Unconditional Applicability of Lehmer’s Measure to the Two-Term Machin-like Formula for π

Lehmer defined a measure depending on numbers beta_i used in a Machin-like formula for pi. When the beta_i 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_i 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.