Bellard's formula is used to calculate the ''n''th digit of π in

base 16 In mathematics and computing, the hexadecimal (also base-16 or simply hex) numeral system is a positional numeral system that represents numbers using a radix (base) of 16. Unlike the decimal system representing numbers using 10 symbols, hexad ...

. Bellard's formula was discovered by

Fabrice Bellard Fabrice Bellard (; born 1972) is a French computer programmer known for writing FFmpeg, QEMU, and the Tiny C Compiler. He developed Bellard's formula for calculating single digits of pi. In 2012, Bellard co-founded Amarisoft, a telecommunication ...

in 1997. It is about 43% faster than the

Bailey–Borwein–Plouffe formula The Bailey–Borwein–Plouffe formula (BBP formula) is a formula for . It was discovered in 1995 by Simon Plouffe and is named after the authors of the article in which it was published, David H. Bailey, Peter Borwein, and Plouffe. Before that ...

(discovered in 1995). It has been used in

PiHex PiHex was a distributed computing project organized by Colin Percival to calculate specific bits of . 1,246 contributors used idle time slices on almost two thousand computers to make its calculations. The software used for the project made use of ...

, the now-completed

distributed computing A distributed system is a system whose components are located on different networked computers, which communicate and coordinate their actions by passing messages to one another from any system. Distributed computing is a field of computer sci ...

project. One important application is verifying computations of all digits of pi performed by other means. Rather than having to compute all of the digits twice by two separate algorithms to ensure that a computation is correct, the final digits of a very long all-digits computation can be verified by the much faster Bellard's formula. Formula: :

\begin
\pi = \frac1 \sum_^\infty \frac \, \left(-\frac \right. &  - \frac1 + \frac - \frac \left.  - \frac - \frac + \frac1 \right)
\end

Notes

External links

Fabrice Bellard's PI pagePiHex web siteDavid Bailey, Peter Borwein, and Simon Plouffe's BBP formula (''On the rapid computation of various polylogarithmic constants'') (PDF)
Distributed computing projects Pi algorithms 1997 introductions {{mathematics-stub