The following algorithm is based on work of Ramanujan and has been used in several world-record calculations of pi.

Initialize a_{0} = 6 – 4 √2 and y_{0} = √2 – 1. Then compute

and

The terms a_{n} form a sequence of approximations to 1/π. The error in each approximation is given by

This says that the number of correct digits roughly *quadruples* after each step. To see this, note that the number of correct decimal places after the *n*th step is the negative of the logarithm base 10 of the error:

[The error term goes to zero so quickly that you cannot (in ordinary
precision) compute the error bound and then take logarithms; the exp(-2 π
4^{n}) term will underflow to zero for n larger than 3. (See why here.) You have to take the log of the error term directly before evaluating numerically.]

The number of correct digits quadruples at each step because the leading term in the numerator above is 4^{n}.

To give a few specifics, a_{1} is accurate to 9 digits, a_{2} is accurate to 41 digits, and a_{3} is accurate to 171 digits. After 10 steps, a_{10} is accurate to over 2.8 million digits.

The algorithm given here was state of the art as of 1989. It was used to set world records in 1986. I don’t know whether it has been used more recently. See more here.

According to this page, π has
been calculated to 6.4 billion digits. You can beat this record if you
can carry out 16 steps of the method described here. a_{16} would be accurate to over 11 billion digits.

**Update**: The algorithm used most recently for world record calculations for pi has been the Chudnovsky algorithm. As of October 2011, the record was over 10 trillion digits.

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