Before I begin, this is going to look atrocious, since I unfortunately cannot write the equations using the <math> tag, so apologies in advance for the clumsiness of the equations. I would have posted this as a comment on the Parameter Bonus (Dark Souls II) page, but I wouldn't have been able to use tabs to reduce the size of the page. Despite this, it's still a very long read, and will get bigger once I've looked into Mundane more, so only continue if you have the time and/or interest. :P
Let's start with the hard stuff, shall we? I've been playing around with this for a little while now, and it would appear that a formula for the amount of extra damage generated by dual-physical stat scaling can be calculated via the following equation:
EDTotal = ( Strength Attack x Strength Contribution ) + ( Dexterity Attack x Dexterity Contribution ) = ( SA x SC ) + ( DA x DC)
Seems simple enough, doesn't it? Unfortunately, this is not the case, as this is what the actual, expanded formula looks like:
EDT (Final) = ( SA x ((EDPhysical (Initial) - (EDPhysical (Final) x DC) / 100) ) + ( DA x (ΔEDPhysical / ΔDA) )
How did we get here, though? Well, to begin with, you need two sets of data from a weapon; the EDT (F) and the DA at two different whole number values (e.g. I used an unupgraded Gyrm Axe, which gives: EDP (I) = 82, DAI = 54 and EDP (F) = 86, DAF = 102). At this point, we are ignoring Strength, as we only want to find the change in EDPhysical (ΔEDP) as a result of changing our Dexterity Attack value (ΔDA). Thus:
Dexterity Contribution = ΔEDP / ΔDA = (EDP (F) - EDP (I)) / (DAF - DAI)
∴ DC = ( 86 - 82 ) / ( 102 - 54 ) = 4 / 48 = 1 / 12 = 0.083
Alright, great. So we know DC now, but what about SC? In this instance, SC cannot be calculated via SC = ΔEDP / ΔSA, since no change in SA has occurred, and anything divided by zero is an indeterminable number. Instead, we must use another formula:
SC = (EDP (I) - (EDP (F) x DC)) / 100
By now, you can probably tell where this is going, but onwards we go:
SC = (82 - (86 x 0.083)) / 100) = 0.763
Ta da! We now know both SC and DC. Here's where the "fun" starts, though, as we can then use these to calculate the EDT (F), and hence the AR of a weapon, at any given point of either Strength or Dexterity Attack! Using the same unupgraded Gyrm Axe, let's say I now have 30 Strength and Dexterity Attack (meaning that SA and DA both equal 102). Now, I already know that at these values, my AR = 276 from the weapon status screen, but will the formula work? Let's see:
ETD (F) = (102 x 0.763) + (102 x 0.083) = 77.8 + 8.5 = 86.3 ∴ AR = BD + ED = 190 + 86.3 = 276.3
When you upgrade stats sometimes, you'll notice that as you get higher up in the levels, you'll see less and less change in EDPhyical. For instance, when I leveled up Dexterity from 28 to 30, the Gyrm Axe's ED remained at 86. This is known as "diminishing returns", but it's never really been apparent why it occurs. Was it a result of a mathematical function, or a deliberate alteration by the devs? However, the calculations I showed you before actually explain it quite well, since it shows that the EDT (F) and AR values are not whole numbers; that they get rounded off for the sake of display.
One thing I should mention is that the formula will only work when a significant change in DA occurs (denoted by ΔDA), such as increasing DA from 54 to 100. A minor change in DA (denoted by δDA) will not provide the correct decimal needed to calculate EDT (F). For instance, let's calculate the change in DC between 28 and 30 Dexterity (98 → 102 DA), and apply the result to re-calculate EDT (F):
DC = δEPD/δDA = (86.3 - 86.1)/(102 - 98) = 0.2 / 4 = 0.1 / 2 = 1 / 20 = 0.05 ∴ ETD (F) = ((102 x (86.1 - (86.3 x 0.05))/100) + (102 x 0.05) = 83.42 + 5.1 = 88.5 ∴ δEPD/δDA cannot be used to calculate DC ∵ ETD (F) = 86.3 when DA = 102
Whilst the previous section covered calulating the scaling for unupgraded weapons, I have not discussed the other side of the spectrum; namely, fully upgraded weapons. Using the example of a fully upgraded Gyrm Axe, my investigations provided some...interesting suggestions. According to the Gamehubbs Character Planner, at 102 SA and DA, a Gyrm Axe +10 should have an EDP (Final) value of 107, whilst at 102 SA and 54 DA, EDP (Initial) = 99. Let's test that, shall we?
DC = ΔEPD/ΔDA = (107 - 99) / (102 - 54) = 8 / 48 = 1 / 6 = 0.167 ∴ ETD(F) = (SA x SC) + (DA x DC) = (SA x ((EPD(I) - (EPD(F) x DC))/100) + (DA x (ΔEPD/ΔDA)) = (102 x (99 - (107 x 0.167)/100)) + (102 x 0.167) = (102 x 0.811) + (102 x 0.167) = 82.75 + 17.03 = 99.78
At this point, you're probably wondering why it isn't working. Well, my theory is that the conventional formula for the Dexterity Contribution provides decimals which are too large to calculate EDT (F) of a fully upgraded weapon (since the DC of a Gyrm Axe +10 is double that of an unupgraded version), thus affecting the Strength Contribution too much. As a result, we must start afresh; however, how will we go about finding DC? Let's write the formula for EDTotal (Final), but with EDTotal (Initial) instead, and without using the original formula for DC:
ETD(I) = (SA x ((EPD(I) - (EPD(F) x DC))/100)) + (DA x DC)
Now, once we solve for DC, we get this:
DC = (((EPD(I) x SA) - (100 x ETD(I))/((EPD(F) x SA) - (100 x DA)))
There are two curious aspects of this formula: a) It suggests that DC is negative on a fully upgraded weapon, and b) It cannot be used to calculate EDT (F) when SA ≠ DA (when SA > DA, EDT (F) is understated; when SA < DA, EDT (F) is overstated). Despite these curiosities, it still works. Using the same numbers as before:
DC = (((EPD(I) x SA) - (100 x ETD(I))/((EPD(F) x SA) - (100 x DA))) = (((99 x 102) - (100 x 107))/((107 x 102) - (100 x 102) = ((10,098 - 10,700)/(10,914 - 10,200)) = - 602 / 714 = - 0.843 ∴ ETD(F) = (SA x ((EPD(I) - (EPD(F) x DC))/100)) + (DA x DC) = (102 x ((99 - (107 x -0.84))/100)) + (102 x - 0.84) = (102 x 1.89) - (102 x 0.84) = 107.1
Now, it's all well and good to solve for a number which is already present in the equation, but what about an (technically) unknown value of EDT (F)? Let's assume I max out my Strength and Dexterity again (meaning SA and DA = 200). According to the Gamehubbs Planner, EDT (F) should equal 210, but let's test it:
DC = - 0.84 ∴ ETD(F) = (SA x ((EPD(I) - (EPD(F) x DC))/100)) + (DA x DC) = (200 x ((99 - (107 x -0.84))/100)) + (200 x - 0.84) = (200 x 1.89) - (200 x 0.84) = 210
The formula predicts EDT (F) as 210, just as the character planner indicated.
If you're reading this section after having read the above two (or maybe eventually three to four) sections, then congratulations for making it through all this rambling! On a more serious note, I have no idea whether this would be useful to anyone; I just figured that it wouldn't hurt to post my findings, and see what people made of them.
Also, I will update this with the single scaling weapon formula fairly soon. Although it's fairly simple, I just haven't quite gotten it to the point where I'm happy with how it's worded.