Certified Roundoff Error Bounds using Bernstein Expansions and Sparse Krivine-Stengle Representations

TL;DR

This paper introduces two novel algorithms using Bernstein expansions and sparse Krivine-Stengle representations to compute rigorous upper bounds of roundoff errors in polynomial and rational programs, enhancing validation of critical embedded software.

Contribution

The paper presents new algorithms based on Bernstein and Krivine-Stengle methods for tight error bounds, along with convergence analysis and software implementations.

Findings

Algorithms achieve competitive performance with existing tools.

Methods provide accurate and rigorous upper bounds.

Software packages FPBern and FPKriSten are released for practical use.

Abstract

Floating point error is a drawback of embedded systems implementation that is difficult to avoid. Computing rigorous upper bounds of roundoff errors is absolutely necessary for the validation of critical software. This problem of computing rigorous upper bounds is even more challenging when addressing non-linear programs. In this paper, we propose and compare two new algorithms based on Bernstein expansions and sparse Krivine-Stengle representations, adapted from the field of the global optimization, to compute upper bounds of roundoff errors for programs implementing polynomial and rational functions. We also provide the convergence rate of these two algorithms. We release two related software package FPBern and FPKriSten, and compare them with the state-of-the-art tools. We show that these two methods achieve competitive performance, while providing accurate upper bounds by comparison with the other tools.