Estimating Pi, $π$

Given the desired depth of calculation

This calculator estimates π to the requested precision. It uses one of the following formulas:

James Gregory and Gottfried Leibniz around 1675. This formula is very slow to converge.

π = 4 - \frac{4}{3} + \frac{4}{5} - \frac{4}{7} + \frac{4}{9} - \frac{4}{11} + ...

John Wallis's formula from 1655. This formula is a little slower than Leibniz's.

\frac{π}{2} = \frac{2}{1} \times \frac{2}{3} \times \frac{4}{3} \times \frac{4}{5} \times \frac{6}{5} \times \frac{6}{7} \times \frac{8}{7} \times \frac{8}{9} \times ...

One of Euler's many formulas. This formula is a little better than Leibniz's.

\frac{π^{2}}{6} = 1 + \frac{1}{2^{2}} + \frac{1}{3^{2}} + \frac{1}{4^{2}} + \frac{1}{5^{2}} + \frac{1}{6^{2}} + ...

Finally, a killer formula from Srinivasa Ramanujan around 1910.

\frac{1}{π} = \frac{2 \sqrt{2}}{3^{4} \times 11^{2}} \sum_{n = 0}^{\infty} \frac{(4 n)!}{{(n!)}^{4}} \times \frac{1103 + 26390 n}{{(4 \times 99)}^{4 n}}

To use the calculator simply enter the depth of the calculation (the number of terms utilized). The display shows the result and the difference from $π$ as defined by javascript's Math.PI. Below $π$ is shown calculated to 50 places.

3.141 592 653 589 793 238 462 643 383 279 502 884 197 169 399 375 11

Enter the number of terms to utilize: Depth of Calculation
Select a formula:

Pi estimate =
Error =

I can only show $π$ to 21 numerals of precision as that is the limit of the Number.precision() function. The Ramanujan function shows 0% error at a depth of 2, while only the first 17 decimals are accurate. This shows the limits of 64-bit math.

Estimating Pi, π

Given the desired depth of calculation

Estimating Pi, $π$