Not worth the effort IMO. I think adding g to pH/pI will correct it in the right direction even if its not exact. It shouldn't be too big of an error.


Solving

1.33*g*k*(1+0.5M)*(1+F(M)*(1+M)) = (1+(1.33*I*g+baseSP)*k)*(0.5+F(M)*(1.5+M))*1.6*z

F(M)*(1.5+M)*1.6 = (0.4+M*(3+M))/(3.2*(1.5+M))*(1.5+M)*1.6 = (0.4+M*(3+M))/2

1.33*g*(1+0.5M)*(1+F(M)*(1+M)) = (1/k + 1.33*I*g+base SP)*(0.4+M*(3+M))*z/2

1/k+1.33*I*g+baseSP = 2.66*g/z*(1+0.5M)*(1+F(M)*(1+M))/(0.4+M*(3+M))

I = 2/z*(1+0.5M)*(1+F(M)*(1+M))/(0.4+M*(3+M)) - (baseSP+1/k)/1.33/g

since (1+F(M)*(1+M))*(1+0.5M) is just the heal coefficient for the optimal level of crit and mastery from base*(1+0.5M)*(1+C*(1+M),

I = 2/z*heal_coeff_opt/(0.4+M*(4+M) - (baseSP+1/k)/1.33/g

not sure if this is correct I will have to test it. This way g is also restricted to the subtracted factor, which might magnify errors if it makes a big difference.

Hmmm, that can't be right, it is the completely the inverse of the result I got. I don't think I could be that far wrong and this result does not make intuitive sense. Lets try going back to the basic principles

dF(x,y,z)/dx = pF/px + pF/py*dy/dx + pF/pz*dz/dx, where p/px denotes partial derivative and d/dx denotes total derivative. Thus the spellpower optimization equation is

dH/dSP = 0 --> pH/pSP + pF/pMR*dMR/dSP + pF/pCR*dCR/dSP = 0

now pF/pMR = pF/pCR and dMR/dSP = dCR/dSP by default, so

pH/pSP = -2pF/pMR*dMR/dSP

using H = base*Heal_Coeff we get

p(base)/pSP*Heal_Coeff = -2base*p(Heal_Coeff)/pMR*dMR/dSP

which leads to

base = -1/2*p(base)/pSP*Heal_Coeff*dSP/dMR/( p(Heal_Coeff)/pMR )

now base is entirely SP dependent so this equation determines everything. The only thing that needs to be calculated here is dSP/dMR

Looking at this equation it is clear that the nominator is 2nd order for MR, while the denominator is 1st order for MR, and since dSP/dMR must be negative the whole thing will be positive, so the optimal base value and hence the optimal spellpower must increase with stats not decrease.

I will solve the equation a little later, once I finish work or someone else can have a crack at it.

Havoc12, your thought would be right if I need to find total derivative. But I want to find partial derivative of CR, Int and MR.

And partial derivative of complex function is:

F( a(x), b(x,y))

pF/px = pF/pa * pa/px + pF/pb * pb/px
there is no need in total derivative when I calculate partial derivatives.

So I think my idea is right, but performing might not =) Ill try to find errors in that wall of formulas -_-

This way your condition is pH/pSP = pH/pCR = pH/pMR, which is not right, that cannot the correct optimisation condition.

The remember that hte optimsation condition we used before for mastery and crit was not pH/pCR = pH/pMR, but rather

pH/pCR = -pH/pMR*dCR/dMR. Because dCR/dMR = -1 due to CR = const - MR we can use the simplified pH/pCR = pH/pMR

Unless you are somehow calculating it indirectly you can't avoid the total derivative because the real optimisation condition for variable x is always dH/dx = 0, the equality of partial derivatives is just a solution of this equation when the two variables scale exactly the same. Your solution must give the same results as mine and just by looking at them you can see that they can't be the same.

when you have spellpower mastery and crit to optimise then 1.33*SP = const - G(MR) - MR, so dSP/dMR = -(1+dG(MR)/dMR)

and this needs to be included in your optimisation equation.

It is not that complicated to solve the formula I posted, since everything in it is easy to calculate separately to find the base and then the base is a linear equation of int.

The equation leads to (1+SP*k) = -1/2*k*Heal_Coeff*dSP/dMR/((1+0.5M*(3+M))*z), since pHeal_coeff/pMR = pHeal_Coeff/pCR = (1+0.5M*(3+M))*z

SP = -1/2*z*Heal_Coeff*dSP/dMR/((1+0.5M*(3+M)) -1/k

and this should accurate

Which I means I was a bit wrong, the nominator is 3rd order in M, while the denominator is 2nd order, but that should still leave us with optimal SP increasing with stats.

SP/1.33 = const-CR-MR needs to be used to calculate dSP/dMR. This is easy since C = F(M), meaning that CR = G(MR)

so dSP/dMR = -(1+dG(MR)/dMR)*1.33, this should not be tough to calculate.

Final formula is SP = 1.33/(2z)*Heal_coeff*(1+dG(MR)/dMR)/(1+0.5M*(3+M)) - 1/k

edit: Fixed the scaling factor

Already solved it pretty much. I just need to calculate dG/dMR later on and my notation is

Heal_Coeff = (1+0.5M)*(1+C*(1+M))

Heal = base*Heal_Coeff.

This is important so we don't get confused.

I hate looking through long lists by eye, trying to find out what the best spellpower is for each value of mastery/crit. That is really annoying.

