A certified infinite norm for the implementation of elementary functions

Sylvain Chevillard (1) , Christoph Lauter (1)

1 LIP - Laboratoire de l'Informatique du Parallélisme

The high-quality floating-point implementation of useful functions f : R -> R, such as exp, sin, erf requires bounding the error eps = (p-f)/f of an approximation p with regard to the function f. This involves bounding the infinite norm ||eps|| 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.

Mots clés: Infinite norm; Optimization; Interval arithmetic; Certified algorithm; Error analysis; Approximation error; Elementary functions

Domaines: Arithmétique des ordinateurs

Sylvain Chevillard, Christoph Lauter. A certified infinite norm for the implementation of elementary functions. Seventh International Conference on Quality Software, Oct 2007, Portland, United States. pp.153 -- 160. ⟨ensl-00119810v2⟩