![The error-tolerant adder type II (ETAII) [Zhu et al. 2009]: the carry propagates through the two shaded blocks.](https://www.researchgate.net/profile/Honglan-Jiang/publication/300630607/figure/fig6/AS:668455405621253@1536383596356/The-error-tolerant-adder-type-II-ETAII-Zhu-et-al-2009-the-carry-propagates-through_Q320.jpg)
![The broken-array multiplier (BAM) with 4 vertical lines and 2 horizontal lines omitted [Mahdiani et al. 2010]. : a carry-save adder cell.](https://www.researchgate.net/profile/Honglan-Jiang/publication/300630607/figure/fig3/AS:581682399268864@1515695299962/The-broken-array-multiplier-BAM-with-4-vertical-lines-and-2-horizontal-lines-omitted_Q320.jpg)
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60
A Review, Classification and Comparative Evaluation of Approximate
Arithmetic Circuits
HONGLAN JIANG, University of Alberta
CONG LIU, University of Alberta
LEIBO LIU, Tsinghua University
FABRIZIO LOMBARDI, Northeastern University
JIE HAN, University of Alberta
Often as the most important arithmetic modules in a processor, adders, multipliers and dividers determine
the performance and the energy efficiency of many computing tasks. The demand of higher speed and power
efficiency, as well as the feature of error resilience in many applications (e.g., multimedia, recognition and
data analytics), have driven the development of approximate arithmetic design. In this article, a review
and classification are presented for the current designs of approximate arithmetic circuits including adders,
multipliers and dividers. A comprehensive and comparative evaluation of their error and circuit character-
istics is performed for understanding the features of various designs. By using approximate multipliers and
adders, the circuit for an image processing application consumes as little as 47% of the power and 36% of
the power-delay product of an accurate design while achieving a similar image processing quality. Improve-
ments in delay, power and area are obtained for the detection of differences in images by using approximate
dividers.
CCS Concepts: •General and reference →Surveys and overviews; Evaluation; Measurement;
•Hardware →Arithmetic and datapath circuits; Combinational circuits;
Additional Key Words and Phrases: Approximate computing, approximate circuit, adder, multiplier, divider,
hardware, accuracy, image processing.
1. INTRODUCTION
While computational errors are in general not desirable, applications such as multi-
media, wireless communication, recognition, and data mining are tolerant of the oc-
currence of some errors [Han and Orshansky 2013]. Due to the perceptual limitations
of humans, these errors do not make an obvious difference in applications such as im-
age, audio and video processing. Moreover, in many digital signal processing (DSP)
systems, inputs from the outside world are noisy, so there is a limit in precision or
accuracy in the computed results. Many applications are based on statistical or prob-
abilistic computation, such as classification and recognition algorithms. Due to the
nature of these applications, trivial errors in computation do not result in a significan-
t performance degradation. Therefore, approximate computing is applicable in many
applications that can tolerate the loss of certain accuracy [Venkatesan et al. 2010].
This work was partly supported by the University of Alberta and the Natural Sciences and Engineering
Research Council (NSERC) of Canada.
Author’s addresses: H. Jiang, C. Liu and J. Han, Department of Electrical and Computer Engineering, U-
niversity of Alberta, Edmonton, AB T6G 1H9, Canada; email: {honglan, cong4, jhan8}@ualberta.ca; L. Liu,
Institute of Microelectronics, Tsinghua University, Beijing 100084, China; email: liulb@tsinghua.edu.cn;
F. Lombardi, Department of Electrical and Computer Engineering, Northeastern University, Boston, MA
02115, USA; email: lombardi@ece.neu.edu.
ACM acknowledges that this contribution was co-authored by an affiliate of the national government of
Canada. As such, the Crown in Right of Canada retains an equal interest in the copyright. Reprints must
include clear attribution to ACM and the author’s government agency affiliation. Permission to make digital
or hard copies for personal or classroom use is granted. Copies must bear this notice and the full citation on
the first page. Copyrights for components of this work owned by others than ACM must be honored. To copy
otherwise, distribute, republish, or post, requires prior specific permission and/or a fee. Request permissions
from permissions@acm.org.
© 2017 ACM. 1550-4832/2017/07-ART60 $15.00
DOI:
http://dx.doi.org/10.1145/3094124
ACM Journal on Emerging Technologies in Computing Systems, Vol. 13, No. 4, Article 60, Pub. date: July 2017.
60:2 H. Jiang et al.
Past research on approximate computing has spanned from circuits to programming
languages [Han 2016]. In [Datla et al. 2009], an approximate squaring circuit is pro-
posed. A new logic synthesis approach is introduced to reduce the area of a synthe-
sized circuit for a given threshold of error rate in [Shin 2010]. In [Chippa et al. 2010],
a scalable-effort design approach is proposed to implement highly efficient hardware
for error-resilient applications. Automated design processes have been proposed for
approximate digital circuit design using Cartesian genetic programming [Vasicek and
Sekanina 2015; Mrazek et al. 2016]. Sampson et al. have developed the EnerJ lan-
guage, an extension to Java [Sampson et al. 2011]. This language supports approx-
imate data types for low-power computation. Various computing and memory archi-
tectures have been proposed for supporting approximate computing applications [Es-
maeilzadeh et al. 2012; Miguel et al. 2015]. In this article, we focus on approximate
circuit designs and particularly approximate arithmetic circuits of adders, multipliers
and dividers.
Design metrics and analytical approaches have been proposed for the evaluation of
approximate adders [Liang et al. 2013; Huang et al. 2012; Miao et al. 2012; Venkatesan
et al. 2010; Liu et al. 2015; Mazahir et al. 2017]. Monte Carlo simulation has been
employed to acquire data for analysis. In this article, the error rate (ER), the error
distance (ED) and the average error are used to evaluate the error characteristics
of the approximate designs. Hardware related figures of merit including critical path
delay, circuit area and power dissipation, as well as compound metrics including the
power-delay product (PDP) and area-delay product (ADP), are utilized to assess the
circuit characteristics of these designs.
Image processing has been essential in diverse applications including multimedia,
biomedical imaging and pattern recognition [Acharya and Ray 2005]. Taking advan-
tage of its inherent error resilience, image processing can be efficiently implemented
by using approximate arithmetic circuits. Therefore, image sharpening and change
detection are considered for further evaluation of the approximate circuits in addition
to the evaluation using design metrics. The simulation results show that the image
sharpening circuit using approximate adders and multipliers saves as much as 53% of
power and 58% of area compared to an accurate design with a similar accuracy. The
change detection circuit using approximate dividers achieves as much as 40% improve-
ment in speed and 25% improvement in power compared with an accurate design at a
similar accuracy. While some preliminary results have been presented in [Jiang et al.
2015] and [Jiang et al. 2016b], this article makes the following new contributions.
1. A larger set of approximate adders that include more recent designs are consid-
ered in this article. Moreover, a new error metric (average error) is used to measure
the output bias of an approximate design, which is very important in an accumulative
operation.
2. Approximate Booth multipliers for signed multiplication are included in the eval-
uation, while only unsigned approximate multipliers are considered in [Jiang et al.
2016b].
3. Current approximate dividers are reviewed and their features are discussed with
respect to performance, accuracy and hardware consumption.
4. The STM CMOS 28nm process is used in the circuit synthesis throughout the
work in this article, while an older 65nm technology was used in [Jiang et al. 2015].
5. The considered approximate arithmetic circuits of adders, multipliers and dividers
are applied to two image processing applications for further evaluation. The accuracy
and circuit characteristics are obtained by simulation and synthesis.
ACM Journal on Emerging Technologies in Computing Systems, Vol. 13, No. 4, Article 60, Pub. date: July 2017.
A Review, Classification and Comparative Evaluation of Approximate Arithmetic Circuits 60:3
FA
a
0
b
0
c
0
s
0
FA
a
1
b
1
c
1
s
1
FA
a
i
b
i
c
i
s
i
c
i+1
c
2
......
FA
a
n-1
b
n-1
c
n-1
s
n-1
c
out
Critical Path
Fig. 1. The n-bit ripple-carry adder (RCA). FA: a 1-bit full adder.
SPG
a
0
b
0
c
0
SPG
a
1
b
1
c
1
SPG
a
i
b
i
c
i
..... .
SPG
a
n-1
b
n-1
s
n-1
c
out
Carry Lookahe ad Generator
s
0
p
0
g
0
s
1
p
1
g
1
s
i
p
i
g
i
p
n-1
g
n-1
Critical Path
c
n-1
Fig. 2. The n-bit carry lookahead adder (CLA). SPG: the cell used to produce the sum,
generate (gi=aibi) and propagate (pi=ai+bi) signals.
2. APPROXIMATE ADDERS
An adder performs the addition of two binary numbers. Two basic adders are the
ripple-carry adder (RCA) (Figure 1) and the carry lookahead adder (CLA) (Figure 2).
In an n-bit RCA, the carry of each full adder (FA) is propagated to the next FA, thus
the delay and circuit complexity increase proportionally with n(or O(n)). An n-bit
CLA consists of nunits that operate in parallel to produce the sum and the generate
(gi=aibi) and propagate (pi=ai+bi) signals for generating the lookahead carries.
The delay of CLA is logarithmic in n(or O(log(n))), thus significantly shorter than for
RCA. However, a CLA requires a larger circuit area (in O(nlog(n))), incurring a higher
power dissipation.
Many approximation schemes have been proposed by reducing the critical path and
hardware complexity of an accurate adder. An early methodology is based on a specu-
lative operation [Lu 2004; Verma et al. 2008]. In an n-bit speculative adder, each sum
bit is predicted by its previous kless significant bits (LSBs) (k < n). As the carry chain
is shorter than n, a speculative adder is faster than a conventional design. A segment-
ed adder is implemented by several smaller adders operating in parallel [Mohapatra
et al. 2011; Zhu et al. 2009; Kahng and Kang 2012; Yang et al. 2016]. Hence, the carry
propagation chain is truncated into shorter segments. Segmentation is also utilized
in [Du et al. 2012; Kim et al. 2013; Ye et al. 2013; Lin et al. 2015; Li and Zhou 2014;
Hu and Qian 2015; Miao et al. 2012; Camus et al. 2015; 2016], but the carry input for
each sub-adder is selected differently. This type of adder is referred to as a carry select
adder. Another method for reducing the critical path delay and power dissipation is
by approximating a full adder [Mahdiani et al. 2010; Gupta et al. 2013; Yang et al.
2013; Cai et al. 2016; Angizi et al. 2017]. The approximate full adder is then used to
implement the LSBs in an accurate adder. Thus, approximate adders are divided into
four categories, as briefly summarized below.
ACM Journal on Emerging Technologies in Computing Systems, Vol. 13, No. 4, Article 60, Pub. date: July 2017.
60:4 H. Jiang et al.
