[Paper Review] Error Analysis of Approximate Array Multipliers

Mahmoud Masadeh, Osman Hasan+1 more|arXiv (Cornell University)|Aug 4, 2019

Low-power high-performance VLSI design7 references4 citations

TL;DR

This paper presents a comprehensive error analysis of 20 distinct 8-bit approximate array multiplier designs, evaluating their accuracy across all possible input combinations. It reveals a strong input-dependent error behavior, where individual input pairs can cause significantly higher errors (e.g., PSNR < 25 dB) despite acceptable average performance, highlighting the need for input-aware quality control in approximate computing systems.

ABSTRACT

Approximate computing is a nascent energy-efficient computing paradigm suitable for error-tolerant applications. However, the value of approximation error depends on the applied inputs where individual output error may reach intolerable level while the average output error is acceptable. Thus, it is critical to show the response of approximate design for various applied inputs where understanding the interdependence between the inputs and the approximation error could be utilized to control the output quality. In this report, we exhaustively analyze the accuracy of 20 different designs of 8-bit approximate array multipliers. We designed these multipliers based on various configurations including the type of approximable component and how much of the result to approximate, i.e., approximation degree. Accuracy analysis shows a strong correlation between the applied inputs and the magnitude of the observed error, i.e., error distance, normalized error distance and peak-to-signal-noise ratio. We may utilize this input-dependency of approximation error, in controlling the quality of approximate computing by eliminating large magnitude errors and improving the quality of the final results.

Motivation & Objective

• To investigate the relationship between input values and approximation error in approximate multipliers, as average error metrics can mask severe individual errors.
• To identify specific input combinations that lead to unacceptably high error levels, even when average output quality is deemed acceptable.
• To evaluate the impact of different approximation configurations (type of approximate full adder and approximation degree) on error distribution across input space.
• To provide a foundation for input-aware quality control in approximate computing by quantifying the dependency of error metrics on input data.
• To support the design of high-quality approximate computing systems by identifying and eliminating input-dependent error outliers.

Proposed method

• Constructed a library of 20 distinct 8-bit approximate array multiplier designs using five types of approximate full adders (AMA1–AMA5) and four approximation degrees (D1–D4), varying the number of approximated output bits.
• Performed exhaustive simulation over all 256 input combinations (0 to 255) for each design to capture the full input space behavior.
• Applied three key accuracy metrics: Error Distance (ED), Normalized Error Distance (NED), and Peak Signal-to-Noise Ratio (PSNR) to quantify error magnitude and quality.
• Used PSNR as a quality-of-service metric, with a threshold of 25 dB to define 'acceptable' output quality for image processing applications.
• Analyzed the distribution of error metrics across input clusters to identify input-dependent error outliers.
• Correlated error behavior with design parameters (add type and approximation degree) to understand the root causes of high error spikes.

Experimental results

Research questions

• RQ1 How does the approximation error of 8-bit array multipliers vary across the full range of input values?
• RQ2 Which specific input combinations lead to unacceptably high error levels (e.g., PSNR < 25 dB) despite acceptable average error?
• RQ3 What is the correlation between the choice of approximate full adder (e.g., AMA1–AMA5) and the resulting error distribution?
• RQ4 How does the approximation degree (D1–D4) influence the likelihood and magnitude of high-error input pairs?
• RQ5 Can input-dependent error behavior be leveraged to improve the quality of results in approximate computing systems?

Key findings

• For D4-based designs (16-bit approximation), all input combinations produced PSNR < 25 dB, indicating universally poor output quality, with average PSNR as low as 8.8 dB (AMA5-based design).
• The AMA1-based D1 design had 19 out of 256 input combinations with PSNR < 25 dB, indicating that even low-approximation designs can produce high-error outliers.
• The AMA4-based D1 design showed the best robustness, with only 6 out of 256 input combinations having PSNR < 25 dB, and an average PSNR of 45.6 dB.
• The AMA5-based D1 design achieved the highest average PSNR (46.8 dB) and only 5 input combinations had PSNR < 25 dB, indicating superior input resilience.
• For designs with D3 (9-bit approximation), over 100 input clusters (up to 151) had PSNR < 25 dB, showing that moderate approximation still introduces significant quality variation.
• A strong correlation was observed between input values and error magnitude, with some inputs causing PSNR to drop below 10 dB, even in otherwise high-quality designs.