  We study the properties of the avoided or hidden quantum critical point
(AQCP) in three dimensional Dirac and Weyl semi-metals in the presence of short
range potential disorder. By computing the averaged density of states (along
with its second and fourth derivative at zero energy) with the kernel
polynomial method (KPM) we systematically tune the effective length scale that
eventually rounds out the transition and leads to an AQCP. We show how to
determine the strength of the avoidance, establishing that it is not controlled
by the long wavelength component of the disorder. Instead, the amount of
avoidance can be adjusted via the tails of the probability distribution of the
local random potentials. A binary distribution with no tails produces much less
avoidance than a Gaussian distribution. We introduce a double Gaussian
distribution to interpolate between these two limits. As a result we are able
to make the length scale of the avoidance sufficiently large so that we can
accurately study the properties of the underlying transition (that is
eventually rounded out), unambiguously identify its location, and provide
accurate estimates of the critical exponents $\nu=1.01\pm0.06$ and
$z=1.50\pm0.04$. We also show that the KPM expansion order introduces an
effective length scale that can also round out the transition in the scaling
regime near the AQCP.