... a
1
b
1
a
2
b
2
a
3
b
3
a
4
b
4
a
n-1
b
n-1
a
n-2
b
n-2
a
n-3
b
n-3
a
n-4
b
n-4
k k
...
a
0
b
0
s
4
s
5
s
n-1
s
n-2
...
a
n-5
b
n-5
a
5
b
5
...
Fig. 3. The almost correct adder (ACA). : the carry propagation path of the sum.
k-bit Adder k-bit Adder k-bit Adder l-bit Adder
...
...
...
a
l-1:0
b
l-1:0
a
n-k-1:n-2k
b
n-k-1:n-2k
a
n-1:n-k
b
n-1:n-k
s
l-1:0
s
n-1:n-k
s
n-k-1:n-2k
a
k+l-1:l
b
k+l-1:l
s
k+l-1:l
Fig. 4. The equal segmentation adder (ESA). k: the maximum carry chain length; l: the
size of the first sub-adder (l≤k).
2.1. Classification of Approximate Adders
2.1.1. Speculative adders. The almost correct adder (ACA) [Verma et al. 2008] is based
on the speculative adder design of [Lu 2004]. In an n-bit ACA, kLSBs are used to
predict the carry for each sum bit (n>k), as shown in Figure 3. Therefore, the critical
path delay is reduced to O(log(k)) (for a parallel implementation such as CLA, the
same below). The design in [Lu 2004] requires (n−k)k-bit sub-carry generators in an
n-bit adder and thus, the hardware consumption is rather high (in O((n−k)klog(k))).
This overhead is reduced in [Verma et al. 2008] by sharing some components among
the sub-carry generators.
2.1.2. Segmented adders. The equal segmentation adder (ESA) divides an n-bit adder
into a number of smaller k-bit sub-adders operating in parallel with fixed carry inputs,
so no carry is propagated among the sub-adders (Figure 4) [Mohapatra et al. 2011].
The delay of ESA is O(log(k)) and the circuit complexity is O(nlog(k)). Its hardware
overhead is significantly lower than ACA.
The error-tolerant adder type II (ETAII) consists of parallel carry generators and
sum generators [Zhu et al. 2009], as shown in Figure 5. The carry signal from the pre-
vious carry generator propagates to the next sum generator. Therefore, ETAII utilizes
more information to predict the carry and thus it is more accurate than ESA for the
same k. The circuit of ETAII is more complex than that of ESA, and its delay is larger
due to the longer critical path (2k).
In an n-bit accuracy-configurable approximate adder (ACAA), n
k−12k-bit sub-
adders are required [Kahng and Kang 2012]. Each sub-adder adds 2kconsecutive bits
ACM Journal on Emerging Technologies in Computing Systems, Vol. 13, No. 4, Article 60, Pub. date: July 2017.
A Review, Classification and Comparative Evaluation of Approximate Arithmetic Circuits 60:5
Carry
Generator
Carry
Generator
Sum
Generator
Sum
Generator
...
...
...
Sum
Generator
Carry
Generator
s
k-1:0
s
n-1:n-k
s
n-k-1:n-2k
a
k-1:0
b
k-1:0
a
n-k-1:n-2k
b
n-k-1:n-2k
a
n-1:n-k
b
n-1:n-k
Fig. 5. The error-tolerant adder type II (ETAII) [Zhu et al. 2009]: the carry propagates
through the two shaded blocks.
Fig. 6. The speculative carry selection adder (SCSA).
with an overlap of kbits and all 2k-bit sub-adders operate in parallel to reduce the
delay to O(log(k)). In each sub-adder, half of the most significant sum bits is selected
as the partial sum. The accuracy of ACAA can be configured at runtime. Moreover,
ACAA has the same carry propagation path as ETAII for each sum, so they are equally
accurate for the same k.
The dithering adder divides an adder into an accurate, more significant sub-adder
and a less significant sub-adder with upper and lower bounding modules [Miao et al.
2012]. The output of the less significant sub-adder is conditionally selected. An effec-
tive “Dither Control” enables a smaller variance in the overall error.
To reduce the error distance, an error control and compensation method is proposed
for a segmented adder in [Yang et al. 2016]. This method employs a multistage laten-
cy to compensate the carry prediction error in a more significant segmentation, thus
trading off computing efficiency for an improved accuracy.
The delays of the segmented adders are O(log(k)) and the circuit complexities are
O(nlog(k)) for ESA and ETAII, and O((n−k)log(k)) for ACAA.
2.1.3. Carry select adders. In a carry select adder, several signals are commonly used.
For the ith block, generate gi,j =ai,jbi,j , propagate pi,j =ai,j ⊕bi,j , and Pi=
k−1
Q
j=0
pi,j ,
where ai,j and bi,j are the jth LSBs of the input operands. Pi= 1 indicates that all k
propagate signals in the ith block are true.
An n-bit speculative carry select adder (SCSA) consists of m=n
ksub-adders (or
window adders) [Du et al. 2012]. Each sub-adder is made of two k-bit adders: adder0
ACM Journal on Emerging Technologies in Computing Systems, Vol. 13, No. 4, Article 60, Pub. date: July 2017.
60:6 H. Jiang et al.
with carry-in “0” and adder1 with carry-in “1”. The carry-out of adder0 is connected
to a multiplexer to select the addition result as part of the final result, as shown in
Figure 6. SCSA and ETAII achieve the same accuracy for the same value of kdue to
the same carry predict function, while SCSA uses an additional adder and multiplexer
in each block.
Similar to SCSA, an n-bit adder is divided into n
kblocks in the carry skip adder
(CSA) [Kim et al. 2013]. Each block in CSA consists of a sub-carry generator and a sub-
adder. The carry-in of the (i+ 1)th sub-adder is determined by the propagate signals of
the ith block: it is the carry-out of the (i−1)th sub-carry generator when all propagate
signals are true (Pi= 1), otherwise it is the carry-out of the ith sub-carry generator.
Therefore, the critical path delay of CSA is O(log(k)). This carry select scheme im-
proves the carry prediction accuracy.
Different from SCSA, the carry speculative adder (CSPA) in [Lin et al. 2015] contains
one sum generator, two internal carry generators (one with carry-0 and one with carry-
1) and one carry predictor in each block. The output of the ith carry predictor is used
to select carry signals for the (i+ 1)th sum generator. linput bits (rather than k, l < k)
in a block are used in a carry predictor. Therefore, the hardware overhead is reduced
compared to SCSA.
The consistent carry approximate adder (CCA) is similar to SCSA in that each block
of CCA consists of adders with carry-0 and carry-1 [Li and Zhou 2014]. The select
signal of a multiplexer is determined by the propagate signal, i.e., Si= (Pi+Pi−1)SC +
(Pi+Pi−1)Ci−1, where Ci−1is the carry-out of the (i−1)th adder0 and SC is a global
speculative carry. In CCA, the carry prediction depends not only on its LSBs, but also
on the higher bits; its critical path delay is similar to that of SCSA.
The generate signals-exploited carry speculation adder (GCSA) has a similar struc-
ture as CSA and uses the generate signals for carry speculation [Hu and Qian 2015].
The difference between them lies in the carry selection; the carry-in for the (i+ 1)th
sub-adder is selected by its own propagate signals rather than its previous block. The
carry-in is the most significant generate signal gi,k−1of the ith block if Pi= 1, or else it
is the carry-out of the ith sub-carry generator. This carry selection scheme effectively
controls the maximum relative error.
In the gracefully-degrading accuracy-configurable adder (GDA) [Ye et al. 2013], the
control signals are used to configure the accuracy by selecting an accurate or approx-
imate carry-in signal using a multiplexer for each sub-adder. The delay of GDA is
determined by the carry propagation and thus by the control signals to the multiplex-
ers.
In the carry cut-back adder (CCBA) [Camus et al. 2016], the full carry propagation
is prevented by a controlled multiplexer or an OR gate for a high-speed operation. The
multiplexer is controlled by a carry propagate block at a higher-significance position
to cut the carry propagation at a lower-significance position. The delay and accuracy of
the CCBA largely depend on the distance between the propagate block and the cutting
multiplexer, thus allowing a high accuracy with a marginal overhead.
The critical path delays of the carry select adders are given by O(log(k)), where kis
the size of the sub-adder.
2.1.4. Approximate full adders. In this type of design, approximate full adders are im-
plemented in the LSBs of a multibit adder. It includes the simple use of OR gates (and
one AND gate for carry propagation) in the so-called lower-part-OR adder (LOA) (Fig-
ure 7) [Mahdiani et al. 2010], the approximate designs of the mirror adder (AMAs)
[Gupta et al. 2013] and the approximate XOR/XNOR-based full adders (AXAs) [Yang
et al. 2013]. Additionally, emerging technologies such as magnetic tunnel junctions
ACM Journal on Emerging Technologies in Computing Systems, Vol. 13, No. 4, Article 60, Pub. date: July 2017.
A Review, Classification and Comparative Evaluation of Approximate Arithmetic Circuits 60:7
a0 b0
al-1 bl-1
...
al-1 bl-1
l-bit OR-based Sub-Adder
(n-l)-bit Accurate
Sub-Adder
al-1:0
bl-1:0
an-1:l
bn-1:l
s0
sl-1
Cin
sl-1:0
sn-1:l
Cout
Fig. 7. The lower-part-OR adder (LOA).
have been considered for the design of approximate full adders for a shorter delay, a
smaller area and a lower power consumption [Cai et al. 2016; Angizi et al. 2017].
The critical path of this type of adders depends on its approximation scheme. For
LOA, it is approximately O(log(n−l)), where lis the number of bits in the lower part
of an adder. In the evaluation, LOA is selected as the reference design because the
other designs require customized layouts at the transistor level; hence, they are not
comparable with the other types of approximate adders that are approximated at the
logic gate level. Finally, an adder with the LSBs truncated is referred to as a truncated
adder that works with a lower precision. It is considered as a baseline design.
2.2. Evaluation of Approximate Adders
2.2.1. Error Characteristics. Monte Carlo simulation is performed to evaluate the accu-
racy of the approximate adders. The error distance (ED) and the relative error dis-
tance (RED) are calculated as: ED =|M0−M|and RE D =ED
M, where M0is the
approximate result and Mis the accurate result [Liang et al. 2013]. The mean error
distance (MED) is the mean of all possible EDs. The error rate (ER, the probability
of producing an incorrect result), the normalized MED (NM ED, the normalization
of MED by the maximum output of the accurate design) and the mean relative error
distance (MRED, the average value of all possible REDs) are used to assess the error
characteristics of the approximate designs. Moreover, the average error (the mean of
all possible errors (M0−M)) is used to evaluate the bias of an approximate arithmetic
design.
The functions of 16−bit approximate adders are simulated by MATLAB using 10
million uniformly distributed random input combinations. Table I shows the simula-
tion results. The size of the carry predictor for CSPA is dk/2ein this evaluation. The
global speculative carry SC for CCA is “0”, which is proved to be more accurate than
using “1”. Additionally, the adder with kLSBs truncated (TruA-k) is simulated for
comparison.
