The Brent-Salamin formula, also called the Gauss-Salamin formula or Salamin formula, is a formula that uses the arithmetic-geometric mean to compute pi. It has quadratic convergence. Let
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(2)
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(3)
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and define the initial conditions to be , . Then iterating and gives the arithmetic-geometric mean , and is given by
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King (1924) showed that this formula and the Legendre relation are equivalent and that either may be derived from the other.