[Mathcraft] +20% ATK/MAG vs. +20% DMG: What you need to know about soft caps

With Steal Power's +50% ATK and Lulu's stackable +20% MAG at hand, the ATK/MAG soft caps are more relevant than they've ever been before. Most people know that +20% ATK/MAG is significantly better than +20% DMG as long as you're well below the soft cap (and significantly worse once you're over it), but exactly at what point does it become better to switch from +20% ATK/MAG RMs (like Devotion (+20% MAG) or Dragoon's Determination (+20% ATK, referring to this as just Determination from now on)) to +20% DMG RMs (like Viera Virtuoso (+20% bow damage) or Attunement II (+20% DMG against weaknesses)?

First, a refresher on the damage formulas.

• Physical damage: (Ability Multiplier) * (MIN(ATK, 590))^1.3 * (ATK)^0.5 / DEF^0.5
• Magical damage: (Ability Multiplier) * (MIN(MAG, 742))^1.15 * (MAG)^0.5 / RES^0.5

So as long as you're below 590 ATK/742 MAG after all buffs (including Devotion/Determination), here's the percentage difference you get from Determination (+20% ATK, or 1.2x ATK multiplier):

• ((Ability Multiplier) * (MIN(1.2*ATK, 590))^1.3 * (1.2*ATK)^0.5 / DEF^0.5) / ((Ability Multiplier) * (MIN(ATK, 590))^1.3 * (ATK)^0.5 / DEF^0.5)
• Cancel out common factors of Ability Multiplier and DEF^0.5: ((MIN(1.2*ATK, 590))^1.3 * (1.2*ATK)^0.5) / ((MIN(ATK, 590))^1.3 * (ATK)^0.5)
• We're assuming you're below 590 ATK after Determination's buff, so MIN(1.2*ATK, 590) is just 1.2* ATK, and MIN(ATK, 590) is just ATK: ((1.2*ATK)^1.3 * (1.2*ATK)^0.5) / ((ATK)^1.3 * (ATK)^0.5)
• x^a * x^b = x^(a+b) : (1.2*ATK)^1.8 / (ATK)^1.8
• Factor out 1.2, simplify ATK^1.8 factors: 1.2^1.8 = 1.3884, or +38.84%

The same approach gives you the DMG boost you get from +20% MAG below the soft cap: 1.2^1.65 = 1.3510, or +35.10%

Let's do it again, but assume you're over 590 ATK/742 MAG before Devotion/Determination:

• The first two steps above are the same, giving you: ((MIN(1.2*ATK, 590))^1.3 * (1.2*ATK)^0.5) / ((MIN(ATK, 590))^1.3 * (ATK)^0.5)
• You're above 590 ATK before Determination's buff, so MIN(1.2*ATK, 590) and MIN(ATK, 590) are both just 590: (590^1.3 * (1.2*ATK)^0.5) / (590^1.3 * (ATK)^0.5)
• Factor out 1.2, simplify 590^1.3 and ATK^0.5 factors: 1.2^0.5 = 1.0954, or +9.54% (same answer for both ATK and MAG)

Okay, we're almost done! So if you're like me, you'd like to know exactly how much ATK/MAG you need (after buffs, but without counting the RM) for the +20% ATK/MAG buff RMs to buff your damage by exactly 20%; less base ATK/MAG means the +20% ATK/MAG buff RMs are better, and more base ATK/MAG means the +20% DMG buff RMs are better. Or, in equation form (for ATK):

• Solve for ATK: ((MIN(1.2*ATK, 590))^1.3 * (1.2*ATK)^0.5) / ((MIN(ATK, 590))^1.3 * (ATK)^0.5) = 1.2

I don't know how to solve that algebraically, but plugging in values gives me 545 ATK, 685 MAG. Above that point, Devotion/Determination give less than a +20% DMG boost, and you should use a +20% DMG RM instead if possible (and if you expect to be above that point for the majority of the battle).

Finally, a couple comments:

• Before anyone says "That's impossibly high ATK/MAG, who cares!", remember: Steal Power/Planet Protector (+50% ATK) lowers that required ATK to 363, and Lulu's Hairpin (+20%) plus Vivi's/Ashe's default SB (+50% +30%) lowers that required MAG to 380 439, easily achievable with RS. (EDIT: Fixed Vivi/Ashe SB error; 439 MAG is still achievable but obviously significantly harder than 380.)
• Once you're above the soft cap, it's really hard to pump a lot more DMG out of your attacks by simply raising ATK. For example, let's see what it would take to hit 40k per Barrage against Soulcage+++ (break resist, 550 DEF): Let's assume you have the maximum possible ATK buffs and DEF debuffs, that's Viera Virtuoso for +20% DMG (way better than Determination since you're miles away from the 590 ATK soft cap here), Advance for stackable +150% ATK, Power Steal for +50% ATK, Armor Breakdown for -40%/2 DEF, and Full Break for stackable -30%/2 DEF: 1.2 (DMG multiplier) * 2.8 (Barrage) * 590^1.3 (ATK soft cap) * (ATK * 2.5 (Advance) * 1.5 (Power Steal))^0.5 / (550 (DEF) * (1-.4/2 (Armor Breakdown)) * (1-.3/2 (Full Break))^0.5 > 39998 (9999 per hit, 4 hits) when base ATK exceeds 884. So once you're above the ATK soft cap, the best way to get significantly more damage is to increase the multipliers by switching from Barrage/Thief's Raid to stronger multi-hit SSBs.