Shadowburn vs. Chaos Bolt - a mathemathical comparison

[Desos 18]

In General/Disclaimer

 

Because I saw Zagam´s post in the other thread, I just want to say something before someone points it out:

These calculations are just a lenghty and maybe detailed version of knowledge already present in guides. I just wanted to know, if there is some reachable treshold for Haste.

Thumb of rule is still: sub 20%-> Shadowburn.

Gains by usage of CB sub 20% are easily outperformed by improvements in other areas of gameplay.

 

As requested from Liquidsteel, I want to show some calculations I made to find out, if there is a point, where Chaos Bolt can outperform Shadowburn. Someone who does not find numbercrunching fun, can skip to the result at the bottom.

 

Basic Explanation:

I wanted to find out at which circumstances Chaos Bolt´s damage could be higher than Shadowburn´s damage, that means for a fix point in time with given values for Spell Power, Mastery, Crit and Haste. 

Therefore we need the formulas used for calculating their damage and compare them. At one point, I used a little trick, which could lead  to some difference of maximum 0.0002%, which is very negligible. 

 

Used Formulas found on wowhead:

 

CB = (2488(+258.75% SP))*1.24*2*M*C

SB = (3365(+350% SP))*1,24*M*C

 

Legend:

CB = Chaos bolt

SB = Shadowburn

SP = Spellpower

M = Mastery 

C = Crit

H = Haste

 

Some notes regarding the formulas:

Both formulas really differ only in base daage, SP coefficient and the guaranteed crit chance, which is represented by the extra "*2" in the formula of CB. Some could say that it belongs into C but it ends in the same result. Plus, this form makes it easier to equalise both formulas t answer the given question:

 

(2488(+258.75% SP))*1.24*2*M*C ? (3365(+350% SP))*1,24*M*C

 

Now with all these parameters, we could draw some crazy geometric forms to decide, but as I said, we want to compare the damage under the same given values, therefore we can eliminate M, C and "1.24":

 

(2488(+258.75% SP))*2 ? (3365(+350% SP))

 

Now we have the problem with the base damage. We cannot really say, how much influence both base damage values have. Of course their influence weakens for higher Spell Power but to be able to further compare Chaos Bolt and Shadowburn, we would have to normalize their contribution. That would take up some further space or we could use some trick first to find out how much they actually contribute. this is easily down by solving both terms with real world numbers for Spell Power. I took tresholds at 50k, 70k, 80k and 100 k Spell Power, or so to speak: No Procs, KTT/PBI, BBoY and PBI+BBoY.

 

As example I take 50k:

Base Damage CB = 2488 + 2.5875*50000 = 131863

Now the contribution in percentage would be :

2488/131863 = 1.8868 %

The same for Shadowburn gives:

3365+3.5*50000 = 178365

3365/ 178365 = 1.8866%

 

The difference between both values of contribution is as I said very negligible since we want damage numbers for a MMO and not numbers for building bridges.

 

This assumption leads us to this:

 

258.75% SP*2 ? 350% SP  or

517.5 % SP ? 350% SP

 

From a DPE perspective, Chaos Bolt would win now with some serious difference. But for a real comparison, we have to measure the influence of Haste/cast time. We now "normalize" both spells with their cast time. To get rid of the ugly percentage symbol and to just handle with one parameter, we eliminate them together with the SP parameter, which should be the same for both:

 

517,5/(3/1+H) = 350/1

 

Now it is easy to figure out the Haste Value for which Chaos Bolt hits for the same:

 

Result

 

(1+H)*172,5 = 350

1+H = 350/172,5

H = 350/172,5 -1

 

H ~ 1.029 or 102.9% Haste

 

Yeah, we found it. But for a almost useful calculation, we have to measure, if there is a common situation, where we can cast Chaos Bolt under these circumstances without having to sacrifice other stats to reach this. Let´s look at common haste buffs, that help with reaching this:

5% buff provided by different classes/specs like shadows, boomkins, eles and hunter pets

20% by troll racial

30% buff by Heroism/Bloodlust or LMG

*I don´t count Lifeblood because for Destro there are more useful professions.

