Bailey–Borwein–Plouffe formula
The Bailey–Borwein–Plouffe formula (BBP formula) is a spigot algorithm for computing the nth binary digit of pi (symbol: π) using base 16 math. The formula can directly calculate the value of any given digit of π without calculating the preceding digits. The BBP is a summation-style formula that was discovered in 1995 by Simon Plouffe and was named after the authors of the paper in which the formula was published, David H. Bailey, Peter Borwein, and Simon Plouffe. Before that paper, it had been published by Plouffe on his own site. The formula is
The discovery of this formula came as a surprise. For centuries it had been assumed that there was no way to compute the nth digit of π without calculating all of the preceding n − 1 digits.
Since this discovery, many formulas for other irrational constants have been discovered of the general form
where α is the constant and p and q are polynomials in integer coefficients and b ≥ 2 is an integer base.
Formulas in this form are known as BBP-type formulas. Certain combinations of specific p, q, and b result in well-known constants, but there is no systematic algorithm for finding the appropriate combinations; known formulas are discovered experimentally.
A specialization of the general formula that has produced many results is
where s, b, and m are integers and is a vector of integers. The P function leads to a compact notation for some solutions. For example, the original BBP formula can be written as
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