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

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.

arXiv comment30 pages

References in corpus (5)

• A formula for pi involving nested radicals
• Three essays on Machin's type formulas
• 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
• An iterative method for computing $\pi$ by argument reduction of the tangent function

Cited by in corpus (1)

• Application of a new iterative formula for computing $\pi$ and nested radicals with roots of $2$