Machin-like formulae
M000009130
$$\pi=4\arctan\left(\frac{1}{2}\right)-4\arctan\left(\frac{1}{4}\right)+8\arctan\left(\frac{1}{5}\right)+4\arctan\left(\frac{1}{8}\right)+4\arctan\left(\frac{1}{21}\right)$$
Compact formula
4[2] - 4[4] + 8[5] + 4[8] + 4[21]
Lehmer's measure
8.27718