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Thread: Impale vs. Cruelty comparison

  1. #1
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    Impale vs. Cruelty comparison

    Update: Adding white damage to formula. This is a little more impactful than I originally assumed.

    First time poster, so please be gentle :-).

    I hope this isn't reptitious but I haven't seen anyone post this yet so here goes:

    Assumptions:
    New ones added in bold
    -Only abilities that can crit and are affected by Impale are used.
    -Cruelty is assumed to increase crit for all yellow attacks and white attacks.
    -Benefits to snap threat (bigger individual crits) from Impale are ignored.
    -Talents that add crit (Incite, S&B, Critical Block) are assumed to be additive.
    -Talent cost is not considered a factor

    Formulas:
    This whole section was overhauled, previous contents was added to the end.
    Formulas:
    TD = (1 + C) * D

    Where TD is the total damage (yellow and white), C is the crit rate, and D is the sum of all the damage of all abilities. We will model D as follows:
    D = D(X)+ D(1-X) where X represents the percentage of yellow damage.

    This gives us the following:
    TD = (1 + C)(X)(D) + (1+C)(1-X)(D)

    Note we havenít actually done anything interesting yet, just expanded the formula into two terms, one for white and one for yellow damage.

    With Cruelty:
    TD = (1.05 + C)(X)(D) + (1.05 + C)(1 - X)(D)
    TD = 1.05XD + CXD + 1.05D + CD Ė 1.05XD Ė CXD
    TD = CD + 1.05D
    TD = (C + 1.05)(D)

    As you may have noticed, the Cruelty formula allows us to completely ignore the % damage thatís yellow vs. the % damage thatís white because it effects both damage components equally.

    With Impale:
    TD = (1.2C + 1)(X)(D) + (1 + C)(1 Ė X)(D)
    TD = 1.2CXD + XD + D + CD - XD Ė CXD
    TD = 0.2CXD + D + CD
    TD = (0.2CX + C + 1)(D)

    Notice what happens when we substitute 1 for X:
    TD = (0.2C + C + 1)(D)
    TD = (1.2C + 1)(D)

    This represents total damage using Impale if 100% of our attacks were yellow damage. Now letís look at the formula again:
    TD = 0.2CXD + D + CD

    D + CD is the standard formula for damage. 0.2CXD represents the contribution from Impale and can be read as:
    The extra damage from Impale is the product of the Impale Bonus Percentage (0.2), the amount we crit (C), the Damage dealt D, and the percentage of that Damage that represents yellow damage (X). This seems like a reasonable model for Impale that takes into account white damage. Now to see how it stacks up to Cruelty.

    TDImpale(C, X) = (0.2CX + C + 1)(D)
    TDCruelty(C) = (C + 1.05)(D)

    We canít really find an intersecting line for these two functions. The first one has two independent variables; the second one only has one. We have to pick a fixed value for X, and then find the intersection point. The shape of the two lines in question are still the same; Cruelty has a higher y-intercept and smaller slope, while Impale starts at 0 but has a bigger slope (with the exception of the cases where X = 0). So whatever intersection point we find, values of C below it will favor Cruelty, values above it will favor Impale. Instead of substituting for X immediately, we will simplify the equation first, so that we can come up with a simple way to test different yellow to white ratios:

    (0.2CX + C + 1)(D) = (C + 1.05)(D)
    0.2CX + C + 1 = C + 1.05
    0.2CX = 0.05
    CX = 0.05 / 0.2
    C = (0.05 / 0.2) / X
    C = 0.25 / X

    Notice again how if we substitute 1 for X to indicate that 100% yellow damage matches up with 25% crit.

    Conclusion:


    My initial assumption that white damage was not significant enough to affect Impale was invalidated to some degree. The new formula for Impale vs. Cruelty is contingent on the ratio of yellow damage to white damage:

    C = 0.25 / X

    C is the minimum crit rating required (with talents & buffs) for Impale to surpass Cruelty in Damage. X is the percentage (as a number between 0-1) that indicates the percentage of total damage that is yellow damge.

    Some examples:

    X = 0.9 (90% of all damage is yellow damage)
    C = 27.7%
    X = 0.85 (85% of all damage is yellow damage)
    C = 29.4%
    X = 0.75 (75% of all damage is yellow damage)
    C = 33%

    Initial post formulas, these are only for yellow damage:

    The average damage contribution from crit can be modeled as:
    TD = C * D

    where C is the crit rate expressed as a number between 0 and 1, and D is the damage of a particular attack, and TD is total damage from crit.

