Convergents of the pi continued fractions are the simplest approximants to π. The first few are given by 3, 22/7, 333/106, 355/113, 103993/33102, 104348/33215, ... (OEIS A002485 and A002486), which are good to 0, 2, 4, 6, 9, 9, 9, 10, 11, 11, 12, 13, ... (OEIS A114526) decimal digits, respectively. Two approximations follow from the near-identity function 3sin x/(2 + cos x) evaluated at x = π/4 and π/8, giving π | ≈ | 12/7(2sqrt(2) - 1) | ≈ | (24sqrt(2 - sqrt(2)))/(4 + sqrt(2 + sqrt(2))), which are good to 2 and 3 digits, respectively.

almost integer | pi | pi formulas