Fast, Sound and Precise Narrowing of the Exponential Function

In this paper we present an algorithm for narrowing the constraint y = e x . The algorithm has been designed to be fast by using only IEEE multiplication. The main difficulty is to design algorithms which soundly, rapidly, and precisely compute upper and lower bounds on e x and ln(y). We prove that our algorithms are correct and produce upper and lower bounds which differ by at most 2 ULP. The method we describe is a modification of the standard range reduction algorithm found in the literature, but is considerably more complex due to the necessity of guaranteeing that the algorithm computes tight upper and lower bounds for all inputs x. The algorithm assumes that the arithmetic operations (+,-,*,/) are implemented in hardware and that rounding control can be specified by the user (e.g. the IEEE 754 specifications of single, double, and extended precision arithmetic all satisfy these assumptions). The bounds for exp and log are computed using the native floating point arithmetic ..