A continued-fraction analysis of trigonometric argument reduction

The calculation of a trigonometric function of a large argument x is effectively carried out by finding the integer N and 0<α<1 such that x=(N+α)π/4. This reduction module π/4 makes it possible to calculate a trigonometric function of a reduced argument, either απ/4 or (1-α)π/4, which lies in the interval (0, π/4). Payne and Hanek (1983) described an efficient algorithm for computing or to a predetermined level of accuracy. They noted that if x differs only slightly from an integral multiple of π/2, the reduction must be carried out quite accurately to avoid a large loss of significance in the reduced argument. We present a simple method using continued fractions for determining, for all numbers r represented in an IEEE floating-point format, the specific x for which the greatest number of insignificant leading bits occur. Applications are made to IEEE single-precision and double-precision formats and two extended-precision formats