Why is SSE scalar sqrt(x) slower than rsqrt(x) * x?

Answer 1

sqrtss gives a correctly rounded result. rsqrtss gives an approximation to the reciprocal, accurate to about 11 bits.

sqrtss is generating a far more accurate result, for when accuracy is required. rsqrtss exists for the cases when an approximation suffices, but speed is required. If you read Intel's documentation, you will also find an instruction sequence (reciprocal square-root approximation followed by a single Newton-Raphson step) that gives nearly full precision (~23 bits of accuracy, if I remember properly), and is still somewhat faster than sqrtss.

edit: If speed is critical, and you're really calling this in a loop for many values, you should be using the vectorized versions of these instructions, rsqrtps or sqrtps, both of which process four floats per instruction.

Answer 2

Instead of supplying an answer, that actually might be incorrect (I'm also not going to check or argue about cache and other stuff, let's say they are identical) I'll try to point you to the source that can answer your question.
The difference might lie in how sqrt and rsqrt are computed. You can read more here http://www.intel.com/products/processor/manuals/. I'd suggest to start from reading about processor functions you are using, there are some info, especially about rsqrt (cpu is using internal lookup table with huge approximation, which makes it much simpler to get the result). It may seem, that rsqrt is so much faster than sqrt, that 1 additional mul operation (which isn't to costly) might not change the situation here.

Edit: Few facts that might be worth mentioning:
1. Once I was doing some micro optimalizations for my graphics library and I've used rsqrt for computing length of vectors. (instead of sqrt, I've multiplied my sum of squared by rsqrt of it, which is exactly what you've done in your tests), and it performed better.
2. Computing rsqrt using simple lookup table might be easier, as for rsqrt, when x goes to infinity, 1/sqrt(x) goes to 0, so for small x's the function values doesn't change (a lot), whereas for sqrt - it goes to infinity, so it's that simple case ;).

Also, clarification: I'm not sure where I've found it in books I've linked, but I'm pretty sure I've read that rsqrt is using some lookup table, and it should be used only, when the result doesn't need to be exact, although - I might be wrong as well, as it was some time ago :).