Fury rotation comparision

[GronkerLonker]

Quietsch suggested to move the discussion about two possible Fury Warrior rotations to a new thread, to keep Thegreatmes GuideThred clean.
I think he's right, as the guide is pretty good and this is more a theorycraft experiment, which should not further polute the thread and the more general QA growing there.

So what happened so far?
The guide contains the currently most reasonable rotation for a fury warrior:
0s:WW-> 1.5s:BT-> 3s:2.5s gab for Slam or other stuff -> 5.5s:BT -> 7s: -> 1s gap not usable -> 8s:end of rotation and next WW

blauberri innitiated this (it's all your fault ) with mentioning a rotation using an eventual BS proc after the WW:
0s:WW-> 1.5s:Slam-> 3s:BT-> 4.5s:2.5s gab for Slam or other stuff -> 7s:BT -> 8.5s:end of rotation and next WW

At first glance I thought it a good idea to trade in the slightly longer duration of the rotation (.5s) for an additional Slam.
So I made a little pic comparing the both rotations:


However Thegreatme and Quietsch made me aware of two things I didn't take into account:
1) Over the course of long fight the rotation delay sums up more and more.
2) There is a good chance of BS not procing after the first BT, which would lead to a rotation duration of 8.5s anyway if you Slamed after WW but not giving the advantage of beeing able to Two-Slam.

And now the fun begins as we try to figure out under which circumstances and if at all, the rotation using Slam after WW could be a good choice.

I try to summarize now what was calculated on this issue so far:
I tried to compare the rotations in regard to how much more damage the 2 Slam rotation would have to give to replace the one Slam rotation.

So I set average damage of a rotation with one slam (no matter if after WW or BT) with WW + 2*BT + Slam = x and the average damage of the rotation with 2 Slams with WW + 2*BT + 2*Slam = y.

We have now 2 rotation sets. Lets call them RotA (including rotation with one Slam after BT if possible and rotation with no Slam if not possible) and RotB (RotA plus rotation with 2 Slams if possible and rotation with one Slam after WW and none after BT).
Using only the first rotation you would always have a duration of 8s no matter if Slam procs or not after the BT. That would make the average "DPS" of the rotation x/8s.

So: x/8s = RotA

Using the proc chance of BS, I assumed, that you would have the followng scenario if you use instant Slam after WW too if it procs:
I thought 80% of the time you would have exactly RotA as BS would not proc after WW leaving you with no choice.
This is not correct, as Quietsch pointed out, that WW has a 40% chance to proc BS so we have RotA only 60% of the time
(Note that the proc of BS after BT if Slam was not used after WW is already included in the average damage of RotA):

0.6*x/8s

Now of the remaining 40% when BS procs after WW, we have another 20% chance of BS procing after BT too and an 80% chance that we cannot use Slam a second time.
However the duration is now .5s longer no matter if BS procs a second time or not.

So we add to the damage above:
0.4*0.2*y/8.5s + 0.4*0.8*x/8.5s

That means 20% of 40% a rotation with y damage (2 Slams) and 8.5s duration and 80% of 40% a rotation with x damage (1 Slam and no Slam) and a 8.5s duration. The overall average damage of RotB beeing:

0.6*x/8s + 0.4*0.2*y/8.5s + 0.4*0.8*x/8.5s = RotB

So now we set both rotations equal to get the relative dps of both and get:

0.6*x/8s + 0.4*0.2*y/8.5s + 0.4*0.8*x/8.5s = x/8s

Crunching the number through the calc we get:

0.075*x/s + 0.08*y/8.5s + 0.32*x/8.5s = x/8s

get x to one and y to the other side:
0.08*y/8.5s = x/8s - 0.075*x/s - 0.32*x/8.5s

extract x:
0.08*y/8.5s = x*(1/8s - 0.075/s - 0.32/8.5s)

get y to be alone:

y = x*((1/8s - 0.075/s - 0.32/8.5s)*8.5s)/0.08

crunch again and we finally get:

y = 1.3125*x

That means the damage of y (remember: WW + 2*BT + 2*Slam) has to be 1.3125 time bigger than that of x (WW + 2*BT + Slam) to be better.
(This time I posted the detailed steps of the calculation to bette lay any mistakes open I may have made)

By thinking this through again I think I realize the following:
1) The delay of BT itself is not an issue, as we look at the rotation of it's own.
2) The rotation using a Slam after WW has a longer duration and thus can not fit as many times into a specific combat timespan (say a battle of 5 minutes).
Again that is not an issue, as we consider this by taking the longer duration of the rotation into account using 8.5s as dps modifier for the case of a Slam after WW and 8s as dps modifier for the case of no Slam after WW.
So we already considered the diffrent durations.
This gets clearer by looking at the units. the unit of x and y is damage / second and thus it already takes time into account.

