Machin-like formulae
M000415
$$\pi=8\arctan\left(\frac{1}{5}\right)+12\arctan\left(\frac{1}{8}\right)+4\arctan\left(\frac{1}{57}\right)$$
Compact formula
8[5] + 12[8] + 4[57]
Lehmer's measure
3.10750
References
STØRMER (1896)
(full reference coming soon)