A Certified Infinite Norm for the Implementation of Elementary Functions

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