Now we can detail it even further by inserting the specific skills for x and y:
WW + 2*BT + 2*Slam = 1.3125(WW + 2*BT + Slam)
Huddel it up and we get:
0.6875*Slam = 0.3125*WW+0.625*BT

Divided by 0.6875 and rounded:

Slam = 0.46*WW + 0.9*BT

So to reach an equal dps with both rotation sets, your Slam would have to have an average damage of almost half your Whirlwind plus 0.9 time your Bloodthirst. I don't think this is ever gonna happen with decent equip or am I wrong?

Thus it would prove to me after all, that you can relax, and use slam only after the first BT not meddling around with a second WW -> Slam rotation.
I really wouldn't have thought that. And hey math really is worth something after leaving school .

Now this does not take into account the BS procs from HS. The next step may be to have a look at how much HS would have to improve the chance to trigger an additional Slam after the WW Slam (as everything else would not improve the damage of that rotation).
Thegreatme did begin with an HS proc calculation, maybe this could fit in here too.

And as a last note, I think the Slam after WW can be worth it if you already know you'll be using something else in the long gab after the first BT (what about a WW->Slam->BT->Execute->BT rotation if Execute is available?).

[Thegreatme]

Modeling HS's effect on comparing the 1 slam rotation vs the 2 slam rotation.

Round 2, FIGHT:


let's try this again, but with more accuracy.


Assumptions:
400 second fight
every MH swing is a HS
full haste buffs, no haste rating, 100% flurry up time.
the first HS hits at T=0.0
I will be citing a bunch of stuff from my first post because I am lazy and don't want to type it out again. and more imprtantly, no one said the math was false so I am going to assume it is correct until proven otherwise.


400s= 50 WW, and up to 100 BT

frequency of MH hits:
as pointed out by quietsch my original calculation was blatantly false
Using the proper calculation:

3.6/(1.25*1.03*1.2)=2.33

Notable HS hits:
0.0
2.33
4.66

Now let’s see how that interacts with WW at T = 0.0
Possible proc outcomes:
Double proc: 0.2*0.2=0.04 (4%)
No procs: 0.8*0.8=0.64 (64%)
Single proc: 1-(0.64+0.04)=0.32 (32%)

Total chance for a proc: 0.32+0.04=0.36 (36%)

0.36*50=18 two slam rotations
18*1.5=27s of BT pushback
27/4=6.75 BT's lost

Now any slam proc that occurs after T=1.5s and before T=5.5s can be used without causing any interference with the BT at T=7.0s

Times of possible bloodsurge procs:
2.33 (HS)
3.0 (BT)
4.66 (HS)

Possible proc outcomes:
Triple proc: 0.2*0.2*0.2=0.008 (0.8%)
Double proc (3 combinations): 3(0.2*0.2)=0.12 (12%)
No procs: 0.8*0.8*0.8=0.512 (51.2%)
Single proc: 1-(0.512+0.12+0.008)=0.36 (36%)

Total chance for a proc: 0.36+0.008+0.12=0.488 (48.8%)

0.488*18= 8.784 rotations that end with using slam twice.


Now because bloodsurge procs last 5s, any procs off WW/HS at t=0.0 will still be able to be used T=3.0 which is where you slam in the single slam rotation.


this means that if you use the double slam rotation you gain 8.784 slams at the cost of losing 6.75 BT's

this means that slam would have to do damage equal to 6.75/8.784=0.7684 (76.84%) of BT's damage for the two rotations to be equal.

now lets see if that could be the case, I will be using the overall data from this parse:
World of Logs - Real Time Raid Analysis
to determine the average damage of slam and BT.
(note I will not be factoring crit in, just regular hits)

Average slam hit damage: 3426.4
Average BT hit damage: 4717.5

3426.4/4714.5= 0.7263 (72.63%)

so according to that parse slam does damage equal to 72.63% of BT's damage, and therefore makes the double slam rotation a DPS loss.

