A few days ago I wrote a post about how to not go about writing an arbitrary precision data type in C to calculate pi. If you read the article, I talked about how a friend and I were trying to accomplish that task in 24 hours. Needless to say, it didn’t work and I resorted to using a library that was already available. Namely, MPFR. After a little research on Wikipedia about the best approximations to pi, and a couple of days of off and on work, I had a pretty good solution up and running.

First, let’s talk about the math behind this. There are a bunch of approximations to pi; some older, some newer, some faster, some slower. At first, I used Newton’s approximation to calculate pi.

This worked, but was slow (I didn’t record exact execution times). As everyone knows, factorials are huge numbers and grow very rapidly. In this case, the numbers were just too big to efficiently accomplish the task at hand. Could have I done something like Sterling’s approximation? Sure, but there’s better ways to calculate pi. No use in wasting time.

Next up, I gave the cubic convergence version of Borwein’s algorithm mainly because there were no factorials in it. This worked pretty well actually. It calculated pi within a reasonable amount of time (more details below), but because it was a recurrance, I would not be able to multithread it.

Now with multithreading in mind, I turned my attention to the 1993 version of Borwein’s algorithm, which was a summation.

On the up side, it was a summation, which is easy to multithread. On the downside, look at all those factorials. Long story short, I hit the same with this approach as I did with Newton’s approximation above; it worked, it was just too slow.