    With Cruelty:
    TDwC = (C + 0.05) * D

    With Impale:
    TDwI = (1.2 * C) * D

    where TDwC = Total Damage w/ Cruelty, and TDwI = Total Damage w/ Impale.

    Both of these are simple linear functions. The goal is to figure out which one has a bigger result at a particular crit level. The first thing we can do is remove D from both equations, since they are common between the two equations. Essentially, we're figuring out which value is a bigger multiplier for D. This leaves us with:

    (C + 0.05)
    and
    (1.2 * C)

    Now we want to find where these two intersect:
    C + 0.05 = 1.2 * C
    0.05 = C * 0.2
    0.05 / 0.2 = C
    C = 0.25

    If these formulas were drawn as two lines, they would intersect at C = 0.25. At C < 0.25, (C + 0.05) has a bigger value than (1.2 * C). At C > 0.25, (1.2 * C) wins out.
    Last edited by Maelbolgia; 10-08-2008 at 08:30 PM.

  2. #2
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    Yes, if they were both Tier 1 talents that you could reach with no additional points, wasted, this would be a valid comparison.

  3. #3
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    Edit: please read the entire thread if you read this post. I concede this later.

    Take the following as constructive criticism; I would hate to see someone deterred from doing math around here:

    This is an interesting approach. However, I think it may be inherently flawed in the way you calculate damage output. This changes the equations pretty significantly. Think about it this way:

    You have an attack with base damage D. Critical Hits do damage 2*D base, and 2.2*D with impale. You do crits and non-crits. With Crit chance C, you can estimate the average damage of your attack as follows. Comments are in [Brackets] and are not part of the formula:

    TD = (1-C) * D [the percentage of hits that are not crits, time their damage] + C * 2 * D [crit chance times crit damage: your formula].
    Simplified, without either talent:
    TD = (1-C) * D + C * 2 * D = D - DC + 2CD = CD + D

    With Impale, we have:
    TD = (1-C) * D + C * 2.2 * D = D - DC + 2.2CD
    TD = 1.2CD + D
    TD = (1.2C + 1)D

    With Cruelty:
    TD = (1 - (C + .05)) * D + (C + .05) * 2 * D = D - DC - .05D + 2CD + .1D
    TD = CD + 1.05D
    TD = (C + 1.05)D

    You also can't simply drop the D from these equations and end up with reasonable values. With respect, I think your approach simply doesn't model damage output as we see it in the game. Take a look at Naka's spreadsheet here:

    http://www.tankspot.com/forums/f14/4...readsheet.html

    I made something similar myself that you can search for if you want, but I won't directly plug myself here. Spend some time looking at the formulas in there and I think you'll gain a bit more insight about a better way to model damage (and threat). Definitely keep at it though. The more ways we can think of approach the problem, the better our data and thus our conclusions will be.
    Last edited by mero12513; 10-07-2008 at 01:43 PM.
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  4. #4
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    Thanks for the feedback Mero. But I'm also going to have to respectfully disagree with the fact that D makes a difference in this issue. I haven't had a chance to look at the formulas in the spreadsheets yet, but looking at your post:

    The two formulas you listed (which I agree with for total damage):
    TD(C, D) = (1.2C + 1) * D
    TD(C, D) = (C + 1.05) * D

    So far we are in agreement. Now, I'm only comparing Impale and Cruelty. Neither of these effect the base damage of an attack. So while the above formulas use D as a variable, I am setting it to a constant for the purposes of comparing Impale and Cruelty. (I think we can both agree that neither impacts the base damage of an attack). This gives us:

    TD(C) = (1.2C + 1) * D
    TD(C) = (C + 1.05) * D

    Where D is some constant value. Now we can solve for the intersection point:

    (1.2C + 1) * D = (C + 1.05) * D
    (1.2C + 1) = (C + 1.05) [divide by D]
    (0.2C + 1) = 1.05 [subtract C]
    0.2C = 0.05 [subtract 1]
    C = 0.05 / 0.2 [divide by 0.2]
    C = 0.25

    Basically, I am not shooting to model total damage, just the difference between Cruelty and Impale. To put it another way:

    TD(C, D) = TD'(C) * D

    TD'(C) represents the Impale function (1.2C + 1) or the Cruelty (C + 1.05)function. I want to know which one produces bigger values since both represent a scalar value that multiplies D to produce Total Damage.

