is used in the proposed multipliers to

implement the adder tree. It also serves as the base for a novel decimal adder/subtractor

necessary for the design of the partial product generators.

3. Decimal Multiplication in FPGA

Processors with dedicated decimal hardware multipliers implement them with iterative algorithms

Processors with dedicated decimal hardware multipliers implement them with iterative

algorithms

^[27][28] to reduce the size of the arithmetic unit. However, iterative algorithms are slow compared to parallel implementations due to its iterative nature. for

fast execution, parallel decimal multiplication consists of partial product generation for each multiplier digit followed by partial product addition. Partial product generation of a N×N multiplication can be implemented with N×N small digit by digit multipliers or N digit by multiplicand multipliers. A digit by digit multiplier can be implemented with logic or with look-up tables

to reduce the size of the arithmetic unit. However, iterative

algorithms are slow compared to parallel implementations due to its iterative nature. for

fast execution, parallel decimal multiplication consists of partial product generation for

each multiplier digit followed by partial product addition. Partial product generation of a

N×N multiplication can be implemented with N×N small digit by digit multipliers or N

digit by multiplicand multipliers. A digit by digit multiplier can be implemented with logic

or with look-up tables

^[29][30][31]), for fast and compact design. However, given the quadratic number of digit by digit multipliers necessary to implement a multiplication, these solutions are viable only for small operand sizes. The proposal in

), for fast and compact design. However, given the quadratic

number of digit by digit multipliers necessary to implement a multiplication, these solutions

are viable only for small operand sizes. The proposal in

^[4] considered recoding of operands to simplify digit by digit multiplication for partial product generation. However, the performance and area of the decimal multiplier based on digit by digit multiplication is still worst than a multiplier with a partial product for each multiplier digit.

The approach followed to implement a 1×N multiplier is to determine the decimal multiples of the multiplier. A direct approach to a design a decimal multiplier based on multiples generates all multiples of the multiplicand. Then, selects the required multiples according to the multiplier digits. The generated multiples are then shifted and added to generate the final product. While simple, the method requires a large multiplexer with all multiplies for each multiplier digit and the generation of all multiples from A to 9A. Knowing that the generation of some multiples are not carry-free, this solution degrades the performance of the multiplier.

Therefore, authors started to consider only a limited set of multiples. In

considered recoding of

operands to simplify digit by digit multiplication for partial product generation. However,

the performance and area of the decimal multiplier based on digit by digit multiplication is

still worst than a multiplier with a partial product for each multiplier digit.

The approach followed to implement a 1×N multiplier is to determine the decimal

multiples of the multiplier. A direct approach to a design a decimal multiplier based on

multiples generates all multiples of the multiplicand. Then, selects the required multiples

according to the multiplier digits. The generated multiples are then shifted and added to

generate the final product. While simple, the method requires a large multiplexer with

all multiplies for each multiplier digit and the generation of all multiples from A to 9A.

Knowing that the generation of some multiples are not carry-free, this solution degrades

the performance of the multiplier.

Therefore, authors started to consider only a limited set of multiples. In

^[32] only multiples A, 2A, 4A, 5A are used, since they can be generated without carry propagation (multiple 4A is generated from 2A in sequence as 2×2A). The other multiples are obtained by adding two of these multiples. Since multiple 4A cannot be generated in a single carryfree step, it has been removed from the set of base multiples in

only

multiples A, 2A, 4A, 5A are used, since they can be generated without carry propagation

(multiple 4A is generated from 2A in sequence as 2×2A). The other multiples are obtained

by adding two of these multiples. Since multiple 4A cannot be generated in a single carryfree

step, it has been removed from the set of base multiples in

^[7]. The other multiples are obtained by adding a multiple from the set {0, 5X, 10X} and a multiple from the set {-2X, -X, 0, X, 2X}. For fast selection of multiples, digits of the multiplier are first recoded, but the solution requires a large multiplexer for each multiplier digit.

Since then, other sets of multiples and encodings were considered. In

. The other multiples

are obtained by adding a multiple from the set {0, 5X, 10X} and a multiple from the set

{-2X, -X, 0, X, 2X}. For fast selection of multiples, digits of the multiplier are first recoded,

but the solution requires a large multiplexer for each multiplier digit.

