Machin's formula (M000001) is a formula for π that was discovered by John Machin in 1706: $$\pi=4\arctan\left(\frac15\right)-16\arctan\left(\frac1{239}\right)$$ There are a huge number of Machin-like formulae that have since been discovered, all of which are of the form: $$\pi=a_0\arctan(b_0)+a_1\arctan(b_1)+a_2\arctan(b_2)+\dots$$ where the \(a\)s and \(b\)s are rational numbers. This website contains 485 Machin-like formulae. You can:

• view the full list of Machin-like formulae
• view the full list of Machin-like formulae with integer \(b\)s
• view the list of named Machin-like formulae
• view the full list of Machin-like formulae sorted by Lehmer's measure
• view a random Machin-like formula
• download a CSV file containing all the formulae

Machin-like formulae

On the page for each Machin-like formula, you will find information about the formula, including:

• The formula in compact notation where a[b] represents \(a\arctan(1/b)\).
• Lehmer's measure.
• References where the formula was first found.

You can find more information about Machin-like formulae on the Machin-like formulae page.

Contributing

machin-like.org is generated from the source code hosted in the mscroggs/machin GitHub repository. You can contribute by opening issues or making pull requests with suggested changes. You can find more information on the contributing page

Acknowledgements

A large number of the Machin-like formulae included on this website first appeared on Michael Roby Wetherfield & Hwang Chien-lih's Machination site, which is now only available via the Wayback machine.