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

Sanjar M. Abrarov; Rehan Siddiqui; Rajinder Kumar Jagpal; Brendan M. Quine

doi:10.3390/appliedmath4030047

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arXiv · 2406.08510

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

Sanjar M. Abrarov·Rehan Siddiqui·Rajinder Kumar Jagpal·Brendan M. Quine

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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.

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Sanjar M. Abrarov, Rehan Siddiqui, Rajinder Kumar Jagpal, Brendan M. Quine. 2024-06-07. A rational approximation of the two-term Machin-like formula for $\pi$. https://doi.org/10.3390/appliedmath4030047

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