Machin-like formula

In mathematics, Machin-like formulas are a popular technique for computing π (the ratio of the circumference to the diameter of a circle) to a large number of digits. They are generalizations of John Machin's formula from 1706:

${\displaystyle {\frac {\pi }{4}}=4\arctan {\frac {1}{5}}-\arctan {\frac {1}{239}}}$

which he used to compute π to 100 decimal places.^[1][2]

Machin-like formulas have the form

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

(1)

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

These formulas are used in conjunction with Gregory's series, the Taylor series expansion for arctangent:

${\displaystyle \arctan x=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{2n+1}}x^{2n+1}=x-{\frac {x^{3}}{3}}+{\frac {x^{5}}{5}}-{\frac {x^{7}}{7}}+\cdots }$

(2)