If sqrt is cheap, then if x>sqrt(2), we can calculate arccos instead. That is if x>0.7071, then do this range reduction:
All the methods described above can be combined with this, and they will all work better because of the smaller interval, which avoids the singularity at x=1. Rather than list more formulae and error plots, I'll generally just tabulate the maximum errors.
Nevertheless, here is the [3,3] minimax rational approximation.
