Fast and accurate approximation algorithms for computing floating point square root

Abstract

The square root is one of the most used functions in many different engineering and scientific applications. We propose new methods for calculating the square root function that are based on the Newton–Raphson method with Heron iteration. A modification of Heron’s formula combined with an improved selection of the magic constants enables a significant reduction of the maximum relative error (MRE). Simple modifications to the Newton–Raphson formula and the magic number method enable implementation on platforms with limited hardware resources, such as microcontrollers and FPGAs, with variable accuracy. Implementations of new approximation algorithms in the C programming language were carefully tested and evaluated against their software and hardware counterparts on the most popular platforms, e.g., CPUs from Intel, AMD and ARM, GPU from Nvidia and IPU from Graphcore. The proposed numerical algorithms are shown to be superior in terms of computational time, the number of clock cycles, accuracy, MRE, and root mean square deviation.