Fast calculation of inverse square root with the use of magic constant $-$ analytical approach

TL;DR

This paper provides a mathematical analysis of the inverse square root calculation method, deriving optimal magic constants to minimize errors in a fast, single-precision floating-point algorithm.

Contribution

It introduces a systematic approach to determine optimal magic constants for inverse square root calculations, improving accuracy.

Findings

Derived optimal magic constants for minimal errors.

Provided a systematic analytical framework.

Enhanced understanding of the algorithm's precision.

Abstract

We present a mathematical analysis of transformations used in fast calculation of inverse square root for single-precision floating-point numbers. Optimal values of the so called magic constants are derived in a systematic way, minimizing either absolute or relative errors at subsequent stages of the discussed algorithm.