Arctangent Formulas and Pi

A method to reduce the Lehmer measure in a multi-term Machin-like formula for $π$

“…However, the arguments in the last two arctangent function arguments are not reciprocals of integers and, consequently, practical application of equation ( 14) may be questionable. In order to eliminate these two quotients in arctangent function arguments we can use the following identity Substituting these two arctangent function relations into equation ( 14) results in equation (11). Consider now how our approach based on identity (13) can be used to eliminate quotients in (14) [17].…”

“…Furthermore, it seems that computer-aided algorithms based on Todd's process still cannot generate the Machin-like formulas for π containing only reciprocals of integers with µ below 1. For example, the following Machin-like formula for π containing only reciprocals of integers (see [16] (11) has the Lehmer measure µ ≈ 1.34085. To the best of our knowledge this is the smallest Lehmer measure ever reported for the Machin-like formulas for π containing only reciprocals of integers.…”

A Method to Reduce the Lehmer Measure in a Multi-Term Machin-Like Formula for π

“…Consider now how our approach based on identity (13) can be used to eliminate quotients in ( 14) [19]. Specifically, applying Equation (13)…”

“…will not be high and further we can apply identity (13). Thus, taking 14717113539487181 715391356779 z = − and substituting it into in identity (13)…”