As shown in Table I, ETAII, ACAA and SCSA have the same error characteristics (in
ER,N MED and MRED) due to the same carry propagation chain for each sum bit.
The NM ED and M RED show the same trend, so only M RED and ER are considered
in the comparison, as shown in Figure 8. An equivalent carry propagation chain is
selected for the considered approximate adders i.e., the parameter kfor ACA, ESA,
LOA and TruA is 8, while it is 4 for CSA, GCSA, ETAII, ACAA, SCSA, CCA and CSPA.
These approximate adders are considered as equivalent approximate adders.
ACM Journal on Emerging Technologies in Computing Systems, Vol. 13, No. 4, Article 60, Pub. date: July 2017.
60:8 H. Jiang et al.
Table I. Simulation Results of the Error Characteristics for the Approximate Adders
Adder Type ER (%) N MED (10−3)MRED (10−3) Average Error
Speculative Adders
ACA-4 16.66 7.80 18.90 -1024.6
ACA-5 7.76 3.90 9.60 -511.7
Segmented Adders
ESA-4 85.07 15.70 40.40 -2047.5
ESA-5 80.03 7.80 20.80 -1023.4
ETAII-4 5.85 0.97 2.60 -127.5
ETAII-5 2.28 0.24 0.65 -31.6
ACAA-4 5.85 0.97 2.60 -127.5
ACAA-5 2.29 0.24 0.65 -31.6
Carry Select Adders
SCSA-4 5.85 0.97 2.60 -127.5
SCSA-5 2.28 0.24 0.65 -31.6
CSA-4 0.18 0.06 0.15 -7.4
CSA-5 0.02 0.004 0.01 -0.5
CSPA-4 29.82 3.90 10.40 -511.4
CSPA-5 11.31 0.98 2.70 -128.3
CCA-4 8.71 0.98 2.00 -128.3
CCA-5 3.78 0.25 0.49 -32.2
GCSA-4 4.26 0.48 0.98 -63.2
GCSA-5 1.52 0.12 0.25 -16.1
Approximate Full Adders
LOA-6 82.19 0.09 0.25 0.2
LOA-8 89.99 0.37 1.00 0.2
Truncated Adders
TruA-6 99.98 0.48 1.30 -63.0
TruA-8 100.0 1.95 5.40 -255.0
Note: The number following the name of each approximate adder is the number
of LSBs used for the carry speculation in the speculative adders, the length of
the segmentation in the segmented adders, and the number of approximated
and truncated LSBs in the approximate full adder-based and truncated adders.
Among these approximate adders, CSA is the most accurate, and GCSA is the sec-
ond most accurate in terms of MRED. LOA has a different structure from the other
approximate adders. Its more significant part is fully accurate, while the approximate
part is less significant. Therefore, the MRED of LOA is rather small, but its E R is
very large. For a similar reason, TruA has the highest ER and very large MRED. The
information used to predict each carry in ESA and CSPA is rather limited, so the ER
and the MRED of ESA and CSPA are larger than most of the other approximate de-
signs. Compared with the other approximate adders, CCA, ETAII, SCSA and ACAA
show moderate ER and M RED. In terms of average error, LOA has the lowest value
because it produces both positive and negative errors that can compensate each oth-
er, while errors are accumulated for the other approximate adders since only negative
errors are generated. Therefore, LOA is suitable for an accumulative operation.
In summary, the carry select adders and the speculative adder (ACA) are very accu-
rate with small values of ER and M RED (except for CSPA using a small number of
bits for carry prediction). Represented by LOA, an approximate full adder based adder
has a moderate MRED, the lowest average error but a very large ER. The segmented
ACM Journal on Emerging Technologies in Computing Systems, Vol. 13, No. 4, Article 60, Pub. date: July 2017.
A Review, Classification and Comparative Evaluation of Approximate Arithmetic Circuits 60:9
0.18 0.68 4.26 5.85 5.85 5.85 8.71
29.82
49.83
89.99
100.00
0.00
20.00
40.00
60.00
80.00
100.00
120.00
CSA ACA GCSA ETAII ACAA SCSA CCA CSPA ESA LOA TruA
ER (%)
(a)
0.1
1
10
CSA GCSA LOA ACA CCA ETAII SCSA ACAA ESA TruA CSPA
MRED (%)
(b)
Fig. 8. A comparison of error characteristics of the approximate adders. (a) ER and (b)
MRED. The parameter, k, is 4 for CSA, GCSA, ETAII, ACAA, SCSA, CCA and CSPA,
and it is 8for ACA, ESA, LOA and TruA for an equivalent carry propagation chain.
adders are not very accurate in terms of NM ED and M RED. With very large values
of ER and M RED, the truncated adder is the least accurate among the equivalent
designs. Three different types of approximate adders, ETAII, ACAA and SCSA, have
the same error characteristics.
2.2.2. Circuit Characteristics. To assess the circuit characteristics, 16-bit approximate
adders and the accurate CLA are implemented in VHDL and synthesized using the
Synopsys Design Compiler (DC) based on an STM CMOS 28 nm process with a supply
voltage of 1.0 V at a temperature of 25◦C. For a fair comparison, all designs use the
same process, voltage and temperature with the same optimization option. Both the
ACM Journal on Emerging Technologies in Computing Systems, Vol. 13, No. 4, Article 60, Pub. date: July 2017.
60:10 H. Jiang et al.
0
200
400
600
800
1000
1200
Delay (ps)
(a)
0.00
20.00
40.00
60.00
80.00
100.00
120.00
140.00
160.00
180.00
200.00
Power (uW)
(b)
0.00
10.00
20.00
30.00
40.00
50.00
60.00
70.00
PDP (fJ)
(c)
Fig. 9. A comparison of delay and power of the approximate adders (a) delay, (b) power
and (c) power-delay product.
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A Review, Classification and Comparative Evaluation of Approximate Arithmetic Circuits 60:11
Table II. Circuit Characteristics of the Approximate Adders
Adder Type Delay
(ps)
Power
(uW )
PDP
(fJ)
Area
(um2)
ADP
(um2.ns)
CLAC 1000 65.9 65.9 60.7 60.7
CLAG 570 105.4 60.1 84.2 48.0
Speculative Adders
ACA-4 250 118.4 29.6 73.8 18.5
ACA-5 270 119.4 32.2 71.8 19.4
Segmented Adders
ESA-4 260 47.0 12.2 49.9 13.0
ESA-5 310 50.6 15.7 51.7 16.0
ETAII-4 550 80.6 44.3 71.6 39.4
ETAII-5 670 78.5 52.6 70.2 47.0
ACAA-4 550 80.9 44.5 70.8 38.9
ACAA-5 650 87.3 56.8 74.6 48.5
Carry Select Adders
SCSA-4 320 134.5 43.0 109.2 34.9
SCSA-5 400 163.0 65.2 126.2 50.5
CSA-4 390 97.8 38.1 142.5 55.6
CSA-5 420 94.3 39.6 131.2 55.1
CSPA-4 300 89.2 26.8 83.7 25.1
CSPA-5 370 117.6 43.5 100.7 37.3
CCA-4 320 172.6 55.2 131.4 42.0
CCA-5 420 209.5 88.0 155.0 65.1
GCSA-4 380 109.7 41.7 74.3 28.2
GCSA-5 460 113.6 52.3 73.3 33.7
Approximate Full Adders
LOA-6 440 75.1 33.0 58.8 25.9
LOA-8 390 66.9 26.1 53.2 20.8
Truncated Adders
TruA-6 390 67.9 26.5 52.4 20.4
TruA-81350 64.2 22.5 46.2 16.2
1TruA-8 is synthesized at a medium mapping effort, different
from the high mapping effort used for the other designs. In
this case, TruA-8 attains a shorter critical path delay, but a
similar PDP and ADP as the results synthesized using the
high mapping effort.
P-channel and the N-channel transistors used in the designs have a typical design
corner with a regular threshold voltage. The critical path delay and area are reported
by the Synopsys DC. Power dissipation is measured by the PrimeTime-PX tool at 1 ns
clock period with 10 million random input combinations. All adders and sub-adders
are implemented as CLA in this article, unless otherwise noted.
Table II reports the results for the delay, power dissipation, power-delay product
(PDP), circuit area and area-delay product (ADP) of the considered adders. Two struc-
tures of the accurate CLA are implemented: CLAC is realized by four cascaded 4-bit
CLAs, while CLAG is realized by four parallel 4-bit CLAs and a carry look-ahead gen-
erator. Among ETAII, SCSA and ACAA (with the same error characteristics when the
same value of kis selected), SCSA, albeit being the fastest, incurs the largest power
dissipation and area because two sub-adders and one multiplexer are utilized in each
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60:12 H. Jiang et al.
10 20 30 40 50 60 70 80 90 100
PDP (fJ)
10-2
10-1
100
101
102
ER (%)
LOA
ESA
ETAII
GCSA
CSA
CSPA
CCA
LOA
TruA
CSPA
CSA
GCSA
ACA
CCA
LOA
ESA
ETAII
TruA
(a)
0 10 20 30 40 50 60 70 80 90 100
PDP (fJ)
10-2
10-1
100
101
102
MRED (10-3)
ACA
ESA
ETAII
GCSA
CSA
CSPA
CCA
LOA
TruA
CCA
ACA
CSPA
ESA
ETAII
GCSA
LOA
CSA
TruA
(b)
Fig. 10. A comprehensive comparison of the approximate adders: (a) ER and PDP and
(b) MRED and PDP. The parameter kfor LOA and TruA ranges from 9 down to 3 from
left to right, it is 3 to 8 for ESA and ACA, and it is from 3 to 6 for the other adders from
left to right. The adders marked by circles are equivalent in terms of carry propagation
and are thus representatives of different designs.
block. ACAA is very slow due to its long critical path. The block of ETAII (a carry gen-
erator and a sum generator) is significantly simpler than those of SCSA and ACAA.
Therefore, ETAII consumes less power and requires a smaller area than SCSA and
ACAA.
In general a circuit with a larger area is likely to consume more power except for CSA
with a relatively low power dissipation but a large area. This is due to its short critical
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A Review, Classification and Comparative Evaluation of Approximate Arithmetic Circuits 60:13
path and enhanced carry select that results in a complex wiring. Figure 9 shows the
delay, power and PDP of the equivalent adders. As expected, the accurate CLAC has
the longest delay among all adders, but not the highest power dissipation. Compared
to CLAC, CLAG is significantly faster and consumes more power and area. TruA is not
very fast, but it is the most power and area efficient design. LOA is also very power and
area efficient compared with the other approximate adders. ESA is the slowest, but it
is very power and area efficient due to its simple segmentation structure. CCA is very
fast but is the most power and area consuming design due to its complex speculative
circuit. Both CSPA and GCSA have moderate power dissipations, but CSPA is faster
and GCSA uses a smaller area. Both the speed and power dissipation of CSA are in the
medium range. In terms of PDP and ADP (shown in Table II), they show very similar
trend. TruA, LOA and CSPA have very small values of PDP and ADP, while these
values are relatively large for ACA and CCA (shown in Figure 9(c)).
