So, I recently took a vacation, and was mostly bored out of my mind. Luckily, I had some downtime to do something more fun: MATH^{cue flash of lightning.} Without further ado, I bring you...

Math - Multi-Elements, Uneven Imperils, and You

I've been lurking and occassionally answering in The Daily Help threads, and noticed that every once in a while, a question regarding whether it is better to equip two different elemental weapons to dual-element chain even with uneven imperils. The difference between the two is that the former reaches the chain cap faster, the latter maintains high imperil damage. The general rule of thumb is to use only the element with the stronger imperil, partly because of friend units. Going forward, as our various main chaining families grow in elemental coverage, I expect this question to appear more frequently.

However, it got me thinking: is it always better damage-wise, and how much better is it?

The Basics: E = mc²

Before we can do any form of evaluation, we must first establish an objective baseline to compare against. Luckily, our current damage formula (for GL) has been well calculated, and is represented below:

Dmg = (ATK^2 / DEF^2) * (Skill Modifier) * (Level Correction) * (Weapon Variance) * (Final Variance) * (Chaining) * (Killers) * (Elemental)

Now for the purpose of our comparisons, we only care about the chaining and elemental modifiers. As such, we can simplify our equation to...

D = N * (Chaining) * (Elemental)
   where N = static & partial values we don't care about

Going back to my introduction, I stated that the general rule is that one element is better for damage when dealing with uneven imperils. What does this look like in math form? Well...

N * (Chaining.1) * (Elemental.1) > N * (Chaining.2) * (avg.Elemental.2)
N * (Chaining.1) * (Elemental.1) / (N * (Chaining.2) * (avg.Elemental.2)) > 1
(Chaining.1 / Chaining.2) * (Elemental.1 / Elemental.2) > 1
   where 1-Elemental Damage is better than 2-Elemental Damage

OR

Ratio = (Chaining.1 / Chaining.2) * (Elemental.1 / Elemental.2)
   where Ratio = Damage comparison of 1-Elemental compared to 2-Elemental

Also, WTB subscripts

Disfunctional Families and hYou

For the purpose of the comparisons, I'm only going to look at the current QT and HE families, and Hyou. Also, spark chaining comparisons only, because the differences is more apparent on those.

Holy Explosion/Divine Ruination Family

Quick Hit Family

Chaining Average

^{Dear future me: If the multi-line table is broken, welcome to Formatting Hell, and fix it. Otherwise, ignore this message.}

Comparison: Dripping Raw

Remember that ratio math equation in the two sections previously? We'll be combining that with the charts above to see if currently the single elemental is always better against an enemy with no resistances or weaknesses. We'll also assume that the imperils are already applied at this point of the calculations. Remember: x > 1 in order for the single element to be better.

Holy Explosion/Divine Ruination Ratio (Double)

Holy Explosion/Divine Ruination Ratio (Triple)

Quick Hit Ratio (Single)

Quick Hit Ratio (Double)

In general..
... long chains and large imperil differences favor 1-element.
... short chains and small imperil differences favor 2-elements.

Comparison: Refining

One thing I didn't mention yet is that some skills are weighted unevenly. What does that mean? Well, when you do damage through multiple hits, damage is split across the various hits. However, many times, the damage is not evenly distributed through the individual hits.

What does this mean for us and our calculations? Well, this means that the above chart is less reliable for us as it was under the assumption that all the damage is evenly distributed. If, for example, a particular skill has most of the damage dealt at the very end (aka: backloaded), then a significant amount of the damage will use the highest chaining modifier, which in turn means that the single-element will be significantly more advantageous.

First, what is the weight of each chain? Well...

Holy Explosion/Divine Ruination Family

Quick Hit Family

As you can clearly see, every one of these chaining moves are back-loaded, which means that the ratio should be shifted upwards. With the exception of enhanced Sepherioth, the back-loaded-ness is so slight that the raw ratio is mostly correct (if a bit under), and not worth the additional, and significantly more painful amount of math to get the exact number.

Conclusion

Unless you're trying to min-max your damage output, you don't have to sweat out the details between chaining and imperils as long as your difference in imperil strength is close enough.

Also: Don't dropout of school, cause MATH is awesome.

^{Math Status: Satisfied}