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 Post subject: CS 101: Haste (LONG - get some Gin before reading)
PostPosted: Tue Jan 12, 2010 8:16 pm  
Boomkin
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Hello again fellow Balance Druids. This is Caster Scaling 101; an indepth look into the fundamental concepts that drive caster DPS and how they interact with various talents, gear, spells, and effects. The focus of this topic will be on Haste. Particularly, what Haste does, or, rather, how Haste directly influences DPS in various ways.

The Basics: How Haste directly effects DPS as it relates to spell cast time.


Spells come in many different flavors, but, for the most part, there are three major cast time sections - 1.5, 2, and 3 seconds. There are a few spells that also fit within the 2.5 cast time section, such as Arcane Blast, but most of these are fairly rare. Anyway, let us assume an equal damage spread between all of these spells. That is to say, a 1.5 second spell deals 555 damage, a 2 second spell deals 740 damage, and a 3 second spell deals 1,110 damage; giving all of the spells a DPS value of 370. Now, apply a moderate, say 10%, level of Haste to each of the cast times.

The basic cast time formula is fairly simple, being Base Cast Time / (1 + (Haste Percentage / 100)). Thus, it follows:

1.5 / (1 + (10 / 100) = 1.3636

2 / (1 + (10 / 100) = 1.8181

3 / (1 + (10 / 100) = 2.7272

Haste does not alter the damage of spells, therefore the damage of all these spells remains unchanged. Now, to calculate out the DPS of a spell, you merely divide the damage done by the time it took to deal the damage. Thus, it follows:

555 / 1.3636 = 407.010854

740 / 1.8181 = 407.018316

1110 / 2.7272 = 407.010854

The damage remains relatively the same, in fact it is exactly the same for a 1.5 and a 3 second cast, but is slightly increased for a 2 second cast. The difference is difficult to see at this level of Haste, so let us up the numbers a bit, shall we? Now, let us assume that we are dealing with 40% Haste.


1.5 / (1 + (40 / 100) = 1.0714

2 / (1 + (40 / 100) = 1.4285

3 / (1 + (40 / 100) = 2.1428

Now for the DPS values:

555 / 1.0714 = 518.013814

740 / 1.4285 = 518.025901

1110 / 2.1428 = 518.013814

As you can see, the difference is very small, but that is to be expected. Haste, as a base stat, does not significantly favor any cast time.

The Next Step: How Haste effects DPS as it relates to Spell Power Coefficients.

Spell Power Coefficients are a different creature in of themselves, and I will discuss it in depth at a later date, so, for now, here is a simple run down of the basics. Spell Power Coefficients are modeled after a simple equation of Base Cast Time / 3.5. There are a few exceptions to this rule found in specific spells, but that is not related to this topic.

Although most classes deal with cast times that are either 1.5, 2, or 3 seconds, a majority of those are modified cast times via a talent which reduces it by a factor of .5. Most spells are actually a 2, 2.5, and 3.5 second cast spell, but there will be more to that in a moment. This means that a base 2 second spell would have a Coefficient of 57.1428%, a base 2.5 second spell would have a Coefficient of 71.4285% and a base 3.5 second spell would have a Coefficient of 100%.

From a base setting, there exists a complete equality between the Coefficient of all spells. Before Haste is put into play, all spells gain 28.5714% Spell Power per second. Haste slightly alters this, however.

2 / (1 + (10 / 100) = 1.8181
57.1428 / 1.8181 = 31.4299543%

2.5 / (1 + (10 / 100) = 2.2727
71.4285 / 2.2727 = 31.4289171%

3.5 / (1 + (10 / 100) = 3.1818
100 / 3.1818 = 31.428751%

The difference is not very large, but it is there. As base cast time increases the amount the Haste appreciates the Spell Power Coefficient of a spell decreases. While this isn't very significant at these variables of Haste, it becomes more significant as Haste increases. Generally, all Haste increases are going to be equalized across the spectrum, however that isn't always true. As I said, many casters have talents which reduce the base cast times of their spells be a specific amount, generally by .5 seconds. While the gain to each spell is static, the actual effect is now. Reducing the cast time of a 2 second spell to 1.5 seconds requires more Haste than reducing a 3.5 second spell to 3 seconds. Thus, the lower the initial cast time of a spell, the higher than gain. Looking at the Spell Power per Second of these spells after their respective reductions is put into play is a different story.

1.5 / (1 + (10 / 100) = 1.3636
57.1428 / 1.3636 = 41.9058375%

2 / (1 + (10 / 100) = 1.8181
71.4285 / 1.8181 = 39.2874429%

3 / (1 + (10 / 100) = 2.7272
100 / 2.7272 = 36.6676445%

Where once there was only a minor, somewhat insignificant difference, there is now a very significant difference to the Coefficient values of each spell. This is because talents which reduce the cast time of a spell by a flat amount are essentially nothing more than Haste gains, but they are much larger Haste gains for faster casting spells which causes the initially small discrepancy to grow exponentially.

With the addition of Spell Power, Haste begins to lose it's impartial DPS gains and begins to favor faster casting spells more so than slower casting spells. We see this very clearly with Balance Druids and Wrath/Starfire. When talented with Starlight Wrath (reduces the cast time of Starfire/Wrath by .5) and Wrath of Cenarius (increases the Spell Power Coefficient of Wrath by 10% and Starfire by 20%,) Wrath gains 44.7618667% Spell Power per Second and Starfire gains 40% Spell Power per Second. As Haste values increase, the gap of Spell Power gains between to two spells also increases. With an additional 50% Haste (Giving Wrath a 1 second cast time and Starfire a 2 second cast time,) Wrath gains 67.1428% Spell Power per Second and Starfire gains 60% Spell Power per Second; an increase of approximately 62.5%.

