Calculation of Pi (ATAN 4-Term Series)

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.

With the development of calculus in the 17th century, many mathematicians attempted to calculate pi using the method of series expansion for atan(x).

\(\normalsize Gregory\ series\hspace{10px}tan^{\tiny-1}x=x-{\normalsize\frac{1}{3}}x^3+{\normalsize\frac{1}{5}}x^5-{\normalsize\frac{1}{7}}x^7+\cdot\cdot\cdot\\ (1)\ Gauss\ 1863\\ \hspace{10px}{\small{\normalsize\frac{\pi}{4}}=12tan^{\tiny-1}{\normalsize\frac{1}{38}}+20tan^{\tiny-1}{\normalsize\frac{1}{57}}+7tan^{\tiny-1}{\normalsize\frac{1}{239}}+24tan^{\tiny-1}{\normalsize\frac{1}{268}}}\\ (2)\ Stormer\ 1896\\ \hspace{10px}{\small{\normalsize\frac{\pi}{4}}=44tan^{\tiny-1}{\normalsize\frac{1}{57}}+7tan^{\tiny-1}{\normalsize\frac{1}{239}}-12tan^{\tiny-1}{\normalsize\frac{1}{682}}+24tan^{\tiny-1}{\normalsize\frac{1}{12943}}}\\ (3)\ Escott\ 1896\\ \hspace{10px}{\small{\normalsize\frac{\pi}{4}}=22tan^{\tiny-1}{\normalsize\frac{1}{28}}+2tan^{\tiny-1}{\normalsize\frac{1}{443}}-5tan^{\tiny-1}{\normalsize\frac{1}{1393}}-10tan^{\tiny-1}{\normalsize\frac{1}{11018}}}\\ (4)\ K.\ Takano\ 1982\\ \hspace{10px} {\small{\normalsize\frac{\pi}{4}}=12tan^{\tiny-1}{\normalsize\frac{1}{49}}+32tan^{\tiny-1}{\normalsize\frac{1}{57}}-5tan^{\tiny-1}{\normalsize\frac{1}{239}}+12tan^{\tiny-1}{\normalsize\frac{1}{110443}}}\\ (5)\ T.\ Murata\ 1982\\ \hspace{10px}{\small{\normalsize\frac{\pi}{4}}=24tan^{\tiny-1}{\normalsize\frac{1}{36}}+8tan^{\tiny-1}{\normalsize\frac{1}{57}}-5tan^{\tiny-1}{\normalsize\frac{1}{239}}-12tan^{\tiny-1}{\normalsize\frac{1}{23382}}}\\ (6)\ A.\ Shibata\ 1983\\ \hspace{10px}{\small{\normalsize\frac{\pi}{4}}=24tan^{\tiny-1}{\normalsize\frac{1}{43}}+20tan^{\tiny-1}{\normalsize\frac{1}{57}}-12tan^{\tiny-1}{\normalsize\frac{1}{117}}-5tan^{\tiny-1}{\normalsize\frac{1}{239}}}\\\)