Bellard's formula

Bellard's formula, as used by PiHex, the now-completed distributed computing project, is used to calculate the nth digit of π in base 2. It is a faster version (about 43% faster^[1]) of the Bailey–Borwein–Plouffe formula.

Bellard's formula was discovered by Fabrice Bellard in 1997.

Formula

$\begin{align} \pi = \frac1{2^6} \sum_{n=0}^\infty \frac{(-1)^n}{2^{10n}} \, \left(-\frac{2^5}{4n+1} \right. & {} - \frac1{4n+3} + \frac{2^8}{10n+1} - \frac{2^6}{10n+3} \left. {} - \frac{2^2}{10n+5} - \frac{2^2}{10n+7} + \frac1{10n+9} \right) \end{align}$

Notes

1. PiHex Credits

External links

• Fabrice Bellard's PI page
• PiHex web site
• David Bailey, Peter Borwein, and Simon Plouffe's BBP formula (On the rapid computation of various polylogarithmic constants) (PDF)