How to Ensure a Faithful Polynomial Evaluation with the Compensated Horner Algorithm

The compensated Horner algorithm improves the accuracy of polynomial evaluation in IEEE-754 floating point arithmetic: the computed result is as accurate as if it was computed with the classic Horner algorithm in twice the working precision. Since the condition number still governs the accuracy of this computation, it may return an arbitrary number of inexact digits. We address here how to compute a faithfully rounded result, that is one of the two floating point neighbors of the exact evaluation. We propose an a priori sufficient condition on the condition number to ensure that the compensated evaluation is faithfully rounded. We also propose a validated and dynamic method to test at the running time if the compensated result is actually faithfully rounded. Numerical experiments illustrate the behavior of these two conditions and that the associated running time over-cost is really interesting.