Gauss-Legendre Algorithm

The Gauss-Legendre algorithm computes the digits of pi and is notable for converging rapidly, producing 45 million correct digits of pi after only 25 iterations, though it is computer memory intensive, which is why record-breaking pi calculations have mostly used the Chudnovsky algorithm instead. The method is based on the individual work of Carl Friedrich Gauss and Adrien-Marie Legendre combined with modern algorithms for multiplication and square roots, repeatedly replacing two numbers with their arithmetic and geometric mean; the version in common use, also called the Gauss-Euler or Brent-Salamin algorithm, was independently discovered in 1975 by Richard Brent and Eugene Salamin, and was used in 1999 to compute the first 200 billion decimal digits of pi.