I have prepared a new paper containing a catalogue of 72 summation formulas, integral formulas and iterative algorithms for Pi. The catalogue contains both classical and modern formulas, ranging from Archimedes’ 2200-year-old algorithm to intriguing formulas found by Ramanujan and the quadratic, cubic, quartic and nonic algorithms of Jonathan Borwein and Peter Borwein, the latter of which double, triple, quadruple and nine-times, respectively, the number of correct digits with each iteration.
The catalogue of formulas and iterative algorithm is followed by results of carefully designed computer implementations, which enable one to compare the relative speed of these formulas. In particular, timing results are presented for 10,000-digit evaluations of the infinite series formulas, 2,000-digit evaluations for the integral formulas, and 100,000-digit evaluations of the iterative algorithms.
The paper is available HERE. Enjoy!