Is SSE floating-point arithmetic reproducible?

SSE is fully specified*. Muller is an expert in floating point arithmetic; who are you going to trust, him or some guy on a gamedev forum?

(*) there are actually a few exceptions for non-IEEE-754 operations like rsqrtss, where Intel never fully specified the behavior, but that doesn't effect the IEEE-754 basic operations, and more importantly their behavior can't actually change at this point because it would break binary compatibility for too many things, so they're as good as specified.

As Stephen noted, results produced by a given piece of SSE assembly code will be reproducible; you feed the same code the same input and you get the same output at the end. (That is, John Watte's quote is flat-out wrong.)

You threw the word "compilers" in there, though. That's a different ball game entirely. Many compilers are still quite bad at preserving the correctness of floating-point code. (The ATLAS errata page makes mention that clang "fails to produce correct code for some operations.") If you use special functions in your code, you're also, to some extent, at the mercy of whoever implemented your math library.