C = (0.4+M(3+M))/(3.2*(1.5+M)) = F(M)

G(CR) = C*z = F(M)*z

dG(MR)/dMR = dC/dM*dM/dMR*z = dC/dM*1.6/z*z = dC/dM*1.6

dC/dM = 1/3.2*((3+2M)*(1.5+M) - (0.4+M*(3+M)) )/((1.5+M)^2) = 1/3.2*(2 - (0.4+M*(3+M))/((1.5+M)^2) )

dG/dMR = 0.5*(2 - (0.4+M*(3+M))/((1.5+M)^2) )

edit: Fixed dG/dMR I forgot a z, which was supposed to cancel out things.

SP = 1.33/(2z)*Heal_coeff*(1+dG/dMR)/(1+0.5M*(3+M)) - 1/k

edit: Fixed the values and now the equation should work better.


The variation in heal_Coeff/(1+0.5M*(3+M), turns out to be tiny. This whole thing is effectively a constant. Because this is a complex polynomial its value oscilates and it ends up having a minimum at 47% mastery, then it increases again. Of course 47% mastery is unreachable.

As you can see it barely varies at all at relevant values of mastery. The maximum variation is like 2%. The only thing that remains is to properly calculate dG/dMR.

I fixed the equation and this is a list for dG/dMR

so the strong scaling of spellpower drops relative to the other stats.

Using the formula I calculated a list of the optimal values of spellpower, mastery and crit

I had tested it using the 2nd value in the list and it's turned out to be 7.7k off the correct value, so I still have an error somewhere. In other words the 2nd value is actually 71k spellpower

dropping 100 mastery and ~81 crit and increasing spellpower by ~241 increases healing by 0.03% and doing the opposite i.e. dropping ~241 spellpower and adding 100 mastery and ~81 crit decreases healing by 0.03% with the values in the table.

Basically what this tells is that spellpower is always better no matter the gear since, we won't get to these values this expansion and even that value is ~7.7k off the mark.

Your data very similar to what I get here
Disc Priest - Mists of Pandaria
And I got optimal SP there nearly constant too.
The question is - is that right?

I really think now that empirical way - right way.

I updated the post above, check again. What you calculated in that post had the correct behaviour because it is exactly what you should get from your approach, which effectively ignores dG/dMR. Adding dG/dMR changes things again, so the drop is a little larger and it does constantly drop, but the values turned out to be much higher that what you calculated.

There is still an error there and I will find it, because I am sure it is in the way I calculated dG/dMR.

Empirically correcting the formula tells us that the optimal value of intellect is ~70k at reasonable mastery values, so int on gear is always better than mastery and crit.

==========================================

Important addition. I used the formula to find out whether 320 mastery is equal to 160 intellect and found that 160 intellect is better than mastery up to 37900 spellpower. Past that point it ends up being better gemming mastery.

I did not include reoptimization of mastery and crit, but the effect should be negligible.

The order for gems is thus spirit > int > mastery +crit assuming you can use the extra mana. Otherwise spirit is out of the equation. After 37900 spellpower mastery and crit become better than intellect.

The breakpoint depends on your mastery/crit value, higher mastery/crit suppresses slightly the breakpoint

Mastery: 0.268 crit: 0.225506222 sp 37900, (1+k*SP)*heal_coeff = 6.788799706

remove 320 mastery and add 160*1.33 = 212.8 intellect

mastery 0.259466667, Crit 0.226392888, (1+k*SP)*heal_coeff = 6.788807919

Change: 1.00000121

remove 212.8 int and add 320 mastery instead:

Mastery: 0.276533333 Crit 0.224619555, (1+k*SP)*heal_coeff = 6.788480925

Change: 0.999953043

If you add another 100 sp to 38000 then the int gem dips below the mastery gem

Crit is the same because these are optimised crit/mastery values.

Doing the same with a higher mastery value and crit gives you a lower sp threshold

For example for mastery 0.288 crit: 0.23541387, the threshold is 35800 sp

While I started this whole calculus discussion, I haven't actually had time to review the recent math posts yet. Let me just add that based on my spreadsheet (other thread), I agree with the basic point that seems to be implied in all this, that a crit rating and a half-Int are very similar in value.

The only reason I haven't gotten too far into the crit/mastery/Int breakpoint calculus on my own is that the primary factor making one stat better than another isn't the slight scaling effects of your current stats, but rather your spell distribution (PWS favors mastery) and the relative effectiveness of heals vs. aegis (aegis favors crit).

Also, if you're looking so simplify analytic discussion of Int scaling: rather than worrying about individual spells, an handy rule of thumb is that the base heal amount is usually/always equal to just about 11,000 times the spellpower coefficient. So you can basically assume that spellpower is additive with a constant base term of 11000 and behaves proportionally otherwise (10,000 might be better figure to use if you want to account for the raid spellpower buff). I put a longer explanation of that here a little while ago: Healing Theory, Part 2: A Tour of Your Character Sheet | It's Dangerous to Go Alone

I tested SS on PTR with various stats, with 3% meta and w/o.
It works good (finally) with ±1 precision compared to theory prediction.


Without 3% meta gem:
SS = AvgHeal * (1+M/2) * (1+C*(1+M))
With 3% meta gem:
SS = AvgHeal * (1+M/2) * (1+0.03*C+1.03*C*(1+M))

AvgHeal=Base*(1+k*SP)
C = CritChance%
M = MasteryPoints*0.016

BEWARE. Tooltips on PTR shows value with mastery healing effect!

So you should just use value from tooltip and keep in mind that this is AvgHeal*(1+M/2)
or manually calculate AvgHeal with your SP number.

maybe so =) I just warn you to calculate it right way.

And after we know that SS works correctly -> all analysis that we made works perfectly for SS. So to maximize spirit shell: Int before SP 35-38k (clarification in progress) then Crit and Mastery balanced.