Digging Deeper: Haste as it effects % gain talents.

Another staple of many caster talents is the flat % gain in damage done. Most casters have a 5 point talent which will increase the damage of certain spells or effects by 10%. Going back to the original DPS value figures where we had all spells dealing an standard 370, we will now increase all of the damage done by these spells by 10%. This means that our 1.5 second spell is now dealing 610.5 damage, our 2 second spell is dealing 814 damage, and our 3 second spell is dealing 1221 damage. This increases the DPS of all of our spells to 407 - the DPS is still equalized. Now, throw in the Haste.

1.5 / (1 + (10 / 100) = 1.3636
610.5 / 1.3636 = 447.711939

2 / (1 + (10 / 100) = 1.8181
814 / 1.8181 = 447.720147

3 / (1 + (10 / 100) = 2.7272
1221 / 2.7272 = 447.711939

The equal values of the 1.5 and the 3 second spells remains the same, such as it was when comparing base cast times. Also, the value of the 2 second cast spell was 0.0018% higher than the gains seen by the 1.5 and 3 second spells, which, again, was the same as before. In short, Haste does not have any impact on the value of flat % damage gain modifiers.

Two for the Price of One: Haste as it effects the DPS value of Crit

Again returning to our standard of normalized DPS, let us now, briefly, look into how Haste effects Crit. We'll go a bit of an 'extreme' in giving both of our spells a 50% chance to crit; for double damage of course. This would mean that a 1.5 second cast spell would deal an average damage of 832.5, a 2 second spell would deal 1,110 damage, and a 3 second spell would deal 1,665 damage. All of the spells would retain a normalized DPS of 555, showing that Crit, at a base level, does not favor any casting speed. Now, to look at the numbers with Haste to see if that normalization is retained.


1.5 / (1 + (10 / 100) = 1.3636
832.5 / 1.3636 = 610.51628

2 / (1 + (10 / 100) = 1.8181
1110 / 1.8181 = 610.527474

3 / (1 + (10 / 100) = 2.7272
1665 / 2.7272 = 610.51628

As shown, Haste has no true influence on the value of Crit nor does Crit influence Haste at any base-level. A 1.5 and 3 second cast spell retain the exact same DPS values, while a 2 second cast spell is still 0.0018% higher.

The Balance Equation - How Haste interacts with the culmination of Balance Druid talents.

All of these figures are well and good for understanding the basic principles of Haste, and knowing the basics of any concept is a wonderful foundation. Builds, however, go well beyond their foundations, so too does Haste go beyond what is shown here thus far. Here is a quick re-cap of what we know about Haste:

1. Haste does not hold any direct correlation to base-cast time. All casting speeds are effected equally under Haste effects.
2. Haste does hold a direct correlation to Spell Power Coefficients. Haste benefits spells with a lower base-cast time more than those with a higher base-cast time.
3. Haste does not hold any direct correlation to %damage modifiers. All casting speeds are effected equally under Haste effects.
4. Haste does not hold any direct correlation to Crit. All casting speeds are effected equally under Haste effects.

Before we can begin to go into the depths of Balance Druid Haste scaling, we must first look at the base values of both Wrath and Starfire. Previously we've been working under assumptions of spells with differing cast times but equal DPS out-put. This isn't how Wrath and Starfire function from a base perspective though. Here are the base damage models.

Wrath deals an average of ((627 - 557) /2) + 557 = 592 damage. It has a base cast time of 2 seconds, which would normally yield a DPS value of 296, however, Starlight Wrath reduces this cast time by .5. At a 1.5 second cast speed, Wrath's base-line DPS is 394.666.

Starfire deals an average of ((1222 - 1038) /2) + 1038 = 1130 damage. It has a base cast time of 3.5 seconds, which would normally yield a DPS value of 322.857143, however, Starlight Wrath reduces this cast time by .5. At a 3 second cast speed, Starfire's base-line DPS is 376.666

The next important talent that we come across is Nature's Grace, which increases spell casting speed by 20% for 3 seconds on Critical Strike. Normally, without factoring any Haste effects, this would mean you get 1 Starfire and 2 Wrath casts per Nature's Grace proc. However since Nature's Grace is a Haste effect in of itself, it allows for you to get 2 Starfires and 3 Wraths off per cast. Due to how Nature's Grace functions, the general formula used to calculate out it's effects is:

(1 + (Nature's Grace Effect * (1 - (1 - Critical Strike Chance)^Number of Casts))

Although Nature's Grace essentially acts as a function of Critical Strike, the amount of average Haste gained from the talent is determined by Critical Strike up to a maximum of 20% Haste at 100% up time which theoretically only occurs at 100% Critical Strike Chance, we are solely interested in the Haste portion of Nature's Grace. For this reason, we are going to set Nature's Grace up time to 100% automatically and merely apply a 20% Haste boost. This would effect the base DPS of Wrath and Starfire like this:

Wrath
1.5 / 1.2 = 1.25
592 / 1.25 = 473.6

Starfire
3 / 1.2 = 2.5
1130 / 2.5 = 452

From Nature's Grace, Wrath went from dealing 394.666 DPS to dealing 473.6 DPS. This is a 20.0002027% gain in DPS. Starfire went from dealing 376.666 DPS to dealing 452 DPS, which is a 20.0002124%. Normally, these would both be rounded simply to a 20% increase in DPS, however that would hide an underlying factor. Starfire gained a slightly higher DPS gain than Wrath did. The amount is hardly significant, but it is there and other factors could potentially exacerbate it. This, however, comes later. Let us continue on.