 

All combined would give more haste than you need but the chance is really small for it to happen sub 20%

1.05*1.2*1.3*1.3=2.1294 or 112.94 %

 

If we take one 30% buff away we would need: 10145 haste rating

 

If we take the troll racial away, we would need: 6096 haste rating

 

If we take one 30% buff and the troll racial away, we would need : 20673 haste rating.

The last one is of course not favorible anymore.

 

 

Usage:

 

I made up my mind, when this could be useful. No one would want to build up another WA for some arbitrary combination of freaking 3 haste procs/buffs. So the fight/part of the fight has to have the condition:

BL towards or at sub 20%

 

Norushen:

Most groups use Bloodlust towards the last third of the fight. Of course there are adds, for which a Havoc´ed SB is favorable. But there are times where no add is present.

 

Sha:

Actually a stricter case than Norushen with reflections, adds and those prisms, where Havoc and Shadowburn are more attractive.

 

Galakras:

Good Example, if you don´t have to run.

 

Shamans:

Assuming 3 tank /separated groups tactic.

 

Nazgrim:

If your group wants to power/BL through Def stance and ignore adds. Also if you don´t want to paddle the meters by Havoc´ed SB´s.

 

Garrosh:

Most groups use BL to fasten the transition between P2-P3 or P3-P4.

 

 

Feel free to criticize / ask questions / point out errors.

 

[Zagam 1,982]

Only issue with your math is you're only looking for a Haste breakpoint while ignoring all Crit measures.  CB scales with Crit, so you'd have to factor that in, while SB scales at a higher base rate but only IF it crits.  Even if you had 1000% Haste, SB would outweigh CB if you had a guaranteed crit chance, so a Haste breakpoint is not sufficient to say CB > SB under a certain Haste threshold. 

 

For example, if you had 4pc, Dark Soul, Expanded Mind, and BBoY up along with Tempus Repit sub 20%, even if you're like me and stack 16k Haste, SB is going to (on average) come out ahead of CB casts because of the massive crit chance. 

 

One thing to keep in mind is crit damage, such as that from a Warrior's Crit Banner.  If that is down, CB is going to be more reliable.  The reason being that SB wouldn't benefit UNLESS it crit.

 

The TL;DR version is if you're trying to decide between CB and SB for optimal damage during execute, you're likely skilled enough that the variance in your DPS will be low regardless of what you choose.  There is a VERY small gain to be made if you correctly use CB under execute phase, but it's not worth scratching your head over unless you've COMPLETELY mastered every other facet of the spec.

[Desos 18]

Yeah, it´s  theoretically and RNG can still screw you.

I thought I actually baked the crit component into the comparison well enough. Theoretically the 2 doubles the damage of Chaos Bolt in the formula and then Crit scales it up. Therefore on average it works for the formula analog to Shadow Burn.

 

Let´s assume a crit chance of 30% and 10 attacks for an amount of 10

Scaling or Critting doesn´t matter in the long run as long as you keep it at or below 100%. After that scaling is "better".

SB model:

10+10+20+10+10+20+10+10+10+20 =130

CB model:

13*10 = 130

 

Edited July 17, 2014 by check0790

[Zagam 1,982]

Just a heads up - since everyone uses PBI, you might want to bake in 2.09 in instead of 2 since PBI inflates crit damage by a percentage indicated on the trinket.  This is important for both formulas since if you have a 588 version, it's 9.7% which is no fluff.  You're exactly right about the 'on average' part - even if you do it properly, you may not see a damage change.  If you ONLY used CB or ONLY used SB, you might see more damage either way - it's all about what happens.  This is one thing I'm looking forward to in WoD - the streamline of damage dealing seems like it'll be far less variable.  The variables in all of this shit is enough to make one go crazy. 

 

Not criticising your work - all theorycrafting is welcome and encouraged here.  I also got my Invisalign put in yesterday and I'm in so much pain right now, that I almost want kicked in the dick just to distract me from the mouth pain.  So if I seem cranky or wrong, just give me a Snickers.  Actually don't - I can't chew the fucking thing.

