A unified algorithm for elementary functions

Approximate computing is an attractive design methodology to achieve low power, high performance (low delay) and reduced circuit complexity by relaxing the requirement of accuracy. In this paper, approximate Booth multipliers are designed based on approximate radix-4 modified Booth encoding (MBE) algorithms and a regular partial product array that employs an approximate Wallace tree. Two approximate Booth encoders are proposed and analyzed for error-tolerant computing. The error characteristics are analyzed with respect to the so-called approximation factor that is related to the inexact bit width of the Booth multipliers. Simulation results at 45 nm feature size in CMOS for delay, area and power consumption are also provided. The results show that the proposed 16-bit approximate radix-4 Booth multipliers with approximate factors of 12 and 14 are more accurate than existing approximate Booth multipliers with moderate power consumption. The proposed R4ABM2 multiplier with an approximation factor of 14 is the most efficient design when considering both power-delay product and the error metric NMED. Case studies for image processing show the validity of the proposed approximate radix-4 Booth multipliers.

Design of Approximate Radix-4 Booth Multipliers for Error-Tolerant Computing

Often as the most important arithmetic modules in a processor, adders, multipliers, and dividers determine the performance and 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 characteristics 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 similar image processing quality. Improvements in delay, power, and area are obtained for the detection of differences in images by using approximate dividers.

A Review, Classification, and Comparative Evaluation of Approximate Arithmetic Circuits

As an important arithmetic module, the adder plays a key role in determining the speed and power consumption of a digital signal processing (DSP) system. The demands of high speed and power efficiency as well as the fault tolerance nature of some applications have promoted the development of approximate adders. This paper reviews current approximate adder designs and provides a comparative evaluation in terms of both error and circuit characteristics. Simulation results show that the equal segmentation adder (ESA) is the most hardware-efficient design, but it has the lowest accuracy in terms of error rate (ER) and mean relative error distance (MRED). The error-tolerant adder type II (ETAII), the speculative carry select adder (SCSA) and the accuracy-configurable approximate adder (ACAA) are equally accurate (provided that the same parameters are used), however ETATII incurs the lowest power-delay-product (PDP) among them. The almost correct adder (ACA) is the most power consuming scheme with a moderate accuracy. The lower-part-OR adder (LOA) is the slowest, but it is highly efficient in power dissipation.

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A Comparative Review and Evaluation of Approximate Adders

Approximate computing has emerged as a new paradigm for high-performance and energy-efficient design of circuits and systems. For the many approximate arithmetic circuits proposed, it has become critical to understand a design or approximation technique for a specific application to improve performance and energy efficiency with a minimal loss in accuracy. This article aims to provide a comprehensive survey and a comparative evaluation of recently developed approximate arithmetic circuits under different design constraints. Specifically, approximate adders, multipliers, and dividers are synthesized and characterized under optimizations for performance and area. The error and circuit characteristics are then generalized for different classes of designs. The applications of these circuits in image processing and deep neural networks indicate that the circuits with lower error rates or error biases perform better in simple computations, such as the sum of products, whereas more complex accumulative computations that involve multiple matrix multiplications and convolutions are vulnerable to single-sided errors that lead to a large error bias in the computed result. Such complex computations are more sensitive to errors in addition than those in multiplication, so a larger approximation can be tolerated in multipliers than in adders. The use of approximate arithmetic circuits can improve the quality of image processing and deep learning in addition to the benefits in performance and power consumption for these applications.

Approximate Arithmetic Circuits: A Survey, Characterization, and Recent Applications

Multiple valued logic (MVL) circuits are particularly attractive for nanoscale implementation as advantages in information density and operating speed can be harvested using emerging technologies. In this paper, a new family of MVL gates is proposed for implementation using carbon nanotube field-effect transistors (CNTFETs). The proposed designs use pseudo N-type CNTFETs and no resistor is utilized for their operation. This approach exploits threshold voltage control of the P-type and N-type transistors, while ensuring correct MVL operation for both ternary and quaternary logic gates. This paper provides a detailed assessment of several figures of merit, such as static power consumption, switching power consumption, propagation delay and the power-delay product (PDP). Compared with resistor-loaded designs, the proposed pseudo-NCNTFET MVL gates show advantages in circuit area, power consumption and energy efficiency, while still incurring a comparable propagation delay. Compared to a complementary logic family, the pseudo-NCNTFET MVL logic family requires a smaller circuit area with a similar propagation delay on average, albeit with a larger PDP and static power consumption. A design methodology and a discussion of issues related to leakage and yield are also provided for the proposed MVL logic family.

