Fast algorithm to calculate Pi in parallel
I am starting to learn CUDA and I think calculating long digits of pi would be a nice, introductory project.
I have already implemented the simple Monte Carlo method which is easily parallelize-able. I simply have each thread randomly generate points on the unit square, figure out how many lie within the unit circle, and tally up the results using a reduction operation.
But that is certainly not the fastest algorithm for calculating the constant. Before, when I did this exercise on a single threaded CPU, I used Machin-like formulae to do the calculation for far faster convergence. For those interested, this involves expressing pi as the sum of arctangents and using Taylor series to evaluate the expression.
An example of such a formula:
Unfortunately, I found that parallelizing this technique to thousands of GPU threads is not easy. The problem is that the majority of the operations are simply doing high precision math as opposed to doing floating point operations on long vectors of data.
So I'm wondering, what is the most efficient way to calculate arbitrarily long digits of pi on a GPU?
You should use the Bailey–Borwein–Plouffe formula
Why? First of all, you need an algorithm that can be broken down. So, the first thing that came to my mind is having a representation of pi as an infinite sum. Then, each processor just computes one term, and you sum them all in the end.
Then, it is preferable that each processor manipulates small-precision values, as opposed to very high precision ones. For example, if you want one billion decimals, and you use some of the expressions used here, like the Chudnovsky algorithm, each of your processor will need to manipulate a billion long number. That's simply not the appropriate method for a GPU.
So, all in all, the BBP formula will allow you to compute the digits of pi separately (the algorithm is very cool), and with "low precision" processors! Read the "BBP digit-extraction algorithm for π"
Advantages of the BBP algorithm for computing π This algorithm computes π without requiring custom data types having thousands or even millions of digits . The method calculates the nth digit without calculating the first n − 1 digits, and can use small, efficient data types . The algorithm is the fastest way to compute the nth digit (or a few digits in a neighborhood of the nth), but π-computing algorithms using large data types remain faster when the goal is to compute all the digits from 1 to n.
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