# 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.
Download (© Springer-Verlag, 2006):
pdf.

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