The next talent of significance is Wrath of Cenarius. Although there are many talents before this one, such as Vengeance, Moonfury, and Improved Insect Swarm, none of these should hold any baring on the scaling nature of Haste as they are either Crit effects or %damage increases which we already know is not impacted by Haste. Spell Power Coefficients, however, are. As was already mentioned under the comparison of Haste and Spell Power Coefficients, Haste favors faster casting spells in increasing their Spell Power gains more so than slower casting spells. When you only consider Starlight Wrath in conjunction with Wrath of Cenarius, Wrath gains 44.7618667% Spell Power per Second while Starfire gains 40% Spell Power per Second. As Haste increases, the disparity continues to grow, until it maxes out at 50% Haste where Wrath gains 67.1428% Spell Power per Second and Starfire gains 60% Spell Power per Second.

At that point, however, Wrath ceases to gain any additional Spell Power per Second as Haste increases due to the fact that Wrath's cast time essentially cannot go lower than 1 second while Starfire's can go below 2 seconds. Therefore, up until the point wherein you hit 50% Haste, Haste will benefit Wrath more than it will benefit Starfire, while Haste will benefit Starfire more than Wrath at all points beyond 50% Haste.

50% Haste is a lot, however, and isn't a value that you are going to get through raw gearing. This does not make it impossible to hit, however, because there are other Haste increasing effects beyond gear in the game. One of which, Nature's Grace, we've already covered. Beyond Nature's Grace, there is also Wrath of Air Totem for 5% Haste and Celestial Focus for 3% Haste. These additional effects, however, are not additive, they are multiplicative with one another. Therefore, having all of these effects up would effect our spell's cast times as follows:

Wrath
((1.5 / 1.03) / 1.05) /1.2 = 1.1558

Starfire
((3 / 1.03) / 1.05) /1.2 = 2.3116

Essentially, this places Wrath gaining 58.0920% Spell Power per second and Starfire gaining 51.91209% Spell Power per Second. This is all under the assumption of 100% Nature's Grace uptime, however. Under normal circumstances, Nature's Grace will mostly likely not have a 100% uptime. Practically speak from a Balance perspective, Nature's Grace uptime is a concept which only applies to Wrath and is irrelevant to Starfire. Starfire will be at a 100% Critical Strike Chance during a Lunar Eclipse and thus have a 100% up time, and you only need to get one Crit outside of a Lunar Eclipse in order to proc a Solar Eclipse which means that Nature's Grace is either still up from Lunar or it isn't up at all. Therefore, we will act under the assumption that Starfire retains a 100% Nature's Grace uptime while Wrath does not. For modesty sake, we'll operate under a 50% Crit chance for Wrath, which is about the current average.

1 + (0.2 * (1 - (1 - .5)^3)) = 1.175

At this level, Nature's Grace is essentially worth 17.5% Haste for Wrath. This would alter our previous cast time to:

((1.5 / 1.03) / 1.05) /1.175 = 1.1803

At a 1.1803 cast time Wrath gains 56.8862% Spell Power per Second. In order to find the arbitrary point break of Haste scaling for Wrath, we would use the formula ((Base Cast Time / Desired Cast Time) - 1) * 32.79 * 100. In this case, our Base Cast Time is 1.1803 and our Desired Cast Time is 1. Therefore we would need 591.2037 Haste Rating in order to achieve this, since you cannot have factions of Haste, the value would actually be 592 or 18.0542% Haste. At this level of Haste, we know that Wrath will have a Spell Power per Second of 67.1428%, Starfire would have:

2.3116 / 1.180542 = 1.9580
120 / 1.9580 = 61.2870% Spell Power per Second

Now, to configure it so that we are at a point wherein Starfire gains the same Spell Power per Second as Wrath, we would need to calculate out the Haste requirement.

120 / X = 67.1482
120 = 67.1482X
120 / 67.1482 = X
1.7870 = X

Starfire needs to have a cast time of 1.7870 in order to meet the Spell Power per Second of Wrath. Therefore ((2.3116 / 1.787) - 1) * 32.79 * 100 = 962.5984. At 963 Haste Rating, Starfire would have the same Spell Power per Second as Wrath. Prior to this point, Wrath would scale better with Spell Power than Starfire, after it, the roles are reversed. This is an important function of the scaling nature of Haste. Haste initially favors faster casting speeds, until you meet their arbitrary limitations. At that arbitrary value, Haste then begins to favor longer cast times. There is no effective method to balance out Haste scaling; the scaling value of Haste would only be equal for specific spells at specific arbitrary values, before which point, they will favor shorter casting spells, and after which Haste will favor faster casting spells.

How does this all play into the 'actual' DPS of Wrath and Starfire? It all depends on the relative values of other stats. Haste does not inherently favor faster or slower casting speeds; from a base value all spells are effected by Haste equally. Haste only effects the scaling nature of Spell Power, therefore the results of Haste scaling are going to be a direct correlation to approximate Spell Power values. Although Spell Power is not the topic of this write-up, I will still continue on to illustrate some points. I'll continue to use 50% Crit as a variable and pick an arbitrary Spell Power value of 3,800.

