Arcsin Approximation: Specific Recommendations

1. It's unknown whether these recommendations generalize to other numbers of degrees of freedom.
2. Instead of fixing the d.f. and finding the max error, it might be better to fix the max error and find the necessary d.f. In fact, this is what the Chebyshev approximation does. However, for all the other methods, this would require a search for the d.f. giving the desired error.
3. It was quite surprising that division never helped, that is, the Pade or Chebyshev-Pade approximation was never the best. I've never seen this before, and would welcome input as to why it happened here.
4. Nevertheless, a minimax rational expression should be much better than a minimax polynomial. Unfortunately, neither Maple nor Mathematica can calculate it in most cases. Maple usually fails with a message that the function doesn't oscillate sufficiently. Possibly, playing with critical points in Remez might help.
5. In fact, rational approximations appear to be a largely unexplored area, altho Newman (ref to come) 's monograph is quite tantalizing. There are assorted existence theorems, but no engineering. That is, there are no blackbox toolkits to use.