Worst Cases of a Periodic Function for Large Arguments

Guillaume Hanrot, Vincent Lefèvre, Damien Stehlé and Paul Zimmermann

Abstract: One considers the problem of finding hard to round cases of a periodic function for large floating-point inputs, more precisely when the function cannot be efficiently approximated by a polynomial. This is one of the last few issues that prevents from guaranteeing an efficient computation of correctly rounded transcendentals for the whole IEEE-754 double precision format. The first non-naive algorithm for that problem is presented, with an heuristic complexity of O(2^{0.676 p}) for a precision of p bits. The efficiency of the algorithm is shown on the largest IEEE-754 double precision binade for the sine function, and some corresponding bad cases are given. We can hope that all the worst cases of the trigonometric functions in their whole domain will be found within a few years, a task that was considered out of reach until now.