A Simple Test Qualifying the Accuracy of Horner'S Rule for Polynomials

Abstract

Polynomials are used in many applications and hidden in libraries such as libm. Whereas the accuracy of the functions used by linear algebra have long been studied, little is available to decide on one scheme to evaluate a polynomial. Common knowledge solely emphasizes that Horner's rule is a good scheme unless the indeterminate is close to one of the polynomial's roots. We propose here a criterion for one step of Horner's scheme to be faithful. A result is defined to be faithful when it was correctly rounded whereas the rounding mode (up, down or to the nearest) cannot be known by the user. Our criterion is checked against the IEEE standard for floating point arithmetic using the Coq automatic proof checker. We then present three programs in Maple, Java and C that check the criterion for a whole polynomial associated with a domain for the indeterminate and a possible truncation error. An example of use is given with the approximation of elementary functions. Key-words:

Provably faithful evaluation of polynomials

“…It was proved in [4] that this computation is faithful over the interval [−1/16; 0]. Here, we can prove that this computation is faithful over the larger interval [−1/4; 0].…”

“…The basic ideas of this application were originally developed in Coq [4]. Due to the proof automation features provided by PVS, the results presented here are significantly better than the original ones.…”

“…Figures 3 and 2 illustrate both situations. The darker strip corresponds to the set of reals r such that |r − f | < ulp(f ), the lighter strip corresponds to the set of real r such that f is a faithful rounding of r. Figure 3: Ulp and faithful rounding when f is a positive power of the radix Therefore, we have two different criteria to guarantee that a float is a faithful rounding of a real value (see [4] for more details). If 0 < f and |f − r| < ulp(f − ) then f is a faithful rounding of r (Faithful1).…”

“…Our work is based on a generic floating-point arithmetic library 1 designed by Daumas, Rideau, and Théry [8], and implemented in Coq. The application of a floating-point formalization to polynomial evaluation was already studied in [4] using the Coqbased floating-point library. However, the results obtained in our PVS formalization are significantly better than the original ones.…”

Formal Verification of Floating-Point Programs

“…A small Taylor polynomial is evaluated using Horner's rule, namely 1 + x(1 + x/2). In this case, the computation will be faithful [4,5] so the rounding and total error can be bounded. More precisely, the source code can be annotated as follows:…”