[Lockybalboa 618]

Math.

 

Mathy math.

 

Mathy math mathers.

 

MEGA MATH MATHY

 

SUPER UBER MATHERS

 

MIND BLOWING MATHY MATH

 

 

 

 

 

....Well, I don't understand half of what you said about the math because...well I suck at math. But the application of the theory looks sound. Very well done, I tip my cap to you.

 

 

[Zagam 1,982]

Also, crit chance is going to be formulated with a probability function, but not a discrete one like you mention.  Basically, you'd want to find N, which is the number of casts done of either CB or SB.  I would imagine under 20%, most bosses would see what, 6-10 casts of your execute?  Maybe more?  Depends on the DPS around you.  Regardless, you'll build a function where c = crit chance, n = casts, and p would be the likelihood of the event.  This probability formula is called a binomial distribution (binom.dist in Excel).

 

If you assign arbitrary crit ratings, you can calculate the probability of obtaining so many crits.  For example, let's assume c = 0.3 for a 30% crit rate and n = 10 for number of events, or casts.  Next, you'd want to calculate the probability of obtaining 0 crits, 1 crit, 2 crits, etc.

 

For this, you can do something like...

 

P(0) = probability of obtaining 0 crits in 10 casts with a 30% crit chance. 

P(0) = 0.028 = 2.8%

P(1) = 0.121 = 12.1%

P(2) = 0.233 = 23.3%

P(3) = 0.267 = 26.7%

P(4) = 0.200 = 20.0%

P(5) = 0.102 = 10.2%

P(6) = 0.037 = 3.7%

P(7) = 0.009 = 0.9%

 

This would be for calculating Shadowburn only since CB always crits.  Now, notice that you could safely assume that you'd get 2, 3, or 4 crits with a total probability of 70%.  Not solid, but good enough for quick calculations.  However, since we said we'd never cast CB without 4pc bonus, let's add 15% to our c.  Fuck it, let's add another 30% since we're feeling awesome.  So now c = 0.3 + .15 + .3 = .75.  Rerunning binom.dist...

 

P(0) = 0

P(1) = 0

P(2) = 0%

P(3) = 0.03%

P(4) = 1.6%

P(5) = 5.8%

P(6) = 14.6%

P(7) = 25.0%

P(8) = 28.1%

P(9) = 18.8%

P(10) = 5.6%

 

Here, you can see that you can pretty much bank on getting a series of crits with a 75% crit chance.  In fact, you have a 92% of getting at least 6.  You can use this to factor in where the actual DPET breakpoint is.  I'll give you a hint - it's pretty close to saying "if you got procs, SB, otherwise, CB."

[Zagam 1,982]

If you like binom.dist, it will help you with SO many things in WoW.  You can use this to simulate probabilities of events with relative ease.  Most people hate statistics, but if you think about it and just understand basic principles, you can apply those principles to more complex events and make sense of it.

 

Everyone thinks it's ridiculously unlucky to not have PBI proc until after BBoY procs.  What about if it doesn't proc until Dark Soul wears off?  Want to find out how unlucky you are if you have either event happen to you?  Binom.dist, bitches.

 

Let's assume you get 20 spell hits off in 10 seconds and 40 off in 20 seconds (may vary, substitute your own numbers in - everything else is constant).  PBI has a proc rate of 15% outside of its internal CD.  So let's binom.dist it.

 

P(20) = probability of landing 20 spells without getting PBI to proc with a 15% = 3.8% 

P(40) = 0.15%

 

Curious for other values?  Plug in a different number for "trials", which would relate to spell casts in this scenario. You can see that 4% of the time, you will likely not get PBI within BBoY.  Shitty, but 4% isn't really crazy bad odds.  This is 4x as likely as you will see Ashes or some other 1% mount drop.  Not seeing PBI until DS wears off with 40 casts?  Crazy low odds - 1/6 of 1%.  Can it happen?  You betcha. 

[Desos 18]

Yeah that would be actually more accurate. It´s the same lazy trick as with the base damage shortcut. But I would like to see someone that gets 6 cast of either into one 4p proc

You could also calculate the uptime of it during DS, which is only in the best case 50% due to need to trigger ember exactly after ICD. 

