You can do it with just 3 instructions for IEEE 754-2019 minimumNumber (ignores NaN):

        vminpd          ymm2, ymm1, ymm0
        vcmpunordpd     ymm0, ymm0, ymm0
        vblendvpd       ymm0, ymm2, ymm1, ymm0

        vminpd          ymm1, ymm0, ymm1
        vbroadcastsd    ymm2, qword ptr [rip + .LCPI0_0]
        vandpd          ymm2, ymm0, ymm2
        vorpd           ymm1, ymm2, ymm1
        vcmpunordpd     ymm2, ymm0, ymm0
        vblendvpd       ymm0, ymm1, ymm0, ymm2

If you want propagating NaNs but don't care about signed zero or NaN payload/sign, you can use

        vminpd  ymm2, ymm0, ymm1
        vminpd  ymm1, ymm1, ymm0
        vorpd   ymm0, ymm1, ymm2

    if (self < other) | self.is_nan() { self } else { other}

        vcmpltpd        ymm2, ymm0, ymm1
        vcmpunordpd     ymm3, ymm0, ymm0
        vorpd           ymm2, ymm3, ymm2
        vblendvpd       ymm0, ymm1, ymm0, ymm2

That vminpd+vminpd+vorpd actually does handle signed zero properly! Screws up NaN payloads to the max though. Can be easily extended to canonicalize the NaN with 2 instrs + constant though of course. (which ends up at the same number of instrs as your proper impl (albeit with worse port distribution and latency), but you get to have a canonical NaN!)

Hit upon https://github.com/llvm/llvm-project/issues/217376 while playing around with proper minimumnum, failing to SMT-verify whatever version of LLVM I had; did find a funky working 6-instr (+ constant) version though:

    vpandn      ymm2, ymm1, ymm0
    vpcmpeqd    ymm2, ymm2, 0x80000000 # whether ymm0 is -0 and ymm1 is +0 (or other cases that magically don't cause issues)
    vcmpltpd    ymm3, ymm0, ymm1
    vcmpunordpd ymm3, ymm3, ymm1 # regular NaN-is-larger ymm0<ymm1
    vpor        ymm2, ymm2, ymm3
    vpblendvb   ymm0, ymm1, ymm0, ymm2