Black Mage Sanctuary - A Guide to DPS

[qivis]

B4 > F3 > F4x2 > F1 > T3P

I'm acutually trying to get around with using SC for Thunder, but when not under Ley Lines, I cannot get T3P after F1 (B4 > F3 > F4x2 > T3P > F1). Is that still a win in DPS?

[Aloise]

Progress made on the Thunder front. Given that you can get an infinite number of procs, your expected potency from casting your initial T1 approaches around 1194 potency. Here are my assumptions, with the math following that. Credit mostly to my older brother who doesn't play this game but is an expert mathematician.

Assumption 1) You cast T1. It ticks fully, regardless of whether you get a proc or not.
Assumption 2) When starting with a freshly applied T3 proc (24 seconds on the dot timer), the timer will run for 18 seconds, and then you will reapply a proc if you got one.
Assumption 3) If the timer runs for more than 18 seconds, you either didn't get a proc or aren't using it.

These assumptions help us keep the rotation and calculations sane. If we spiral into what-ifs, the well is far too deep to conceivably ever get out of. Additionally, by setting the bar relatively high for this, we can safely say that the majority of situations you find yourself in are going to be worse than this. If, when doing comparisons to a spell like Fire 4, F4 and T3P chains come out roughly equal, we can say that a T3P chain might be worse than a F4. If this theoretical T3P chain is noticeably worse, we can say that a realistic T3P chain is almost certainly worse. Now, onto the math.

For the time being, we're going to assume that you've cast T1, gotten a proc, and used it. We'll go back and visit that probability of that happening later. Once you're in your T3P loop, each proc has the same probability to get another proc. I'm calling it p(T3PQ), which is in this case roughly 0.468 (1-0.9^6). Each T3P that leads into another T3P also does the same damage. Your probability of getting a full T3P, and thus ending the proc chain, is 1-p(T3PQ), which I'm calling p(T3PF). Let us assume T1 has just procced and we've used T3P. Our expected damage from the duration of this first proc is:

p(T3PQ)*630 (Henceforth T3PQ) + p(T3PF)*710 (Henceforth T3PF)

However, T3PQ chains into another proc. The expected damage of our SECOND proc is:

p(T3PQ)*(p(T3PQ)*T3PQ + p(T3PF)*T3PF), which distributes to (p(T3PQ)^2)*T3PQ + p(T3PQ)*p(T3PF)*T3PF

You can see how this works. The probabilities must be multiplied by the probability that we even get to the second proc. The more procs you get, the more you must multiply by p(T3PQ). The total expected damage is the sum of all of these probabilities of doing damage. This can be expressed in sigma notation:



You have two options at this point. You can write this as a closed-form equation, or just evaluate it for high numbers and see what it comes out to. I'm going to forego the former (though it has been done and I can provide the closed-form equation if needed), and choose the latter. I wrote a small C program that takes an input of how many procs are allowed at most, and outputs the evaluation of that sigma expression for that number. Worth noting is that the program does have a slight inaccuracy for the very last proc, as the potency for the last one should always be 710, but at high proc counts (and even low ones), the difference is negligible. However, we must also consider the probability that T1 procs to even require this equation. This is done simply by multiplying the entire sigma expression by p(T3PQ) (it's not always p(T3PQ), just a happy coincidence here. Really it's p(T1 procs in x ticks)), and adding p(T3PF)*270 (T1 potency), with a special case that for 0 procs you always get 270 potency. With all this in place, here are some of the results:

Expected potency assuming we can get 0 procs maximum: 270
Expected potency assuming 1 proc max: 602.3
2 proc max: 917
3 proc max: 1064.3
4 proc max: 1133.3
5 proc max: 1165.5
10 proc max: 1193.3

A quick sanity check confirms these numbers make sense. You're going to be doing 270 potency a bit more than half the time. The slightly more than half the time you do get one, you'll be getting only 710 potency extra. Slightly less often than THAT, you'll be getting 630 potency + 710 or 630 + 630 + etc. Over half the time being only 270 potency weighs our average down severely.

After 10 procs it begins asymptoting hard, but we've got a good spread of situations here. What I am currently working on is adapting my program and the maths to put this in a PPS perspective. I think I'm close but I'm not exactly there and it's driving me nuts. I will definitely post again once I get it done.

Final statement: If you have the time to apply around 10+ procs (which is more than 3 minutes), expect the average potency resulting from an initial T1 cast to approach 1194. PPS numbers pending more pounding away at a keyboard.

