Fast approximate reciprocal approximations for iterative algorithms

TL;DR

This paper introduces a low-complexity, non-iterative approximation method for the reciprocal function, enabling fast computation suitable for real-time iterative algorithms with minimal performance loss.

Contribution

It proposes a combinatorial logic-based reciprocal approximation that is low in area and high in speed, with adjustable error characteristics and demonstrated effectiveness in practical applications.

Findings

Implementation with low area and high clock frequency

Error can be optimized by adjusting a constant parameter

Negligible performance degradation in application scenarios

Abstract

The reciprocal function, 1/x, is important for many real-time algorithms. It is used in a large variety of algorithms from areas ranging from iterative estimation to machine learning. Many of these algorithms are iterative in nature and require the online computation of the reciprocal. Such an iterative structure often prevents effective use of pipelining for implementation of the reciprocal. For this reason, a reciprocal algorithm requiring only a low amount of clock cycles is desired. Many real-time algorithms, often being of approximate nature, can tolerate the use of only an approximate solution of the reciprocal. For this reason, we present a low complexity non-iterative approximation of the reciprocal function. This approximation can be calculated using only combinatorial logic. We present synthesis results showing that the proposed approach can be implemented with low area requirements at high clock frequencies. We analytically describe the error of the approximation and show that by optimizing a constant value used in the approximation, different variants with different error behaviors can be obtained. We furthermore present performance results of application examples that, when using our proposed method, show only negligible performance degradation compared to when using the exact reciprocal function, demonstrating the versatility of our proposed approach.