Optimizing polynomials for floating-point implementation

TL;DR

This paper introduces a numerical algorithm that optimizes polynomial approximations for floating-point functions by zeroing small coefficients, reducing complexity and avoiding catastrophic cancellations.

Contribution

It presents a modified Remez algorithm targeting incomplete monomial bases, enabling more efficient polynomial approximations with fewer coefficients.

Findings

Polynomials with fewer coefficients are achieved compared to traditional methods.

The technique reduces the risk of catastrophic cancellations in floating-point evaluations.

The algorithm is purely numerical and applicable as a black-box to various functions.

Abstract

The floating-point implementation of a function on an interval often reduces to polynomial approximation, the polynomial being typically provided by Remez algorithm. However, the floating-point evaluation of a Remez polynomial sometimes leads to catastrophic cancellations. This happens when some of the polynomial coefficients are very small in magnitude with respects to others. In this case, it is better to force these coefficients to zero, which also reduces the operation count. This technique, classically used for odd or even functions, may be generalized to a much larger class of functions. An algorithm is presented that forces to zero the smaller coefficients of the initial polynomial thanks to a modified Remez algorithm targeting an incomplete monomial basis. One advantage of this technique is that it is purely numerical, the function being used as a numerical black box. This algorithm is implemented within a larger polynomial implementation tool that is demonstrated on a range of examples, resulting in polynomials with less coefficients than those obtained the usual way.