 

OT:The first 2 or 3 days can be a real pain in the ass, where you can only sleep with some pain killers. But the first day is the worst. Atleast it was for me. But it is worth the effort

Edited July 17, 2014 by check0790

[Zagam 1,982]

Yeah that would be actually more accurate. It´s the same lazy trick as with the base damage shortcut. But I would like to see someone that gets 6 cast of either into one 4p proc

I was calculating number of casts in entire execute phase, not just one 4pc proc or any one specific proc.

 

OT:The first 2 or 3 days can be a real pain in the ass, where you can only sleep with some pain killers. But the first day is the worst. Atleast it was for me. But it is worth the effort

I imagine.  Just took them out for lunch and it felt like that first pee in the morning - so relieving.  Put them back in after lunch and it was like Mike Tyson converted to a kickboxer and landed a roundhouse on my entire front upper row of teeth.  The end justifies the journey, but good lord this is going to suck.  Good thing my future wife is an amazing cook and makes a mean chili and chicken noodle soup.

[Gnar 26]

Alright well heres the math I wanted, lol. I won't dispute that it's possible for CB to be better now (very nice math and posts) in an ideal case but this ideal case is really "ideal". No movement sub 20% with nothing killable in sight and haste out your ass. Doesn't help that I play Alliance and don't have berserking to make me even consider this. Regardless, the extra damage difference is hella negligible even in perfect circumstances and if you play the rest of the class perfectly, jusing CB is not gonna make your DPS rise or fall significantly. Although, if you do use CB all the time with no SB, then you are in fact dumb (reference to my post that spawned this mathematical nonsense and theorycrafting, great post all the same).

[Desos 18]

As the thread name says, it is just a mathematical comparison. It also has flaws(PBI/Crit banner not included). But it made fun and was actually something "new", which I can play around with on live. I got no beta invite, but I like to think about mechanics.

[Liquidsteel 279]

My head hurts.

[Gnar 26]

As the thread name says, it is just a mathematical comparison. It also has flaws(PBI/Crit banner not included). But it made fun and was actually something "new", which I can play around with on live. I got no beta invite, but I like to think about mechanics.

Regardless, it is a well thought out post in response to my very simple comment of someone presumably forgetting to use Shadowburn. I will say that my min maxing efforts will go towards other aspect of my play after seeing all this math though. 4pc SB FTW!

[Desos 18]

Try reading Thecks Sacredduty, there are sometimes really hard posts about theorycrafting. Actually try it, there is a good post about beginner theorycrafting

[Gnar 26]

Ha, I'll take a look at it when I get home from work. Maybe then I can scrutinize every cast I make instead of every other cast... the life of a lock.

[Kazistrasz 216]

....Well, I don't understand half of what you said about the math because...well I suck at math. But the application of the theory looks sound. Very well done, I tip my cap to you.

 

 

 

I have a degree in math, and even I TL;DR'd on all of that.

[Desos 18]

Almost forgot: thanks for all replies, either the hattipping or adding ones ;)

[gahhda 95]

Simc has validated this point since the start of the expansion. Shadowburn at best breaks even with Chaos Bolt sub 20% in general, there isn't really a point where either becomes better that is NORMALLY achievable. The primary difference is chaos bolt always crits which means its crit scaling is automagically smoothed. Shadowburn will average out to a similar Damage/Ember if the crit is hyper averaged, but that rarely happens. This tier, however, we have large amounts of crit available and large amounts of bonus crit achievable (DS/set bonuses). That means that, CURRENTLY, you usually actually have relatively high crit rates with shadowburn, meaning it is less subjective to rng.

 

That being said, Shadowburn is normally significantly faster than chaos bolt. That time not spent casting chaos bolt gets spent generating more embers for more shadowburns.

 

Overall, there is an incredibly insignificant difference on a patchwerk fight, but shadowburn allows more control, better cleave, and better movement. No reason to hardcast CB sub 20%.

Edited August 1, 2014 by gahhda