Modern era
Superfast algorithms based on the theory of modular equations and the arithmetico-geometric mean
(XXth century)

Salamin/Brent :

1976 :

a derived algorithm

I haven't tested the following algorithms yet

Quadratic convergence



Cubic convergence

This sequence converges towards the multiple of Pi closest to f₀.

f_n=f_n-1+sin(f_n-1)

J and P.Borwein :

1) 1984 : quadratic convergence (based on the arithmetico-geometric mean)



2) 1987 : quadratic convergence (also based on the AGM)



3) quadratic convergence (based on modular equations as the following equations)





4) quadratic convergence :



5) cubic convergence :



6) quartic convergence :



7) quintic convergence:




8) septic convergence :



9) nonic convergence :



10) "hexadecimalic" (!) convergence (order 16)