Borwein's algorithm

Borwein's algorithm is a name usually given to the most prominent of a number of algorithms to calculate the value of π devised by Jonathan and Peter Borwein.

It works as follows:

• Start out by setting

  $a_{0}=6-4{\sqrt {2}}$

  $y_{0}={\sqrt {2}}-1$

• Then iterate

  $y_{k+1}={\frac {1-(1-y_{k}^{4})^{1/4}}{1+(1-y_{k}^{4})^{1/4}}}$

  $a_{k+1}=a_{k}(1+y_{k+1})^{4}-2^{2k+3}y_{k+1}(1+y_{k+1}+y_{k+1}^{2})$

Then a_k converges quartically against 1/π; that is, each iteration approximately quadruples the number of correct digits of π.