Just interesting how it works in games and other software.
More precisely, I'm asking for a solution in C++.
Something like:
if AMX available -> Use AMX version of the math library
else if AVX-512 available -> Use AVX-512 version of the math library
else if AVX-256 available -> Use AVX-256 version of the math library
etc.
The basic idea I have is to compile the library in different DLLs and swap them on runtime but it seems not to be the best solution for me.
See Are the xgetbv and CPUID checks sufficient to guarantee AVX2 support? which shows how to detect CPU and OS support for new extensions: cpuid and xgetbv, respectively.
ISA extensions that add new/wider registers that need to be saved/restored on context switch also need to be supported and enabled by the OS, not just the CPU. New instructions like AVX-512 will still fault on a CPU that supports them if the OS hasn't set a control-register bit. (Effectively promising that it knows about them and will save/restore them.) Intel designed things so the failure mode is faulting, not silent corruption of registers on CPU migration, or context switch between two programs using the extension.
Extensions that added new or wider registers are AVX, AVX-512F, and AMX. OSes need to know about them. (AMX is very new, and adds a large amount of state: 8 tile registers T0-T7 of 1KiB each. Apparently OSes need to know about AMX for power-management to work properly.)
OSes don't need to know about AVX2/FMA3 (still YMM0-15), or any of the various AVX-512 extensions which still use k0-k7 and ZMM0-31.
There's no OS-independent way to detect OS support of SSE, but fortunately it's old enough that these days you don't have to. It and SSE2 are baseline for x86-64. Everything up to SSE4.2 uses the same register state (XMM0-15) so OS support for SSE1 is sufficient for user-space to use SSE4.2. SSE1 was new in 1999, with Pentium 3.
Different compilers have different ways of doing CPUID and xgetbv detection. See does gcc's __builtin_cpu_supports check for OS support? - unfortunately no, only CPUID, at least when that was asked. I'd consider that a GCC bug, but IDK if it ever got reported or fixed.
Typically setting function pointers to selected versions of some important functions. Inlining through function pointers isn't generally possible, so make sure you choose the boundaries appropriately, like an AVX-512 version of a function that includes a loop, not just a single vector.
GCC's function multi-versioning can automate that for you, transparently compiling multiple versions and hooking some function-pointer setup.
There have been some previous Q&As about this with different compilers, search for "CPU dispatch avx" or something like that, along with other search terms.
See The Effect of Architecture When Using SSE / AVX Intrinisics to understand the difference between GCC/clang's model for intrinsics where you have to enable -march=skylake or whatever, or manually -mavx2, before you can use an intrinsic. vs. MSVC and classic ICC where you could use any intrinsic anywhere, even to emit instructions the compiler wouldn't be able to auto-vectorize with. (Those compilers can't or don't optimize intrinsics much at all, perhaps because that could lead to them getting hoisted out of if(cpu) statements.)