Since then, other sets of multiples and encodings were considered. In

^[14] two different decimal encodings (4221 and 5211) are used to generate and reduce the partial products with two different architectures. In one the architectures the multiplier is recoded into a signed-digit (SD) set [-5, 5], while in the other the multiplier is encoded as A = YU5 +YL like in [7], where Y

two different

decimal encodings (4221 and 5211) are used to generate and reduce the partial products with two different architectures. In one the architectures the multiplier is recoded into a

signed-digit (SD) set [-5, 5], while in the other the multiplier is encoded as A = YU5 +YL

like in [7], where Y

∈ {0, 1, 2} and Y

^L ∈ {-2, -1, 0, 1, 2}. Signed-digit (SD) recoding of the multiplier in the set [-5, 5] was adopted by several authors for the implementation of a decimal multiplier

∈ {-2, -1, 0, 1, 2}. Signed-digit (SD) recoding of

the multiplier in the set [-5, 5] was adopted by several authors for the implementation

of a decimal multiplier

^[33][34]. The base architecture generates multiples {0, X, 2X, 3X, 4X, 5X}. These are selected for each partial product and the output is complemented to obtain the negative of the multiple. The partial products are then reduced with a partial product reduction module. Different representations are used to improve the generation of complements and the decimal addition. The radix-5 algorithm proposed in

. The base architecture generates multiples {0, X, 2X, 3X,

4X, 5X}. These are selected for each partial product and the output is complemented to

obtain the negative of the multiple. The partial products are then reduced with a partial

product reduction module. Different representations are used to improve the generation

of complements and the decimal addition. The radix-5 algorithm proposed in

^[14] was followed by

was

followed by

^[35]

but using an hybrid 8421–5421 representation.

^[15] a decimal multiplier is proposed using a redundant decimal addition algorithm based on a weighted bit-set encoding. The method generates double BCD (Binary-Coded Decimal) numbers using decimal multiples 2X, 4X, and 5X. The redundant decimal adder is used to reduce the generated 2n BCD partial products to a redundant number in the range of [0, 15]. The final redundant product is then converted to BCD encoding.

The special case of constant decimal multiplication was considered in

a decimal multiplier is proposed using a redundant decimal addition algorithm

based on a weighted bit-set encoding. The method generates double BCD (Binary-Coded

Decimal) numbers using decimal multiples 2X, 4X, and 5X. The redundant decimal adder

is used to reduce the generated 2n BCD partial products to a redundant number in the

range of [0, 15]. The final redundant product is then converted to BCD encoding.

The special case of constant decimal multiplication was considered in

^[36]. Constant decimal multiplication is widely used in economic and financial applications. The authors address this problem to design a solution with smaller area, power and delay compared to constant decimal multiplication implemented with a general decimal multiplier. The work proposes a new redundant digit set in {0, 18} and a 3:1 compressor. The results show an improvement in the area up to 89%.

Partial products are then added is a step known as partial product reduction using decimal adders. Partial product reduction can be designed with an adder tree or with a multioperand adder. An adder tree successively reduces pairs of partial products until a final result. Multioperand addition takes into account that multiple partials have to be reduced into a single value. In

. Constant

decimal multiplication is widely used in economic and financial applications. The authors

address this problem to design a solution with smaller area, power and delay compared to

constant decimal multiplication implemented with a general decimal multiplier. The work

proposes a new redundant digit set in {0, 18} and a 3:1 compressor. The results show an

improvement in the area up to 89%.

Partial products are then added is a step known as partial product reduction using

decimal adders. Partial product reduction can be designed with an adder tree or with a

multioperand adder. An adder tree successively reduces pairs of partial products until

a final result. Multioperand addition takes into account that multiple partials have to be

reduced into a single value. In

^[13] three techniques were proposed for multioperand decimal addition. Two of the approaches consider speculative addition that speculates about BCD correction values which are corrected while adding the operands. The other technique uses a binary adder that produces a binary sum which is then corrected. This last technique achieved the best area-delay results. A mixed binary and BCD multioperand addition was proposed in

three techniques were proposed for multioperand

decimal addition. Two of the approaches consider speculative addition that speculates

about BCD correction values which are corrected while adding the operands. The other

technique uses a binary adder that produces a binary sum which is then corrected. This

last technique achieved the best area-delay results. A mixed binary and BCD multioperand

addition was proposed in

^[37]. Digits in a column are all added in binary, converted to decimal and finally added with decimal adders.

. Digits in a column are all added in binary, converted to

decimal and finally added with decimal adders.

^[7][32]

the adder tree is implemented with decimal carry look-ahead adders. In

^[38] partial products are recoded to 4221. This codification simplifies addition since it avoids he correction step. The method reduces three partial products to two equally weighted 4221 decimal digits. These two operands are then converted to BCD add added to generate the final result.

A different approach for decimal multiplication considers binary multipliers as the base arithmetic unit

partial products are recoded to 4221. This codification simplifies addition since it avoids he

correction step. The method reduces three partial products to two equally weighted 4221

decimal digits. These two operands are then converted to BCD add added to generate the

final result.

