Probabilistic Error Analysis of Limited-Precision Stochastic Rounding: Horner's Algorithm and Pairwise Summation

TL;DR

This paper analyzes the accuracy and hardware trade-offs of limited-precision stochastic rounding in numerical algorithms like Horner's method and pairwise summation, combining theoretical and experimental insights.

Contribution

It extends previous work by providing a detailed probabilistic error analysis of limited-precision stochastic rounding applied to specific algorithms.

Findings

Limited-precision SR can effectively reduce errors in Horner's algorithm.

Trade-offs between accuracy and hardware resources are characterized.

Experimental results validate theoretical error bounds.

Abstract

Stochastic rounding (SR) is a probabilistic rounding mode that mitigates errors in large-scale numerical computations, especially when prone to stagnation effects. Beyond numerical analysis, SR has shown significant benefits in practical applications such as deep learning and climate modelling. The definition of classical SR requires that results of arithmetic operations are known with infinite precision. This is often not possible, and when it is, the resulting hardware implementation can become prohibitively expensive in terms of energy, area, and latency. A more practical alternative is limited-precision SR, which only requires that the outputs of arithmetic operations are available in higher, finite, precision. We extend previous work on limited-precision SR presented in [El Arar et al., SIAM J. Sci. Comput. 47(5) (2025), B1227-B1249], which developed a framework to evaluate the trade-off between accuracy and hardware resource cost in SR implementations. Within this framework, we study the Horner algorithm and pairwise summation, providing both theoretical insights and practical experiments in these settings when using limited-precision SR.