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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-likeformula for $\pi$. When the $\beta_j$ are integers, Lehmer's measure can beused to determine the computational efficiency of the given Machin-like formulafor $\pi$. However, because the computations are complicated, it is unclear ifLehmer's measure applies when one or more of the $\beta_j$ are rational. Inthis 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 tangentfunction.

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