> What's the x86-64 opcode for arcsin?

Not required. ATAN and SQRTS(S|D) are sufficient, the half-angle approach in the article is the recommended way.

> People have gotten PhDs for smaller optimizations. I know. I've worked with them.

I understand the can, not sure about the should. Not trying to be snarky, we just seem to be producing PhDs with the slimmest of justifications. The bar needs to be higher.