Precise error estimate of the Brent-McMillan algorithm for the computation of Euler's constant

TL;DR

This paper provides a precise, explicit error estimate for the Brent-McMillan algorithm used to compute Euler's constant, improving understanding of its accuracy and confirming conjectures about its error behavior.

Contribution

The paper derives an explicit expression for the optimal error estimate of the Brent-McMillan algorithm, with a complete proof and improved bounds.

Findings

Explicit error formula for the algorithm's minimal term

Proof confirming the conjectured error magnitude

Enhanced error bounds for the computation of Euler's constant

Abstract

Brent and McMillan introduced in 1980 a new algorithm for the computation of Euler's constant $\gamma$, based on the use of the Bessel functions I\_0(x) and K\_0(x). It is the fastest known algorithm for the computation of $\gamma$. The time complexity can still be improved by evaluating a certain divergent asymptotic expansion up to its minimal term. Brent-McMillan conjectured in 1980 that the error is of the same magnitude as the last computed term, and Brent-Johansson partially proved it in 2015. They also gave some numerical evidence for a more precise estimate of the error term. We find here an explicit expression of that optimal estimate, along with a complete self-contained formal proof and an even more precise error bound.