### Computing π

Here is a scheme for computing π that is fairly
fast and mathematically very simple.
There are certainly faster ways.
Initialize x to some approximation of π, say 3.
Each iteration of the following triples the number of significant digits:
x ← x+sin x

This works well because
if |x − π| < ε ∧ x>0 then |x+sin x − π| < ε^{3}.

How, do we compute sin?
The Taylor series is the only way I have thought of.
Commercial grade software cheats and assumes the value of π.

Here is a fanciful way to compute √π.

See 10^{6} digits of π.
See GMP for some code.

And then musically.

2010 March 14 record

2010 August 6 record

2011 Oct (10^{13} decimal digits);

π;

The quadrillionth bit of π is ‘0’;

### Computing π with divq

We compute sin x = x − x^{3}/3! + x^{5}/5! − x^{7}/7! …

= x(1−x^{2}/(2∙3))(1−x^{2}/(4∙5))(1−x^{2}/(6∙7))(1−