possible skewing elements:
currently these numbers are assuming that HS always lands on a WW, which is not the case, so it could skew the numbers in favor of the double slam rotation. the only way to fix this would be to individually model every HS hit, which is a waste of time for the small amount of accuracy gained in my opinion.
These numbers are assuming every single MH swing is a HS, which doesn't happen.
They are also assuming that HS hit every time WW does, which doesn't happen. and by calculating it this way it is adding extra HS hits. at a 2.33 speed you would have 171.67 MH hits over a 400s fight. in my calculations there would be 4.43 hits over each 8 second rotation. which means my calculations would assume there to be 221.67 MH hits over 400 seconds (IE 50 extra hits), which adds 10 extra slams to both rotations.
they assume 100% flurry up time, which doesn't happen and will affect the number of HS's that land

what this means is that really the gap between the 1 slam rotation vs. the 2 slam roation is much bigger than my calculations indicate.

Again please rip my math apart, be hypercritical of my calculations, etc.

[Superspy23]

If Thegreatme's numbers are right then that means that they are really very close in total damage output. (The 4.21% deficiency in slam damage vs BT seems to represents fractions of a percent in overall damage) However that near equality is based on having 100% HS uptime to provide greater frequency of Slam procs.

In short it seems that best case has the 2 slam rotation lacking by less than 1%. Best case in theory has a 5 second buff procing 48% of the time in an 8 second rotation. Check me here but that should yield a 30% uptime. Has anyone ever seen a log with that high of a Slam! uptime?

Basically we won't ever see this perfect scenario and if we did we'd loose a fraction of a percent of damage. Am I analyzing this right?

[Thegreatme]

My numbers are *mostly* right, but the difference isn't anything game breaking but my numbers are working in favor of the double slam rotation, so really the gap is larger than my calculations indicate.

as far as 30% up time on the Slam! buff, that would be the case in an ideal situation if you never used slam because when you slam it removes the buff, reducing its uptime.

we won't ever see this scenerio under normal circumstances. It *might* be possible in some odd cases like maybe having a heroic justicebringer but every thing else being naxx gear (really hard hitting weapon but low attackpower for high slam damage and low BT damage)

and yes, your analysis is correct.

[Quietsch]

Again please rip my math apart, be hypercritical of my calculations, etc.

Again, i'm not sure, but afaik it is the case, that WW has two hits, which both could procc BS.
(Apart from this your maths look right.)

Math for comparing BT & Slam, if you have neither 4pcT9, nor 4pcT8, would be
((0.5*x)*1.1)=((Wpndps*Wpnspd+(x/14)*Wpnspd)+250)
So even if you got heroic Justicebringer, with "normal" ap at this gearlvl (which should be around 7000 minimum), you would have ((0.5*7000)*1.1)=((281*3.6+(7000/14)*3.6)+250) --> 3850 = 3061.6, and every more ap would grant a bigger increase for BT, as for Slam.

[Thegreatme]

Again, i'm not sure, but afaik it is the case, that WW has two hits, which both could procc BS.

I am unsure of this as well and as such I left it out to be safe.

[Quietsch]

But if it is right, your maths would be really wrong. (Even if it wouldn't change the result of it.)

/e btw, maybe someone of you could test it?

[GronkerLonker]

For me this looks all pretty correct, so the HS can not really push the 2 Slam rotation ahead in either case (WW with 20% or 40% BS chance).
The WW thing I will definetly check just out of curiosity.
Doing the math above I realized the funny thing, that assuming a 40% proc chance of BS with WW, this would disfavour the 2 Slam rotation, as you have a smaller number of 1 Slam rotations over the course of battle. This fact underlines the damage loss of the 2 Slam rotation logically I think (more 1 Slam rotations = better than more 2 Slam rotations).

Man but it would have been great to discover something new .

[Thegreatme]

But if it is right, your maths would be really wrong. (Even if it wouldn't change the result of it.)

it wouldn't change the math too much in a way because you already have to calculate the outcomes of 3 possible procs and could use the same numbers pretty much:

.488*50=24.4

.488*24.4=11.9072 gained slams instead of 8.784 slams.

I am currently lurking EJ's fury thread to see if any testing has been done on it... so far only people asking what we don't know, and no test results, if I find nothing I will spend some time tonight and test it my self.

