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

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.

35 pages

References in corpus (2)

• A generalized series expansion of the arctangent function based on the enhanced midpoint integration
• Algorithmic determination of a large integer in the two-term Machin-like formula for pi

Cited by in corpus (2)

• A rational approximation of the two-term Machin-like formula for $\pi$
• Application of a new iterative formula for computing $\pi$ and nested radicals with roots of $2$