On Thunder Math
This Document has been updated for the changes to F4 in 4.4 which affects the average potency per GCD. The result didn’t change though.
My name is Fürst Blumier and I did some calculations on an old problem that every BLM faces: In which cases do I hardcast a T3 in UI after I got still a dot ticking from the previous AF phase?
The Problem
Imagine you got a T3P in your AF phase and use it there in the second half at some point. Depending on procs, movement or other delays the dot is still ticking for some time as you are in UI and are about to hardcast T3. What do you do? Pray for a T3P? Hardcast a T3 to be safe? What maximizes my potency for which dot time? Those questions will be answered in this paper. We’ll begin with the 2 variants of what a BLM could do and how the potency of that is calculated.
The Basics
First of all, refreshing a T3 dot while the old one is still running diminishes the value of this GCD. Using a GCD to cast a dot has a certain value based on how many ticks you get out of it. Cutting a dot short by a few ticks makes this GCD lose those ticks in value. To give an easy example: imagine your T3P has 6s left and you refresh it with another T3. That gives the T3P GCD the potency of
390+640s(SpS)
If it would tick down completely it would get its full value obviously. Here, s(SpS) is a scalar value that depends on your SpS and makes your dots stronger. The values for this were calculated with those parts of the damage formula.
Variant 1: Praying for a proc
Depending on how much time on a dot is left, the probability for a proc changes. The probability to get a proc within the next n ticks is 1 – the probability to get no procs. The probability to get no procs in the next n ticks is just 0.9n. That leads us to:
p(proc) = 1-0.9n
Because of this we can’t say for certain that will get a proc. We now assume an expected potency for this proc. Which is just the proc potency weighted by the probability:
Eproc=p(proc)(390+840s(SpS))=(1-0.9n)(390+840s(SpS))
But what happens if we don’t get a proc? Well our dot will tick down and we will “save” a GCD. For the value of this saved GCD that we didn’t use for either a T3P or a T3 we will use the average potency per GCD, called AvgPot. Foul is purposely taken out of the calculation of the average GCD, since no matter how you decide, the saved GCD will not give you any additional Fouls in your encounter. The number of Fouls is a constant that only depends on the encounter length, which is why this makes sense. This leads to:
Eproc=(1-0.9n)(390+840s(SpS))+0.9nAvgPot
Now for the last step imagine you got lucky and you got your T3P. This dot itself has now the probability to proc again. We will now look at the expected potency of a future GCD. This dot will be assumed to run through completely since it will be casted sometime at the beginning of AF after the old dot ran out. This leads us to the complete expected potency of a T3P depending on ticks, while taking the reproc chance into account:
Eproc=(1-0.9n)(390+840s(SpS)+(1-0.98)390)+0.9nAvgPot
The value of the reproc T3P’s dot is taken out of the calculation since if you imagine, this reproc T3P would again be at the end of your AF and would lead again to the beginning of the calculation and to the question “Should I hardcast or not?”. Now we are nearly done. The only thing left is again the value of the average GCD, if the reproc doesn’t happen. This leads to the final expected potency of:
Eproc=(1-0.9n)(390+840s(SpS)+(1-0.98)390+0.98AvgPot)+0.9nAvgPot
So our overall potency of this variant results in the sum of the old end-of-AF-T3P potency and Eproc.
Variant 2: Hard casting T3
This variant is way easier to calculate. If you are hardcasting a T3 you will cut the old dot short. On top of that, since we want to compare the same number of GCDs in both variants, we will also take the reproc chance of a T3 into account. This leads to the expected potency:
Eproc=70+n40s(SpS)+(1-0.98)390+0.98AvgPot)
Where n is the number of ticks. The overall potency is again the the sum of the short cutted end-of-AF-T3P potency and Eproc.
The Results
Now for the results. I wrote a little Google Doc that did the calculations for me that are illustrated above and just calculated the difference of both variants for each number of ticks left on the the old T3P. You can play around with it here, just make a copy and put in your own SpS value to see the differences. All in all though, using this model the result is:
If you have less than 12s left on your dot, refresh it with a hardcast T3.
Problems and Outlook
This model uses the average pot/GCD of a BLM in its calculations. This is to be honest a very bad value in my opinion, since the variance is so high. But to get a better value instead of the average pot/GCD we would need a sim. So it’s as good as we can get.
On top of that certain assumptions in terms of dots running through in terms of reprocs were made. These assumptions aren’t true in a few fringe cases (eg. reproc happens super early, so it literally cannot run through, since proc time is 18s and dot length is 24s).This would lead to less potency in these cases but the error should be small. For all intents and purposes the assumptions and approximations that were made in these calculations are good enough. With a functioning sim, we can expect better values.
