A Certified Infinite Norm for the Implementation of Elementary Functions

Abstract

The high-quality floating-point implementation of useful functions f : R → R, such as exp, sin, erf requires bounding the error ε = p−f f of an approximation p with regard to the function f . This involves bounding the infinite norm ε ∞ of the error function. Its value must not be underestimated when implementations must be safe. Previous approaches for computing infinite norm are shown to be either unsafe, not sufficiently tight or too tedious in manual work. We present a safe and self-validating algorithm for automatically upper-and lower-bounding infinite norms of error functions. The algorithm is based on enhanced interval arithmetic. It can overcome high cancellation and high condition number around points where the error function is defined only by continuous extension. The given algorithm is implemented in a software tool. It can generate a proof of correctness for each instance on which it is run. Keywords: Certified infinite norm, interval arithmetic, elementary functionsRésumé Pour garantir la qualité de l'implémentation en arithmétique flottante de fonctions usuelles f : R → R telles que exp, sin, erf, il faut borner l'erreur ε = p−f f commise entre f et une approximation p. Cela implique de borner la norme infinie ε ∞ de la fonction d'erreur. Si on veut que l'implémentation soit sure, on ne doit en aucun cas renvoyer une borne inférieureà la valeur exacte. Nous montrons que les approches précédentes visantà calculer la norme infinie ne sont pas satisfaisantes : soit elles ne sont pas sures, soit pas assez précises, soit elles nécessitent un travail manuel trop fastidieux. Nous présentons un algorithme sûr, qui fournit une preuve de sa propre correction, et qui minore et majore automatiquement la norme infinie de fonctions d'erreur. Cet algorithme est fondé sur une version améliorée d'arithmétique d'intervalle. Il peut contourner les difficultés duesà une grande cancellation et un mauvais conditionnement autour de points où la fonction d'erreur n'est définie que par continuité. L'algorithme proposé aété implémenté dans un outil logiciel. Il peut générer une preuve de correction pour toute instance sur laquelle il est exécuté.

Efficient and accurate computation of upper bounds of approximation errors

“…In fact, in [15] authors of this article tried to use a similar recursive technique, but they observed that the derivative of the error, the second derivative of the error and so on and so forth are all prone to the same phenomenon of very high overestimation. That is why, for higher degree of p (higher than 10) the number of splittings needed to eliminate the correlation problem is still unfeasible.…”

Certifying the Floating-Point Implementation of an Elementary Function Using Gappa

“…In addition, the polynomial is an approximation to the sine function, with a relative error bound of ǫ approx which is supposed known (how it was obtained it is out of the scope of this paper [18]).…”

Sollya: An Environment for the Development of Numerical Codes

“…This rigor is achieved through an extensive use of Interval Arithmetic [9], extended to cover functions with false singularities [4]. Interval Arithmetic is of course available at the Sollya interface, too.…”

“…-Supremum norms of approximation error functions: Sollya can compute safe bounds on the supremum norm of the error ε = p/f − 1 made when replacing a function f by a particular polynomial p in a bounded domain I. Such computations are a requirement for code certification for mathematical functions [4]. -Support for code generation in IEEE 754 arithmetic: Sollya offers extensive support for simulating IEEE 754 arithmetic.…”