    Now to plug in some numbers. While one cannot prove by example, let's take two extreme cases of D and two cases of C such that one is below 25% and one is above 25%:

    TD(C, D) = (1.2C + 1) * D
    TD(C, D) = (C + 1.05) * D

    //15% crit, 85 damage Impale
    TD(0.15, 85) = 100.3
    //15% crit, 85 damage Cruelty
    TD(0.15, 85) = 102.3

    //15% crit, 10000 damage Impale
    TD(0.15, 10000 ) = 11800
    //15% crit, 10000 damage Cruelty
    TD(0.15, 10000 ) = 12000


    //35% crit, 85 damage Impale
    TD(0.35, 85) = 120.7
    //35% crit, 85 damage Cruelty
    TD(0.35, 85) = 118.9

    //35% crit, 10000 damage Impale
    TD(0.35, 10000 ) = 14200
    //35% crit, 10000 damage Cruelty
    TD(0.35, 10000 ) = 13999.9

    All the results coincide with the assertion that Cruely is better at C < 25% and Impale is better at C > 25%, even for VERY high values of D. I am not asserting the above examples as proof of course, but I welcome you to come up with an example that violates this rule.

  5. #5
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    Witness a rare occurrence: me sticking my foot in my mouth.

    You're absolutely right. I went back and looked at my own equations, and they come to the same conclusion you do. I made the mistake of basing my argument on my conclusion that Impale is better than cruelty, which it is, but only because 1 point in Impale is better than 1 point in Cruelty. In realistic situations, 5/5 Cruelty beats 2/2 Impale because our crit chance isn't high enough on average (even with talents).

    In fact, if I take my own spreadsheet and plug in 25% crit chance for every ability, Impale and Cruelty both give a 3.31% TPS increase. Thanks for pointing it out my error.
    I. Am. Warrior.

  6. #6
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    Very nice, this is a great answer that I'm sure ALOT of warriors are debating between.

    Personally, I will still probably take impale, for the spike threat aspect, that will allow me time to get a TC/Demo/Shout down/up.

    The real debate is about the talent points.

    You need 10 in arms minimal, to reach impale, and you need 0 in fury.

    Now, I will be taking the 5 parry and 3 heroic strike already, reguardless, so the debate comes between 2/2 charge+2/2 impale for 4 talent cost, or 5/5 cruelty, costing 5 points. Decisions decisions.
    November 23, 2004 8:27:03 AM - Glomgore 10+ Years of tanking? Priceless

  7. #7
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    thank you for the math. I felt that impale was going to be inferior to 5/5 cruelty for prot warriors (with our dismal crit levels), having it backed up is good.

  8. #8
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    Quote Originally Posted by Shadevarr View Post
    thank you for the math. I felt that impale was going to be inferior to 5/5 cruelty for prot warriors (with our dismal crit levels), having it backed up is good.
    I'd slow down with that assumption. Our crit levels are actually quite good. Although base crit is pretty bad, Incite, Sword and Board, and Critical block bring up the crit levels of our base abilities by a great deal.
    I. Am. Warrior.

  9. #9
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    My warrior has 7% crit with no Cruelty on the Ptr. Add 15% to Dev, SS, HS from talents, 5% from Rampage/LotP and 3% from Master Poisoner/Heart of the Crusader, and I'm looking at 30% crit chance on Dev, SS, and HS. Revenge is still at 15%.

    To be honest, there's a reason I didn't include any subjective remarks on the issue in the original post: I'm still trying to decide between the two, as well as other talents in the prot tree. I would like to spec Impale, but I need to be able to justify it over Cruelty. The original analysis gives a good starting point: if you can't make 25% crit chance in the environment/gear in which you are playing your warrior (e.g. solo aoe grinding in full prot gear), Cruelty is a better choice. Once you reach 25%, you can consider going into Impale.
    Last edited by Maelbolgia; 10-08-2008 at 10:07 AM. Reason: Accidental submission

  10. #10
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    With the talents giving us the 15% on our important skills, things such as TC, which will be one of our strongest soloing tools, along with a major AoE tool (where we lack) I think the impale still might help us more, as we advance higher into gear levels and gain more crit, and can start favoring AGI a bit more (instead of the massive amounts of STA previously needed in BC), impale becomes even more valuable.