Design and Evaluation of Multiple Valued Logic Gates Using Pseudo N-Type Carbon Nanotube FETs

This paper proposes several designs of approximate restoring dividers; two different levels of approximation (cell and array levels) are employed. Three approximate subtractor cells are utilized for integer subtraction as basic step of division; these cells tend to mitigate accuracy in subtraction with other metrics, such as circuit complexity and power dissipation. At array level, exact cells are either replaced or truncated in the approximate divider designs. A comprehensive evaluation of approximation at both cell- and array (divider) levels is pursued using error analysis and HSPICE simulation; different circuit metrics including complexity and power dissipation are evaluated. Different applications are investigated by utilizing the proposed approximate arithmetic circuits. The simulation results show that with extensive savings for power dissipation and circuit complexity, the proposed designs offer better error tolerant capabilities for quotient oriented applications (image processing) than remainder oriented application (modulo operations). The proposed approximate restoring divider is significantly better than the approximate non-restoring scheme presented in the technical literature.

On the Design of Approximate Restoring Dividers for Error-Tolerant Applications

This paper proposes several approximate divider designs; two different levels of approximation (cell and array levels) are investigated for non-restoring division. Three approximate subtractor cells are proposed and designed for the basic subtraction; these cells mitigate accuracy in subtraction with other metrics, such as circuit complexity and power dissipation. At array level, by considering the exact cells, both replacement and truncation schemes are introduced for approximate array divider design. A comprehensive evaluation of approximation at both cell and divider level is pursued. Different circuit metrics including complexity and power dissipation are evaluated by HSPICE simulation. Mean error distance (MED), normalized error distance (NED) and MED-power product (MPP) are provided to substantiate the accuracy and power trade-off of inexact computing. Different applications in image processing are investigated by utilizing the proposed approximate arithmetic circuits.

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Design of Approximate Unsigned Integer Non-restoring Divider for Inexact Computing

Power has become a key constraint in nanoscale integrated circuit design due to the increasing demands for mobile computing and higher integration density. As an emerging computational paradigm, an inexact circuit offers a promising approach to significantly reduce both dynamic and static power dissipation for error-tolerant applications. In this paper, an inexact floating-point adder is proposed by approximately designing an exponent subtractor and mantissa adder. Related operations such as normalization and rounding are also dealt with in terms of inexact computing. An upper bound error analysis for the average case is presented to guide the inexact design; it shows that the inexact floating-point adder design is dependent on the application data range. High dynamic range images are then processed using the proposed inexact floating-point adders to show the validity of the inexact design; comparison results show that the proposed inexact floating-point adders can improve the power consumption and power-delay product by 29.98 and 39.60 percent, respectively.

Design and Analysis of Inexact Floating-Point Adders

Approximate high radix dividers (HR-AXDs) are proposed and investigated in this paper. High-radix division is reviewed and inexact computing is introduced at different levels. Design parameters such as number of bits (N) and radix (r) are considered in the analysis; the replacement of exact cells with inexact cells in a binary signed-digit adder is introduced by utilizing different replacement schemes. Cell truncation and error compensation are also proposed to further extend inexact computation. Circuit-level performance and the error characteristics of the inexact high radix dividers are analyzed for the proposed designs. The combined assessment of the normal error distance, power dissipation, and delay is investigated and applications of approximate high-radix dividers are treated in detail. The simulation results show that the proposed approximate dividers offer extensive saving in terms of power dissipation, circuit complexity, and delay, while only incurring in a small degradation in accuracy thus making them possibly suitable and interesting to some applications and domains such as low power/mobile computing.

Linbin Chen

Papers

Design and Evaluation of Multiple Valued Logic Gates Using Pseudo N-Type Carbon Nanotube FETs

On the Design of Approximate Restoring Dividers for Error-Tolerant Applications

Design of Approximate Unsigned Integer Non-restoring Divider for Inexact Computing

Design and Analysis of Inexact Floating-Point Adders

Design, Evaluation and Application of Approximate High-Radix Dividers