Wrath
(592 + (3800 * .671)) * 1.1 * 1.03 * 1.04 * (1 + ((.50 * 1.09) * 1)) * 1.13 * 1.03 * 1.03 = 6856.82651

Starfire
(1130 + (3800 * 1.2)) * 1.1 * 1.03 * 1.04 * (1 + ((.53 * 1.09) * 1)) * 1.13 * 1.03 = 12311.6315

Now we'll look at the DPS of these spells are variable levels of Haste. First will be to look at the base times we had just configured. That is not taking into account any Haste from gear, but only the Haste from Wrath of Air Totem, Celestial Focus, and Nature's Grace. To keep the results more practical, I'll continue to assume a 100% Nature's Grace uptime for Starfire and use the previous calculations for Wrath.

Wrath
6856.82651 / 1.1803 = 5809.39296

Starfire
12311.6315 / 2.3116 = 5326.02159

Before the addition of Crit and Spell Power, there was a 2.6% difference between Wrath and Starfire, favoring the former. With the addition of Spell Power and Crit (well, and raid buffs, but those are flat modifiers which are unaffected by Haste,) there is a 9.0756% difference in DPS between Wrath and Starfire, still favoring the former. Assuming that both spells scale with Crit at the same variation, and by all accounts they should, then we can see that Spell Power causes this additional gap. It is important, however, to determine how much of this increase in the discrepancy is due to Spell Power scaling and how much is due to Haste scaling. Therefore, we will remove all Haste effects including Starlight Wrath from the equation- keep in mind that, at a base level, Starfire deals more DPS than Wrath without Starlight Wrath.

6856.82651 / 2 = 3428.41325

12311.6315 / 3.5 = 3517.609

As we can see, Starfire is once again back to dealing higher levels of DPS than Wrath does. This is to be expected though, as I said, without Starlight Wrath, Starfire deals more DPS than Wrath at a base level, therefore unless Wrath scaled significantly better from Spell Power or Crit, it should remain lower. However, without Starlight Wrath, Starfire actually ends up, initially at least, scaling better with Spell Power than Wrath; with Wrath gaining 33.5741% Spell Power per Second and Starfire gaining 34.2857143% Spell Power per Second.

Now to look at the data from a point wherein you cap Wrath at a 1 second cast time. In our theoretical data, this would occur at 592 Haste.

Wrath
6856.82651 / 1 = 6856.82651

Starfire
12311.6315 / 1.9580 = 6287.86083

The DPS disparity between Wrath and Starfire maintains close to the original disparity hovering at 9.04%. The loss of 0.03% in the disparity is attributed to the raising of the numerical data - the damage data does not scale in direct proportion to the original function, thus disparities will always shrink (you can see the same effect by testing the spells with nothing but Spell Power and no Haste effects. Starfire will always out DPS Wrath, but the disparity will continue to shirk because you start comparing them at a 100's DPS level and end up comparing them in a 10,000's DPS level, however, even at 1,000,000 Spell Power, Wrath will never out-DPS Starfire.)

This final test shows exactly what every test has been showing from the beginning - Haste scaling does not hold any direct correlation to spell cast time. Adding in additional Haste will not cause the variable DPS of the two spells to alter, it will always remain approximately the same. What will change is Spell Power. If we had increased Spell Power instead of Haste, then Wrath would continue to grow the disparity between it and Starfire - provided that the aforementioned natural drift does not obfuscate the data a bit; but the disparity of 7% should be enough for more variable levels of data, it's merely a matter of being relative towards your base value (ie. going from 100 Spell Power to 10,000 Spell Power might not show as large of a disparity as going from 9,000 Spell Power to 10,000 Spell Power.))

This trend and correlation of Haste/Spell Power will continue on causing Wrath to out-DPS Starfire until you reach an arbitrary value of Haste - the amount will be directly influenced by Critical Strike Rating (due to Nature's Grace, not an underlying mechanic of Crit itself) and Spell Power.

Wrap Up

Haste scaling in of itself is a relatively constant concept across all spells when it comes to direct damage spells. The disparities within Haste scaling between classes come from predominately two sources; amount of non-Haste effected abilities (IE. certain DoTs, Spells with cooldowns) and talents/effects which offer a disparity of Haste values (IE Starlight Wrath.)

The only current draw back to Haste within the game is the ability for select abilities to reach the effective cap; reducing their cast time below the GCD. Up until that point, all direct damage spells will scale roughly equal with Haste. This is not to say that every spec will necessarily scale with Haste exactly the same, again refer back to the above, but that their Haste scaling should be within a close approximation. Therefore, the mechanic wherein spells are capable of being able to reach below the GCD needs to be addressed for all classes in order for scaling disparities between classes to be corrected. Predominately, this is caused by an over-abundance of Haste effects, not necessarily Haste from gear. Effects such as Backdraft, Nature's Grace, Wrath of Air Totem, and Bloodlust/Heroism need to have their benefits re-addressed in order for better equality in scaling. In a perfect setting, it should never be possible for a spell to drop below the GCD in order for scaling to remain balanced.

Edit- I should add that, Haste doesn't technically increase DPS. I mean, it does, but it doesn't do it in the same scaling method that Crit or Spell Power does. Instead, the effects of Haste are directly tied into Spell Power such that, additional Haste merely increases the relative value of your current Spell Power by allowing spells to gain a larger Spell Power per Second benefit. This is why there are always theory-crafted results at various levels which will state at Spell Power/Haste X, Haste is worth more than Spell Power and at Spell Power/Haste Y, Spell Power is worth more than Haste. The oft quoted phrase that, for Balance Druids, Haste is a higher DPS yield than Spell Power below 400 Haste is only true above a specific variable of Spell Power. This is because additional Haste acts as a %function modifier instead of a flat damage function.


