IEEE754 Binary32 Floating-Point Logarithmic Algorithms based on Taylor-Series Expansion with Mantissa Region Conversion and Division

Abstract

In this paper, an algorithm based on Taylor series expansion is proposed to calculate the logarithm (log₂x of IEEE754 binary32 accuracy floating-point number by a multi-domain partitioning method. The general mantissa (1≤x<2) is multiplied by 2, 4, 8, … (or equivalently left-shifted by 1, 2, 3, … bits , the regions of (2≤x<4), (4≤x<8) , (8≤x<16),… are considered, and Taylor-series expansion is applied. In those regions, the slope of f(x)=log₂x with respect to x is gentle compared to the region of (1≤x<2), which reduces the required number of terms. We also consider the trade-offs among the numbers of additions, subtractions, and multiplications and Look-Up Table (LUT size in hardware to select the best algorithm for the engineer's design and build the best hardware device