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

If you want proper IEEE 754-2019 minimum (propagate NaN, -0.0 < +0.0, NaN bitpattern picked in the usual way) you can do it in 6:

        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

I personally find this a load of nonsense I don't care about.

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

What I do in Polars is a bit different, there for propagating NaNs I do

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

this isn't fully optimal on x86-64 but it's fairly simple and autovectorizes decently on various platforms, here's AVX2:

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