A Fast Self-correcting π Algorithm
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We have rediscovered a simple algorithm to compute the mathematical constant π=3.14159265⋯. The algorithm had been known for a long time but it might not be recognized as a fast, practical algorithm. The time complexity of it can be proved to be O(M(n)log^2 n) bit operations for computing π with error O(2^-n), where M(n) is the time complexity to multiply two n-bit integers. We conjecture that the algorithm actually runs in O(M(n)log n). The algorithm is self-correcting in the sense that, given an approximated value of π as an input, it can compute a more accurate approximation of π with cubic convergence.
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