from a pair of bounds obtained from the polygon method. Thus, more accurate results were obtained from polygons with fewer sides. Viète's formula, published by François Viète in 1593, was derived by Viète using a closely related polygonal method, but with areas rather than perimeters of polygons whose numbers of sides are powers of two. The last major attempt to compute by this method was carried out by Grienberger in 1630 who calculated 39 decimal places of using Snell's refinement. Machin-like formula For fast calculations, one may use formulae such as Machin's: together with the Taylor series expansion of the function arctan(x). This formula is most easily verified using polar coordinates of complex numbers, producing: ({,} = {239, 132} is a solution to the Pell equation 2−22 = −1.) Formulae of this kind are known as Machin-like formulae. Machin's particular formula was used well into the computer era for calculating record numbers of digits of , but more recently other similar formulae have been used as well. For instance, Shanks and his team used the following Machin-like formula in 1961 to compute the first 100,000 digits of : and they used another Machin-like formula, as a check. The record as of December 2002 by Yasumasa Kanada of Tokyo University stood at 1,241,100,000,000 digits. The following Machin-like formulae were used for this: K. Takano (1982). F. C. M. Størmer (1896). Other classical formulae Other formulae that have been used to compute estimates of include: Liu Hui (see also Viète's formula): Madhava: Newton / Euler Convergence Transformation: where is the double factorial, the product of the positive integers up to with the same parity. Euler: (Evaluated using the preceding series for ) Ramanujan: David Chudnovsky and Gregory Chudnovsky: Ramanujan's work is the basis for the Chudnovsky algorithm, the fastest algorithms used, as of the turn of the millennium, to calculate . Modern algorithms Extremely long decimal expansions of are typically computed with iterative formulae like the Gauss–Legendre algorithm and Borwein's algorithm. The latter, found in 1985 by Jonathan and Peter Borwein, converges extremely quickly: For and where , the sequence converges quartically to , giving about 100 digits in three steps and over a trillion digits after 20 steps. The Gauss–Legendre algorithm (with time complexity , using Harvey–Hoeven multiplication algorithm) is asymptotically faster than the Chudnovsky algorithm (with time complexity ) – but which of these algorithms is faster in practice for