You don't have to stop using MMX to use AVX, although you might benefit from using equivalent SSE2 instructions when available (might as well, who doesn't have an SSE2 processor now, you have to go pre-P4 for that), and that gets a tiny little mention near the end.
I think he's saying that the MMX versions of these functions actually perform worse than what the compiler could produce from the plain C code, because the compiler would be able to use SSE2 for all floating-point math. Hence, these elaborately hand-optimized code paths are actually "pessimizations".
I don't think he's saying that you need to stop using MMX to use AVX. I think he's saying that MMX is old and SSE is much better, especially on x86-64 where you're guaranteed to have at least SSE2.
You don't have to stop using MMX to use AVX, although you might benefit from using equivalent SSE2 instructions when available (might as well, who doesn't have an SSE2 processor now, you have to go pre-P4 for that), and that gets a tiny little mention near the end.