Effective Quadratic Error Bounds for Floating-Point Algorithms Computing the Hypotenuse Function

TL;DR

This paper introduces a method to derive tight, explicit quadratic error bounds for floating-point algorithms computing the hypotenuse, improving error analysis especially in low-precision contexts.

Contribution

It develops a novel approach using computer algebra to obtain explicit quadratic error bounds for floating-point hypotenuse algorithms, capturing error correlations.

Findings

Derived explicit quadratic error bounds for hypotenuse algorithms

Compared five algorithms with varying complexity

Bounds are sharp and particularly relevant for low-precision formats

Abstract

We provide tools to help automate the error analysis of algorithms that evaluate simple functions over the floating-point numbers. The aim is to obtain tight relative error bounds for these algorithms, expressed as a function of the unit round-off. Due to the discrete nature of the set of floating-point numbers, the largest errors are often intrinsically "arithmetic" in the sense that their appearance may depend on specific bit patterns in the binary representations of intermediate variables, which may be present only for some precisions. We focus on generic (i.e., parameterized by the precision) and analytic over-estimations that still capture the correlations between the errors made at each step of the algorithms. Using methods from computer algebra, which we adapt to the particular structure of the polynomial systems that encode the errors, we obtain bounds with a linear term in the unit round-off that is sharp in manycases. An explicit quadratic bound is given, rather than the $O()$-estimate that is more common in this area. This is particularly important when using low precision formats, which are increasingly common in modern processors. Using this approach, we compare five algorithms for computing the hypotenuse function, ranging from elementary to quite challenging.