Haste vs Expertise hard-cap?

Hi,

assumed we have 7,5% hit and 7,5% exp and therefore, have reached spell hit cap and melee hit cap, and our melee attacks can't be dodged anymore:
Could it be possible that haste gives more Shield of the Righteous uptime now than exp?

Jugdement always hits and gains nothing from expertise, so the only HP generator exp influences is CS (+Grand Crusader procs).
But haste improves BOTH, CS and Judgement, by the same rate each, that exp improves CS.
My conclusion is, haste>exp(after exp-softcap), but that differs from the mainstream opinion (getting exp-hardcap).

Am I overlooking something?

Edit: was bad info

Hm, I'm pretty sure Judgement always hits. Just tested it, with some pieces of armor taken off, 60 Judgements, 60 hits on the boss training dummy. My chance to get parried is 7,5% and my chance to get dodged is 4,33% (i took some peices of armor off).The chance of this happening would be below a tenth of a percent ((1-(0,075+0,0433))^60=0,000523902472), if Judgement was on the melee table.

Judgement cannot be parried, but can be judged, making your chance for that run 7.02%.

Nooska wrote:Judgement cannot be parried, but can be judged, making your chance for that run 7.02%.

"can be judged?"

Judgment cannot be dodged or parried, it can only miss. Expertise does absolutely nothing for Judgment.

Expertise is still better than haste for HPG, however. Proof:

Assume that your base (with 0% miss and 0% haste) HP generation rate is B. Your actual HPG rate is then:

A = B*(1-m)*(1+h)

where m is your chance to miss (really the sum of miss+dodge+parry, for this let's assume you're already at 7.5% hit and 7.5% expertise so we're only concerned with parry) and h is your decimal haste (i.e. 10%=0.10).

Adding 1% expertise is the same as turning m into m-0.01, or:

Am = B*(1-m+0.01)*(1+h) = B*(1-m)*(1+h) + 0.01*B*(1+h) = A + 0.01*B*(1+h)

Adding 1% haste turns h into h+0.01, giving:

Ah = B*(1-m)*(1+h+0.01) = A + 0.01*B*(1-m)

Since (1+h) will always be larger than (1-m), you'll always get more HPG out of expertise than haste.

Ahhh, of course. It's an absolute "percentage point" increase, applying it to the smaller coefficient is of course a higher increase overall. I should have thought this through before assuming stuff.

Thank you very much for explaining.


Edit: Actually, your m influenced the complete B, even though a part of B, the Judgement part is not influenced by it. The whole B is influenced by haste, though. So I think your calculations might be incorrect (please read my latest post)

*cough* Judged = dodged - wasn't it dodgeable at some point during the beta then?, or am I mixing it up with something completely different?

Assume that your base (with 0% miss and 0% haste) HP generation rate is B. Your actual HPG rate is then:

A = B*(1-m)*(1+h)

Doesn't this only cover the HPG rate of Crusader Strike? What about the HPG rate of Judgement which is not influenced by m? The factor (1-m) only applies to the part of B which is generated by CS.


If I do the same for Crusader Strike and Judgement I get this:
Assume we have 7.5% hit and 7.5% expertise. Let J be the base HP generation of Judgement and C the base HP generation of Crusader Strike.
J is not influenced by m, because as you confirmed earlier, Judgement can not be parried or dodged, and we have reached the hit cap, so we are only concerned about parry. So assume for the moment m=0,075 (because we have only 7.5% exp) and h=0. C is increased by factor 1.2 to include Grand Crusader.
Our actual HPG rate A is then: A = C*(1-m)*(1+h)*1.2 + J*(1+h) = C*(1-0.075)*1*1.2 + J*1 = 1.11C+1J

So 1.11C+1J is our HPG rate with 7,5%hit, 7,5%expertise and 0%haste.

So here comes the important part:
Now, if we increase expertise by 3 percent points (1020 rating), thus lower m by 0.03 we get the following:
Ae = (C*(1-m+0.03)*(1+h)*1.2) + (J*(1+h)) = (C*(1-0.045)*1*1.2) + (J*1) = 1.146C+1J

If we instead increase haste by the same amount of rating (=2,4%), thus increasing h to 0.024 we get this:
Ah = (C*(1-m)*(1+h)*1.2) + (J*(1+h)) = (C*(1-0.075)*1.024*1.2) + (J*1.024) = 1.13664C+1.024J

==> Ah = 1.13664C+1.024J
==> Ae = 1.146C+1J

Which is higher? J = 1 HP every 6 seconds. C = 1HP every 4.5 seconds. C = 13.3333333 per minute and J = 10 per minute.

