Updated for 2.3 live release.
Read these first:
http://us.battle.net/d3/en/forum/topic/18300018238#1
and
http://us.battle.net/d3/en/forum/topic/18704215805#1
When comparing equipment, skills and gems we often ask the question "how much increased damage does gear A do over gear B?". The purpose of this analysis is to quantify this number for the new Bane of the Stricken gem. Bane of the Stricken
- Each attack you make against an enemy increases the damage it takes from your attacks by (0.80% + 0.01% * rank).
Gain 25% increased damage against Rift Guardians and bosses at rank 25.
- The primary bonus increases damage by (0.80% + 0.01%*rank), additively with itself, multiplicatively with everything else (ie. it's in its own damage category)
- The primary bonus appears to have an ICD of 1/APS.
- The secondary bonus is multiplicative, so 1.25x damage vs. rift guardians.
Since the primary bonus gets stronger and stronger the longer the fight lasts, the total % increased damage contributed by the gem will be a function of the damage of each strike and the target's total HP.
Let's define some terms...
d = damage per strike
h = target's total HP
r = 0.80% + 0.01% * rank (stricken's primary damage modifier)
The damage sequence of each strike against the target with the Stricken gem follows an arithmetic sequence. ex, if d = 1 and r = 1.00%:
strike 1 does 1.00 damage
strike 2 does 1.01 damage
strike 3 does 1.02 damage
...
strike n does d * (1 + (n-1) ) damage
The total sum of an arithmetic sequence is given by
Sn = n * (a1 + an) / 2
where n = sequence length, a1 is the first term in the sequence, an is the last term in the sequence. an can be expressed as:
an = a1 + (n - 1) * D
where D is the common difference between all terms (in this case, it's r as defined above)
To simplify calculations, let's make d constant, so each strike does the same baseline damage without stricken's modifier. Then, the number of strikes to kill the target (without stricken) will be given by h/d.
let x = h/d
How about when Stricken is added? To solve this, we need to solve for n (# of strikes) in the Sn equation given above. So when Sn >= h, the target is dead.
h <= n * (a1 + an) / 2
h <= d * n * (1 + (1 + r * (n - 1) ) / 2
h/d <= n * (2 + r * (n - 1) )
expand, rearrange terms, sub x=h/d...
0 <= r*n^2 + (2 - r)n - 2x
Use the quadratic equation to find the roots:
n = ( -(2 - r) +/- sqrt( (2 - r)^2 + 8rx) ) / 2r
Ignore the negative root since it doesn't mean anything, then, simplifying a bit...
n = ( (r - 2) / 2r ) + sqrt( ( (2 - r)^2 + 8rx) / 4r^2 )
This gives us n = # of strikes to kill the target, with stricken equipped. Technically to account for the last hit we need to round up to the nearest whole digit, so we can use the ceiling function.
n = CEILING( ( (r - 2) / 2r ) + sqrt( ( (2 - r)^2 + 8rx) / 4r^2 ) )
Now, determining how much % increased damage contributed by Stricken, it's easy.
+% damage = (x - n) / n
Now let's throw some numbers in:
damage per hit (d) = 1b
target hp (h) = 100b
stricken rank = 20 (for a nice round number)
r = 0.8% + 0.01%*rank = 1%
x = h/d = 100 (# strikes to kill target, without stricken)
n = CEILING( ( (1% - 2) / 2*1% ) + sqrt( ( (2 - 1%)^2 + 8*1%*100) / 4*(1%)^2 ) )
n = 74
Stricken's %increased damage = (x - n) / n = 35.14%
Now a table! HotA builds are currently seeing roughly 4b crits normally so let's use that as the baseline.
%increased damage by Stricken, target HP vs. gem rank
damage per strike = 4b
_________| ___________stricken rank_______________ |
target hp | 30 | 40 | 50 | 60 | 70 | 80
6.00E+10 | 7.14% | 7.14% | 7.14% | 7.14% | 7.14% | 7.14%
7.00E+10 | 5.88% | 5.88% | 5.88% | 5.88% | 5.88% | 5.88%
8.00E+10 | 5.26% | 5.26% | 5.26% | 11.11% | 11.11% | 11.11%
9.00E+10 | 9.52% | 9.52% | 9.52% | 9.52% | 9.52% | 15.00%
1.00E+11 | 8.70% | 8.70% | 13.64% | 13.64% | 13.64% | 13.64%
2.00E+11 | 21.95% | 21.95% | 25.00% | 25.00% | 28.21% | 28.21%
3.00E+11 | 29.31% | 31.58% | 33.93% | 36.36% | 38.89% | 41.51%
4.00E+11 | 38.89% | 40.85% | 42.86% | 44.93% | 49.25% | 51.52%
5.00E+11 | 45.35% | 48.81% | 52.44% | 54.32% | 58.23% | 60.26%
6.00E+11 | 53.06% | 56.25% | 59.57% | 63.04% | 66.67% | 68.54%
7.00E+11 | 59.09% | 63.55% | 66.67% | 69.90% | 73.27% | 76.77%
8.00E+11 | 65.29% | 69.49% | 73.91% | 76.99% | 80.18% | 85.19%
9.00E+11 | 70.45% | 75.78% | 80.00% | 84.43% | 87.50% | 92.31%
1.00E+12 | 76.06% | 81.16% | 85.19% | 90.84% | 93.80% | 98.41%
2.00E+12 | 122.22% | 129.36% | 135.85% | 142.72% | 148.76% | 155.10%
3.00E+12 | 158.62% | 166.90% | 175.74% | 184.09% | 191.83% | 198.80%
4.00E+12 | 189.02% | 199.40% | 208.64% | 218.47% | 227.87% | 236.70%
5.00E+12 | 216.46% | 227.23% | 238.75% | 249.16% | 259.20% | 268.73%
Important note: While the performance numbers look pretty imba at first glance, note that the primary effect only stacks on the first target hit. Thus, performance in AoE situations will be much lower compared to single-target.
