When exactly are Dark Lances better than Disintegrators?

Here's some functions:

1. To wound roll
You have this many chances out of six to wound.
this is W roll =1+UnitStep(2X-(Y+1))+UnitStep(X-Y)+UnitStep(X-(Y+1))+UnitStep(X-Y*2)
because 6 always wounds, if str > 1/2 toughness 5+ wounds, if str = toughness 4+ wounds, if str>toughness 3+ wounds, if str >= 2* toughness 2+ wounds
Unitstep defines H(0)=1.

It looks like below.

2. To wound probability is the above / 6

3. If damage = 2 then damage is min(W, D). If damage = d6 then damage is (1+min(w,2)+min(w,3)+min(w,4)+min(w,5)+min(w,6))/6. It looks like below.

4. The wounds multiplier vs AP is given by min(1-(7-Sv-AP)/6,1) which simplifies to min((SV+AP-1)/6,1) and looks like this for Sv3+ with variable AP.

5. Then the entire expression for wounds at bs3+ is
Dissie:
2/3 (to hit) * 3 (shots) * (1+UnitStep(10-(Y+1))+UnitStep(5-Y)+UnitStep(5-(Y+1))+UnitStep(5-Y*2))/6 * min(W,2) * min((SV+2)/6,1)

DL:
2/3 (to hit) * (1+UnitStep(16-(Y+1))+UnitStep(8-Y)+UnitStep(8-(Y+1))+UnitStep(8-Y*2))/6 * (1+min(w,2)+min(w,3)+min(w,4)+min(w,5)+min(w,6))/6 * min((SV+3)/6,1)

and we can table these for, say Sv3+ and 6 wounds and here is what that would look like.



So that gives us the general equation to solve which is

2/3*(1+UnitStep(16-(Y+1))+UnitStep(8-Y)+UnitStep(8-(Y+1))+UnitStep(8-Y*2))/6*(1+min(w,2)+min(w,3)+min(w,4)+min(w,5)+min(w,6))/6*min((SV+3)/6,1) > 2/3*3*(1+UnitStep(10-(Y+1))+UnitStep(5-Y)+UnitStep(5-(Y+1))+UnitStep(5-Y*2))/6*min(w,2)*min((SV+2)/6,1)


Let's consider first SV = 3

then it simplifies to

(1+UnitStep(16-(Y+1))+UnitStep(8-Y)+UnitStep(8-(Y+1))+UnitStep(8-Y*2))/6*(1+min(w,2)+min(w,3)+min(w,4)+min(w,5)+min(w,6))/6 > 3*(1+UnitStep(10-(Y+1))+UnitStep(5-Y)+UnitStep(5-(Y+1))+UnitStep(5-Y*2))/6*min(w,2)*5/6

and further to

(1+UnitStep(16-Y-1)+UnitStep(8-Y)+UnitStep(8-Y-1)+UnitStep(8-Y*2))*(1+min(w,2)+min(w,3)+min(w,4)+min(w,5)+min(w,6)) > (1+UnitStep(10-Y-1)+UnitStep(5-Y)+UnitStep(5-Y-1)+UnitStep(5-Y*2))*min(w,2)*15

unfortunately this was too complex for Wolfram Alpha still, so assumed w>=2 and solved instead as

(1+UnitStep(16-(Y+1))+UnitStep(8-Y)+UnitStep(8-(Y+1))+UnitStep(8-Y*2))*(1+2+min(w,3)+min(w,4)+min(w,5)+min(w,6)) > (1+UnitStep(10-(Y+1))+
UnitStep(5-Y)+UnitStep(5-(Y+1))+UnitStep(5-Y*2))*30, w>=2, Y>0

which yields the following solutions

1. w>5, 5 < Y <= 8
2. w>5, 9 < Y <= 15
3. 3<w<=5, 5 < Y <= 7
4. 3<w<=5, 9 < Y <= 15

there are no solutions for w = 1 so the assumption was justified.

Then let's consider Sv4+. we can assume w>=2 since this is a situation where lance is worse. it's now

(1+UnitStep(16-(Y+1))+UnitStep(8-Y)+UnitStep(8-(Y+1))+UnitStep(8-Y*2))*(1+2+min(w,3)+min(w,4)+min(w,5)+min(w,6)) > (1+UnitStep(10-(Y+1))+
UnitStep(5-Y)+UnitStep(5-(Y+1))+UnitStep(5-Y*2))*30*6/5, w>=2, Y>0

and solutions are
1. w>4, 5<Y<=7
2. w>4, 9<Y<=15

Finally Sv2+. Here it's modifying from above
(1+UnitStep(16-(Y+1))+UnitStep(8-Y)+UnitStep(8-(Y+1))+UnitStep(8-Y*2))*(1+2+min(w,3)+min(w,4)+min(w,5)+min(w,6))*5/6 > (1+UnitStep(10-(Y+1))+
UnitStep(5-Y)+UnitStep(5-(Y+1))+UnitStep(5-Y*2))*30*6/5*2/3, w>=2, Y>0

and solutions are
1. w > 23/5, 5 < Y <= 8
2. w > 23/5, 9 < Y <= 15
3. 57/20 < w < 23/5, 5 < Y <= 7
4. 57/20 < w < 23/5, 9 < Y <= 15

these can be put as
1. w > 4, 5 < Y <= 8
2. w > 4, 9 < Y <= 15
3. 2 < w < 5, 5 < Y <= 7
4. 2 < w < 5, 9 < Y <= 15

seems like a result that you should check so let's do a table of results. Results 1-2 can be seen to be true comparing tables for w6 and sv3+. Let's check the difference sv2+ for dissie and DL at 4 wounds.

Substituting, it is
2/3  * (1+UnitStep(16-(Y+1))+UnitStep(8-Y)+UnitStep(8-(Y+1))+UnitStep(8-Y*2))/6 * (1+2+3+4+4+4)/6 * 5/6 - 2/3 *3 * (1+UnitStep(10-(Y+1))+UnitStep(5-Y)+UnitStep(5-(Y+1))+UnitStep(5-Y*2))/6 * 2 * 2/3 and then we do a table for it with Y from 1 to 16.

And sure enough this is positive only at Y=6 and Y=7 and Y > 10 to 15.


If you're good at math, I would be grateful for your help in checking this!

Images:
1. Wound roll function, S5

2. Wound roll function, S8

3. Damage d6 function

4. Armor save function, at sv3+ and ap value = a

5. Dissie cannon damage as function of toughness at sv3+

6. Same for Dark Lance

7. The difference of Dark Lance and Disintegrator cannon expected wounds for a W4 Sv2+ target as a function of toughness

Dark Lance wounds - Dissie Wounds, T6 or T7, Sv3+, W= wounds remaining on target

Table[2/3×4/6 (1/6 (1 + min(w, 2) + min(w, 3) + min(w, 4) + min(w, 5) + min(w, 6))) - 2/3×3×2/6 min(w, 2)×5/6, {w, 0, 7}]



Dark Lance wounds - Dissie Wounds, T8, Sv3+, W= wounds remaining on target

Not really, though at this point I doubt anyone would be surprised.