Yeah, but the people talking about finding their own interpolating polynomials and Pade approximations are looking for those approximations in the regular commo

Yeah, but the people talking about finding their own interpolating polynomials and Pade approximations are looking for those approximations in the regular common cases, and omitting the special cases happens almost by accident. You could just take the same approximation used by openlibm or whatever and it's likely better than anything you yourself can come up with.