Some Eclipse / Nature's Grace 3.1 Math

[Sunkin]

Nature's Grace and Eclipse 3.1 MathCraft

I. Nature's Grace

A. Effect on Haste Rating
Nature's Grace will now impose a soft haste rating cap, or a point at which increasing haste rating further will yield a diminished dps return due to a hard cap of 1 second on the GCD.

The total haste required to dip the wrath cast time under 1 second is 50%, so the haste rating cap can found using the following maths:

NG * WoA * Imp Moonkin Aura * CF * (1 + [haste rating] / lvl 80 conversion) = 1.5

1.2 * 1.05 * 1.03 * 1.03 * (1+[haste rating]/3279) = 1.5

By algebra, [haste rating] = 400!

However, this is no where near as strict as the hit rating cap. Unlike increasing hit past the cap, adding more haste will still yield an overall increase in dps; the increase just wont be as large per point as under 400. How much this soft cap affects your gearing decisions has a lot to do with your current gear and rotation. The diminished return will be felt least by moonkins with starfire-heavy rotations, and lower crit chance. However, hypothetically, if a geared moonkin procced Lunar Eclipse with the first wrath crit every time, the diminished return of 400+ haste will not be felt at all!

As a side note, this diminished value of haste is constant per each point in the range 400 to 1136, assuming other stats remain constant.

B. Effect on Crit Rating
Due to NG now having a 3 second duration, landing a crit during a NG proc is less beneficial than landing a crit outside of the NG proc, as NG proc time is wasted in the first case. So, a natural question is how much crit is needed to be confident that NG will be up all the time, but won't be wasted?

To answer that question, its necessary to know how many casts we can get off during the NG proc. Wraths can never be casted more than 1 second apart, but with any amount of haste, 3 wraths should be able to be casted within the duration.
Likewise, with any amount of haste, 2 starfires can be casted, but is it possible to get in 3?

The haste required to reduce starfire cast to 1.5 seconds is 100%
1.2 * 1.05 * 1.03 * 1.03 * (1+[haste rating]/3279) = 2, yielding 1626 as the haste rating required. I don't see that as being possible any time soon.

How about during blood lust/heroism?
1.3 * 1.2 * 1.05 * 1.03 * 1.03 * (1+[haste rating]/3279) = 2, yielding 495 as the minimum haste rating required. With latency, a realistic value might be higher, but it is definitely reachable.

So back to the original question. To maximize NG effectiveness, the lowest crit chance that maintains the NG proc is desired. So for 3 wraths, 33% crit chance is needed for the expected number of crits during NG proc to be exactly 1.

Similarly, for starfire, 50% crit chance is needed to get the expected number of crits to 1. However, it makes no sense to be chaining starfires after the first has crit except during an eclipse, so we can reduce the crit chance required by 30% (the eclipse buff) , down to 20%.

In a smart 25-man, you should be getting the following crit chance bonuses:
5% (Moonkin Aura), 5% (Winter's Chill / Scorch), 3% (Master Poisoner/Heart of the Crusader), 4% (Nature's Majesty), 3% (Improved Faerie Fire) = 20%y So, its probably safe to assume at least 35% crit raid buffed with decent gear.

/*Math will Be Reworked Soon*/
Since 35% surpasses the necessary crit values calculated, the expected uptime for Nature's Grace for any raiding boomkin is near 100%! It seems fairly obvious that blizz means this talent to be essentially a 3-point passive 20% increase in haste!

Another important thing to note is that its almost useless to try to itemize crit solely on the basis of efficient use of NG procs as even with very low crit on gear, NG will most likely be up 100% of the time.

C. Summary of Conclusions:
-At haste values over 400, wrath casts will encounter a GCD wall during NG procs.
-Expected uptime of NG in a 25man raid setting us 100% with almost any gear.

-ECLIPSE-

II. Eclipse
A. Proc Time Comparison

There is much speculation and not a lot of math addressing which Eclipse takes longer to proc.

Lunar Eclipse Proc Time:

Chance on wrath hit to proc = .66 * crit

How many wraths needed to proc LE on average? 1 / (.66 * crit).
Assuming raid buffs of 20% crit:
Crit chance on gear Wraths required to proc LE
0% 6 wraths
10% 5 wraths
17.8% 4 wraths
30.5% 3 wraths

Note: the expected number of wraths to proc LE does scale continuously with crit; these are just some example values.