*edit*: most of this has been tested on EJ and with relatively similar results, they were also using different processes of doing the comparisons so I think the answer is pretty much set in stone of 1 slam roation > 2 slam Rotation

[Quietsch]

it wouldn't change the math too much in a way because you already have to calculate the outcomes of 3 possible procs and could use the same numbers pretty much:

.488*50=24.4

.488*24.4=11.9072 gained slams instead of 8.784 slams.

Which would lead to a 56.6% damage ratio between BT/slam, which is really often achieved.
(Not to missunderstand this: Yes, with this calculation, 2-slam-rotation would look better, but there is the fact, Thegreatme mentioned, that there are way too much HS included, so that 1-slam-rotation would still be better.)

[Superspy23]

I know this is likely to throw a wrench in but what about the T8 2 piece bonus or the T9 4 piece bonus which give benefits to slam (and HS)?

How much is that set bonus going to influence which rotation you go with?

It seems to me that the set bonus is likely to weigh more heavily than anything thus far as it seems the difference is less than 1% damage difference (best case) between the rotations.

Also what's "worst case" (no HS) difference in damage between the rotations?

[GronkerLonker]

Slam = 0.46*WW + 0.9*BT

So to reach an equal dps with both rotation sets, your Slam would have to have an average damage of almost half your Whirlwind plus 0.9 time your Bloodthirst. I don't think this is ever gonna happen with decent equip or am I wrong?

If my calculations are correct, the above calculation would be the worst case scenario (I know my approach is somewhat different from yours with the BT delay, but I think it is accurate too).

Having thought this through, I have a question concerning your calculation Thegreatme:
You assume, BT is delayed 1.5s I would think that is not quite correct, as BT is delayed by 1.5s inside of one rotationt, but there is a gap of 1s after the second BT in a rotation where you do not use Slam after WW.
This gap is filled by the Slam after WW (no matter if the Slam procs after BT again).
This is where the overall duration of 8.5s for a WW -> Slam ... rotation comes from.
So I would say the delay is only .5s.
BUT it is not only a delay of one BT!
The .5s delay influences the complete next rotation, that means at least 2x BT and one WW and at most 2xBT, 2x Slam and one WW.

Thus you can not only take the demage relation between BT and Slam into Account, but have to use the Slam = 0.46*WW + 0.9*BT formula (at least for the worst case scenario without HS procing BS).

for your calculation, the damage loss is more:

(20% WW)
0.36*50=18 two slam rotations
18*.5s*2=18s of BT pushback
18*.5s=9s of WW pushback
18/4=4.5 BT's lost
9/8=1.125 WW's lost

8.784 gained slams

so SlamDamage = (4.5BT + 1.125WW)/8.784 to have an equal damage

(40% WW)
0.488*50=24.4 two slam rotations
24.4*.5s*2=24.4s of BT pushback
24.4*.5s=12.2s of WW pushback
24.4/4=6.1 BT's lost
12.2/8=1.525 WW's lost

11.9072 gained slams

so SlamDamage = (6.1BT + 1.525WW)/11.9072 to have an equal damage

Again best case scenario here with 100% HS uptime etc.
Also I think it is not completely accurate, as you (odd as it sounds) loose Slams by gaining Slams as each possible following Slam is delayed .5s too if Slamming after WW.

I don't really know how to calculate this, its getting pretty complex at this point. And it may not be necessary to be that precise.


As for the set bonuses, the increased crit chance is "easily" calculate into average damage for Slam, The increase HS damage has no influence on the rotation.

The haste proc chance is way more difficult to take into consideration I think (calculate uptime etc.)

[GronkerLonker]

Out of curiosity I will try to model the additional proc chances calculated by Thegreatme into my approach (is everyone else having so much fun with this or should I see a doctor for it?):

Assuming a 40% proc chance for BS from WW.

The HS at 0.0 boosts the proc chance for a BS after WW by an additional 20% to 60%, so the cahnce for a normal rotation is lowered by the same amount.
Two HS around the first BT give us a total chance of 60% for a second Slam after BT.

0.4*x/8s + 0.6*0.6*y/8.5s + 0.6*0.4*x/8.5s = RotB

Reads:
A 40% chance to not proc BS after WW and a 60%chance, that it does.
Of this 60% we have a 60% chance that BS procs after BT and a 40% chance that it does not.