[StouterTaru]

With all this in place, here are some of the results:

Expected potency assuming we can get 0 procs maximum: 270
Expected potency assuming 1 proc max: 458.4
2 proc max: 605.7
3 proc max: 674.6
4 proc max: 706.9
5 proc max: 722
10 proc max: 735

The numbers don't make sense when you factor in the extra GCDs casting the TC procs. Net potency for a single cast won't go over 710 (or around 740 if you include spell speed DoT effects)

[Aloise]

The numbers don't make sense when you factor in the extra GCDs casting the TC procs. Net potency for a single cast won't go over 710 (or around 740 if you include spell speed DoT effects)

These numbers are not PPS or potency per GCD, they are raw potency. You can expect an average raw TOTAL potency coming from T1 + procs given those number of procs allowed at most at around those number. For example, if you can only get a T1 + use a single proc before the boss goes invulnerable, you can substitute that T1's potency with 602.3, because over an infinite number of samples you won't get a proc slightly over half the time and only net 270 potency from casting, but will get a proc slightly less than half the time and get 270+710 = 980 potency.

For this example you could easily break it down into expected PPS by doing (p(T1 doesn't proc)*(T1 PPS) + (p(T1 does proc)*(Potency of T1 + T3P)/2 GCDs). What I'm working on is applying this to the work I already have so I can get expected PPS in addition to expected total potency.

[Aloise]

I've updated my numbers. I forgot a small detail in my program that bumped the numbers up significantly.

[Aloise]

I've solved it. From a PPS perspective, Thunder 1 is NEVER worth casting if:
1) You can avoid it, AND
2) It deprives you of even a single F4 cast.

Strap in, it's time for some more mathematics. I'm using the same assumptions as in my previous long post:

Assumption 1) You cast T1. It ticks fully, regardless of whether you get a proc or not.
Assumption 2) When starting with a freshly applied T3 proc (24 seconds on the dot timer), the timer will run for 18 seconds, and then you will reapply a proc if you got one.
Assumption 3) If the timer runs for more than 18 seconds, you either didn't get a proc or aren't using it.

This equation was a tad trickier to figure out because you've got to move a lot of numbers around to play nice with the division and the way you need to consider PPS. Basically, it works like this:

Let's say you can get a maximum of 4 T3 procs off under the given assumptions before a boss goes invulnerable. The difficulty with calculating a probabilistic PPS is that you can only consider completed chains, where you get a final 710 potency T3P and no more procs. This is because you need to know how many GCDs you've used to do the entire chain. Running totals don't really work. Therefore, your sigma expression to calculate the expected PPS of a T3P chain with maximum length n is:



This one looks a bit different, so I'll explain it. The first half of the equation is you setting up the probability that your given iteration is i procs deep, AND is a probability chain where you do NOT get the proc. We have to subtract by p(T3PQ)^(i+1) to do this, as subtracting by that leaves only the probability that we DON'T get a further proc. Then, we simply sum up all the potencies along this particular chain, and divide by the total time (GCD * i+1). You will always have 270 potency and 710 potency. You will also have i-1 T3PQ potency attacks. Consider, if you can get 2 procs maximum. Your best possible chain will be T1 > T3PQ > T3PF. 2 procs, 1 T3PQ.

Here are some expected values:
0 procs max: 122.73 (Just T1's PPS)
1 proc: 169.5
2 proc: 174.2
3 proc: 175.3
4 proc: 175.6
5 proc: 175.7
10 proc: 175.7

All across the board, your expected PPS from a Thunder 1 cast is lower than that of a Fire 4, by a minimum of 20 PPS. Therefore, inserting a Thunder 1 cast inside of an Umbral Ice phase when the T1 cast is not needed AND the T1 cast will result in losing a Fire 4 cast is a DPS loss. I'm not particularly concerned about the second condition, because in my mind a GCD spent on something unnecessary is a GCD wasted. The first condition, however, leads me to suggest a minor change in the rotation: Cast Blizzard 4 immediately after Blizzard 3, and Thunder 1 after that only if necessary. This will help reduce uncertainty about whether or not to cast T1.

I welcome any comments and critiques you guys have, but at the time of posting this is looking pretty solved to me. Additionally, I can share my thunder.c file for anyone who wishes to tinker with it or check my numbers for themselves.

