[pclmulqdq]

If you're doing ~20 other FP ops (not including division) per inverse square root and not using a huge dependency chain of inverse square roots, you might not even notice the impact of using an exact inverse square root with this method.

Short of functions like exp and the trig functions, DIY approximations are usually not worth it these days. However, FP division getting faster has made these functions faster, too, since they can now use rational approximations rather than pure polynomials.

[adgjlsfhk1]

polynomials are still king. a 6th degree polynomial is often enough and will be faster than a 2nd degree rational.

[pclmulqdq]

A 6th degree polynomial is firmly in the "fast approximation" category. I think sin/cos are ~15th degree polynomials for 1 ULP of precision.

[adgjlsfhk1]

sin/cos commonly use 2 different 5th degree polynomials (depending on quadrant of the sin graph) https://github.com/JuliaLang/julia/blob/bb83df1d61fb649efd15...

[moonchild]

https://github.com/rutgers-apl/The-RLIBM-Project/blob/main/l... has a lot less and claims to be correctly rounded.

[pclmulqdq]

For 32-bit float, yes. For double precision, not so much.