Computation of Arctan N for −∞ < N < +∞

Abstract

Rational (R) and polynomial (P) approximations to Arctan N are studied with the aim of computing this function, to any prescribed accuracy and without unduly increasing the number PC of stored constants, in a minimum number M of multiplications (and divisions for R approximations). The number Dg of first correct significant digits in principle is not bounded. The results corresponding to the values 8, 10, 18 and 20 of this number are as follows; (Table of results is given for M ranging in value from 4 to 10.) If M is increased, subroutines with smaller PC are easily deduced from our general results. Thus, for instance, rational approximations with Dg = 6 can be obtained in three multiplications only, if PC = 19 (combination m* = 3, q = 10); but the same accuracy Dg = 6 characterizes also the cases M = 4 with PC = 11 and M = 5 with PC = 7 (combinations m* = 4, q = 6 and m = 5, q = 4). If polynomial approximations are used, Dg = 6 is obtained for M = 5, PC = 7, but also for M = 4 and PC = 11. No subroutines with a stored table of values of Arctan x are considered.