Ah= 1.13664*13.3333333 + 1.024*10 = 25.3952
Ae= 1.146*13.33333+10 = 25.2799962

==>Ae < Ah ???

Another Example (+2550 rating: +7.5% exp(hardcap) vs +6% haste): (keep in mind, this is only the rating and expertise which goes beyond the standard of the first calculation)
Ae = 1.2C+1J = 25.999996HP per minute
Ah = C*(1-0.075)*1.2*1.06+1.06J = 1.1766C + 1.06J = 26.28800HP per minute
==> Ah>Ae

Another Example (+170 rating: +0.5%exp vs 0.4%haste):
Ae = C*(1-0.075+0.005)*1.2+J=1.116C+1J=24.88 HP per minute
Ah = C*(1-0.075)*1.2*1.004+1.004J = 1.1544C + 1.004J = 24.8992 HP per minute
==> Ah>Ae



EDIT: Gab rightfully corrected that haste needs 25% more rating than expertise for 1 percent point increase.

Not sure (I'm bad at math) but are you taking into account in your 3% examples that:

1% haste = 425 rating so 3% haste = 1275 rating
and
1% expertise = 340 rating so 3% = 1020 rating

Thus making expertise the supperior choice for HPG?

You are correct, thanks! I forgot that. But it's still not enough difference to change the result, at least not with a 3% example. I corrected it and added other examples (exp hardcap vs 6% haste and 0.5%exp vs 0.4%haste) which show the same results.

Assumed my calculation was correct, which I don't know (I hope that theck will have another look) and haste yields more HPG per minute on average, expertise might still be better because exp hardcap means we can only ever have a certain timewindow without SHoR-Buff, while the haste-approach is not "luck-proof". Also, I don't know if ShotR can be parried and if it still gives the buff when parried.

ShoR can be parried / dodged, but still gives the buff.

I actually wrote up a long reply to this yesterday, but the forum went down before I got to post it. Hopefully it's still saved on my desktop, so I'll try and re-post when I get home.

Hm... you're right, I wasn't being rigorous enough there. It would be accurate for hit rating, but for expertise CS and J should really be treated independently. However, if we're going to go to that level of rigor, we need to do so more accurately than you did.

First, Judgement isn't 1 HP every 6 seconds, it's rotationally constrained to 1 HP every 6.75 seconds.
Second, you get 1% expertise for 340 expertise rating, but it takes 425 haste rating to give you 1% haste.

So, rigorously then:
A = C*(1 - m + dm)*(1+h+dh)*1.2 + J*(1+h+dh)

where
C is your CS rate (1/4.5=0.2222)
J is your J rate (1/6.75=0.1481)
m and h are decimal dodge/parry and haste values (assuming 7.5% hit - the earlier version is enough to prove hit>haste)
dm and dh are added expertise or haste.

Expanding this and ignoring second-order terms (i.e. dh*dm), we get:
A = 1.2*C*(1-m)*(1+h) + J*(1+h) + 1.2*C*(1+h)*dm + (1.2*C*(1-m)+J)*dh

What we're really comparing are those last two terms: Ae=1.2*C*(1+h)*dm and Ah=(1.2*C*(1-m)+J)*dh. But we want to do so in terms of rating r, and since dm=r/340 and dh=r/425, we can write:

Ae=(1.2*C*(1+h)/340)*r
Ah=((1.2*C*(1-m)+J)/425)*r

For h=0, m=0.075, we get

Ae = (7.8431e-4)*r
Ah = (9.2897e-4)*r

So at 0% haste and exp soft-cap, haste is about 18% better for HPG than expertise. Conversely, it takes about 18.44% haste before expertise catches up to haste.

So yes, haste is actually better for holy power generation than expertise. That doesn't really matter though, since the advice to hard-cap expertise isn't based on raw HPG or SotR uptime. It was based on damage smoothing, and in those simulations the hit/exp capped gear set out-performed the soft-capped gear set that stacked haste.

The difference wasn't huge, but it was also a stochastic simulation (i.e. get as much SotR uptime as possible). In practice, a good player is banking and using HP intelligently, and the certainty that your next generator will hit has even more value there. Nothing sucks more than missing a Crusader Strike and dying to the next melee attack because you couldn't refresh SotR.

Ok, thank you for explaining. Everything's clear now.

dm and dh are added mastery or haste

I believe you mean expertise instead of mastery.

Yep, typo from generally using "dm" and "m" for mastery. In this case I used them for a generic "miss." Not the clearest notation, I guess, but it got the point across. I'll edit the other post for clarity.