M000000446

$$\pi=\frac{20}{3}\arctan\left(\frac{1}{2}\right)+\frac{8}{3}\arctan\left(\frac{1}{53}\right)+\frac{4}{3}\arctan\left(\frac{1}{4443}\right)$$
Compact formula20/3[2] + 8/3[53] + 4/3[4443]
Lehmer's measure4.17602
References
Wrench Jr., J. W. The derivation of arctangent equalities (PhD thesis, Yale University), 1938.