How long does it take to proc LE? [expected number of wrath casts] * [expected cast time]
So, 1 / (.66 * crit) * 1.5 / (haste)

Here are some sample numbers to give a feeling for the scaling

--14% crit on gear--
250 haste: 4.65 s
400 haste: 4.46 s
550 haste: 4.28 s

--20% crit on gear--
250 haste: 3.95 s
400 haste: 3.79 s
550 haste: 3.64 s

--26% crit on gear--
250 haste: 3.43 s
400 haste: 3.29 s
550 haste: 3.17 s

Solar Eclipse Proc Time:

How many Starfires needed on average to proc SE? 1 / crit

Assuming raid buffs of 20% crit:
Crit chance on gear SFs required to proc SE
0% 4 SFs
3.3% 3 SFs
20% 2 SFs

How long does it take to proc SE? [expected number of SF casts] * [expected cast time]
T = 1 / crit * 3 / (haste)

Now, lets compare this to the model for LE, which was T = 1 / (.66 * crit) * 1.5 / (haste)
Setting them equal, 3 / (crit * haste) = 2.27 / (crit * haste).

Canceling, we see that on average lunar eclipse should take about 30% less time to proc than solar eclipse, regardless of gear!

Heres another sample breakdown for SE to give a feel for scaling.
--14% crit on gear--
250 haste: 6.13 s
400 haste: 5.88 s
550 haste: 5.65 s

--20% crit on gear--
250 haste: 5.21 s
400 haste: 5 s
550 haste: 4.8 s

--26% crit on gear--
250 haste: 4.53 s
400 haste: 4.35 s
550 haste: 4.18 s

Another thing this breakdown makes clearer is that proccing Solar Eclipse is much more dependent on crit than lunar eclipse.

B. DPS during Proc Time Comparison (ie: SF spam vs. wrath spam)

NOTE: Here's where I have to make some more impactful assumptions. I am going to assume that nature's grace is up through the whole proccing period. This will most likely be the case for proccing lunar eclipse (see the previous section), however its somewhat difficult to deal with the idea that nature's grace has an expected uptime of 100% but the expected length of time to proc Solar Eclipse is greater than 3 seconds, even with over 50% buffed crit. However, as gear gets better, this assumption gets less detrimental to the overall calculation, so I don't feel too bad using it.

I'm going to ignore % damage / spell damage buffs/debuffs from raid as they will scale equally well regardless of rotation.

Raid Buffed/Talented: +280 sp, +20% crit, +3%.+3%,+5%,+20% haste
Moonfury: +10% damage
Wrath of Cenarius: 10% improved spell power coefficient
Typical base numbers of 2700 sp, 20% crit from gear, 400 haste rating

Lunar Eclipse Proc DPS:

Wrath Base Damage: 588
Improved Insect Swarm: +3% damage
DPS = (1.13) * (588 + [sp]*.6714) * (1 + .2 + [crit]) / (1.5 / (1.2 * 1.05 * 1.03 * 1.03 * (1+[haste_r]/3279)))

DPS = (588 + [sp] * .6714) * ( 1.2 + [crit]) * (1 + [haste_r]/3279) * 1.51 / 1.5s

Typical DPS = 3796

Solar Eclipse Proc DPS:

Starfire Base Damage: 1120 + [165]
Improved Insect Swarm: +3% crit

DPS = (1.13) * (1120 + [sp + 165]*1.2) * (1 + .23 + [crit]) / (3 / (1.2 * 1.05 * 1.03 * 1.03 * (1+
[haste_r]/3279)))
cleaning this up:
DPS = (1120 + [sp + 165]*1.2) * (1.23 + [crit]) * (1 + [haste_r]/3279) * 1.51 / 3s

Typical DPS = 3680

Some preliminary analysis shows that wrath spam scales better with crit, and even at 0% crit starfire spam is only better with under 1550 spell power. This is old news for the most part.

C. DPS under the Eclipse Buff

Lunar Eclipse:

DPS = (1120 + [sp + 165]*1.2) * (.3 + 1.23 + [crit]) * (1 + [haste_r]/3279) * 1.51 / 3s
Typical DPS = 4453


Solar Eclipse:

Solar Eclipse:
DPS = (1.13 + .3) * (588 + [sp]*.6714) * (1 + .2 + [crit]) / (1.5 / (1.2 * 1.05 * 1.03 * 1.03 * (1+
[haste_r]/3279)))
DPS = (588 + [sp] * .6714) * ( 1.2 + [crit]) * (1 + [haste_r]/3279) * 1.91 / 1.5s
Typical DPS = 4801

D. Overall DPS comparison

We can ignore the Eclipse cool down period when calculating which rotation is better because its dps and duration is the same regardless of which eclipse buff was proced.

