Machin-like formulae
M000222: Gauss's identity
$$\pi=48\arctan\left(\frac{1}{18}\right)+32\arctan\left(\frac{1}{57}\right)-20\arctan\left(\frac{1}{239}\right)$$
Name
Gauss's identity
Compact formula
48[18] + 32[57] - 20[239]
Lehmer's measure
1.78660
References
GAUSS (1863)(¢¢2.64766, ¢¢2.89392)
(full reference coming soon)