Pi Calculation (ATAN Binary Series)

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

With the development of calculus in the 17th century, many mathematicians attempted to calculate pi using the series expansion method for atan(x).
Leibniz’s simple series for atan(1) converged poorly, yielding only two significant figures even after 1,000 calculations; however, many accelerated formulas were subsequently discovered. ﷐0-1:\normalsize Gregory\ series\\
\hspace{40px}tan^{\tiny-1}x=x-{\large\frac{1}{3}}x^3+{\large\frac{1}{5}}x^5-{\large\frac{1}{7}}x^7+\cdot\cdot\cdot\\
(1)\ Leibniz\ 1671\\
\hspace{40px}{\large\frac{\pi}{4}}=tan^{\tiny-1}\ 1=1-{\large\frac{1}{3}}+{\large\frac{1}{5}}-{\large\frac{1}{7}}+\cdot\cdot\cdot\\
(2)\ Machin\ 1706\\
\hspace{40px}{\large\frac{\pi}{4}}=4tan^{\tiny-1}{\large\frac{1}{5}}-tan^{\tiny-1}{\large\frac{1}{239}}\\
(3)\ Hermann\ 1706\\
\hspace{40px}{\large\frac{\pi}{4}}=2tan^{\tiny-1}{\large\frac{1}{2}}-tan^{\tiny-1}{\large\frac{1}{7}}\\
(4)\ Euler\ 1738\\
\hspace{40px}{\large\frac{\pi}{4}}=tan^{\tiny-1}{\large\frac{1}{2}}+tan^{\tiny-1}{\large\frac{1}{3}}\\
(5)\ Euler\ \&\ Vega\ 1755\\
\hspace{40px}{\lar