  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.
