Holy Theorycrafting: SoR vs. SoW+Consecration

I apologize for the lack of posts recently. Real life plus the upcoming holidays has really put a hard limit on the amount of time I have to spare, and sadly online activities tend to be the first thing on my chopping block in such situations. Anyway, to continue with the topic at hand...

In case you don't know, I am one of the moderators at the wonderful healing-oriented forums known as PlusHeal. Well, it just so happened that one of the forum posters there asked a question regarding which strategy dealt damage faster: using Seal of Righteousness for straight-up damage dealing, or using Seal of Wisdom to facilitate the mana regeneration required to be able to cast Consecration as often as possible.

As a Seal of Righteousness user myself, I found this to be a very interesting question. Since I was in the mood to do a bit of theorycrafting at the time, I went ahead and mathed it out. In addition to posting this on the PlusHeal forums, I figured I'd may as well post it here as I've been somewhat lacking in post content recently, and because I spent a good chunk of time on it.

So, let's start with the general idea. As I understand it, the entire point of using Seal of Wisdom instead of Seal of Righteousness is that the former allows you to spam Consecration on cooldown without running into mana issues, whereas the latter would result in downtime if you use Consecration too often. In other words, the whole idea comes down to the notion of Seal of Righteousness vs. Seal of Wisdom + Consecration.

Question:

Which deals more DPS, Seal of Righteousness without Consecration or Seal of Wisdom with Consecration?

Assumptions:

1. Judgement of Wisdom and Blessing of Wisdom have 100% uptime.
2. Divine Plea is used on cooldown.
3. Judgement of Wisdom is used on cooldown to maximize damage output. (This does not necessarily have to be true with the Seal of Wisdom playstyle, but we'll assume it to make the math slightly easier.)
4. One can maintain Holy Shock and Consecration spamming indefinitely with Seal of Wisdom active, but not with Seal of Righteousness active.
5. Librams are not taken into account.
6. There are no differences in playstyle other than which seal is used and whether or not Consecration is used on cooldown.
7. Seal of Righteousness's Judgement and Seal of Wisdom's Judgement both have the same chance to deal a critical strike, and both deal a 50% bonus damage when they critically strike.
8. Multiple cooldowns completing simultaneously are not taken into account.
   

Abbreviation Key:

• SoR: Seal of Righteousness
• SoW: Seal of Wisdom
• Cons: Consecration
• DPS: damage per second
• MWS: melee weapon speed (calculated after haste effects are taken into account)
• AP: attack power
• SPH: spell power
• CritChance: chance to deal a critical strike in decimal form (i.e. percentage divided by 100) (NOT crit rating)
• Cons#: a damage modifier which depends on which rank of Consecration you cast—rank 6 (level 70) = 72, rank 7 (level 75) = 87, rank 8 (level 80) = 113
• JCD: Judgement cooldown (varies depending on talents)
  

Given:

• Seal of Righteousness damage per swing = MWS * (0.022 * AP + 0.044 * SPH)
• Seal of Righteousness Judgement base damage formula = 1 + 0.2 * AP + 0.32 * SPH
• Consecration total damage = 8 * (Cons# + 0.04 * AP + 0.04 * SPH)
• Seal of Wisdom damage = 0
• Seal of Wisdom Judgement base damage formula = 1 + 0.16 * AP + 0.25 * SPH
  

Theorycrafting:

First, let's calculate the DPS of SoR and Cons:

SoR DPS
= [SoR damage per swing] * [swings / second]
= [MWS * (0.022 * AP + 0.044 * SPH)] * [1 / MWS]
= 0.022 * AP + 0.044 * SPH

Cons DPS
= [Cons damage] / [Cons cooldown]
= [8 * (Cons# + 0.04 * AP + 0.04 * SPH)] / [8]
= Cons# + 0.04 * AP + 0.04 * SPH


Next, let's figure out how much damage Judgement will contribute. Remember that unlike SoR and Cons, Judgements can deal critical strikes, so we should include that into our calculations:

Judgement bonus damage from critical strikes
= [Judgement base damage] * CritChance * 0.5

average Judgement damage
= [Judgement base damage] + [Judgement bonus damage from critical strikes]
= [Judgement base damage] + [[Judgement base damage] * CritChance * 0.5]
= [Judgement base damage] * (1 + (CritChance * 0.5))

average Judgement DPS
= [average Judgement damage] / [JCD]
= [[Judgement base damage] * (1 + (CritChance * 0.5))] / [JCD]


Now that we have the formula for the average Judgement DPS, let's calculate the numbers for both SoR Judgements and SoW Judgements:

average SoR Judgement damage
= [SoR base damage] * (1 + (CritChance * 0.5))
= [1 + 0.2 * AP + 0.32 * SPH] * (1 + (CritChance * 0.5))

average SoR Judgement DPS
= [average SoR Judgement damage] / [JCD]
= [(1 + 0.2 * AP + 0.32 * SPH) * (1 + (CritChance * 0.5))] / [JCD]

average SoW Judgement damage
= [SoW base damage] * (1 + (CritChance * 0.5))
= [1 + 0.16 * AP + 0.25 * SPH] * (1 + (CritChance * 0.5))

average SoW Judgement DPS
= [average SoW Judgement damage] / [JCD]
= [(1 + 0.16 * AP + 0.25 * SPH) * (1 + (CritChance * 0.5))] / [JCD]


