Design and Evaluation of Approximate Logarithmic Multipliers for Low Power Error-Tolerant Applications

Liu, Weiqiang; Xu, Jiahua; Wang, Danye; Wang, Chenghua; Montuschi, Paolo; Lombardi, Fabrizio

doi:10.1109/tcsi.2018.2792902

Cited by 126 publications

(60 citation statements)

References 39 publications

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“…In addition to hardware performance, we compare the normalized mean error distance (NMED) for all evaluated multipliers. The NMED is defined in [19], [38], [39] as average error distance over all input combinations normalized by the maximum output of the exact multiplier.…”

Section: Synthesis Resultsmentioning

confidence: 99%

See 1 more Smart Citation

On the Design of Logarithmic Multiplier Using Radix-4 Booth Encoding

Pilipović

Bulić

2020

IEEE Access

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This paper proposes an energy-efficient approximate multiplier which combines radix-4 Booth encoding and logarithmic product approximation. Additionally, a datapath pruning technique is proposed and studied to reduce the hardware complexity of the multiplier. Various experiments were conducted to evaluate the multiplier's error performance and efficiency in terms of energy and area utilization. The reported results are based on simulations using TSMC-180nm. Also, the applicability of the proposed multiplier is examined in image sharpening and convolutional neural networks. The applicability assessment shows that the proposed multiplier can replace an exact multiplier and deliver up to a 75% reduction in energy consumption and up to a 50% reduction in area utilization. Comparative analysis with the state-of-the-art multipliers indicates the potential of the proposed approach as a novel design strategy for approximate multipliers. When compared to the state-of-the-art approximate non-logarithmic multipliers, the proposed multiplier offers smaller energy consumption with the same level of applicability in image processing and classification applications. On the other hand, some state-of-the-art approximate logarithmic multipliers exhibit lower energy consumption than the proposed multiplier but deliver significant performance degradation for the selected application cases. INDEX TERMS Approximate computing, arithmetic circuit design, booth encoding, logarithmic multipliers, multipliers, power-efficient processing, truncated multipliers.

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Section: Synthesis Resultsmentioning

confidence: 99%

“…Liu et al [19] utilize truncated approximate adders for mantissa addition, which delivers a design with smaller barrel shifters. Their multiplier offers similar accuracy as Mitchell's multiplier and at the same time, delivers a smaller power-delay product.…”

Section: A Approximate Logarithmic Multipliersmentioning

confidence: 99%

On the Design of Logarithmic Multiplier Using Radix-4 Booth Encoding

Pilipović

Bulić

2020

IEEE Access

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“…Exponentials and logarithms are important in deep neural networks [21]. Approximate multipliers that extend Mitchell's technique are presented and analyzed in [22].…”

Section: Some Classical Tricks Of the Tradementioning

confidence: 99%

“…Relative difference between Approximation (55) to √ with the constant 127 • 222 replaced by 532369100 and the actual √ in[1,4].…”

mentioning

confidence: 99%

Elementary Functions and Approximate Computing

Muller

2020

Proc. IEEE

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We review some of the classical methods used for quickly obtaining low-precision approximations to the elementary functions. Then, for each of the three main classes of elementary function algorithms (shift-and-add algorithms, polynomial or rational approximations, table-based methods) and for the additional, specific to approximate computing, "bit-manipulation" techniques, we examine what can be done for obtaining very fast estimates of a function, at the cost of a (controlled) loss in terms of accuracy.

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“…There are many other ways of approximating multiplica-tion that had not been applied to deep CNNs, such as [43], [44], [45] among countless others. While we believe that the studied multiplier designs are the most promising, there are most likely other related opportunities for improving CNNs.…”

Section: Related Workmentioning

confidence: 99%

Low-power implementation of Mitchell's approximate logarithmic multiplication for convolutional neural networks

Kim¹,

Barrio²,

Hermida³

et al. 2018

2018 23rd Asia and South Pacific Design Automation Conference (ASP-DAC)

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This paper analyzes the effects of approximate multiplication when performing inferences on deep convolutional neural networks (CNNs). The approximate multiplication can reduce the cost of underlying circuits so that CNN inferences can be performed more efficiently in hardware accelerators. The study identifies the critical factors in the convolution, fully-connected, and batch normalization layers that allow more accurate CNN predictions despite the errors from approximate multiplication. The same factors also provide an arithmetic explanation of why bfloat16 multiplication performs well on CNNs. The experiments are performed with recognized network architectures to show that the approximate multipliers can produce predictions that are nearly as accurate as the FP32 references, without additional training. For example, the ResNet and Inception-v4 models with Mitch-w6 multiplication produces Top-5 errors that are within 0.2% compared to the FP32 references. A brief cost comparison of Mitch-w6 against bfloat16 is presented, where a MAC operation saves up to 80% of energy compared to the bfloat16 arithmetic. The most far-reaching contribution of this paper is the analytical justification that multiplications can be approximated while additions need to be exact in CNN MAC operations.

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Design and Evaluation of Approximate Logarithmic Multipliers for Low Power Error-Tolerant Applications

Cited by 126 publications

References 39 publications

On the Design of Logarithmic Multiplier Using Radix-4 Booth Encoding

On the Design of Logarithmic Multiplier Using Radix-4 Booth Encoding

Elementary Functions and Approximate Computing

Low-power implementation of Mitchell's approximate logarithmic multiplication for convolutional neural networks

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