Oh, and one final note: This doesn't cover a Sharpcast Thunder 1. That's always a PPS gain. (270 + 710)/4.4 = 222.7 PPS, which is far above F4's PPS.

[Garotte14]

Snip

All the math seems fine, but you're missing something that no one has considered. The argument is, "If you get a fast umbral tick, is it worth it to cast Thunder?"

Well, if you are playing correctly, after you cast Blizzard III, you should have already qued your next cast. So technically, no matter which way you spin it, you should be casting Thunder or Blizzard. Otherwise, you are delaying a cast to see if you get a fast umbral tick. Delaying casting, even slightly will always be a DPS decrease.

The alternative would be to always que Blizzard IV after Blizzard III, but what if you don't get the tick? Casting Thunder at this point is risking Enochian and Fire IVs.

[Aloise]

I've checked my PPS numbers for even the most optimal assumptions possible. Your expected PPS from a T1 cast if everything ticks for max duration is still about 10 PPS less than F4.

All the math seems fine, but you're missing something that no one has considered. The argument is, "If you get a fast umbral tick, is it worth it to cast Thunder?"

Well, if you are playing correctly, after you cast Blizzard III, you should have already qued your next cast. So technically, no matter which way you spin it, you should be casting Thunder or Blizzard. Otherwise, you are delaying a cast to see if you get a fast umbral tick. Delaying casting, even slightly will always be a DPS decrease.

The alternative would be to always que Blizzard IV after Blizzard III, but what if you don't get the tick? Casting Thunder at this point is risking Enochian and Fire IVs.

Let me elaborate on this line, and speak about the rotation in a vacuum while doing so:

The first condition, however, leads me to suggest a minor change in the rotation: Cast Blizzard 4 immediately after Blizzard 3, and Thunder 1 after that only if necessary. This will help reduce uncertainty about whether or not to cast T1.

Casting T1 after B4 has next to no adverse side effects on the rotation. There are exactly two situations in which refreshing Enochian becomes dodgy.

The first is if you refresh Enochian so it is now 20s, then cast T1. You get a T1 proc in the next 2 GCDs. In order to use that proc, you will have to position it before you finish all your F4s, which might lead to Eno falling off right before you finish the fourth F4.

The second is if you refresh Enochian so it is now 25s, then cast T1. You get a T3P during Astral Fire, and get a F3P as well. Your T3P will fall off before you cast B4. With some way to shave off a few fractions of a second (T3P > Swift F4 or Ley Lines), Eno will probably drop.

Outside of these two situations, casting T1 after B4 has no adverse effects on your rotation. This is, of course, in a vacuum. Outside of a vacuum, Swiftcast and Aetherial Manipulation exist, which help cut down on stress and lost time due to mechanics. There isn't a good way to objectively measure these things outside of the vacuum. But on a dummy, B3 > B4 > T1 is superior to B3 > T1 > B4.

[Madrone]

But on a dummy, B3 > B4 > T1 is superior to B3 > T1 > B4.

Wouldn't you have to wait for an mp tick for that B4 (unless piety >= 310), whereas you never wait for the mp to cast T1 if Piety >= 284? So your statement is true if Piety <= 283 or tick timing was perfect. Its not true if Piety is between 284 and 309 and tick timing wasn't perfect. Its true again, if Piety is 310+ or tick timing was perfect.

[Aloise]

Wouldn't you have to wait for an mp tick for that B4 (unless piety >= 310), whereas you never wait for the mp to cast T1 if Piety >= 284? So your statement is true if Piety <= 283 or tick timing was perfect. Its not true if Piety is between 284 and 309 and tick timing wasn't perfect. Its true again, if Piety is 310+ or tick timing was perfect.

Correct. Achieving 310 Piety on any race requires the same number of melds. 5 if no relic, none if relic. Now, that's lost substats. Let's assume the worst case scenario and say you lose out on 5 SpS melds. That's -60 SpS, weighted at -17 AP. BiS values for INT are going to top out right around 1600. 17/1600 = about a 1% AP/DPS decrease. However, having to cast T1 when you don't have to is somewhere in the ballpark of a 25-10 PPS decrease. It's quite hard if not impossible to mathematically prove it, but when the PPS of F4 is hovering around the 190 area, it seems like losing the PPS is worse than losing 1% damage. SSS for A12S is 2347 DPS. 1% of that is 23 DPS. I have a feeling that 10-25 PPS, even if you don't gain that 100% of the time, translates to more than a piddly 23 DPS.