Last edited by Murmurs on Tue Jan 12, 2010 9:46 pm, edited 1 time in total.

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 Post subject: Re: CS 101: Haste (LONG - get some Gin before reading)
PostPosted: Tue Jan 12, 2010 8:18 pm  
Boomkin
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Words I definitely used too much:
Arbitrary
Relative
Variation
Disparity
However


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 Post subject: Re: CS 101: Haste (LONG - get some Gin before reading)
PostPosted: Tue Jan 12, 2010 9:18 pm  
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Crikey, you have been busy there. :dizzy

Some nice information in here. Thanks!

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 Post subject: Re: CS 101: Haste (LONG - get some Gin before reading)
PostPosted: Wed Jan 13, 2010 3:37 pm  
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Pleasant reading for my first post read in almost a year on TMR.

P.S. hey guys!


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 Post subject: Re: CS 101: Haste (LONG - get some Gin before reading)
PostPosted: Wed Jan 13, 2010 3:49 pm  
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Interesting post. I'm digesting. I may (or may not) have something to say about it in the near future.

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 Post subject: Re: CS 101: Haste (LONG - get some Gin before reading)
PostPosted: Wed Jan 13, 2010 4:51 pm  
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One thing that glared at me: you missed Imp. Moonkin Form or the same 3% haste buff from ret pallies.

Very nice write-up. A few grammatical/spelling errors (affect/effect), but a lot of VERY good information.

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 Post subject: Re: CS 101: Haste (LONG - get some Gin before reading)
PostPosted: Wed Jan 13, 2010 5:26 pm  
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Murmurs wrote:
The Next Step: How Haste effects DPS as it relates to Spell Power Coefficients.

Spell Power Coefficients are a different creature in of themselves, and I will discuss it in depth at a later date, so, for now, here is a simple run down of the basics. Spell Power Coefficients are modeled after a simple equation of Base Cast Time / 3.5. There are a few exceptions to this rule found in specific spells, but that is not related to this topic.

Although most classes deal with cast times that are either 1.5, 2, or 3 seconds, a majority of those are modified cast times via a talent which reduces it by a factor of .5. Most spells are actually a 2, 2.5, and 3.5 second cast spell, but there will be more to that in a moment. This means that a base 2 second spell would have a Coefficient of 57.1428%, a base 2.5 second spell would have a Coefficient of 71.4285% and a base 3.5 second spell would have a Coefficient of 100%.

From a base setting, there exists a complete equality between the Coefficient of all spells. Before Haste is put into play, all spells gain 28.5714% Spell Power per second. Haste slightly alters this, however.

2 / (1 + (10 / 100) = 1.8181
57.1428 / 1.8181 = 31.4299543%

2.5 / (1 + (10 / 100) = 2.2727
71.4285 / 2.2727 = 31.4289171%

3.5 / (1 + (10 / 100) = 3.1818
100 / 3.1818 = 31.428751%

The difference is not very large, but it is there. As base cast time increases the amount the Haste appreciates the Spell Power Coefficient of a spell decreases. While this isn't very significant at these variables of Haste, it becomes more significant as Haste increases. Generally, all Haste increases are going to be equalized across the spectrum, however that isn't always true. As I said, many casters have talents which reduce the base cast times of their spells be a specific amount, generally by .5 seconds. While the gain to each spell is static, the actual effect is now. Reducing the cast time of a 2 second spell to 1.5 seconds requires more Haste than reducing a 3.5 second spell to 3 seconds. Thus, the lower the initial cast time of a spell, the higher than gain. Looking at the Spell Power per Second of these spells after their respective reductions is put into play is a different story.


Your differences here are purely rounding error. Put these equations into Excel and take the decimal out to 15 places. The results are the exact same.

Code:
=(3.5/3.5)/(3.5/(1+0.1))
=(2.5/3.5)/(2.5/(1+0.1))
=(1.5/3.5)/(1.5/(1+0.1))


Spell Power Coeffecients do become more significant as cast times decrease due to haste affects, but this does not mean that Haste favors spells with lower cast times over those with higher cast times. As long as the haste effect is applied equally for the various cast times, the Spell Power Coefficients will stay relatively equal for each of the cast times. The excel equations above prove that. It is also important to note that Haste Rating and any talent that provides haste as a percentage will apply the Haste Effect to all Cast times equally.

The difference is when Haste affects are not applied equally. Starlight Wrath is the perfect example. Both Starfire and Wrath have their cast times decreased by 0.5 seconds. For Starfire this is equivallent to 16.6667% haste. For Wrath it is equivalent to 33.3333% haste. Now lets reverse them and do the math.

Here is your math. I'm not trying to call you out but I do want to point out there is rounding error in you values here as well.
Murmurs wrote:
1.5 / (1 + (10 / 100) = 1.3636
57.1428 / 1.3636 = 41.9058375%

3 / (1 + (10 / 100) = 2.7272
100 / 2.7272 = 36.6676445%


To get the real values here are a couple of excel formulas.
Code:
=(2/3.5)/(1.5/(1+(10/100)))
=(3.5/3.5)/(3/(1+(10/100)))


So, if you apply 33.333% haste to a 3.5 second Starfire, it's cast time is reduced to 2.625 seconds. If a 2.0 second Wrath gets 16.6667% haste it's base cast time is now 1.7143 seconds. Starfire still has the longer cast time in this example. Lets apply the same equations you used above.