A different approach for decimal multiplication considers binary multipliers as the

base arithmetic unit

^{[19][20][39][40]}. This permits using binary multipliers that are faster and may already be available in the system. Also, it implements both binary and decimal multiplication in a single module. The method first converts the BCD operands of the multiplication to binary. The converted operands are then multiplied using the binary multiplier. The binary product is then converted to BCD. The main drawback of the binary-based method is the large overhead introduced by the converters

. This permits using binary multipliers that are faster

and may already be available in the system. Also, it implements both binary and decimal

multiplication in a single module. The method first converts the BCD operands of the

multiplication to binary. The converted operands are then multiplied using the binary

multiplier. The binary product is then converted to BCD. The main drawback of the binary-based method is the large overhead introduced by the converters

^[18][19]. A balanced solution was proposed in

. A balanced

solution was proposed in

^[20] that subdivides the multiplier and the multiplicand into smaller blocks and applies the method to each of these sub-blocks. The partials are then aligned and added using decimal adders to generate the final product.

Most works on decimal multiplication target ASICs, but several architectures have been proposed for FPGA and coarse-grained reconfigurable computing

that subdivides the multiplier and the multiplicand into

smaller blocks and applies the method to each of these sub-blocks. The partials are then

aligned and added using decimal adders to generate the final product.

Most works on decimal multiplication target ASICs, but several architectures have

been proposed for FPGA and coarse-grained reconfigurable computing

^[41]. Any of the previous architectures can be directly mapped to FPGA. However, a careful adaptation of the design leads to a more efficient architecture since logic functions in FPGAs are implemented with look-up tables. In

. Any of the

previous architectures can be directly mapped to FPGA. However, a careful adaptation of

the design leads to a more efficient architecture since logic functions in FPGAs are implemented with look-up tables. In

^[42] a parallel implementation of a multiplier was mapped in Virtex-4 FPGA from Xilinx. The architecture obtains the partial products using digit by digit multiplication with a binary multiplier followed by binary to BCD conversion

a parallel implementation of a multiplier was mapped

in Virtex-4 FPGA from Xilinx. The architecture obtains the partial products using digit by digit multiplication with a binary multiplier followed by binary to BCD conversion

^[43]

. The work in

^[22]

described previously was mapped on a 6-input LUT FPGA.

A new optimization of the multiplication algorithm was considered in

^[44] where the application of the Karatsuba-Ofman algorithm reduces the area of the parallel decimal multipliers on FPGA at the cost of an increase in delay. A BCD multiplier using the atomic 1×1 digit multiplier was proposed in

where the

application of the Karatsuba-Ofman algorithm reduces the area of the parallel decimal

multipliers on FPGA at the cost of an increase in delay. A BCD multiplier using the atomic

1×1 digit multiplier was proposed in

^[45]. The effort of the work is on the partial product reduction unit. The two-digit partial products of all 1×1 digit multiplications are correctly aligned to generate the complete partial products. The partial products are then reduced with a mix of binary decimal compressors and decimal adders.

Recently, a new decimal multiplier

. The effort of the work is on the partial product

reduction unit. The two-digit partial products of all 1×1 digit multiplications are correctly

aligned to generate the complete partial products. The partial products are then reduced

with a mix of binary decimal compressors and decimal adders.

Recently, a new decimal multiplier

^[46] improved the area of the best previous decimal multipliers on FPGA by about 20%. The solution considers a new decimal adder based on a mixed BCD/excess-6 representation and a 5221 recoding of the multiplier digits. Partial products are obtained from the addition of a multiple in the set {0, 2X, 5X, 2X+5X} and a multiple in the set {X, 2X}.

Two novel decimal multipliers on FPGA with different area/performance tradeoffs with both multipliers improving the area and performance of state-of-the-art multipliers were proposed in

improved the area of the best previous decimal

multipliers on FPGA by about 20%. The solution considers a new decimal adder based on

a mixed BCD/excess-6 representation and a 5221 recoding of the multiplier digits. Partial

products are obtained from the addition of a multiple in the set {0, 2X, 5X, 2X+5X} and a

multiple in the set {X, 2X}.

Two novel decimal multipliers on FPGA with different area/performance tradeoffs

with both multipliers improving the area and performance of state-of-the-art multipliers

were proposed in

^[47]. Both methods use a new adder/subtractor based on the excess-3 representation of multiples. Two different sets of multiples are considered: {0, X, 2X, 5X, 10X} and {2X, 4X, 5X}. Partial products are obtained by the addition or subtraction of two multiples of the sets. The method permits a very efficient generation of multiples, which considerably reduces the required resources.

. Both methods use a new adder/subtractor based on the excess-3

representation of multiples. Two different sets of multiples are considered: {0, X, 2X, 5X,

10X} and {2X, 4X, 5X}. Partial products are obtained by the addition or subtraction of two

multiples of the sets. The method permits a very efficient generation of multiples, which

considerably reduces the required resources.

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1. Introduction

2. Decimal Addition in FPGAw+z+6≤15

3. Decimal Multiplication in FPGA

References

2. Decimal Addition in FPGA $w + z + 6 \leq 15$