In conclusion, the carry select adders are likely to have large values of power dis-
sipation and area at a moderate performance. The segmented adders are power and
area efficient. A speculative adder is very fast, but it is also very power consuming
with a moderate area. Conversely, the approximate full adder based adder is slow, but
it consumes a low power and area. The approximate full adders are very efficient in
PDP and ADP, while the speculative adders are not. The truncated adder is very power
and area efficient, but with a relatively long delay.
2.2.3. Discussion. Since the ADP shows a similar trend as the PDP, the PDP is con-
sidered for a comprehensive comparison of the approximate adders, as shown in the
two-dimensional (2-D) plots of Figure 10. CSA-6is accurate due to the precise carry
generated for every block, so the ER and M RED of CSA-6 are 0. Therefore, they are
not shown in Figure 10. The equivalent adders are marked by circles. Among adders
with the same accuracy (ETAII, SCSA and ACAA), ETAII is the most efficient in terms
of delay, power and area. Thus, it is shown as a representative in Figure 10. Compared
with the other approximate adders, CCA has the largest PDP and moderate ER and
MRED. Among the schemes with moderate PDPs (CSPA, GCSA and ETAII), ETAII
and GCSA have moderate MREDs and ERs, while CSPA shows slightly higher val-
ues of these measures. ESA has a rather small PDP, but a considerably large ER and
MRED. ACA has a larger PDP than ESA, but it has both lower ER and MRED. A-
mong all approximate adders, CSA shows the best performance with very small PDP,
ER and M RED values.
With the highest ERs, LOA and TruA show the smallest PDPs for a similar M RED
due to their low power dissipation. In fact, these approximate adders show a decent
tradeoff in error distance and hardware efficiency. In particular, they are useful in
applications in which hardware efficiency is of the utmost importance.
3. APPROXIMATE MULTIPLIERS
3.1. Classification of Approximate Multipliers
Generally, a multiplier consists of stages of partial product generation, accumulation
and a final addition, as shown in Figure 11 for a 4×4unsigned multiplication. Let Ai
and Bjbe the ith and jth least significant bits of inputs Aand Brespectively, a partial
product Pj,i is usually generated by an AND gate (i.e., Pj,i =AiBj). The commonly
used partial product accumulation structures include the Wallace, Dadda trees and a
carry-save adder array. The Wallace tree for a 4×4unsigned multiplier is shown in
the dotted box of Figure 11. The adders in each layer operate in parallel without carry
propagation, and the same operation repeats until two rows of partial products are left.
For an n-bit multiplier, log(n)layers are required in a Wallace tree. Therefore, the delay
of the partial product accumulation stage is O(log(n)). Moreover, the adders in Figure
ACM Journal on Emerging Technologies in Computing Systems, Vol. 13, No. 4, Article 60, Pub. date: July 2017.
60:14 H. Jiang et al.
A
B
Multiplicand
Multiplier
Partial
products
×
Products
Wallace
tree
Partial products
generation
Partial products
reduction
Final addition
Fig. 11. The basic arithmetic process of a 4×4unsigned multiplication with possible
truncations to a limited width. : an input, a partial product or an output product; :
a truncated bit; : a full adder or a half adder.
FAFAFA
FAFA
FA
FA
FAFA
P
0,0
P
0,1
P
0,2
P
0,3
P
1,0
P
1,1
P
1,2
P
1,3
P
2,0
P
2,1
P
2,2
P
2,3
P
3,0
P
3,1
P
3,2
P
3,3
FA
C
0,0
C
0,1
C
0,2
S
0,1
S
0,2
C
2,0
C
2,1
C
2,2
S
21
S
2,2
S
0,0
S
2,0
C
3,0
C
3,1
C
3,2
S
3,1
S
3,2
S
3,0
HA
HA
HA
HAHAHA HA
C
0,3
S
0,3
C
1,0
C
1,1
C
1,2
S
1,1
S
1,2
S
1,0
C
1,3
S
1,3
C
2,3
S
2,3
C
3,3
S
3,3
‘0’‘0’‘0’‘0’
Fig. 12. Partial product accumulation of a 4×4unsigned multiplier using a carry-save
adder array. HA: a half adder; FA: a full adder.
11 can be considered as a (3:2) compressor and can be replaced by other counters or
compressors (e.g. a (4:2) compressor) to further reduce the delay. The Dadda tree has
a similar structure as the Wallace tree, but it uses as few adders as possible.
A carry-save adder array is shown in Figure 12; the carry and sum signals generated
by the adders in a row are passed on to the adders in the next row. Adders in a column
operate in series. Hence the partial product accumulation delay of an n-bit multiplier
is approximately O(n), longer than that of the Wallace tree. However, an array requires
a smaller area due to the simple and symmetric structure.
Three main methodologies are used for the approximate design of a multiplier: i)
approximation in generating the partial products [Kulkarni et al. 2011], ii) approxi-
mation (including truncation) in the partial product tree [Mahdiani et al. 2010; Kyaw
et al. 2010; Bhardwaj et al. 2014], and iii) using approximate designs of adders [Liu
et al. 2014], counters [Lin and Lin 2013] or compressors [Ma et al. 2013; Momeni et al.
2015] to accumulate the partial products. For a signed integer operation, Booth multi-
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A Review, Classification and Comparative Evaluation of Approximate Arithmetic Circuits 60:15
Table III. K-Map for the 2×2Underdesigned Multiplier Block.
M2M1M0
B1B0
00 01 11 10
A1A0
00 000 000 000 000
01 000 001 011 010
11 000 011 111 110
10 000 010 110 100
pliers have been widely used due to the fast operation on a reduced number of partial
products. Some recent designs use shifting and addition to obtain the final product
by rounding the inputs to a form of 2m(mis a positive integer) [Hashemi et al. 2015;
Zendegani et al. 2017; Mitchell Jr 1962].
Based on the different schemes in approximation, approximate multipliers are clas-
sified into three unsigned types and signed Booth multipliers. Following this classifi-
cation, existing designs of approximate multipliers are briefly reviewed next.
3.1.1. Approximation in generating partial products. The underdesigned multiplier (UDM)
utilizes an approximate 2×2multiplier obtained by altering a single entry in the Kar-
naugh Map (K-Map) of its function (as highlighted in Table III) [Kulkarni et al. 2011].
Table III shows the K-Map of the approximate 2×2multiplier, where A1A0and B1B0
are the two 2-bit inputs, and M2M1M0is the 3-bit output. In this approximation, the
accurate multiplication result “1001” is simplified to “111” to save one output bit when
both the inputs are “11”. Assuming the value of each input bit is equally likely, the
error rate of the 2×2multiplier is then (1
2)4=1
16 . Larger multipliers can be designed
based on the 2×2multiplier. This multiplier introduces an error when generating the
partial products, however the adder tree remains accurate.
3.1.2. Approximation in the partial product tree. A bio-inspired imprecise multiplier re-
ferred to as a broken-array multiplier (BAM) is proposed in [Mahdiani et al. 2010]. The
BAM operates by omitting some carry-save adders in an array multiplier in both hori-
zontal and vertical directions (Figure 13). A more straightforward approach to trunca-
tion is to truncate some LSBs on the input operands and thus, a smaller multiplier is
sufficient for the remaining MSBs. This truncated multiplier (TruM) is considered as
a baseline design.
The error tolerant multiplier (ETM) is divided into a multiplication section for the
MSBs and a non-multiplication section for the LSBs [Kyaw et al. 2010]. Figure 14
shows the architecture of a 16-bit ETM. A NOR gate based control block is used to
deal with the following two cases: i) if the product of the MSBs is zero, then the upper
accurate 8-bit multiplier is activated to multiply the LSBs without any approximation,
and ii) if the product of the MSBs is nonzero, the non-multiplication section is used
as an approximate multiplier to process the LSBs, while the multiplication section is
activated to accurately multiply the MSBs.
The static segment multiplier (SSM) was proposed using a similar partition scheme
[Narayanamoorthy et al. 2015]. Different from ETM, no approximation is applied to
the LSBs in the SSM. Either the MSBs or the LSBs of the operands are accurately
multiplied depending on whether its MSBs are all zeros. [Liu et al. 2017a] has shown
that only small improvements in accuracy and hardware are achieved compared to
ETM, thus this design is not further considered in this comparison study.
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60:16 H. Jiang et al.
Vertical Break
Level (VBL)
Horizontal Break
Level (VBL)
Horizontally-omitted cell Vertically-omitted cell
Fig. 13. The broken-array multiplier (BAM) with 4 vertical lines and 2 horizontal lines
omitted [Mahdiani et al. 2010]. : a carry-save adder cell.
Standard 8-bit Multiplier
LSB Non-multip lication
block
Standard 8-bit Multiplier
A
0
-A
7
B
0
-B
7
P
0
-P
15
A
0
-A
7
B
0
-B
7
A
8
-A
15
B
8
-B
15
P
0
-P
15
P
16
-P
31
Control Block
A
8
-A
15
:
B
8
-B
15
:
Fig. 14. The 16-bit error-tolerant multiplier (ETM) of [Kyaw et al. 2010].
A power and area-efficient approximate Wallace tree multiplier (AWTM) is based
on a bit-width aware approximate multiplication and a carry-in prediction method [B-
hardwaj et al. 2014]. An n-bit AWTM is implemented by four n/2-bit sub-multipliers,
as shown in Figure 15, where the most significant sub-multiplier AHBHis further im-
plemented by four n/4-bit sub-multipliers. The AWTM is configured into four different
modes by the number of approximate n/4-bit sub-multipliers in the most significant
n/2-bit sub-multiplier, while the other three multipliers (AHBL,ALBHand ALBL) are
approximate. The approximate partial products are then accumulated by a Wallace
tree.
3.1.3. Using approximate counters or compressors in the partial product tree. An approximate
(4:2) counter is proposed in [Lin and Lin 2013] for an inaccurate 4-bit Wallace multi-
plier. Table IV shows the K-Map of the approximate (4:2) counter, where X1· · · X4are
the four input signals of a (4:2) counter (i.e., the partial products in the partial product
tree of a multiplier), Cand Sare the carry and sum, respectively. The values of CS in
the box are approximated as “10” for “100” in the approximate counter when all input
signals are “1.” As the probability of obtaining a partial product of “1” is 1
4, the error
rate of the approximate (4:2) counter is (1
4)4=1
256 . The inaccurate 4-bit multiplier is
then used to construct larger multipliers with error detection and correction circuits.
This approximate multiplier is referred to as ICM.
