acoatoitgs / pi-approximation

Collection of algorithms for computing pi

Pi approximations

This is a collection of algorithms whose objective is to compute pi (π) to high precision. I aim to further expand this list with some other algorithms in the future.

1. Montecarlo method

This method is based on randomness. The idea is to generate 2D points randomly in a square with an inscribed circle in it. Let $I$ be the number of points inside the circle, and $T$ the total number of points, we get that:

$$\frac{\pi}{4} \approx \frac{I}{T}$$

This method works for up to 4 decimal places. After that, it becomes slow and inaccurate. There are much better methods listed here. Compile the program with: g++ montecarlo.cpp -o montecarlo

2. Madhava-Leibniz series

This approach is based on the Madhava-Leibniz convergent series:

$$\pi = 4\sum_{n=0}^{\infty}\frac{(-1)^n}{2n+1}$$

It's able to compute π up to 9 decimal places in about 10 seconds. This is already a pretty good result and it goes beyond any practical use.

Compile the program with: g++ madhava_leibniz.cpp -o madhava_leibniz

3. Borwein's algorithm

This is the best algorithm listed here. It was discovered in 1985 by Jonathan and Peter Borwein. It's an iterative approach that is able to give over a trillion correct digits after only 20 iterations. The given code computes the first million digits in about a second.

$$y_0 = \sqrt{2}-1, a_0 = 6-4\sqrt{2}$$

$$f(y) = (1-y^4)^{\frac{1}{4}}$$

$$y_{k+1} = \frac{1-f(y_k)}{1+f(y_k)}$$

$$a_{k+1} = a_{k}(1+y_{k+1})^4-2^{2k+3}y_{k+1}(1+y_{k+1}+y_{k+1}^2)$$

$$\pi \approx \frac{1}{a_k}$$

Compile the program with: gcc borweins.c -o borweins -lmpfr