[bee_rider]

I wonder to what extent the Newton-Rhapson strategy plays nicer with big fancy reordering/pipelining/superscalar chips. It has more little instructions to shuffle around, so my gut says it should be beneficial, but the gut can be misleading for this kind of stuff.

Also,

-funsafe-math-optimizations

Fun, safe math optimizations should be turned on by default! ;)

[pclmulqdq]

If you have to do this in software, Goldschmidt's algorithm parallelizes a lot better than Newton-Raphson, but isn't always general-purpose (IIRC). It uses a multiplicative update rule instead of an addition like NR. Division, square root, and inverse square root all use that algorithm under the hood (at least in AMD and IBM processors).