In the compressor based multiplier, accurate (3:2) and (4:2) compressors are im-
proved to speed up the partial product accumulation [Baran et al. 2010]. By using the
improved compressors, better energy and delay characteristics are obtained for a mul-
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A Review, Classification and Comparative Evaluation of Approximate Arithmetic Circuits 60:17
A
L
B
L
A
L
B
H
A
H
B
L
Final Product (2n b its)
n bits
n/2 bits
A
HH
B
HH
A
HL
B
HL
A
HL
B
HH
A
HH
B
HL
A
H
B
H
n/4 bits
Fig. 15. The approximate Wallace tree multiplier (AWTM) [Bhardwaj et al. 2014]. n
is the width of the multiplier, AHBH,ALBH,AHBLand ALBLare partial products
generated by the n/2-bit sub-multipliers, AHH BH H ,AHLBHH ,AH H BHL and AH LBH L
are partial products generated by the n/4-bit sub-multipliers.
Table IV. K-Map for the 4 : 2 Approximate Counter
CS X1X0
00 01 11 10
X3X4
00 00 01 10 01
01 01 10 11 10
11 10 11 10 11
10 01 10 11 10
tiplier. To further reduce delay and power, two approximate (4:2) compressor designs
(AC1 and AC2) are presented in [Momeni et al. 2015]; these compressors are used in
a Dadda multiplier with four different schemes. Approximate counters in which the
more significant output bits are ignored, are presented and evaluated in [Kelly et al.
2009]. Several signed multipliers are then implemented using these approximate coun-
ters. The more accurate schemes 3 and 4 of the approximate compressor based mul-
tiplier (referred to as ACM-3 and ACM-4) in [Momeni et al. 2015] are considered for
comparison.
In the approximate multiplier with configurable error recovery, the partial product-
s are accumulated by a novel approximate adder (Figure 16) [Liu et al. 2014]. The
approximate adder utilizes two adjacent inputs to generate a sum and an error bit.
The adder processes data in parallel, thus no carry propagation is required. Two ap-
proximate error accumulation schemes are then proposed to alleviate the error of the
approximate multiplier (due to the approximate adder). OR gates are used in the first
error accumulation stage in scheme 1 (AM1), while in scheme 2 (AM2), both OR gates
and the approximate adders are used. The truncation of 16 LSBs in the partial prod-
ucts in AM1 and AM2 results in TAM1 and TAM2 respectively [Liu 2014].
3.1.4. Approximate Booth multipliers. The Booth recoding algorithm handles binary num-
bers in 2’s complement. The modified (or radix-4) Booth algorithm is commonly used
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60:18 H. Jiang et al.
Fig. 16. The approximate adder cell. Si: the sum bit; Ei: the error bit [Liu et al. 2014].
P
0,0
P
0,2
P
0,4
P
0,6
P
0,1
P
0,3
P
0,5
P
0,7
P
0,8
P
1,0
P
1,2
P
1,4
P
1,6
P
1,1
P
1,3
P
1,5
P
1,7
P
1,8
P
2,0
P
2,2
P
2,4
P
2,6
P
2,1
P
2,3
P
2,5
P
2,7
P
2,8
P
3,0
P
3,2
P
3,4
P
3,6
P
3,1
P
3,3
P
3,5
P
3,7
P
3,8
Main Part (MP) Truncation Part (TP)
n
0
n
1
n
2
n
3
TP
major
TP
minor
exact
carry
approximate
carry
Fig. 17. The partial products for an 8-bit fixed-width modified Booth multiplier [Cho
et al. 2004]. Pi,j is the jth partial product in the ith partial product vector and niis the
sign of the ith partial product vector.
due to its ease in generating partial products. Little work has been reported for
approximate Booth multipliers, whereas the fixed-width Booth multiplier utilizes a
truncation-based approach has been studied for more than a decade. The conventional
post-truncated fixed-width multiplier generates an output with the same width as the
input operand by truncating the lower half of the product. Truncation of half of the
partial products is widely used because the post-truncated scheme does not achieve a
significant circuit advantage over the accurate multiplier. A direct truncation of partial
products incurs a large error, so many error compensation schemes have been proposed
[Cho et al. 2004; Min-An et al. 2007; Wang et al. 2011; Chen and Chang 2012]. Anoth-
er approach is to use an approximate Booth encoder with a simple circuit [Liu et al.
2017b]. Most of the approximate Booth multipliers are based on the modified Booth
algorithm; the partial products are accumulated by an array structure in [Cho et al.
2004; Min-An et al. 2007; Wang et al. 2011; Farshchi et al. 2013] while a parallel carry-
save-adder tree is used in [Chen and Chang 2012; Liu et al. 2017b]. The approximate
Booth multiplier in [Jiang et al. 2016a] is based on the radix-8 Booth algorithm.
Figure 17 shows the partial products of an 8-bit fixed-width modified Booth multi-
plier with error compensation [Cho et al. 2004]. The final product is the addition of the
main part (MP) and the carry signals generated in the truncation part (TP). The carry
signals are approximated by the output of Booth encoders. The approximate carry σis
σ=$2−1(
n/2−2
P
i=0
zeroi+ 1)%, where nis the multiplier width, and zeroiis “1” if the ith
partial product vector is not zero or zeroi= 0 otherwise. This multiplier is referred to
as BM04.
The multiplier in [Min-An et al. 2007] can adaptively compensate the quantization
error by keeping different numbers of the most significant columns of the partial prod-
ucts (ω(ω≥0)). Two types of binary thresholding are proposed for error compensation.
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A Review, Classification and Comparative Evaluation of Approximate Arithmetic Circuits 60:19
Different from BM04, nrather than (n−1) columns of partial products are truncated
for an n-bit multiplier. The error compensation for each type of binary thresholding
varies with the value of ωand the partial products of the ωth column in the truncation
part (from left to right). This multiplier is denoted as BM07.
The multiplier presented in [Wang et al. 2011] uses ncolumns of partial products
in the truncation part for an n-bit multiplier; the most significant one column in the
truncation part is reserved for error compensation. The error compensation using a
simplified sorting network significantly reduces the mean and mean-squared errors by
making the error symmetric as well as centralizing the error distribution around zero.
This design is referred to as BM11.
A fixed-width Booth multiplier was designed based on a probabilistic estimation
bias in [Chen and Chang 2012]. Therefore, this multiplier is referred to as PEBM.
The number of columns of the accumulated partial products varies in accordance
with the desired trade-off between hardware and accuracy. The error compensation
formula is derived from a probability analysis rather than a time-consuming ex-
haustive simulation. The carry generated by the truncation part is approximated by
σ=$2−1(
n/2−1−bω/2c
P
i=0
zi−1)%, where zi=P0,n/2−1+nn/2−1when iis n/2−1and
zi=zeroiotherwise.
Based on BAM, the broken Booth multiplier (BBM) uses a modified Booth algorithm
to generate partial products and omits carry-save adders to the right of a vertical line
[Farshchi et al. 2013]. BBM has a smaller power-delay product (PDP) for the same
mean-squared error compared to BAM.
An approximate recoding adder is proposed in [Jiang et al. 2016a] for calculating
the triple multiplicands to reduce the additional delay encountered in a radix-8 Booth
multiplier. A Wallace tree and a truncation technique are then utilized for partial prod-
uct accumulation to reduce power and delay. The most efficient fixed-width multiplier
ABM2_R15 is considered in this comparison and referred to as ABM2 in this article for
simplicity. In [Liu et al. 2017b], two approximate Radix-4 Booth encoders are designed
for the partial product generation by simplifying the exact K-Map. The generated par-
tial products are then accumulated by using exact 4-2 compressors.
3.2. Evaluation of Approximate Multipliers
3.2.1. Error Characteristics. The considered (16 ×16) approximate multipliers are simu-
lated by MATLAB with 10 million uniformly distributed random input combinations.
The ER,N MED,M RED and average error are obtained and shown in Table V. TruM-
krepresents the truncated multiplier with kLSBs truncated in the input operands.
According to Table V, most of the designs, especially those with truncation, have
large ERs close to 100%. However, ICM has a relatively low ER of 5.45%, because it
uses just one approximate counter in a 4×4sub-multiplier with an error rate of only
1
256 . UDM also shows a lower ER than the other approximate multipliers. In terms of
the average error, ACMs have the smallest value, while the average errors for all the
other approximate unsigned multipliers show the same trend with the NM ED. This
is because ACMs produce both positive and negative errors, but the other approximate
unsigned multipliers produce either negative or positive errors.
Figure 18 shows the NMEDs and M RE Ds of the equivalent approximate multipli-
ers that are configured to have 16-bit accurate MSBs (except for ICM and UDM that
have only one configuration). Thus, the truncated LSBs in the partial product is 16 for
BAM, the number of MSBs used for error compensation is 16 for AM1, AM2, TAM1
and TAM2, the size of the accurate sub-multiplier is 8 for ETM, 8 LSBs are truncat-
ed for TruM, and the mode number of ACM and AWTM is 4. Among the unsigned
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60:20 H. Jiang et al.
Table V. Error Characteristics of the Approximate Multipliers
Multiplier Type ER (%) N MED (10−3)MRED (%) Average Error (104)
Multipliers with Approximation in Generating Partial Products
UDM 80.99 13.92 3.33 -5974.9
Multipliers with Approximation in the Partial Product Tree
BAM-16 99.99 0.06 0.21 -24.6
BAM-17 99.99 0.11 0.36 -49.2
BAM-18 99.99 0.22 0.63 -95.0
BAM-20 100.00 0.79 1.79 -337.5
ETM-7 99.99 0.97 1.56 -413.4
ETM-8 100.00 1.94 2.85 -825.1
AWTM-1 100.00 2.70 75.00 1159.6
AWTM-2 100.00 1.88 39.37 807.4
AWTM-3 99.99 0.12 2.51 51.5
AWTM-4 99.94 0.02 0.33 8.6
Multipliers using Approximate Counters or Compressors
ICM 5.45 0.29 0.06 -124.2
ACM-3 99.99 0.01 0.29 3.95
ACM-4 99.97 0.01 0.26 1.44
AM1-13 99.38 0.90 0.54 -385.5
AM1-16 98.22 0.81 0.34 -347.9
AM2-10 99.64 0.88 1.20 -379.8
AM2-13 99.36 0.35 0.34 -148.9
AM2-16 97.96 0.27 0.13 -115.1
TAM1-13 99.99 1.14 0.77 -488.5
TAM1-16 99.99 1.06 0.58 -457.1
TAM2-10 99.99 0.90 1.27 -384.2
TAM2-13 99.99 0.36 0.41 -153.3
TAM2-16 97.99 0.28 0.22 -121.0
Truncated Unsigned Multipliers
TruM-4 99.61 0.11 0.23 -49.2
TruM-8 100.0 1.94 2.85 -834.1
Approximate Booth Multipliers
PEBM 99.99 0.023 0.27 -1.02
BBM 100.00 0.092 0.57 -9.83
BM11 99.99 0.022 0.18 -0.003
BM07 99.99 0.024 0.16 -1.56
BM04 99.99 0.027 0.48 -2.66
ABM2 99.99 0.034 0.44 -0.614
Note: The parameter kfollows the acronym of each approximate multiplier. For AM1,
AM2, TAM1 and TAM2, this parameter refers to the number of MSBs used for error
reduction and for ETM, the number of LSBs in the inaccurate part. It is the mode
number in AWTM and ACM, and the vertical broken length (VBL) for BAM.
approximate multipliers, UDM has the largest NMED while ACM has the smallest.
