Calculating Pi (Arithmetic-Geometric Mean)

The calculation ends when the current value equals the previous value. Increasing the number of digits used in the calculation improves the precision of π.

In 1976, Salamin and Brent discovered that using Gauss’s formula (1809) causes the value of π to converge quadratically at an extremely high speed.
With this convergence, the number of significant digits doubles with each additional loop—5 digits, 10 digits, and 20 digits.

\(\normalsize Gauss-Legendre\ method\ 1809\\ \hspace{80px}\pi={\large\frac{2\hspace{1px}{\rm AGM}^2\left(1,{\large\frac{1}{\sqrt{2}}}\right)}{1-{\large\displaystyle \sum_{\small k=0}^{\small \infty}}2^kc_k^2}}\\\\ The\ Square\ {\rm AGM}\\ \hspace{80px} by\ \ Salamin\ \&\ Brent,\ 1976\\ (1)\ a_0=1,\hspace{20px}b_0={\large\frac{1}{\sqrt{2}}},\hspace{20px}t_0={\large\frac{1}{4}}\\ (2)\ a_{n+1}={\large\frac{1}{2}}(a_n+b_n),\hspace{20px}b_{n+1}=\sqrt{a_nb_n},\\ \hspace{50px}t_{n+1}=t_n-2^n(a_n-a_{n+1})^2\\ (3)\ \pi=\displaystyle \lim_{n\to\infty}{\large\frac{(a_n+b_n)^2}{4t_n}}\\ \normalsize The\ Quartic\ {\rm AGM}\ by\ J.M.\ Borwein\\ \hspace{120px} \&\ P.B.\ Borwein,\ 1985\\ (1)\ y_0=\sqrt{2}-1,\hspace{20px}a_0=6-4\sqrt{2}\\ (2)\ y_{n}={\large\frac{1-\sqrt[4]{1-y_{n-1}^4}}{1+\sqrt[4]{1-y_{n-1}^4}}},\\ \hspace{10px}a_{n}=(1+y_{n})^4a_{n-1}-2^{2n+1}y_{n}(1+y_{n}+y_{n}^2)\\ (3)\ \pi={\large\displaystyle \lim_{\small n \to\infty}\frac{1}{a_n}}\\ \)