Computation of Sin N, Cos N and m √N using an electronic computer

TLDR: Rational Pade approximations to Sin N and Cos N allow the computation of both functions in 0≤N≤π/2 with the first ten correct significant digits in four multiplications and divisions only.

Abstract: Rational Pade approximations to Sin N in the interval 0≤N≤41π/256 and to Cos N in 0≤N≤ 87π/256 allow the computation of both functions in 0≤N≤π/2 with the first ten correct significant digits in four multiplications and divisions only. If the infinite range 0≤N<∞ is considered, one more multiplication reduces it to the range 0≤N≤π/2 so that the total number of operations is five. The method is flexible and gives any desired accuracy. Thus if eighteen first correct significant digits are required, they are obtained in seven operations for any N in (0, ∞). The same method applied to √N and 3√N yields a very accurate first guess which then is improved by Newton's method. For the radicals m√N with m>4, Newton's method is too slow, and rational Pade approximations studied in this paper yield better subroutines.