Encyclopedia
Please note this is a comparison between Version 2 by Mário Véstias and Version 1 by Mário Véstias.
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 FPGAw+z+6≤15
Decimal addition is implemented by direct manipulation of decimal numbers or using
a binary adder followed decimal correction.
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 [6][7].
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, [8][9] consider a redundant BCD representation to
achieve carry free operations.
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 [10][11][12][13][14][15][16][17].
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 [18] and BCD to binary conversion [19] to use binary adders. The main problem
of this solution is the overhead of binary-BCD conversions [20].
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 [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 [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].
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 [6][25]. In [25] 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.
like carry-save addition can be applied efficiently [6][25]. In [25] 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.
In [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.
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.
23. Decimal Multiplication in FPGA
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 [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 [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.
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 [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 [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.
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 [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.
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 [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 YU ∈ {0, 1, 2} and YL ∈ {-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 [14] was
followed by [35] but using an hybrid 8421–5421 representation.
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 YU ∈ {0, 1, 2} and YL ∈ {-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 [14] was
followed by [35] but using an hybrid 8421–5421 representation.
In [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.
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%.
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 [37]. Digits in a column are all added in binary, converted to
decimal and finally added with decimal adders.
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 [37]. Digits in a column are all added in binary, converted to
decimal and finally added with decimal adders.
In [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.
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 [18][19]. 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.
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 [18][19]. 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 [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 [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 [43]. The work in [22] described previously was mapped on a 6-input LUT FPGA.
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 [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 [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 [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.
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 [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}.
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.
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.
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