Fast medium precision computation of elementary functions

Fast medium precision computation of elementary functions

Low precision methods for the elementary functions ln, exp, cos, sin, etc. are well established in modern computer hardware. Beyond that, ultrahigh precision methods for ln(x) etc. are nowadays based on the AGM-iteration, see e.g. the comprehensive monograph "Pi and the AGM" by Borwein & Borwein. This talk presents a new idea for medium precision evaluation of those functions, within the range of 60-3000 bits, say. Standard domain reductions like x'= x - n.ln 2 for exp, x'= x.2^n in [1,2) for ln, and x'= x - n.\pi/2 for cos + i.sin plus Taylor series approximations are combined with further reductions of similar type by diophantine combinations of incommensurable logarithms, like z = x' -(k.ln 3 - m.ln 2) for exp, later on multiplying with 3^k. Analogous methods apply in the case of the trigonometric functions, where the role of such extra base 3 is then taken by Gaussian primes 2+i, 2-i, --- plus 3+2i, 3-2i for an improved design of that kind.