A Fast Self-correcting $\pi$ Algorithm

TL;DR

This paper presents a rediscovered, fast, self-correcting algorithm for computing π with cubic convergence, improving efficiency and accuracy in high-precision calculations.

Contribution

It introduces a simple, practical algorithm for π that is self-correcting and potentially faster than known methods, with proven and conjectured complexity bounds.

Findings

Algorithm has complexity O(M(n) log^2 n) for error O(2^{-n})

It exhibits cubic convergence, enabling rapid accuracy improvements

Potentially runs in O(M(n) log n) based on conjecture

Abstract

We have rediscovered a simple algorithm to compute the mathematical constant \[ \pi=3.14159265\cdots. \] The algorithm had been known for a long time but it might not be recognized as a fast, practical algorithm. The time complexity of it can be proved to be \[ O(M(n)\log^2 n) \] bit operations for computing $\pi$ with error $O(2^{-n})$, where $M(n)$ is the time complexity to multiply two $n$-bit integers. We conjecture that the algorithm actually runs in \[ O(M(n)\log n). \] The algorithm is \emph{self-correcting} in the sense that, given an approximated value of $\pi$ as an input, it can compute a more accurate approximation of $\pi$ with cubic convergence.