1.7143 / (1 + (10 / 100) = 1.5584
57.1428 / 1.5584 = 36.6667%

2.625 / (1 + (10 / 100) = 2.3863
100 / 2.3863 = 41.9048%

The excel for my equations:
Code:
=(2/3.5)/((2/1.166666666666667)/(1+(10/100)))
=(3.5/3.5)/((3.5/1.333333333333333)/(1+(10/100)))


These are the same values you have as above, but reversed and without the rounding error.

So, it is missleading to say:
Murmurs wrote:
2. Haste does hold a direct correlation to Spell Power Coefficients. Haste benefits spells with a lower base-cast time more than those with a higher base-cast time.


Because, what you are really seeing is that it is better to get more Haste then it is to get less haste.

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 Post subject: Re: CS 101: Haste (LONG - get some Gin before reading)
PostPosted: Wed Jan 13, 2010 8:26 pm  
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Great information in there and a great read. Thank you!

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 Post subject: Re: CS 101: Haste (LONG - get some Gin before reading)
PostPosted: Wed Jan 13, 2010 9:35 pm  
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Quote:
Your differences here are purely rounding error. Put these equations into Excel and take the decimal out to 15 places. The results are the exact sam


Ahh, but the thing of it is that the World of Warcraft engine itself doesn't operate using an excel sheet, it operates on the principle of rounding to the 4th significant digit, which does cause discrepancies due to rounding that inevitably end up favoring specific cast times at specific Haste values. For example, I did make a rounding error in that equation because WoW's engine would round the 2 second cast time spell's cast time to 1.8182 instead of 1.8181 since the next significant digit would be an 8 and thus round the previous one up. That would give the spell a 31.4282257% conversion which is actually less than all of the others.

However, I did that 'mistake' on purpose because I saw this trend and the trend is that shorter cast times will gain a higher benefit from Spell Power than longer in all situations wherein the Haste value causes either all of the cast times to be rounded up or to not be rounded at all. Although, if the Haste value causes one of the cast times to round up and the others to not round up, then the one which does round up will get the lest % gain from Spell Power than the others.

You would be correct if WoW went to 15 significant digits, but it only calculates out to 4, which inherently causes discrepancies that generally favor shorter cast times over longer cast times. And, truth be told, I think you don't even have to go to 15 significant digits in order for the results to be perfectly equalized, you should only have to go out to 8.

Quote:
One thing that glared at me: you missed Imp. Moonkin Form or the same 3% haste buff from ret pallies.

Very nice write-up. A few grammatical/spelling errors (affect/effect), but a lot of VERY good information.


So I did, but that wouldn't really change anything other than the arbitrary value of Haste in which Wrath would start to gain less from Haste than Starfire does.


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 Post subject: Re: CS 101: Haste (LONG - get some Gin before reading)
PostPosted: Wed Jan 13, 2010 11:16 pm  
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Battle of the theorycrafters, go go go!

/popcorn

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 Post subject: Re: CS 101: Haste (LONG - get some Gin before reading)
PostPosted: Wed Jan 13, 2010 11:44 pm  
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Ironically that's what I was doing too (checking values in excel and extending decimal). I am curious to know how we are sure there are 4 sig figs (or do we mean decimal places)?

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 Post subject: Re: CS 101: Haste (LONG - get some Gin before reading)
PostPosted: Thu Jan 14, 2010 9:23 am  
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This thread is rapidly starting to make less sense.

1) How do you know how WoW truncates its values?
2) Even if it does stop at 4 significant figures, nobody cares about a scaling difference on the order or 10^-5.
3) Graylo's right on the math. Haste speeds up all spells by an equal ratio until the GCD cap. Even without doing any math, you can see that it increases the spellpower gains from all spells by an equal ratio.
4) The only interesting thesis you posit: that haste increases the spellpower scaling of short spells more that in does long spells, is wrong. Again, just look at the first code block in Graylo's post.
5) Some thing statements and conclusions in the OP are nonsensical. e.g. "Haste doesn't technically increase DPS."

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 Post subject: Re: CS 101: Haste (LONG - get some Gin before reading)
PostPosted: Thu Jan 14, 2010 9:54 am  
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Murmurs wrote:
Quote:
Your differences here are purely rounding error. Put these equations into Excel and take the decimal out to 15 places. The results are the exact sam


Ahh, but the thing of it is that the World of Warcraft engine itself doesn't operate using an excel sheet, it operates on the principle of rounding to the 4th significant digit, which does cause discrepancies due to rounding that inevitably end up favoring specific cast times at specific Haste values. For example, I did make a rounding error in that equation because WoW's engine would round the 2 second cast time spell's cast time to 1.8182 instead of 1.8181 since the next significant digit would be an 8 and thus round the previous one up. That would give the spell a 31.4282257% conversion which is actually less than all of the others.

However, I did that 'mistake' on purpose because I saw this trend and the trend is that shorter cast times will gain a higher benefit from Spell Power than longer in all situations wherein the Haste value causes either all of the cast times to be rounded up or to not be rounded at all. Although, if the Haste value causes one of the cast times to round up and the others to not round up, then the one which does round up will get the lest % gain from Spell Power than the others.

You would be correct if WoW went to 15 significant digits, but it only calculates out to 4, which inherently causes discrepancies that generally favor shorter cast times over longer cast times. And, truth be told, I think you don't even have to go to 15 significant digits in order for the results to be perfectly equalized, you should only have to go out to 8.