ICM, AM2 and TAM2 have similar values of NMED, however ICM has the small-
est MRED, while the M RED of TAM2 is the largest. Therefore ICM has the highest
accuracy in terms of MRED, while TAM2 is the least accurate among these three ap-
proximate multipliers. This indicates that multipliers with simple truncation tend to
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A Review, Classification and Comparative Evaluation of Approximate Arithmetic Circuits 60:21
0.0001
0.001
0.01
0.1
1
10
NMED (%)
(a)
0.01
0.1
1
10
MRED (%)
(b)
Fig. 18. Comparison of NM ED and MRED of the approximate multipliers with in-
creasing (a) NM ED and (b) M RED. ACM and AWTM represent ACM-4 and AWTM-4,
respectively. The truncated number of LSBs in the partial product is 16 for BAM, the
number of MSBs used for error compensation is 16 for AM1, AM2, TAM1 and TAM2,
and 8 LSBs are truncated for TruM. ETM is ETM-8 in Table V.
have larger MREDs when their N M E Ds are similar. BAM has moderate values of
NM ED and M RED, while ETM and TruM have both large M RE D and N M ED.
Hence, ICM is the most accurate design with the lowest ER,M RED and a moder-
ate NM ED. ACM, AWTM, BAM, AM2 and TAM2 also show good accuracy among all
considered approximate multipliers with both low NM EDs and M REDs. ETM, TruM
and UDM are not very accurate in terms of these metrics.
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60:22 H. Jiang et al.
For the approximate Booth multipliers in Table V, a column of the most significant
partial products in the truncation part (adjacent to the MP part) is kept for PEBM,
BM07 and BM04. 15-bit columns of partial products are truncated in BBM and ABM2
to keep the same width of the output as the other designs. The ERs of all approximate
multipliers are close to 100% due to truncation. Most designs have similar NMEDs
except for BBM. BBM has the largest NM ED and M RED, because there is no error
compensation. BM07 and BM11 have very small MRED values, while PEBM has a
slightly larger value. BM11 has the lowest average error and the average error of
ABM2 is also very small.
In summary, as a multiplier approximated in generating the partial products, UDM
has very large values of NM ED and M RED, and a relatively small ER. The multipli-
ers approximated in the partial product tree mostly have moderate NMEDs and very
large MREDs (except for BAMs with fewer than or equal to 18 truncated bits and
AWTM-4). The multipliers approximated using approximate counters or compressors
have small values of both NMED and MRED, while the multiplier truncated on the
input operands have large values of both metrics (when the truncated number of LSBs
is larger than 4). Among the considered approximate Booth multipliers, BBM shows
the lowest accuracy in terms of both NM ED and M RED. BM11 has the smallest aver-
age error. The other approximate Booth multipliers show similar N MEDs and various
MREDs.
3.2.2. Circuit Characteristics. The 16 ×16 approximate multipliers are implemented in
VHDL and synthesized using the same tool and process as in the simulation of ap-
proximate adders. The only difference is that the clock period is 4 ns for the power es-
timation of the multipliers because of a longer critical path delay. The accurate Wallace
multiplier (WallaceM) optimized for speed [Oklobdzija et al. 1996] and array multiplier
(ArrayM) are also simulated for comparison. To reduce the effect of the final addition,
the same multi-bit adder in the tool library is utilized in all approximate multiplier
designs as the final adder. Table VI shows the critical path delay, area, power, PDP
and ADP of the considered multipliers. TruMA and TruMW are the truncated array
and Wallace multipliers, respectively.
Figure 19 shows the comparison of delay, power and PDP of the equivalent approxi-
mate multipliers. The accurate array multiplier (ArrayM) is the slowest and the Wal-
lace multiplier (WallaceM) consumes more area (as per Table VI); this is consistent
with the theoretical analysis. Due to the expressively fast carry-ignored operation,
AM1/TAM1, AM2/TAM2 have smaller delays compared to most of the other designs.
BAM is significantly slower due to its array structure. AWTM, UDM, ICM and ACM
have larger delays than the other approximate multipliers. BAM consumes a very low
power, the power consumptions of AWTM and ACM are in the medium range, while
UDM and ICM incur a relatively high power consumption. TruMA, TruMW and ETM
have both a short delay and a low power dissipation.
A multiplier with a higher power dissipation usually has a larger area and thus larg-
er PDP and ADP. In terms of power and area, TruMA, TruMW, ETM, TAM1/TAM2 and
BAM are among the best designs. A common feature of these designs is that they all
use truncation, which can significantly affect the MRED while the NM ED may not
be significantly changed. If most of the inputs have large values, the error introduced
by truncation can be tolerated; thus truncation is a useful scheme to save area and
power. Otherwise, truncation-based designs may yield unacceptably inaccurate result-
s. Without truncation, a multiplier whose design is approximated in generating partial
product (e.g. UDM) tends to have a large delay, power and area. These measures for
the multipliers approximated in the partial product tree (e.g. AWTM) are moderate,
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A Review, Classification and Comparative Evaluation of Approximate Arithmetic Circuits 60:23
Table VI. Circuit Characteristics of the Approximate Multipliers
Multiplier Type Delay
(ns)
Power
(uW )
PDP
(fJ)
Area
(um2)
ADP
(um2.ns)
ArrayM 2.58 477.4 1,231.7 921 2,375.7
WallaceM 2.03 461.3 936.4 934 1,896.0
Multipliers with Approximation in Generating Partial Products
UDM 2.01 352.7 708.9 829 1666.7
Multipliers with Approximation in the Partial Product Tree
BAM-16 2.34 221.3 517.8 441 1,031.9
BAM-17 2.17 189.5 411.2 384 833.3
BAM-18 1.99 161.1 320.6 331 658.7
BAM-20 1.65 111.0 183.2 237 390.4
ETM-7 1.57 140.5 220.6 349 547.6
ETM-8 1.50 108.5 162.8 288 431.4
AWTM-1 1.69 247.8 418.8 640 1,081.8
AWTM-2 1.69 259.4 438.4 665 1,123.7
AWTM-3 1.69 270.3 456.8 690 1,165.6
AWTM-4 1.74 280.0 478.2 715 1,243.2
Multipliers using Approximate Counters or Compressors
ICM 1.87 367.4 687.0 937 1,751.4
ACM-3 1.97 279.5 550.6 738 1,454.5
ACM-4 2.00 284.1 568.2 724 1,447.0
AM1-13 1.38 355.4 490.5 819 1,128.8
AM1-16 1.57 380.6 597.5 878 1,378.5
AM2-10 1.29 336.8 434.5 816 1,052.6
AM2-13 1.53 364.2 557.2 919 1,406.1
AM2-16 1.71 400.4 684.7 1,051 1,797.2
TAM1-13 1.31 192.0 251.5 460 602.6
TAM1-16 1.45 214.6 311.2 516 748.2
TAM2-10 1.23 180.2 221.6 477 586.7
TAM2-13 1.48 212.5 314.5 584 864.3
TAM2-16 1.62 244.9 396.7 693 1,122.7
Truncated Unsigned Multipliers
TruMA-4 1.89 243.5 460.2 503 950.6
TruMA-8 1.19 92.1 109.6 211 250.5
TruMW-4 1.62 262.4 425.1 561 908.2
TruMW-8 1.10 98.4 108.2 239 263.0
Approximate Booth Multipliers
PEBM 1.83 264.3 483.7 528 966.2
BBM 1.91 250.3 478.1 487 930.2
BM11 1.96 258.1 505.9 475 931.0
BM07 2.03 270.4 548.9 528 1071.8
BM04 2.05 249.8 512.1 447 916.4
ABM2 2.25 208.0 468.0 424 954.0
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60:24 H. Jiang et al.
0.00
0.50
1.00
1.50
2.00
2.50
3.00
Delay (ns)
(a)
0.00
100.00
200.00
300.00
400.00
500.00
600.00
Power (uW)
(b)
0
200
400
600
800
1000
1200
1400
PDP (fJ)
(c)
Fig. 19. A comparison of delay, power and PDP of the approximate multipliers. (a)
delay, (b) power and (c) power-delay product.
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A Review, Classification and Comparative Evaluation of Approximate Arithmetic Circuits 60:25
100 200 300 400 500 600 700 800
PDP (fJ)
10-1
100
MRED (%)
AM1-16
AM1-13
TAM1
AM2-16
AM2-13
AM2-10
TAM2
ETM-8
ETM-7
AWTM-4
AWTM-3
ICM
UDM
BAM
ACM-3
ACM-4
TruMA
TruMW
Fig. 20. MRED and PDP of the approximate unsigned multipliers. The parameter k
for TruMA and TruMW is from 8 down to 2 from left to right; it is 21 down to 13
for BAM, and 10 to 16 for TAM1 and TAM2. The multipliers marked by circles are
equivalent in terms of the number of accurate MSBs and are thus representatives of
different designs.
while the multipliers using approximate counters or compressors (ICM, ACM, AM1,
AM2) require higher power and area.
In terms of PDP (Figure 19(c)) and ADP, TruMA, TruMW, ETM, TAM1 and TAM2
have very small values, while ICM, UDM and AM2 are the opposite. The values of PDP
and ADP for AM1, ACM, BAM and AWTM are in the medium range.
For the approximate Booth multipliers, PEBM is the fastest, but it is very power and
area consuming due to the use of a carry save adder tree for the parallel accumulation.
ABM2 is the slowest but the most power and area efficient design due to the smaller
number of partial products generated by the time-consuming recoding adder in the
radix-8 algorithm. Thus, it has the smallest PDP and a moderate value of ADP. With
no error compensation, BBM shows a small delay, low power dissipation and small
circuit area, and thus smaller PDP and ADP compared with most of the other designs
(except ABM2 and BM04). BM11 and BM04 have similar values for all circuit metrics.
BM07 has a similar delay, but with a higher power and area and thus a larger PDP
and ADP.
3.2.3. Discussion. MRED and PDP are jointly considered next for an overall evalua-
tion of the approximate multipliers, as shown in Figure 20 and Figure 21.
Figure 20 shows that TruMW has a smaller PDP than TruMA when the same num-
ber of LSBs is truncated. Among the truncation-based designs, the truncated unsigned
multipliers (TruMA and TruMW) are slightly more accurate (in MRED) than BAM
and ETM for a similar PDP. TruMW has a smaller M RED than most other approxi-
mate designs (except TAM1, TAM2 and ICM).
