Exact Lookup Tables for the Evaluation of Trigonometric and Hyperbolic Functions

Abstract

Elementary mathematical functions are pervasively used in many applications such as electronic calculators, computer simulations, or critical embedded systems. Their evaluation is always an approximation, which usually makes use of mathematical properties, precomputed tabulated values, and polynomial approximations. Each step generally combines error of approximation and error of evaluation on finite-precision arithmetic. When they are used, tabulated values generally embed rounding error inherent to the transcendence of elementary functions. In this article, we propose a general method to use error-free values that is worthy when two or more terms have to be tabulated in each table row. For the trigonometric and hyperbolic functions, we show that Pythagorean triples can lead to such tables in little time and memory usage. When targeting correct rounding in double precision for the same functions, we also show that this method saves memory and floating-point operations by up to 29 and 42 percent, respectively.