Unconditional applicability of Lehmer's measure to the two-term Machin-like formula for $\pi$

Abstract

Lehmer defined a measure $$ \mu=\sum\limits_{j=1}^J\frac{1}{\log_{10}\left(\left|\beta_j\right|\right)}, $$ where the $\beta_j$ may be either integers or rational numbers in a Machin-like formula for $\pi$. When the $\beta_j$ are integers, Lehmer's measure can be used to determine the computational efficiency of the given Machin-like formula for $\pi$. However, because the computations are complicated, it is unclear if Lehmer's measure applies when one or more of the $\beta_j$ are rational. In this article, we develop a new algorithm for a two-term Machin-like formula for $\pi$ as an example of the unconditional applicability of Lehmer's measure. This approach does not involve any irrational numbers and may allow calculating $\pi$ rapidly by the Newton$-$Raphson iteration method for the tangent function.