Bailey–Borwein–Plouffe formula

The Bailey–Borwein–Plouffe formula (BBP formula) is a formula for π. 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, it had been published by Plouffe on his own site. The formula is:

${\displaystyle \pi =\sum _{k=0}^{\infty }\left[{\frac {1}{16^{k}}}\left({\frac {4}{8k+1}}-{\frac {2}{8k+4}}-{\frac {1}{8k+5}}-{\frac {1}{8k+6}}\right)\right]}$

The BBP formula gives rise to a spigot algorithm for computing the nth base-16 (hexadecimal) digit of π (and therefore also the 4nth binary digit of π) without computing the preceding digits. This does not compute the nth decimal digit of π (i.e., in base 10). But another formula discovered by Plouffe in 2022 allows extracting the nth digit of π in decimal. BBP and BBP-inspired algorithms have been used in projects such as PiHex for calculating many digits of π using distributed computing. The existence of this formula came as a surprise. It had been widely believed that computing the nth digit of π is just as hard as computing the first n digits.

Since its discovery, formulas of the general form:

${\displaystyle \alpha =\sum _{k=0}^{\infty }\left[{\frac {1}{b^{k}}}{\frac {p(k)}{q(k)}}\right]}$

have been discovered for many other irrational numbers ${\displaystyle \alpha }$, where ${\displaystyle p(k)}$ and ${\displaystyle q(k)}$ are polynomials with integer coefficients and ${\displaystyle b\geq 2}$ is an integer base. Formulas of this form are known as BBP-type formulas. Given a number ${\displaystyle \alpha }$, there is no known systematic algorithm for finding appropriate ${\displaystyle p(k)}$, ${\displaystyle q(k)}$, and ${\displaystyle b}$; such formulas are discovered experimentally.