There is a very fast converging algorithm for computing pi that you can do on a desktop calculator.
- Set x = 3
- Now set x = x + sin(x)
This converges ridiculously fast, after 1 step you get 4 digits right, after 2 steps you get 11 correct, in general we find:
# steps Digits right 1 4 2 11 3 33 4 100 5 301 6 903 7 2708 8 8124
of course on a pocket calculator, you only need to do 2 steps to have an accuracy greater than the calculator can display. To make this chart I had to trick a computer into doing high precision arithmetic, the code is here. Granted, this approximation is really cheating, since sin is a hard function to compute, and basically being able to compute sin means you know what pi is already. Really, this is just Newton's method for computing the root of sin(x) in disguise