Bellard formula

The Bellard formula is a series that can be used to calculate the first digits of the circle number in the hexadecimal system. ${\displaystyle p}$ ${\displaystyle \pi }$

Fabrice Bellard was the first to publish an article on the formula in 1997. It is about 1.43 times as fast as the Bailey-Borwein-Plouffe formula .

An important application of the formula is the verification of the calculations of all first digits by using other methods. Thus, not all digits have to be calculated by two different algorithms , since it is sufficient to check the last digits of a "complete" calculation using the Bellard formula. ${\displaystyle \pi }$

The formula

The formula is:

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

To calculate the first hexadecimal places of , it is sufficient to calculate a partial sum that deviates from the series by at most . ${\displaystyle p}$ ${\displaystyle \pi }$ ${\displaystyle 16^{-p}}$

Web 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)

Individual evidence

1. ↑ A new formula to compute the n'th binary digit of pi. Retrieved October 5, 2019 . 
2. ↑ PiHex credits . In: Center for Experimental and Constructive Mathematics . Simon Fraser University. March 21, 1999. Archived from the original on June 10, 2017. Retrieved March 30, 2018.
3. ↑ Peter Trueb: Hexadecimal Digits are Correct! . October 31, 2016. Archived from the original on November 16, 2016. Retrieved on December 28, 2016.