Bellard's Formula

Bellard's formula is a fast algorithm discovered by Fabrice Bellard for calculating the nth binary or hexadecimal digit of pi without needing to compute the preceding digits. It refines the earlier Bailey-Borwein-Plouffe formula, arranged to reduce the number of terms required and to evaluate roughly 43 percent faster in practice. The formula's digit-extraction property depends on an infinite series for pi whose terms can each be evaluated modulo a power of two, which is what allows a specific digit to be found without deriving all of the digits before it.