A square root algorithm faster than Newton's method for multiprecision numbers, using floating-point arithmetic

TL;DR

This paper introduces an optimized square root algorithm that outperforms Newton's method for multiprecision numbers by leveraging floating-point arithmetic for initial guesses, demonstrated through Java implementation and benchmarking.

Contribution

The paper presents a novel, faster square root algorithm for multiprecision numbers using floating-point initial guesses, improving upon classical methods and standard library implementations.

Findings

The new algorithm is significantly faster than Newton's method in tests.

Implementation in Java confirms practical efficiency gains.

Benchmarking shows improved performance over standard library algorithms.

Abstract

In this paper, an optimized version of classical Bombelli's algorithm for computing integer square roots is presented. In particular, floating-point arithmetic is used to compute the initial guess of each digit of the root, following similar ideas to those used in "The Art of Computer Programming" Vol. 2, p. 4.3.1 for division. A program with an implementation of the algorithm in Java is also presented, and its running time is compared with that of the algorithm provided by the Java standard library, which uses the Newton's method. From tests, the algorithm presented here turns out to be much faster.