BLM AOE Rotation

[Fenli]

Always cast Fire III.
That math is bad; it's missing Crits. You should have around a one in six chance to crit, which is nothing to scoff at, and those crits deal +50% damage. Also missing Direct Hits, which have around a one in four chance to happen, dealing a flat +25% damage, which stacks with crits (should they happen simultaneously).
Those chances are so high, that even with the worst luck in the world, you're dealing one or the other multiple times a fight.
It's simple maths: when you multiply a big number instead of a small one, you get an even bigger one. A crit "cold" Flare is a sad, sad thing, where a crit "hot" Flare is glorious.

Plus, Triplecast (and Manafont) exist. Why wouldn't you be using all the tools at your disposal?

[Rongway]

That math is bad

You didn't provide a quote, so I'm going to assume you're talking about mine.

Plus, Triplecast (and Manafont) exist. Why wouldn't you be using all the tools at your disposal?

I addressed this. They're excluded from the math because they won't change the conclusion. If you use all the same oGCDs in both rotations, all the extra Flares will be performed under the same conditions in both rotations and will increase the numbers in both scenarios in similar ways. Any difference between the two rotations will come down to starting with (Fire III + Flare) vs just casting Flare. I never said not to use them. They just don't matter when the specific question is "should I cast Fire III or not?"

it's missing Crits. You should have around a one in six chance to crit, which is nothing to scoff at, and those crits deal +50% damage. Also missing Direct Hits, which have around a one in four chance to happen, dealing a flat +25% damage, which stacks with crits (should they happen simultaneously).
Those chances are so high, that even with the worst luck in the world, you're dealing one or the other multiple times a fight.
It's simple maths: when you multiply a big number instead of a small one, you get an even bigger one. A crit "cold" Flare is a sad, sad thing, where a crit "hot" Flare is glorious.

These also don't matter when the specific question is "should I cast Fire III or not?"

The expected damage of a Fire III hit is
(critrate * (1+critbonus)) * fire3dmg + (1-critrate)*fire3dmg
=(critrate + critrate*critbonus + 1 - critrate) * fire3dmg
=(1 + critrate * critbonus) * fire3dmg

Each Flare hit is an independent event, and the expected outcome of each independent Flare hit is
(critrate * (1+critbonus)) * flarehit_i + (1-critrate)*flarehit_i
=(critrate + critrate*critbonus + 1 - critrate) * flarehit_i
=(1 + critrate * critbonus) * flarehit_i

The sum of the expected damage from each Flare hit is
(1 + critrate * critbonus) * flaredmg_1 + (1 + critrate * critbonus) * flaredmg_2 + ... + (1 + critrate * critbonus) * flaredmg_n

Or
= (1 + critrate * critbonus) * sum[i=1..n](flaredmg_i)

So if you want to compare the damage between a "cold" Fire III and a "cold" Flare, you can compare them directly without worrying about crit, because inequality relations are preserved under (positive) multiplication and division.
(1 + critrate * critbonus) * fire3dmg < (1 + critrate * critbonus) * sum[i=1..n](flaredmg_i)
===> fire3dmg < sum[i=1..n](flaredmg_i)

Critting a single target attack doesn't make it better than an AoE attack just because of the crit; the AoE attack will crit just as often, on average, and what matters is the total damage across all hits in the cast.

[Fenli]

what matters is the total damage across all hits in the cast.

Correct, which is why you want AF3 crit Flares

[PyurBlue]

It's simple maths: when you multiply a big number instead of a small one, you get an even bigger one.

That's why skipping Fire III can be good. F3 can only ever hit one target so its damage is proportional to potency * 1. Flare can hit many targets so the damage it produces is roughly potency * (potentially big number). Flare is also higher potency than F3 in the fire place, and when used under Umbral Ice they both cast faster than their recast time, so F3 isn't any faster.