Always Chebyshev Interpolation In Elementary Function Computations

A common practice for computing an elementary transcendental function nowadays has two phases: reductions of input arguments to fall into a tiny interval and polynomial approximations for the function within the interval. Typically the interval is made tiny enough so that one won’t have to go for polynomials of very high degrees for accurate approximations. Often approximating polynomials as such are taken to be the best polynomials or any others such as the Chebyshev interpolating polynomials. The best polynomial of degree n has the property that the biggest difference between it and the function is smallest among all possible polynomials of degrees no higher than n. Thus it is natural to choose the best polynomials over others. In this paper, it is proved that the best polynomial can only be more accurate by at most a fractional bit than the Chebyshev interpolating polynomial of the same degree in computing elementary functions, or in the other word the Chebyshev interpolating polynomials will do just as good as the best polynomials (which are harder to compute). Similar results were obtained in 1967 by M. J. D. Powell who, however, did not target at elementary function computations in particular and placed no assumption on the function and remarkably whose results imply accuracy differences of no more than 2 to 3 bits in the context of this paper.