Significant Digits:
The number of significant digits you use is irrelevent. I've never completely bought the "Blizzard only uses 4 significant digits" arguement. The fact that they only use 4 significant digits maybe true (I haven't seen official confirmation), but we don't know how extensive it is. For example, how does the game determine a spells coeffiecent? Does it look it up from a table that says it is 0.5714 or is it a calculation imbedded in an equation?

Maybe I have missed something, but I've never seen anything that details how the 4 significant digits is applied.

However, it still doesn't matter. As I said before, how many significant digits are irrelevent to the arguement. The differences you showed are due to rounding error, not haste favoring shorter cast times. I will redo the math in your format to show you why.

The Math:
Once again, here are your numbers:
Murmurs wrote:
Wrath
1.5 / (1 + (10 / 100) = 1.3636
57.1428 / 1.3636 = 41.9058375%

Starfire
3 / (1 + (10 / 100) = 2.7272
100 / 2.7272 = 36.6676445%


Starlight Wrath takes 0.5 seconds off of both spells cast times. For Starfire this is equal to 16.6667% Haste, and is equal to 33.3333% haste for Wrath. If we reverse these haste affects you get the following:

3.5 / (1 + (33.3333 / 100) = 2.6250 seconds for Starfire

2.0 / (1 + (16.6667 / 100) = 1.7143 seconds for Wrath

As you can see, when you reverse the impacts of Starlight Wrath, Starfire still has a longer cast time then Wrath. Therefore, according to your statements Wrath should benefit from haste more then Starfire when it comes to spell coefficents. Right.

Now, lets redo your math with the Starlight Wrath impact reversed

Wrath
1.7143 / (1 + (10 / 100) = 1.5585
57.1428 / 1.5585 = 36.6652550%

Starfire
2.6250 / (1 + (10 / 100) = 2.3864
100 / 2.3864 = 41.9041234%

Notice how close my Wrath number is to your Starfire number and vice versa. You can clearly see from these numbers that haste does not favor casts with shorter cast times. If that were true then my Wrath number should be bigger then my Starfire number.

What your numbers are showing is that More Haste is better then Less Haste. The difference is purely due to Starlights Wrath applying unequal evels of haste to two spells while the spell coefficent remains constant.

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 Post subject: Re: CS 101: Haste (LONG - get some Gin before reading)
PostPosted: Thu Jan 14, 2010 11:58 am  
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Quote:
Significant Digits:
The number of significant digits you use is irrelevent. I've never completely bought the "Blizzard only uses 4 significant digits" arguement. The fact that they only use 4 significant digits maybe true (I haven't seen official confirmation), but we don't know how extensive it is. For example, how does the game determine a spells coeffiecent? Does it look it up from a table that says it is 0.5714 or is it a calculation imbedded in an equation?

Maybe I have missed something, but I've never seen anything that details how the 4 significant digits is applied.

However, it still doesn't matter. As I said before, how many significant digits are irrelevent to the arguement. The differences you showed are due to rounding error, not haste favoring shorter cast times. I will redo the math in your format to show you why.


Not to be rude Graylo, but this is the exact opposite of what you've said before. I used to go to only 3 significant digits since that is what the game actually displays, you, however, were the one to correct me and say I should be going to 4.

Either way, significant digits do matter, and more so on to which digit it is rounded to.

Theoretically speaking, there is a difference in Haste between a 2 second cast spell casting at a 1.1818 speed and a 1.1819 speed. The difference in Haste value, however, is different that that of a 3 second cast spell that would be casting at a 1.1818 speed and a 1.1819 speed. The lower the base cast time you are dealing with, the greater the impact rounding has because it causes a larger Haste 'discrepancy.' I would take a larger amount of Haste to reduce the cast time of a 2 second spell by 0.0001 second than it would a 3 second spell, therefore faster casting spells benefit more via rounding.

It is this discrepancy that would cause faster casting spells to benefit more from Haste than slower casting spells, because, from the base-line, all spells benefit equally.

Simple rounding simply doesn't account for all of the difference, it is the innate nature of using significant digits that does this as well. The 2.5 comparison to the 3.5 speed show this. There is a very small difference between the 2.5 and 3.5 spells, yet neither of these spells should have been rounded up because the next significant digit would have been 2 and 1 respectively.

As to the Coefficients themselves, all of them are set figures that are also rounded to four significant digits. Due to this, most Coefficients are a numerical XX.XX% figure. For some reason, WoWwiki has been edited with a Hurricane value of 12.898% which is the only Coefficient I know up that has even been theorycrafted to the fifth digit. In all of the theory that I've read on EJ and elsewhere, I've yet to see any other Coefficient be used this way, all of them have been XX.XX% factors, so I haven't a clue why Hurricane would be edited this way. I'm going to have to do some digging into EJ and some in-game testing to see if I can confirm that Coefficient. The only other exception to this rounding that I've ever seen is Frostfire Bolt with Empowered Fire (having a Coefficient of 100.71% when rounding should put it at 100.7%)


Quote:
Notice how close my Wrath number is to your Starfire number and vice versa. You can clearly see from these numbers that haste does not favor casts with shorter cast times. If that were true then my Wrath number should be bigger then my Starfire number.

What your numbers are showing is that More Haste is better then Less Haste. The difference is purely due to Starlights Wrath applying unequal evels of haste to two spells while the spell coefficent remains constant.


