Complete calculation of AP / Crit% conversion

(This guide created by Grunticus, level 70 orc hunter of the Mal'Ganis server. All credit is due to him for this wonderful guide. Date: 04/11/2007)

GRUNT's FORMULA:

Tired of reading people guessing how much crit = AP based on their gear and not having any mathematical means of proving things one way or the other (other than the crude 2 AP = 1 Crit Rating technique), I have done the math for you. "Instinct", "I heard that...", and "but my dps box shows..." are irrelevant statements. There is too much misinformation going around and the only solution is pure crude math.

Note:
This is a mathematical derivation with a proof of concept calculation examples. Everything below is based on a simple steady/autoshot rotation. Because DPS for the most part derives from the total damage output of a steady/autoshot, the formulae below are 100% valid. Haste/quiver speed/ etc are 100% irrelevant to this total damage output (but are rather just linear adjustments to your dps).

(A) Formula for autoshot damage:
AutoShotDamage = DamagePercentageBonus*RWSMod*(AmmoDPS*WeaponSpeed + (RAP/14)*WeaponSpeed + WeaponDamage + ScopeDamage)

(B) Formula for steadyshot damage:
SteadyShotDamage = DamagePercentageBonus*RWSMod*(150 + 2.8*WeaponDamage/WeaponSpeed + 0.2*RAP )

(C) Formula for steady/autoshot rotation damage:
White damage w/o crits:
AutoSteadyDamage = AutoshotDamage + SteadyShotDamage = DamagePercentageBonus*RWSMod*[ AmmoDPS*WeaponSpeed + WeaponSpeed*RAP/14 + WeaponDamage + ScopeDamage + 150 + 2.8*WeaponDamage/WeaponSpeed + 0.2*RAP ]

(D) Now accounting for Crits (remember 22 crit rating = 1% basecrit):
TotalDamage = AutoSteadyDamage * (100 + BaseCrit*MortalShotModifier)/100

(E) Grunt's Formula: (basic multivariable calculus) Deriv(TotalDamage)/deriv(ap) = (weaponSpeed/14 + 0.2) * DamagePercentagebonus * RWSMod * (100 + BaseCrit*MortalShotModifier)/100

Deriv(TotalDamage)/deriv(crit)= DamagePercentagebonus * RWSMod * MortalShotModifier*[weaponSpeed*(AmmoDamage + BaseAP/14) + weaponDamage + scopeDamage + 150 + 2.8*WeaponDamage/Weaponspeed + 0.2*BaseAP]/100

Note: MortalShotModifier = 1.00 w/o Mortal Shots, or 1.30 with Full Mortal Shots (30% crit damage bonus). Humanoid/beast slaying would add even more on those targets.

[Grunt's Formula]
Deriv(ap)/deriv(crit) = MortalShotModifier*[weaponSpeed*(AmmoDamage + BaseAP/14) + weaponDamage + scopeDamage + 150 + 2.8*WeaponDamage/WeaponSpeed + 0.2*BaseAP ] / [ (weaponSpeed/14 + 0.2) * (100 + BaseCrit*MortalShotModifier) ]

(If you don't understand how I computed E, it doesnt matter. 2 Case examples below prove it is correct.)

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Validation of Grunt's Formula (for people who don't know calculus):
If you agree with Formula A, B, C, and D we prove below that formula E is correct.

Example #1: (MM Hunter in PVP type gear)
BaseAP = 2000
BaseCrit = 25% Weapon: Gladiator's Heavy Crossbow
WeaponSpeed = 3.2
WeaponDamage = (202+304)/2 = 253 Scope: +28 crit (we adjust BaseCrit by 28/22 as a result -> baseCrit = 26.27)
ScopeDamage = 0
MortalShotModifier = 1.3 (full 5/5 talents)
AmmoDamage = 37
RWS = 1.05 (5/5 Ranged Weapon Spec)
DamagePercentagebonus = 1.00 (assume no humanoid slaying)


Deriv(ap)/deriv(crit) = 1.3*[3.2*(37+2000/14) + 253 + 0 + 150 + 2.8*253/3.2 + 0.2*2000] / [(3.2/14+0.2)*(100+26.27*1.3)] = 2080/57.5 = 36.17
According to Grunt's Formula, for every 1% crit (22 crit rating), this player gains an effective 36.17 AP.

Lets Prove IT!!!

Ok lets plug in those numbers....
add 5% Crit rating into formula D with our baseStats from Example#1:
TotalDamage = (100 + (26.27+5)*1.3)/100 * 1.05 * [37*3.2 + 3.2*2000/14 + 253 + 0 + 150 + 2.8*253/3.2 + 0.2*2000]
= 1.4065 * 1680 = 2362.9 TotalDamage
Now add 5*36.17 = 181 AP into formula D with our baseStats from Example#1
TotalDamage = (100 + (26.27)*1.3)/100 * 1.05 * [37*3.2 + 3.2*(2000+181)/14 + 253 + 0 + 150 + 2.8*253/3.2 + 0.2*(2000+181)] = 1.3415 * 1761 = 2362.9 TotalDamage

PROOF!!!! 2362.9=2362.9 (181 AP has the same exact TotalDamage as 5% crit (36.17 AP = 1% crit))

FURTHERMORE, because this hunter gets 10% AP bonus from MasterMarksman, 32.88 AP = 1% Crit.

IN OTHER WORDS, 1.5 AP = 1 Crit Rating for this hunter

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Example #2: (AP raid buffer hunter in good gear, say a 0/31/30 type layout)
BaseAP = 2600
BaseCrit = 20%
Weapon: Sunfury
WeaponSpeed = 2.9
WeaponDamage = (157+292)/2 = 224.5 Scope + 12
Damage MortalShotModifier = 1.3 (full 5/5)
AmmoDamage = 37
RWS = 1.0 (0/5)
DamagePercentageBonus = 1.05 (humanoid/beastslaying)

Grunt's Formula produces: 1.3*[2.9*(37+2600/14) + 224.5 + 12 + 150 + 2.8*224.5/2.9 + 0.2*2600] / [(2.9/14 + 0.2)*(100+20*1.3)]
= 2300 / 51.3 = 44.83 AP = 1% Crit

Lets add 5% Crit Rating into formula D with these baseStats:
Total Damage = (100 + (20+5)*1.3)/100 * 1.05 * [37*2.9 + 2.9*2600/14 + 224.5 + 12 + 150 + 2.8*224.5/2.9 + 0.2*2600]
= 1.325 * 1857.6 = 2461 damage
Instead adding 5*44.83 = 224 AP yields:
Total Damage = (100 + 20*1.3)/100 * 1.05 * [37*2.9 + 2.9*(2600+224)/14 + 224.5 + 12 + 150 + 2.8*224.5/2.9 + 0.2*(2600+224)]
= 1.26 * 1953 = 2461 damage

PROOF!!! 2461 = 2461 (224 AP has the same exact TotalDamage as 5% crit (44.83 AP = 1% crit))

More Proof....
So a Raid buffed 0/31/30 hunter would be just as good off adding 5% Crit (110 Crit Rating) as adding 224 AP.

IN OTHER WORDS, 2.03 AP = 1 Crit Rating for this hunter (note how different it is from the 0/43/18 type hunter in example #1)

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Numbers do not lie. Like I said, if you agree with formulae A-D, formula E [Grunt's Formula] is correct as proven above in 2 separate examples. The results also do match the stats blizzard provided us on gear. (the supposed 2AP =1 crit rating thing).