Low precision logarithmic number systems: Beyond base-2

TL;DR

This paper explores the impact of choosing different bases in low-precision logarithmic number systems (LNS), demonstrating how base selection affects error, implementation efficiency, and power consumption in hardware applications.

Contribution

It introduces methods for optimizing base choice in low-precision LNS to reduce errors and hardware complexity, extending beyond the traditional base-2 system.

Findings

Choosing a suitable base reduces average error in addition and subtraction.

Logic-based implementation of LNS operations is more efficient than lookup tables.

Optimized base selection can lower area and power consumption in hardware circuits.

Abstract

Logarithmic number systems (LNS) are used to represent real numbers in many applications using a constant base raised to a fixed-point exponent making its distribution exponential. This greatly simplifies hardware multiply, divide and square root. LNS with base-2 is most common, but in this paper we show that for low-precision LNS the choice of base has a significant impact. We make four main contributions. First, LNS is not closed under addition and subtraction, so the result is approximate. We show that choosing a suitable base can manipulate the distribution to reduce the average error. Second, we show that low-precision LNS addition and subtraction can be implemented efficiently in logic rather than commonly used ROM lookup tables, the complexity of which can be reduced by an appropriate choice of base. A similar effect is shown where the result of arithmetic has greater precision than the input. Third, where input data from external sources is not expected to be in LNS, we can reduce the conversion error by selecting a LNS base to match the expected distribution of the input. Thus, there is no one base which gives the global optimum, and base selection is a trade-off between different factors. Fourth, we show that circuits realized in LNS require lower area and power consumption for short word lengths.