Machin-like formulae
M000010531
$$\pi=4\arctan\left(\frac{1}{2}\right)-8\arctan\left(\frac{1}{4}\right)+12\arctan\left(\frac{1}{5}\right)+4\arctan\left(\frac{1}{8}\right)+8\arctan\left(\frac{1}{21}\right)$$
Compact formula
4[2] - 8[4] + 12[5] + 4[8] + 8[21]
Lehmer's measure
8.27718