In Figure 20, TAM1-13, TAM1-16, TAM2-13, TruMA-6, TruMW-6 and BAM-18 have
both small PDPs and MREDs. Most of the other designs have at least one major short-
coming. ICM and ACM incur a very low error, but their PDPs are very high. Other than
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60:26 H. Jiang et al.
0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58
PDP (pJ)
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
MRED (%)
BM04
BM11
ABM2
PEBM
BM07
BBM
Fig. 21. MRED and PDP of the approximate Booth multipliers.
the truncated designs, ETM-8 has the smallest PDP but with a rather large MRED.
UDM shows a poor performance in both PDP and MRED. Even though some BAM
configurations have small PDPs, their delays are generally large (Figure 19(a)); more-
over, some BAM configurations have low accuracies. AWTMs have large PDPs and only
AWTM-4 has a high accuracy.
As for the approximate Booth multipliers (Figure 21), ABM2 shows the lowest PDP
and a moderate accuracy. BM11 and BM07 are very accurate in terms of M RED but
with relatively poor PDPs. PEBM shows both a moderate PDP and MRED.
4. APPROXIMATE DIVIDERS
4.1. Classification
The divider is a less frequently used arithmetic module compared to the adder and
multiplier; therefore, less research has been pursued on an approximate design.
Two methodologies have been advocated for sequential division, the digit recurren-
t algorithm [Liu and Nannarelli 2012] and the functional iterative algorithm (e.g.,
using the Newton-Raphson algorithm [Flynn 1970]). A sequential divider has a low
hardware complexity, however its delay is considerably longer than an adder and a
multiplier, so it significantly affects the overall performance of a processor. Thus, di-
viders made of combinational logic circuits are discussed in this article. Like multipli-
cation, division can also be implemented by an array structure, in which adder cells
are replaced by subtractor cells. Several approximations are made on the array divider
while retaining a low-power [Chen et al. 2015; 2016; Chen et al. 2017] and high-speed
[Hashemi et al. 2016] operation. In addition, different approximate divider designs
based on rounding [Zendegani et al. 2016] and curve fitting [Low and Jong 2013; Wu
and Jong 2015] are also proposed.
4.1.1. Approximate array dividers. Four types of approximate unsigned integer non-
restoring divider (AXDnr) are presented in [Chen et al. 2015]. Three approximate
subtractors (AXCSs) are designed for the array of an unsigned divider by simplifying
the circuit of an exact subtractor cell. The AXCSs are then used to replace the exact
subtractor cells at the least significant vertical, horizontal, square or triangle cells of
the array divider. Moreover, a truncation scheme is utilized by discarding the approx-
imate subtractors for comparison. Based on the same theory and design, four types
of approximate restoring dividers (AXDr) are further proposed by using the proposed
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A Review, Classification and Comparative Evaluation of Approximate Arithmetic Circuits 60:27
AXCSs [Chen et al. 2016]. It has been shown that AXDrs are slightly more accurate
and consumes lower power than AXDnrs.
To make the remaining subtractor cells more efficient, a dynamic approximate di-
vider (DAXD) is proposed by dynamically selecting the inputs of the subtractor cells
[Hashemi et al. 2016]. DAXD selects a fixed number of bits in the input operands from
the most significant non-zero bit, and then truncates the least significant bits. There-
fore, it can be implemented by two leading-one detectors, two multiplexers, a smaller
array divider and a barrel shifter.
To further improve the performance and power efficiency, an approximate signed-
digit adder is proposed for high-radix dividers [Chen et al. 2017]. Compared to the con-
ventional radix-2 design, the approximate radix-4 and radix-8 dividers show a higher
speed and consume lower power, albeit with a slightly lower accuracy.
4.1.2. Curve fitting based approximate dividers. A widely used methodology to reduce the
hardware overhead of a divider is based on binary logarithms, that is to obtain the
antilogarithmic value of the difference between the logarithmic values of the dividend
and the divisor . Mitchell et al. first developed the logarithm approximation of a binary
number by shifting and counting [Mitchell Jr 1962]. Inspired by it, a new antilogarith-
mic algorithm was proposed using a piecewise linear approximation [Low and Jong
2013]. A high-speed divider (HSD) based on this algorithm was then presented. The
antilogarithmic algorithm is directly approximated from the input operands, thus on-
ly look-up tables and multiplications are required, i.e., no logarithmic or subtraction
operation is needed. HSD achieves a better accuracy and a much higher speed (but at
a larger area) than the divider implemented directly by Mitchell’s algorithm.
A similar curve fitting approach was used for the design of a floating-point divider
(FPD) [Wu and Jong 2015]. FPD partitions the curved surfaces of the quotient into
several square or triangular regions and linearly approximates each region by curve
fitting. Finally, the division of the mantissas is implemented by a comparison module, a
look-up table, shifters and adders. This approximate divider achieves a better accuracy
than that in [Low and Jong 2013] with similar circuit characteristics.
4.1.3. Rounding based approximate dividers. A high-speed and energy-efficient rounding-
based approximate divider (SEERAD) is presented in [Zendegani et al. 2016]. It trans-
forms the division to a smaller multiplication by rounding the divisor Bto a form of
2K+L/D, where Kshows the bit position of the most significant “1” of B(K=blog2Bc),
and Land Dare constant integers found from an exhaustive simulation by the con-
dition of obtaining the lowest mean relative error. Different accuracy levels are con-
sidered to improve the accuracy of SEERAD by varying Dand Lwith combinations of
the more relevant bits of Bafter the most significant “1”. The multiplier in SEERAD
is implemented by several shift units and an adder block. Thus, SEERAD is very fast.
4.2. Discussion
The approximate array dividers, AXDnr, AXDr and DAXD, are designed for the un-
signed n/(n
2)division, where nis the width of the dividend. As the borrow signal pass-
es through all subtractor cells, the critical path of an array divider is O(n2)[Parhami
2000]. Thus, the speed of the approximate array dividers is not very fast, but the area
and power dissipation are relatively low among the approximate designs. The accuracy
of the array dividers depends on the number of replaced cells (AXDnr and AXDr) and
the size of the sub-divider (DAXD).
The approximate dividers based on curve fitting are very accurate, e.g, the maximum
relative error distance of a 16/16 HSD and FPD are 0.20% and 0.14%, respectively.
Although the curve fitting using software is complex, the hardware implementation
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60:28 H. Jiang et al.
consisting of look-up tables, smaller multipliers and adders is very simple. Thus, curve
fitting based dividers are usually fast. However, the look-up tables used for storing the
constant coefficients make the approximate dividers area and power consuming.
The rounding based approximate divider has a maximum relative error distance of
6.25% (for the design with the highest accuracy level) in a 16/16 divider and hence, it
is not very accurate. Its hardware implementation is simpler than those of the curve
fitting based dividers. Therefore, the rounding based approximate divider has a very
high speed and relatively large area and power dissipation due to the use of look-up
tables.
5. IMAGE PROCESSING APPLICATION
Low power dissipation and small circuit area are basic requirements for consumer
electronic products, especially for mobile devices with stringent battery restrictions.
Therefore, approximate designs have been pervasively considered for implementation-
s in image processing. Approximate multipliers have been utilized for image sharp-
ening [Jiang et al. 2016b]. In this article, both the adder and multiplier in the image
sharpening algorithm are replaced by approximate designs. Moreover, approximate di-
viders are used to detect the difference between two images to show the changes. This
application is known as change detection.
5.1. Image Sharpening
The image sharpening algorithm computes R(x, y)=2I(x, y)−S(x, y)[Lau et al. 2009],
where Iis the input image, Ris the sharpened image, and Sis given by
S(x, y) = 1
4368
2
X
m=−2
2
X
n=−2
G(m+ 3, n + 3)I(x−m, y −n),(1)
where G is a 5×5matrix given by
G=
16 64 112 64 16
64 256 416 256 64
112 416 656 416 112
64 256 416 256 64
16 64 112 64 16
.(2)
Simulation results in [Jiang et al. 2016b] show that AM2-15, AM1-15, TAM2-16,
TAM1-16, BAM-16, AM2-13, AM1-13, ACM-4, ACM-3, TAM2-13, TAM1-13, BAM-17,
AWTM-4 and BAM-18 achieve visually acceptable image sharpening results. Among
these multipliers, the ones with a moderate hardware overhead (AM1-13, TAM2-13,
TAM1-16, TAM1-13, BAM-17 and BAM-18) are selected in this article for image sharp-
ening. Likewise, the approximate adders LOA, CSA, ETAII and CSPA are selected.
As the multiplication result of a 16 ×16 multiplier is 32-bit wide, 32-bit approximate
adders are used for image sharpening. The value of parameter kis 8 for CSA, ETAII
and CSPA, and 16 for LOA.
The results for image sharpening using the selected approximate multipliers and
adders are given in Table VII, while the accurate result is shown in Figure 22. The
images sharpened using CSPA have unacceptable defects and some defects (white dots)
can be seen in the image sharpened by AM1-13 and ETAII-8 when zooming into the
images in Table VII. Other images processed by the approximate designs show similar
quality with the accurate result. This is also confirmed by the peak signal-to-noise ratio
(PSNR), as shown in Table VIII. The PSNRs of the images sharpened by a truncation
based multiplier (i.e., TAM1-16, TAM2-13, TAM1-13, BAM-17 or BAM-18) are fixed
as the adder is changed among LOA-16, CSA-8 and ETAII-8. This occurs because the
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A Review, Classification and Comparative Evaluation of Approximate Arithmetic Circuits 60:29
Fig. 22. The image sharpened using an accurate multiplier and an accurate adder.
lower 16 bits of the multiplication results generated by these multipliers are zeros and
hence, only the higher half of an approximate adder (as an accurate 16-bit adder for
LOA-16, CSA-8 or ETAII-8) is used.
The image sharpening algorithm is implemented in VHDL by using the selected
approximate adders and multipliers. In the implementation, no pipelining or memory
unit is used for exclusively showing the hardware characteristics of the approximate
arithmetic circuits. It is synthesized by Synopsys DC using the same process, voltage
and temperature as in the simulation of the approximate adders. The 512 ×512 pixel
values of the image shown in Table VII are used as inputs for assessing the power
dissipation using the PrimeTime-PX tool at a clock period of 10 ns.
Table IX shows the circuit characteristics of image sharpening using approximate
multipliers and adders. Using the same multiplier, the image sharpening implementa-
tions with LOA-16 and ETAII-8 show similar values for the delay, power and area (ex-
cept for AM1-13), while the implementations using CSA-8 have relatively larger values
for these metrics. Likewise, the image sharpening circuits have similar characteristics
using the same adder except that AM1-13, BAM-17 and BAM-18 based schemes show
slightly larger values.
