Machin-like formula

In 1706, William Jones published a contribution from John Machin of the first 100 decimal digits of the circle constant π = 3.14159..., at that time a record for accuracy, along with the unexplained and surprising formula used to calculate them,

This formula is an expanded form of the equation now called Machin's formula,

where the arctangents of ⁠${\displaystyle {\tfrac {1}{5}}}$⁠ and ⁠${\displaystyle {\tfrac {1}{239}}}$⁠ have been expanded using the arctangent series,

When applied directly to find ⁠${\displaystyle {\tfrac {1}{4}}\pi =\arctan 1}$⁠, the arctangent series converges extremely slowly, requiring five billion terms to obtain 10 correct decimal digits. Machin's formula is dramatically more practical, needing only six terms to obtain 10 correct digits.

Several other well-known mathematicians immediately set to work making sense of Machin's formula, developing their own variants and extensions, now in general called Machin-like formulas. These have the form

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 ${\displaystyle a_{n}<b_{n}}$.

The angle addition formula for arctangent asserts that

if $${\displaystyle -{\frac {\pi }{2}}<\arctan {\frac {a_{1}}{b_{1}}}+\arctan {\frac {a_{2}}{b_{2}}}<{\frac {\pi }{2}}.}$$ All of the Machin-like formulas can be derived by repeated application of equation 3. As an example, we show the derivation of Machin's original formula. One has: $${\displaystyle {\begin{aligned}2\arctan {\frac {1}{5}}&=\arctan {\frac {1}{5}}+\arctan {\frac {1}{5}}\\&=\arctan {\frac {1\cdot 5+1\cdot 5}{5\cdot 5-1\cdot 1}}\\&=\arctan {\frac {10}{24}}\\&=\arctan {\frac {5}{12}},\end{aligned}}}$$ and consequently $${\displaystyle {\begin{aligned}4\arctan {\frac {1}{5}}&=2\arctan {\frac {1}{5}}+2\arctan {\frac {1}{5}}\\&=\arctan {\frac {5}{12}}+\arctan {\frac {5}{12}}\\&=\arctan {\frac {5\cdot 12+5\cdot 12}{12\cdot 12-5\cdot 5}}\\&=\arctan {\frac {120}{119}}.\end{aligned}}}$$ Therefore also $${\displaystyle {\begin{aligned}4\arctan {\frac {1}{5}}-{\frac {\pi }{4}}&=4\arctan {\frac {1}{5}}-\arctan {\frac {1}{1}}\\&=4\arctan {\frac {1}{5}}+\arctan {\frac {-1}{1}}\\&=\arctan {\frac {120}{119}}+\arctan {\frac {-1}{1}}\\&=\arctan {\frac {120\cdot 1+(-1)\cdot 119}{119\cdot 1-120\cdot (-1)}}\\&=\arctan {\frac {1}{239}},\end{aligned}}}$$ and so finally $${\displaystyle {\frac {\pi }{4}}=4\arctan {\frac {1}{5}}-\arctan {\frac {1}{239}}.}$$