by infinite sum of arctangent functions as and where is the n-th Fibonacci number. However, these two formulae for are much slower in convergence because of set of arctangent functions that are involved in computation. Arcsine Observing an equilateral triangle and noting that yields with a convergence such that each additional five terms yields at least three more digits. Digit extraction methods The Bailey–Borwein–Plouffe formula (BBP) for calculating was discovered in 1995 by Simon Plouffe. Using math, the formula can compute any particular digit of —returning the hexadecimal value of the digit—without having to compute the intervening digits (digit extraction). In 1996, Simon Plouffe derived an algorithm to extract the th decimal digit of (using base10 math to extract a base10 digit), and which can do so with an improved speed of time. The algorithm requires virtually no memory for the storage of an array or matrix so the one-millionth digit of can be computed using a pocket calculator. However, it would be quite tedious and impractical to do so. The calculation speed of Plouffe's formula was improved to by Fabrice Bellard, who derived an alternative formula (albeit only in base2 math) for computing . Efficient methods Many other expressions for were developed and published by Indian mathematician Srinivasa Ramanujan. He worked with mathematician Godfrey Harold Hardy in England for a number of years. Extremely long decimal expansions of are typically computed with the Gauss–Legendre algorithm and Borwein's algorithm; the Salamin–Brent algorithm, which was invented in 1976, has also been used. In 1997, David H. Bailey, Peter Borwein and Simon Plouffe published a paper (Bailey, 1997) on a new formula for as an infinite series: This formula permits one to fairly readily compute the kth binary or hexadecimal digit of , without having to compute the preceding k − 1 digits. Bailey's website contains the derivation as well as implementations in various programming languages. The PiHex project computed 64 bits around the quadrillionth bit of (which turns out to be 0). Fabrice Bellard further improved on BBP with his formula: Other formulae that have been used to compute estimates of include: Newton. Srinivasa Ramanujan. This converges extraordinarily rapidly. Ramanujan's work is the basis for the fastest algorithms used, as of the turn of the millennium, to calculate . In 1988, David Chudnovsky and Gregory Chudnovsky found an even faster-converging series (the Chudnovsky algorithm): . The speed of various algorithms for