Faithful Polynomial Evaluation with Compensated Horner Algorithm

TL;DR

This paper introduces two conditions to guarantee faithful polynomial evaluation in IEEE-754 floating point arithmetic, using an improved algorithm and runtime validation, with promising experimental results.

Contribution

It provides the first sufficient conditions for faithful polynomial evaluation with the compensated Horner algorithm, including an a priori bound and a dynamic runtime check.

Findings

The a priori bound effectively predicts faithfulness.

The dynamic check accurately verifies faithfulness during computation.

Experimental results show the approach is computationally efficient.

Abstract

This paper presents two sufficient conditions to ensure a faithful evaluation of polynomial in IEEE-754 floating point arithmetic. Faithfulness means that the computed value is one of the two floating point neighbours of the exact result; it can be satisfied using a more accurate algorithm than the classic Horner scheme. One condition here provided is an apriori bound of the polynomial condition number derived from the error analysis of the compensated Horner algorithm. The second condition is both dynamic and validated to check at the running time the faithfulness of a given evaluation. Numerical experiments illustrate the behavior of these two conditions and that associated running time over-cost is really interesting.