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 $ u=1.01pm0.06$ and $z=1.50pm0.04$. 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.
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