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

Published May 24, 2021

Lehmer defined a measure

where the may be either integers or rational numbers in a Machin-like formula for . When the are integers, Lehmer’s measure can be used to determine the computational efficiency of the given Machin-like formula for . However, because the computations are complicated, it is unclear if Lehmer’s measure applies when one or more of the are rational. In this article, we develop a new algorithm for a two-term Machin-like formula for as an example of the unconditional applicability of Lehmer’s measure. This approach does not involve any irrational numbers and may allow calculating rapidly by the Newton–Raphson iteration method for the tangent function. Read More »