A Fast Self-correcting $\pi$ Algorithm

We have rediscovered a simple algorithm to compute the mathematical constant $$\pi=3.14159265\cdots.$$ 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 $\pi$ 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 \emph{self-correcting} in the sense that, given an approximated value of $\pi$ as an input, it can compute a more accurate approximation of $\pi$ with cubic convergence.