Innovative, simple and efficient algorithms for logarithms with adjustable accuracy

Abstract

The computation of logarithmic functions is essential for many computational applications in science and engineering. Various computational techniques have been developed to approximate these functions. In this article, we present a simple and efficient algorithm for calculating the logarithmic function using IEEE 754 standard arithmetic types that minimises the number of arithmetic operations compared to previous implementations. Our starting point is Mitchell’s linear approximation, which reduces the computation of the logarithm to just the addition of the exponent and the significand. We developed a parameterised framework for choosing the number of segments in a piecewise polynomial approximation scheme in combination with polynomial degree and bit manipulation techniques, which allows a tradeoff between the accuracy and efficiency. We measured a speedup by a factor of 3--20 with respect to the C/C++ library software implementation while keeping the absolute error at a low level. This article presents and discusses our proposed implementation, compares it to other logarithm realisations and tests these realisations on various platforms.