[RPG] How much damage does a Pick do on a critical hit

Smite damage is not precision damage, and since it is extra points not extra dice of damage, it is definitely multiplied. The only things not multiplied on crits are extra dice of damage, such as precision damage or weapon effects like flaming.

Let:

$$\begin{align} m&\text{: chance of a miss} \\ h&\text{: chance of a hit} \\ c&\text{: chance of a critical hit} \\ D_m=0&\text{: damage on a miss} \\ D_h&\text{: damage on a hit} \\ D_c&\text{: damage on a critical hit} \\ D = mD_m+hD_h+cD_c&\text{: damage} \\ H \ge 2&\text{: the lowest number on the d20 which is a hit} \\ T \ge H&\text{: the lowest number on the d20 which is a threat} \\ C &\text{: the lowest number on the d20 which confirms a critical} \\ \end{align}$$

Then:

$$\begin{align} m&= {H-1\over 20} \\ h&= {1-m-c} \\ c&= {21-T\over 20}\times{21-C\over 20} \\ \end{align} $$

In general \$H=AC-\text{Attack mod}\$, however, this cannot be less than 2 or greater than 20. Normally, \$C=H\$ but because of Battle Ardor, in this case \$H=AC-\text{Attack mod}-2\$, again not less than 2 or greater than 20. We therefore need to consider a range of \$AC-\text{Attack mod}\$ from 2 to 22.

This spreadsheet crunches the numbers. For all of them Weapon Specialization is better, doing 4.3 more mean damage when you need a 2 to hit dropping to 0.2 when you need a 20.