Compared with the image sharpening circuit using accurate multipliers and adders,
the approximate designs using CSA-8 or AM1-13 achieve small improvement in terms
of delay and area because CSA and AM1 are less efficient in delay and area compared
with the other approximate designs. By using LOA-16, ETAII-8, TAM2-13, BAM-17
or BAM-18, the circuit can be 23% faster and saves as much as 53% in power, 58% in
area, 64% in PDP and 62% in ADP compared to the accurate design.
5.2. Change Detection
In the application of change detection, the ratio between two corresponding pixel val-
ues is calculated by a divider [Chen et al. 2016]. The changes in two images are then
highlighted by normalizing the pixel ratios. In this section, 16/8divider designs are
used to calculate the division of two 8-bit gray-level images as shown in Figure 23(a)
and (b). To ensure a higher accuracy, the pixel values of the first image are multiplied
by 64. As HSD and FPD are designed for floating-point division, AXDr, DAXD and
SEERAD are selected for the change detection. Among the four types of AXDr, the
triangle replacement has been shown to achieve the best results for image processing
[Chen et al. 2016]. Therefore, three designs of AXDr with the triangle replacement of
depth 8, AXDr1 (using approximate subtractor 1), AXDr2 (using approximate subtrac-
tor 2) and AXDr3 (using approximate subtractor 3), are used for the change detection.
For DAXD, 8/4and 10/5accurate dividers are utilized in DAXD8 and DAXD10, re-
spectively. Four accuracy levels are considered in SEERAD; they are referred to as
SEERAD1, SEERAD2, SEERAD3 and SEERAD4. Moreover, the accurate array di-
vider (denoted as ArrayD) is simulated for comparison.
Figure 23 shows the change detection results by the dividers and the obtained P-
SNR value is shown in the parentheses. It is clear that AXDr1, AXDr3 and SEERAD4
ACM Journal on Emerging Technologies in Computing Systems, Vol. 13, No. 4, Article 60, Pub. date: July 2017.
60:30 H. Jiang et al.
Table VII. Images Sharpened using Different Approximate Adder and Multiplier
Pairs
Approximate
design LOA-16 CSA-8 ETAII-8 CSPA-8
TAM1-16
AM1-13
TAM2-13
TAM1-13
BAM-17
BAM-18
perform very well in the application of change detection, while the results by the other
designs are of lower quality.
The characteristics of the change detection circuits are obtained by using the same
tool and process as in the simulation of approximate adders. The clock period is 6 ns,
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A Review, Classification and Comparative Evaluation of Approximate Arithmetic Circuits 60:31
Table VIII. PSNRs of the Sharpened Images (dB)
```````````
Multiplier
Adder LOA-16 CSA-8 ETAII-8 CSPA-8
TAM1-16 46.97 46.97 46.97 25.01
AM1-13 45.21 45.06 36.86 24.20
TAM2-13 41.87 41.87 41.87 24.32
TAM1-13 41.42 41.42 41.42 24.35
BAM-17 40.09 40.09 40.09 25.19
BAM-18 33.99 33.99 33.99 24.21
Table IX. Delay, Power, and Area of Image Sharpening using Approximate
Multipliers and Adders
Multiplier Adder Delay
(ns)
Power
(mW )
PDP
(pJ)
Area
(um2)
ADP
(um2.ns)
ArrayM CLAG 6.74 1.995 13.45 31,183.9 210,179.5
TAM1-16 LOA-16 5.36 0.9723 5.21 18,139.0 97,225.0
TAM1-16 CSA-8 7.45 1.032 7.69 23,652.1 176,208
TAM1-16 ETAII-8 5.34 0.9643 5.15 18,056.8 96,423.3
AM1-13 LOA-16 5.41 1.193 6.45 26,644.0 144,144
AM1-13 CSA-8 7.41 1.377 10.20 30,586.5 226,646
AM1-13 ETAII-8 6.40 1.369 8.76 28,214.7 180,574
TAM2-13 LOA-16 5.25 1.055 5.54 17,057.8 89,553.5
TAM2-13 CSA-8 6.43 1.053 6.77 20,526.6 131,986
TAM2-13 ETAII-8 5.22 1.041 5.43 16,975.6 88,612.6
TAM1-13 LOA-16 5.25 0.9467 4.97 17,221.0 90,410.3
TAM1-13 CSA-8 7.45 0.9942 7.41 22,734.1 169,369
TAM1-13 ETAII-8 5.34 0.9350 4.88 17,138.8 89,464.5
BAM-17 LOA-16 6.14 1.226 7.53 14,993.8 92,061.9
BAM-17 CSA-8 7.36 1.247 9.17 16,533.0 121,683
BAM-17 ETAII-8 6.13 1.211 7.42 14,868.5 91,143.9
BAM-18 LOA-16 5.97 1.097 6.55 13,285.3 79,313.2
BAM-18 CSA-8 6.89 1.117 7.70 16,901.8 116,453
BAM-18 ETAII-8 5.96 1.076 6.41 13,156.0 78,409.8
ACM Journal on Emerging Technologies in Computing Systems, Vol. 13, No. 4, Article 60, Pub. date: July 2017.
60:32 H. Jiang et al.
(a) Input image 1 (b) Input image 2 (c) Accurate output
(d) AXDr1 (40.46 dB) (e) AXDr2 (22.91 dB) (f) AXDr3 (41.71 dB)
(g) DAXD8 (25.22 dB) (h) DAXD10 (24.33 dB) (i) SEERAD1 (22.08 dB)
(j) SEERAD2 (28.15 dB) (k) SEERAD3 (33.17 dB ) (l) SEERAD4 (36.61 dB)
Fig. 23. Change detection using different approximate dividers.
and the input combinations for the power evaluation are the two images shown in
Figure 23(a) and (b) with 384 ×507 pixels. The synthesis results are shown in Table X.
To be consistent with the other designs, AXDr1, AXDr2 and AXDr3 are implemented
at the gate level rather than at the transistor level as in [Chen et al. 2016].
Table X shows that the array-based dividers (ArrayD, AXDrs and DAXD) are more
power and area efficient than the rounding based approximate dividers (SEERADs).
However, they are very slow except for DAXD that uses a smaller accurate array di-
vider. SEERADs are very fast, but they consume more power and area due to the use
of look-up tables and the large output size (including the fractional part). In terms of
ACM Journal on Emerging Technologies in Computing Systems, Vol. 13, No. 4, Article 60, Pub. date: July 2017.
A Review, Classification and Comparative Evaluation of Approximate Arithmetic Circuits 60:33
PDP and ADP, DAXD10 has the smallest values followed by SEERAD3. Among the
approximate dividers that produce excellent change detection results, SEERAD4 has
the smallest PDP but the largest ADP. AXDr1 results in the largest PDP and the s-
mallest ADP, while AXDr3 has both moderate PDP and ADP. To be more specific, the
change detection using AXDr3 saves 25% of power and 12% of area compared with the
accurate design. It is 40% faster with 18% reduction in PDP by using SEERAD4.
Table X. Accuracy and Circuit Characteristics of the Change Detection using
Approximate Dividers.
Divider PSNR (dB)Delay
(ns)
Power
(uW )
PDP
(fJ)
Area
(um2)
ADP
(um2.ns)
ArrayD – 4.08 54.29 221.50 425.8 1,737.2
AXDr1 40.46 3.85 50.71 195.23 415.5 1,599.7
AXDr2 22.91 4.36 54.08 235.79 408.2 1,779.6
AXDr3 41.71 4.58 40.55 185.72 376.2 1,722.9
DAXD10 24.33 2.43 40.25 97.84 375.7 912.9
SEERAD3 33.17 1.83 60.27 110.29 615.8 1,126.8
SEERAD4 36.61 2.43 70.62 181.33 765.4 1,859.9
6. CONCLUSION
In this article, designs of approximate arithmetic circuits are reviewed. Their error
and circuit characteristics are evaluated using functional simulation and hardware
synthesis with an industrial 28nm technology library.
Approximate Adders: In general, approximate speculative adders show high accura-
cy and relatively small PDPs. The approximate adders using approximate full adders
in the LSBs are slow, but they are power efficient with high ERs (due to the approx-
imate LSBs), low average error and moderate NMED and MRED values (due to the
accurate MSBs). The error and circuit characteristics of the segmented and carry select
adders vary with the prediction of the carry signals.
A truncated adder has a smaller MRED (an indicator of a smaller error magnitude)
than most approximate designs at a similar PDP except for LOA and CSA. However,
it has a lower performance and a significantly higher ER compared with the other
approximate designs. As a result, a simple truncation of the LSBs in an adder causes a
high ER and does not significantly improve the performance of the adder, though with
a relatively small error distance.
Approximate Multipliers: For approximate multipliers, truncation of part of the par-
tial products is an effective scheme to reduce hardware complexity, while preserving a
moderate NM ED and M RED. Similarly, truncating some LSBs of the input operands
can efficiently reduce the hardware overhead of a multiplier and result in a moderate
MRED (an indicator of the error magnitude) that is smaller than most other approxi-
mate designs, except for TAM1, TAM2 and ICM, for a similar PDP.
Albeit with a relatively low ER, UDM shows a low accuracy in terms of the error
distance and a relatively high circuit overhead, because the 2×2approximate multi-
ACM Journal on Emerging Technologies in Computing Systems, Vol. 13, No. 4, Article 60, Pub. date: July 2017.
60:34 H. Jiang et al.
plier is used to compute the most significant bits and accurate adders are utilized to
accumulate the generated partial products. ICM has the lowest ER among all designs.
When truncation is not used, multipliers approximated in the partial product tree tend
to have a poor accuracy (except AWTM-3 and AWTM-4) and moderate hardware con-
sumption, while multipliers using approximate counters or compressors are usually
very accurate with relatively high power dissipation and hardware consumption. The
approximate Booth multipliers show different characteristics in hardware efficiency
and accuracy.
Approximate Dividers: For the dividers, the approximate array dividers are slow,
but they are hardware efficient with variable accuracy depending on the approxima-
tion parameters. The dividers based on curve fitting are very accurate and fast but
they require a large area and high power dissipation due to the utilization of look-up
tables. The rounding based approximate dividers have a very high speed, large area
and power dissipation, with a relatively low accuracy.
Application: Image sharpening is implemented using the selected approximate mul-
tipliers and adders. The accuracy and circuit characteristics of the image sharpening
obtained by simulation indicate significant savings in hardware, while producing sim-
ilar sharpening quality as the accurate design. On average, the designs using approxi-
mate adders and multipliers with an acceptable accuracy consumes only 55% of power,
62% of area, 51% of PDP and 57% of ADP compared to the accurate design.
Approximate dividers are utilized in the change detection of images. The simulation
and synthesis results show that the change detection circuits using the approximate
array divider (AXDr1 and AXDr3) are power and area efficient but very slow, whereas
the one using the rounding based approximate divider (SEERAD4) consumes more
power and area with a high performance for an excellent detection accuracy.
ACKNOWLEDGMENT
The authors would like to thank Vincent Camus from EPFL for his contributions in
proof reading the article.
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