This is the correct Doppler formula for all moving
bodies,

f' = Exp(-v/c) f

where,
f is the ffrequency at the source, and
f' is the observed frequency.

If the observer and the source are moving directly
away from each other, the velocity is v 0, and v 0
if they are approaching.

Consider now, the relativistic Doppler formula

f_r' = sqrt((1 - v/c)/(1 + v/c)) f

We can expand both equations, f' and f_r', in their
respective power series until any arbitrary order, and
see the ratio f'/f_r is


f'/f_r = 1 - x^3/3 - x^5/5 + x^6 /18 - x^7/7 + ...

where x = v/c.

So, in any experimental test for that ratio, you'd have
to deal with measurement accuracies that could at
least make the term x^3/3 significant.