Again RotA = RotB, some mathamagic and we get:

y=1.1042x

Again putting in the single skills:

WW + 2*BT + 2*Slam = 1.1042(WW + 2*BT + Slam)
=>
Slam = 0.1042*WW+0.2084*BT

So the damage of Slam has to be 0.1042 time the damage of your WW plus 0.2084 times the damage of your BT.

It's getting closer now, but this again is the best case scenario, that will most likely not happen in general.

I have to note here, that the two different calculations differ in their result. WW part is mostly equal, but BT part is more then two times apart.

The endresult of my calculation with Thegreatmes approach would be

Slam = 0.1282*WW + 0.5126*BT

[Quietsch]

Reads:
A 40% chance to not proc BS after WW and a 60%chance, that it does.
Of this 60% we have a 60% chance that BS procs after BT and a 40% chance that it does not.

60% after BT?
0.0 HS & WW
1.5 Slam
2.33 HS
3.0 BT
So from where did you get 60%?
It's only 40%, if i haven't made a mistake.

[GronkerLonker]

In the rotation with a Slam after the WW, the second BT is triggered at 7s.
So you have a window of 1.5s (first Slam triggerd) to 5.5s (1.5s before you would want to trigger the second BT) where a HS (or the BT at 3s) can proc a Slam.
Following Thegreatme (if I interpreted it correctly, what not necessary is the case ) there is a HS at 4.66s, that could trigger a BS proc.

So its:
0.0 HS & WW
1.5 Slam
2.33 HS
3.0 BT
4.66 HS + latest possibility of a BS proc
^ if proc at this
4.66 Slam (I don't think one could do that, but the timespan should be enough)
6.16 you're of GCD again and have .84s left for the next BT.

/e some polishing...

[GronkerLonker]

I think I will run a calculation tomorrow with a 40% chance for the first Slam (no HS) and one HS in the window till the second BT, that should be a reasonable scenario don't you think?

[Quietsch]

Ah, that was your meaning. Ok, than it's right.
(And your 2nd post also.)

[GronkerLonker]

So here we go for another scenario. We assume you have enough rage to HS once in your rotation.
If you consider using a two Slam rotation, you should do it after the first Slam and at least 1.5s before your second BT.
That means somewhere between 1.5s and 5.5s, a window of opportunity of 4 seconds. That should be enough no matter what your weapon speed is (unless you're debuffed there, but then you may not have enough rage to HS anyways).
You would not want to to it at another time, as you want to reduce the number of rotations where you can only Slam after WW and not after the first BT, so increasing the chance to Slam after BT is the key point.
This would also improve the proc chance in a rotation where you do not Slam after WW, so its the propper place to HS there too.

Ok let's go:

RotA = x/8s again

RotB = 0.6*x/8s + 0.4*04*y/8.5s + 0.4*0.6*x/8.5s

Reads:
A 60% chance that BS does not proc after WW (normal rotation) and a 40% chance that it does.
Of these 40% a 40% chance, that BS procs after the first BT (20% from BT and 20% from the HS) and a 60% chance, that it doesn't.

Set them equal, calculate:
y = 1.15625x

Put in the detailed Skills:
WW + 2*BT + 2*Slam = 1.15625(WW + 2*BT + Slam)
Slam = 0.15625*WW + 0.31250*BT

So if I am not completely mistaken, in that scenario tha average damage of your Slam has to equal 0.15625 times the damage of your WW and 0.31250 times the damage of your BT.

Assuiming a 20% proc chance of WW we get the same result.
So that's a bit surprising at first glance, but I think it just states, that it reduces the damage gain of the 2 Slam rotation (WW->Slam->BT->Slam->BT) with the damage loss of the 1 Slam after WW rotation (WW->Slam->BT->something else->BT) by the same percentage.
It makes sense that only the distribution of the percent chance between damage loss and damage gain within the rotation is important.

[GronkerLonker]

I just discovered an important mistake in my reasoning. I didn't take into account, that the WW in the 1 Slam rotation can proc BS as well and so the chance to Slam after BT in that rotation is 20% to 40% (depending on the proc chance of WW) higher then in the 2 Slam rotation if Slam does not proc after WW.
That could be a big amount of damage, looks like I have to do it all over again *sigh*.

[Thegreatme]

the values you are using for WW's chance to proc BS are off. this is because you are not taking in to account that there could be a proc off both hits, but that would still only count as 1 proc. so in reality the theoretical proc rate would be 36% and not 40%