I am looking for a formula/algorithm to calculate PI~3.14 in a given precision.
The formula/algorithm must have only very basic arithmetic as
because I want to implement these operations in C++ and want to keep the implementation as simple as possible (no bignum library is allowed).
I have found that this formula for calculating Pi is pretty simple:
Pi/4 = 1 - 1/3 + 1/5 - 1/7 + ... = sum( (-1)^(k+1)/(2*k-1) , k=1..inf )
(note that (-1)^(k+1) can be implemented easily by above operators).
But the problem about this formula is the inability to specify the number of digits to calculate. In other words, there is no direct way to determine when to stop the calculation.
Maybe a workaround to this problem is calculating the difference between n-1th and nth calculated term and considering it as the current error.
Anyway, I am looking for a formula/algorithm that have these properties and also converges faster to Pi
In your original (slowly converging) example, the error term can be computed because this is an alternating series; see http://en.wikipedia.org/wiki/Alternating_series#Approximating_Sums
Essentially, the next uncomputed term is a bound on the error.