x ← x+sin x
This works well because if |x − π| < ε ∧ x>0 then |x+sin x − π| < ε3.
How, do we compute sin? The Taylor series is the only way I have thought of. See this. Commercial grade software cheats and assumes the value of π.
Here is a fanciful way to compute √π.
See 106 digits of π. See GMP for some code.
And then musically.
2010 March 14 record
2010 August 6 record
2011 Oct (1013 decimal digits);
The quadrillionth bit of π is ‘0’;