A rational approximation of the two-term Machin-like formula for $\pi$

TL;DR

This paper introduces a rational approximation of the two-term Machin-like formula for pi, enabling efficient digit computation with quadratic convergence without trigonometric functions.

Contribution

It develops an algorithm using binary representation of 1/π for rational approximation, improving pi digit calculation methods.

Findings

Quadratic convergence in pi digit computation

Elimination of trigonometric functions and surds

Implementation examples in Mathematica

Abstract

In this work, we consider the properties of the two-term Machin-like formula and develop an algorithm for computing digits of $\pi$ by using its rational approximation. In this approximation, both terms are constructed by using a representation of $1/\pi$ in the binary form. This approach provides the squared convergence in computing digits of $\pi$ without any trigonometric functions and surd numbers. The Mathematica codes showing some examples are presented.