A Rapidly Converging Machin-like Formula for $\pi$

TL;DR

This paper introduces a simple, rapidly converging recurrent formula for generating Machin-like formulas for calculating pi, allowing flexible and efficient computation with controllable precision.

Contribution

The paper presents a novel recurrent method for constructing Machin-like formulas for pi, enabling finite decompositions with arbitrarily small Lehmer's measure and practical implementation considerations.

Findings

The method produces rapidly converging Machin-like formulas.

The approach allows for partial formulas tailored to desired precision.

A Python program is provided for computing the formulas and Lehmer's measure.

Abstract

We present a simple recurrent formula to generate the Machin-like expression for calculating $\pi/4$. The method works for any denominator in the starting term and always provides a finite decomposition. We show that the terms in the Machin-like formula decrease so rapidly that the Lehmer's measure can be made arbitrarily small only by selecting the first term. We introduce the concept of the partial Machin-like formula. While the growth of the integer numbers may quickly render the computer implementation impractical, the same reason restricts the total contribution of the high terms. If the required precision is known in advance, the subset of the expression may be selected to satisfy it. We also present the Python program to compute the terms of the Machin-like formula (full and partial), and its Lehmer's measure.