    Some possible parsing and actual WWS might be the key here for some real answers.

    With Imp revenge, we already get our bonus 20% damage ^_^
    November 23, 2004 8:27:03 AM - Glomgore 10+ Years of tanking? Priceless

  11. #11
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    Quote Originally Posted by mero12513 View Post
    Witness a rare occurrence: me sticking my foot in my mouth.

    You're absolutely right. I went back and looked at my own equations, and they come to the same conclusion you do. I made the mistake of basing my argument on my conclusion that Impale is better than cruelty, which it is, but only because 1 point in Impale is better than 1 point in Cruelty. In realistic situations, 5/5 Cruelty beats 2/2 Impale because our crit chance isn't high enough on average (even with talents).

    In fact, if I take my own spreadsheet and plug in 25% crit chance for every ability, Impale and Cruelty both give a 3.31% TPS increase. Thanks for pointing it out my error.
    So since realistically, we won't hit 25% crit with tank gear... we should always spec for Cruelty over Impale?

  12. #12
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    well, we won't get 25% crit unbuffed, but with the +15% crit talents stuck to our prot talents, I can see us reaching 25% crit buffed easily.

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  13. #13
    You seem to have forgotten a vital point.

    HUGE crits are so much more fun then lower constant crits
    Go go pew pew Impale.

  14. #14
    Are you factoring in the fact that Impale only works on yellow damage?

  15. #15
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    Quote Originally Posted by intolerantape View Post
    Are you factoring in the fact that Impale only works on yellow damage?
    No. The original post listed several assumptions about this comparison. I am speculating that white damage will be a small component of our damage compared to yellow damage, but I haven't ran any numbers on that one yet, so if it's a 50/50 split for example, Impale would lose a lot of its value. Definately a good thing to verify.

    At this point the only solid conclusion I can come to based on these numbers is this:

    If, after your talents and buffs your crit chance is less than 25%, then Cruelty is better than Impale, pretty much all-around. After 25% however, Impale is better for yellow damage, but there are other factors that people have pointed out that may sway them one way or the other: Impale is deeper in arms making the cost unappealing; white damage is only affected by Cruelty, not Impale; it is fun to have Impale and have big crits. (As a main tank I am not in a position to consider the last one valid, but I certainly love big SS crits :-)).

  16. #16
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    Hmm, some quick math is suggesting that a 75 to 25 Yellow to White damage ratio is pushing Impale to the 33% crit mark. I think I will try to work out white damage into this, and get rid of one of the assumptions. Does anyone know the ratio of white to yellow damage in wotlk?

  17. #17
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    Posts like this are the reason Tankspot is the best WoW site. Period.

    So if this holds up on the white vs. yellow damage analysis, when we have the ability to "bank" 2 specs, we could have one as our solo grinding spec with Impale, since our dps gear could likely get us to 25% crit. And a 5-man instance spec might have Cruelty since our more traditional tanking gear is, as was noted above, less likely to get us to 25% crit.

  18. #18
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    I've updated the original post with the new information factoring in white damage. I'm not sure what the exact ratio between white damage and yellow damage is, so I made a general formula. I really appreciate the feedback I've been getting from all you guys. If you spot any errors in my math please let me know.

  19. #19
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    Depends heavily on how often you use Heroic Strike, of course.
    As per my updated spreadsheet, even untalented and with 50% Heroic Strikes, white damage is only roughly 10% of your threat. So with talents and 75-100% HS, it should become pretty negligible.

  20. #20
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    Quote Originally Posted by Naka View Post
    As per my updated spreadsheet, even untalented and with 50% Heroic Strikes, white damage is only roughly 10% of your threat.

    10% of threat =/= 10% of damage

    A Shield slam that hits for 800 will do more threat than a white attack for 1000, solely because Shield Slam has innate threat attached to it.

    I think this thread is more focused on the damage part of it. The increase in threat should (in theory) be equal to the increase in damage as both Impale and Cruelty are based only off of (damage done x multipliers) and no innate numbers


    White attacks are the only attack (as far as I know for a Prot Warrior) that do not have a flat amount of threat associated with them. Just a straight dmg x 1.45.
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