Three essays on Machin's type formulas

Abstract

We study three questions related to Machin's type formulas. The first one gives all two terms Machin formulas where both arctangent functions are evaluated $2$-integers, that is values of the form $b/2^a$ for some integers $a$ and~$b$. These formulas are computationally useful because multiplication or division by a power of two is a very fast operation for most computers. The second one presents a method for finding infinitely many formulas with $N$ terms. In the particular case $N=2$ the method is quite useful. It recovers most known formulas, gives some new ones, and allows to prove in an easy way that there are two terms Machin formulas with Lehmer measure as small as desired. Finally, we correct an oversight from previous result and give all Machin's type formulas with two terms involving arctangents of powers of the golden section.

arXiv comment22 pages, 3 figures

Cited by in corpus (4)

• A generalized series expansion of the arctangent function based on the enhanced midpoint integration
• An iterative method for computing $\pi$ by argument reduction of the tangent function
• 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 $2$