Machin-like formulas

In mathematics, Machin-like formulae are a popular technique for computing π to a large number of digits. They are generalizations of John Machin's formula from 1706:

which he used to compute π to 100 decimal places.

Machin-like formulas have the form

${\displaystyle c_{0}{\frac {\pi }{4}}=\sum _{n=1}^{N}c_{n}\arctan {\frac {a_{n}}{b_{n}}}}$

 

 

 

 

()

where ${\displaystyle a_{n}}$ and ${\displaystyle b_{n}}$ are positive integers such that ${\displaystyle a_{n}<b_{n}}$, ${\displaystyle c_{n}}$ is a signed non-zero integer, and ${\displaystyle c_{0}}$ is a positive integer.

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