For the asinf libcall on macOS/x86, my former colleague Eric Postpischil invented the novel (at least at the time, I believe) technique of using a Remez-optimized refinement polynomial following rsqrtss instead of the standard Newton-Raphson iteration coefficients, which allowed him to squeeze out just enough extra precision to make the function achieve sub-ulp accuracy. One of my favorite tricks.

We didn't carry that algorithm forward to arm64, sadly, because Apple's architects made fsqrt fast enough that it wasn't worth it in scalar contexts.