Bailey, Borwein, and Plouffe in the article ``On the Rapid Computation of Various Polylogarithmic Constants'' give the following formula for $\pi$, which allows the computation of an individual binary digit in the binary expansion of $\pi$ with small storage

\begin{displaymath}\pi = \sum_{i=0}^{\infty }\frac{1}{16^i}\left(\frac{4}{8\,i+1}-\frac{2}{8\,i+4}
-\frac{1}{8\,i+5}-\frac{1}{8\,i+6} \right)

Erich Kaltofen and C. Ryan Vinroot, following their integer relation approach (re)-discovered the following alternate formula:

\begin{displaymath}2\pi = \sum_{i=0}^{\infty }\frac{1}{16^i}\left(\frac{8}{8\,i+2}+\frac{4}{8\,i+3}
+\frac{4}{8\,i+4}-\frac{1}{8\,i+7} \right)

A Maple V.4 session showing our derivation is here. This session can be loaded as Maple text and executed.

Adamchik and Wagon [Am. Math. Monthly, Nov. 1997; url] give the following pretty variant:

\begin{displaymath}\pi = \sum_{j=0}^{\infty }\frac{(-1)^j}{4^j}\left(\frac{2}{4\,j+1}+\frac{2}{4\,j+2} +\frac{1}{4\,j+3} \right)

Their solution is dependent on the two given above as follows:
\begin{align*}4\pi = & \sum_{k=0}^{\infty }\frac{1}{4^{2k}}\left(\frac{8}{8\,k+1...
+6}+\frac{1}{8\,k+7} \right)
which is 2 times BBP plus 1 times our variant.

Fabrice Bellard has given a formula for base 210, which allows a faster algorithm for computing the hexadecimal digits of $\pi$.

About this document ...

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The translation was initiated by Erich Kaltofen on 1998-12-10

Erich Kaltofen