We now have all the numbers we really need for our final calculations, but before we get to the really heavy math, let's go ahead and do something that will make later calculations a bit easier. Since SoR Judgements have greater AP and SPH coefficients than SoW Judgements and they both have the same CritChance and JCD, we know for a fact that SoR Judgements will deal more damage than their SoW counterparts. So, let's just go ahead and figure out exactly how much more DPS we get from SoR Judgements than SoW Judgements on average.

difference between SoR and SoW Judgement DPS
= [average SoR Judgement DPS] - [average SoW Judgement DPS]
= [((1 + 0.2 * AP + 0.32 * SPH) * (1 + (CritChance * 0.5))) / JCD] - [((1 + 0.16 * AP + 0.25 * SPH) * (1 + (CritChance * 0.5))) / JCD]
= ((1 + (CritChance * 0.5)) * ((1 + 0.2 * AP + 0.32 * SPH) - (1 + 0.16 * AP + 0.25 * SPH))) / JCD
= ((1 + (CritChance * 0.5)) * ((1 - 1) + (0.2 - 0.16) * AP + (0.32 - 0.25) * SPH)) / JCD
= ((1 + (CritChance * 0.5)) * (0.04 * AP + 0.07 * SPH)) / JCD


And now it's finally time for the really heavy math. Since we're trying to figure out which playstyle deals greater DPS, I would feel uncomfortable equating the two and then mathing it out. So, we'll just use an arbitrary symbol to separate the two sides while we figure it out. Let's go with ">?<" since we're trying to figure out which is greater than and which is less than the other.

[SoR DPS] + [average SoR Judgement DPS] >?< [Cons DPS] + [average SoW Judgement DPS]

[SoR DPS] + [average SoR Judgement DPS] - [average SoW Judgement DPS] >?< [Cons DPS]

[SoR DPS] + [difference between SoR and SoW Judgement DPS] >?< [Cons DPS]

[0.022 * AP + 0.044 * SPH] + [((1 + (CritChance * 0.5)) * (0.04 * AP + 0.07 * SPH)) / JCD] >?< [Cons# + 0.04 * AP + 0.04 * SPH]


That's pretty much as far as my rusty math skills can get me. Now there's nothing left but to plug in the numbers and see what comes out. I'm going to go ahead and input my numbers as an example.

(0.022 * AP + 0.044 * SPH) + (((1 + (CritChance * 0.5)) * (0.04 * AP + 0.07 * SPH)) / JCD) >?< (Cons# + 0.04 * AP + 0.04 * SPH)

(0.022 * 474 + 0.044 * 1071) + (((1 + (0.2201 * 0.5)) * (0.04 * 474 + 0.07 * 1071)) / 10) >?< (72 + 0.04 * 474 + 0.04 * 1071)

(10.428 + 47.124) + (((1 + 0.11005) * (18.96 + 74.97)) / 10 >?< (72 + 18.96 + 42.84)

57.552 + (1.11005 * 93.93) / 10 >?< 133.8

57.552 + 104.2669965 / 10 >?< 133.8

57.552 + 10.42669965 >?< 133.8

67.97869965 < 133.8


Alternately, you can skip the "calculate the difference between SoR and SoW Judgement DPS" step, which yields the actual DPS potential of both playstyles:

[SoR DPS] + [average SoR Judgement DPS] >?< [Cons DPS] + [average SoW Judgement DPS]

[0.022 * AP + 0.044 * SPH] + [((1 + 0.2 * AP + 0.32 * SPH) * (1 + (CritChance * 0.5))) / JCD] >?< [Cons# + 0.04 * AP + 0.04 * SPH] + [((1 + 0.16 * AP + 0.25 * SPH) * (1 + (CritChance * 0.5))) / JCD]

(0.022 * 474 + 0.044 * 1071) + (((1 + 0.2 * 474 + 0.32 * 1071) * (1 + (0.2201 * 0.5))) / 10) >?< (72 + 0.04 * 474 + 0.04 * 1071) + (((1 + 0.16 * 474 + 0.25 * 1071) * (1 + (0.2201 * 0.5))) / 10)
(10.428 + 47.124) + ((1 + 94.8 + 342.72) * (1 + 0.11005) / 10) >?< (72 + 18.96 + 42.84) + ((1 + 75.84 + 267.75) * (1 + 0.11005) / 10)

57.552 + (438.52 * 1.11005 / 10) >?< 133.8 + (344.59 * 1.11005 / 10)

57.552 + (438.52 * 0.111005) >?< 133.8 + (344.59 * 0.111005)

57.552 + 48.6779126 >?< 133.8 + 38.25121295

106.2299126 < 172.05121295

So, it seems that using Seal of Wisdomand Consecration yields more DPS than Seal of Righteousness without Consecration. Naturally, using Seal of Righteousness with Consecration would deal more than either of the alternatives, but this is not a sustainable DPS cycle (which was the goal).