Machin's formula (M000000001) is a formula for π that was discovered by John Machin in 1706: $$\pi=16\arctan\left(\frac15\right)-4\arctan\left(\frac1{239}\right)$$ 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.
In many sources, Machin-like formulae are written as formulae for computing \(\pi/4\), and so Machin's formula would be written as $$\frac\pi4=4\arctan\left(\frac15\right)-\arctan\left(\frac1{239}\right).$$ On this website, we follow the convention of the formulae always being equal to \(\pi\).
On the page for each Machin-like formula, you will find the formula in compact notation. In this notation, a[b] represents \(a\arctan(1/b)\). For example, the compact formula 16[5] - 4[239] represents Machin's formula, ie $$\pi=16\arctan\left(\frac15\right)-4\arctan\left(\frac1{239}\right)$$
Lehmer's measure as introduced in 1938[1] as a measure of how computationally efficient a Machin-like formula is, with lower values corresponding to more efficient formulae.
For the general Machin-like formula $$\pi=a_0\arctan(b_0)+a_1\arctan(b_1)+a_2\arctan(b_2)+\dots$$ Lehmer's measure is computed using $$\sum_i\frac1{\log_{10}\left(\frac1{b_1}\right)}.$$ For example, for Machin's formula, Lehmer's measure is $$\frac1{\log_{10}(5)}+\frac1{\log_{10}(239)}=1.85112$$