An Exchange Algorithm for Optimizing Both Approximation and Finite-Precision Evaluation Errors in Polynomial Approximations

The finite precision implementation of mathematical functions frequently depends on polynomial approximations. A key characteristic of this approach is that rounding errors occur both when representing the coefficients of the polynomial on a finite number of bits, and when evaluating it in finite precision arithmetic. Hence, to find a best polynomial, for a given fixed degree, norm, and interval, it is necessary to account for both the approximation error and the floating-point evaluation error. While efficient algorithms were already developed for taking into account the approximation error, the evaluation part is usually a posteriori handled, in an ad hoc manner. Here, we formulate a semi-infinite linear optimization problem whose solution is a best polynomial with respect to the supremum norm of the sum of both errors. This problem is then solved with an iterative exchange algorithm, which can be seen as an extension of the well-known Remez exchange algorithm. An open source C implementation using the Sollya library is presented and tested on several examples, which are then analyzed and compared against state-of-the-art Sollya routines.