An iterative method for computing $\pi$ by argument reduction of the tangent function

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

doi:10.3390/mca29020017

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arXiv · 2312.05413

An iterative method for computing $\pi$ by argument reduction of the tangent function

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

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Abstract

In this work, we develop a new iterative method for computing the digits of $\pi$ by argument reduction of the tangent function. This method combines a modified version of the iterative formula for $\pi$ with squared convergence that we proposed in a previous work and a leading arctangent term from the Machin-like formula. The computational test we performed shows that algorithmic implementation can provide more than $17$ digits of $\pi$ per increment. Mathematica codes, showing the convergence rate for computing the digits of $\pi$, are presented.

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Sanjar M. Abrarov, Rehan Siddiqui, Rajinder Kumar Jagpal, Brendan M. Quine. 2023-12-08. An iterative method for computing $\pi$ by argument reduction of the tangent function. https://doi.org/10.3390/mca29020017

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