But that is just it - they are close by not exact, all of which is an attribute due to rounding. In that example of Haste levels, Wrath's cast time was rounded up - Starfire's should have been too, but doing that only changes the value to 36.66636 which still shows a slightly favoritism towards a longer cast time. But that is how it would function in rounding, since, by rounding up, the shorter cast time is in effect 'losing' more Haste than the longer cast time.

It's the rounding nature of the functions in of themselves which cause slight differences within all of the values, but the rounding differences lean more towards favoring shorter cast times over longer cast times simply due to the fact that rounding holds a much larger impact for those abilities. This is generally why Haste tends to show a real instead of theoretical favoritism towards faster casting spells. If rounding were not a function of the game, then there would be no favoritism and Haste would effect each spell exactly the same, however, the game does use rounding. Specifically, the rule would be that Haste favors any spell cast time which is not rounded at that specific Haste value - IE, at 10% Haste, the 2 second spell should be rounded up but the others should not be, however at 12% Haste, the others should be rounded up and the 2 second spell should not be. If all spell cast times are rounded at a specific value, then the longer cast time should, theoretically, gain the most from Haste since it took the least amount of 'loss' due to rounding. If none of spells are rounded, then faster cast times are favored due to nature of the significant digits used.

In a practical game-sense, however, Haste would favor all faster casting abilities due to talents such as Starlight Wrath given that all casting classes have these talents, with Arcane, IIRC, being the only one that doesn't- Destruction Warlocks, too, for that matter, since Incinerate only gains a .25 reduction, but that is their only spammable nuke.


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 Post subject: Re: CS 101: Haste (LONG - get some Gin before reading)
PostPosted: Thu Jan 14, 2010 2:28 pm  
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Murmurs wrote:
Not to be rude Graylo, but this is the exact opposite of what you've said before. I used to go to only 3 significant digits since that is what the game actually displays, you, however, were the one to correct me and say I should be going to 4.
Fair enough. I do remember saying that, but I think you are missunderstanding what I said then and what I'm saying now.

I've never critisized anyones work for using more significant digits then they needed to. At the time I think we were talking about spell DPS and how different stats affected that DPS. When you are looking at the marginal value of Stats, the number of significant digits is important. If you don't have enough then the rounding will skew your results one way or another. In my opinion, 3 is not enough. Four is better, but I personally like a lot more when I do theorycrafting.

All that a side, significant digits are irrelevent to my arguement. More on that below.

Murmurs wrote:
Theoretically speaking, there is a difference in Haste between a 2 second cast spell casting at a 1.1818 speed and a 1.1819 speed. The difference in Haste value, however, is different that that of a 3 second cast spell that would be casting at a 1.1818 speed and a 1.1819 speed. The lower the base cast time you are dealing with, the greater the impact rounding has because it causes a larger Haste 'discrepancy.' I would take a larger amount of Haste to reduce the cast time of a 2 second spell by 0.0001 second than it would a 3 second spell, therefore faster casting spells benefit more via rounding.

It is this discrepancy that would cause faster casting spells to benefit more from Haste than slower casting spells, because, from the base-line, all spells benefit equally.

Simple rounding simply doesn't account for all of the difference, it is the innate nature of using significant digits that does this as well. The 2.5 comparison to the 3.5 speed show this. There is a very small difference between the 2.5 and 3.5 spells, yet neither of these spells should have been rounded up because the next significant digit would have been 2 and 1 respectively.
You're missing a key component of rounding. Rounding cuts both ways. It can both help and hurt.

Rounding up may help a spell with a shorter cast time more, but rounding down will also hurt a spell with a shorter cast time more. It averages out.


Murmurs wrote:
But that is just it - they are close by not exact, all of which is an attribute due to rounding. In that example of Haste levels, Wrath's cast time was rounded up - Starfire's should have been too, but doing that only changes the value to 36.66636 which still shows a slightly favoritism towards a longer cast time. But that is how it would function in rounding, since, by rounding up, the shorter cast time is in effect 'losing' more Haste than the longer cast time.

It's the rounding nature of the functions in of themselves which cause slight differences within all of the values, but the rounding differences lean more towards favoring shorter cast times over longer cast times simply due to the fact that rounding holds a much larger impact for those abilities. This is generally why Haste tends to show a real instead of theoretical favoritism towards faster casting spells. If rounding were not a function of the game, then there would be no favoritism and Haste would effect each spell exactly the same, however, the game does use rounding. Specifically, the rule would be that Haste favors any spell cast time which is not rounded at that specific Haste value - IE, at 10% Haste, the 2 second spell should be rounded up but the others should not be, however at 12% Haste, the others should be rounded up and the 2 second spell should not be. If all spell cast times are rounded at a specific value, then the longer cast time should, theoretically, gain the most from Haste since it took the least amount of 'loss' due to rounding. If none of spells are rounded, then faster cast times are favored due to nature of the significant digits used.
Again, you're only rounding up, and not concidering the opposite side. The way rounding would skew the results would average out over a variety of values.

Murmurs wrote:
In a practical game-sense, however, Haste would favor all faster casting abilities due to talents such as Starlight Wrath given that all casting classes have these talents, with Arcane, IIRC, being the only one that doesn't- Destruction Warlocks, too, for that matter, since Incinerate only gains a .25 reduction, but that is their only spammable nuke.


No, you're looking at this from the wrong perspective.

Assuming you have static Spell Coeffients, Spell Power will favor the spell with a greater amount of Haste. Talents like Starlight Wrath that apply greater amounts of haste to specific spells, cause those spells to favor Spell Power, not haste.

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