[kmill]

The algorithm is computing an alternating series, with the largest n being 5x10^7. The error in an alternating series (when the sequence is strictly decreasing in absolute value) is the absolute value of the next term. This is 4/(2*(5x10^7 + 1) + 1), which is a bit less than 4x10^-8, so we would only expect 7 digits to be correct. In fact, it is the 8th digit which goes awry (not the 10th): it should be a 5 rather than a 7.

This particular series, which comes from a series for the arctan of 1, has very slow convergence. An easily better one is Machin's formula, which converges faster than a geometric series (that is, the number of terms needed is at most linear in the number of wanted digits).