Given (from previous data) that it takes 5s to proc Solar Eclipse, at 3680 DPS, and the eclipse buff lasts for 15s at 4801 DPS, then the DPS by choosing Solar Eclipse is:
(5 * 3680 + 15 * 4801 ) / 20 = 4521 DPS

Given (from previous data) that it takes 3.79s to proc Solar Eclipse, at 3796 DPS, he eclipse buff lasts for 15s at 4453 DPS, then the DPS by choosing Lunar Eclipse is:
(3.79 * 3796 + 15 * 4453 ) / 18.79 = 4320 DPS

E. Summary of Conclusions
-Lunar Eclipse takes 30% less time to proc on average than Solar Eclipse regardless of gear!
-For maximum DPS, Solar Eclipse should be chosen.

[Shwaam]

World of Warcraft, the game which became a complicated algebra.

Nice post though, keep the good job up.

[Thepropphet]

lol this looks like my math book.

[rabbimojo]

...
So back to the original question. To maximize NG effectiveness, the lowest crit chance that maintains the NG proc is desired. So for 3 wraths, 33% crit chance is needed for the expected number of crits during NG proc to be exactly 1.

Similarly, for starfire, 50% crit chance is needed to get the expected number of crits to 1. However, it makes no sense to be chaining starfires after the first has crit except during an eclipse, so we can reduce the crit chance required by 30% (the eclipse buff) , down to 20%.

In a smart 25-man, you should be getting the following crit chance bonuses:
5% (Moonkin Aura), 5% (Winter's Chill / Scorch), 3% (Master Poisoner/Heart of the Crusader), 4% (Nature's Majesty), 3% (Improved Faerie Fire) = 20%y So, its probably safe to assume at least 35% crit raid buffed with decent gear.

Since 35% surpasses the necessary crit values calculated, the expected uptime for Nature's Grace for any raiding boomkin is near 100%! It seems fairly obvious that blizz means this talent to be essentially a 3-point passive 20% increase in haste!
...

I think I'm going to call this awry. I love to see math, but I think this is a wonky application of probability.
IE, "So for 3 wraths, 33% crit chance is needed for the expected number of crits during NG proc to be exactly 1." is (I think) right, but that doesn't mean uptime is anywhere near 100%!
The problem is that sometimes you get a refresh, sometimes you get a drought.

The math is a bit tricky, and I'd have to get out a stat reference to make sure I got it right. Since I'm in a rush atm, I recommend looking up information from BC on ISB uptime. It had the exact same issue, where people thought "expected # of crits = 1 comes out to nearly 100% uptime", when in truth a raid full of ~33% crit ISBers would see an uptime more like 75-85%.

ISB and NG are different in the sense that ISB had charges and NG does not, but the uptime calculations are quite similar. Anyways, I highly recommend reading about it!

[Mobutu]

Pretty much what rabbi just said.  The chance that you're going to see (at least) 1 crit out of three casts with 33% crit chance is only 70%, and 80% chance to get 1 out of four casts to crit.

In order to get, say, a 95% chance to get (at least) 1 crit out of 3 casts, you'd need a 63% crit chance.  These kinds of probabilities aren't additive.

Other than that, looks like some good info and some solid work put in. All that does is increase the value of higher crit for NG uptime.

[Sunkin]

Good points. I assumed that expected value of 1 crit in 3 seconds would mean 100% uptime. I will look into the statistics behind it more, and see if I can change it. It may mean some assumptions ive made will have to be reworked, because assuming 100% NG uptime really helps the theorycrafting.

Thanks for your comments guys!

[Bigbluedruid]

I'm not going to hurt my brain with lots o math , but I can speak of my experience on the PTR. My gear gives me about 36 percent crit (with talents, tier, and iff) and about 520 haste (with food buff). Messing around with a target dummy, it seemed like NG was up almost 100 percent of the time, I didn't really "test" it so it could have been more like 90 percent, but it was there most of the time for sure. It seemed to reduce my starfire cast time by about .4 seconds. Old (live) NG reduces your next spell by .5 seconds and has also seemed to proc very often. The NG lasting over 3 seconds as opposed to the next spell may be a plus, but the loss of .1 second on my starfire seems kinda nerfish for me idk.

[Sunkin]

Ok, I think a better fit for the dps calculations than assuming 100% uptime, is to give a constant portion of the 20% NG buff, where that percentage is (1 - (1-crit)^n) with n being the number of casts during the NG buff.

[Deleted]

Most of what i've seen up till know from EJ still seemed to put Lunar Eclipse over Solar one. Probably because they are taking higher haste values (T7 bis gear puts you around 600), SF idol, and clipping into account.