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

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

Notes

1. PiHex Credits Archived July 21, 2006, at the Wayback Machine.

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)