For those just here for the formula:
(Base Damage * TYPE A ( + Scorching RPMs + Stainless Steel Bear + Desperate Measures + Action Skill Damage ) * TYPE B ( + Drowning in Brass + Phalanx Doctrine) * Specialist Bear * (Splash Damage Bonuses when applicable) * Critical Hit (2 * 1+( Scorching RPM’s) * Elemental Multipliers (+Stoke the Embers if fire) + (Bonus fire damage (Fire in the Skag Den, Cloud of Lead, Experimental Munitions).
TYPE B only works with the left hand.
The rest of the op is testing/proof.
Video test that I will be referencing for this section:
Testing was done on TVHM no mayhem, using the basic V-35, on a technical. So damage is being lowered by 20% based on elemental multipliers.
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The initial damage shown @:12s is 1274 (The actual damage for V-35 is ~1592, so rounding is being done.)
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@:23 seconds I put on a relic that boosts AOE damage by 33%. This is the same as splash damage. Shooting the car gives 1694 damage (1274 * 1.33 = 1694)
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@:33 seconds I put on a blast master com fully charged + it has 28% more splash damage. The number now is 3324 (1274 * (1+1+.33+.28)=3325 …but the game is rounding)
TLDR; IB benefits from splash damage boosts. Obviously only if the weapon does splash damage…
For the record, Iron Bear’s weapons that do splash damage are:
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Exploding Bullets
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Molten Roar
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Every Grenade Launcher/ Rocket Pod
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Corrosive Sabot Round
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The “Random” bonus elemental damage on Wild Swing
The rest of the vid is me shooting the car at different timings. The grenade launcher normally shoots at ~1 shot a second, so it maintains a low number of combo stacks (namely 1) , but it still illustrates that that IB benefits from C-c-combo (3390 is 2% more than 3324)… which is weird but we take those.
Now onto the next video:
Specialist Bear increases IB’s damage by 25% if you use two of the same weapon; only the diamonds count so feel free to mix chevrons.
1274 * (1+.25) =1592
Putting back on the relic we get a number of 2118, which is spicy. This is 1274 * (1+.25) * (1+.33)=2118. Specialist Bear and Scorching RPMs are additive, and splash damage boosts are multiplicative.
While on the subject, the Shaped Charge Augment gives a 35% boost on an direct impact. Not only is this boost multiplicative ( Shooting a car with this and specialist bear gives 2149 ( 1274 * 1.25 * 1.35 ) ) but it applies after splash damage boosts (the same example but with the relic is 2859 (1274 * 1.25 * 1.33 * 1.35)
Finally Critical hit damage. Gearbox, add a testing dummy 
This test was done with the minigun. Base Damage is shown at 414.
Base Critical Hit Damage is shown as 827, a little less than 2 * 414. Which shows IB doesn’t get boosted by GR crit damage.
Despite that, IB does gain Scorching RPM’s crit boost; As soon as I slot it in it goes to 993 damage. (414 * (2 * (1+.20)) = 993.6… again rounding.
12/13/19 - Recent patches changed some aspects of IB’s damage. Gearbox made Desperate measures, and later Drowning in Brass and Phalanx Doctrine work with IB, and they are on a separate part of the formula, but they only work for the left gun 
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On top of this they also made Specialist Bear apply on a separate part of the formula as well.
The damage on flesh for the railgun is 7401
SSB is a 20% additive boost (7401 * 1.2 = 8881)
Then Desperate Measures is added multiplicitavely (7401 * 1.2 * 1.5 = 13321.8)
Then with one stack of DiB you get 15k (7401 * 1.2 * (1+.2+.5))
Finally, Phalanx Doctrine + SSB (7401 * 1.2 * 1.1 = 9769)
Now for a similar test for Specialist Bear:
First Scorching RPMs (7401 * 1.25 = 9251)
Then with specialist bear and desperate measures for the left gun:
7401 * 1.25 * 1.25 = 11564 ( * 1.5 = 17346)
12/19/2019
Desperate measures now works with both hands… but is now a TYPE A boost.
Using the same test as above:
base damage 7401
7401 * (1+.5+.25) * 1.25 = 16k, as apposed to the 17k we got before.
Because of this I’m separating the base damage boosts into two categories: TYPE A (Desperate Measures, Stainless Steel Bear and Scorching RPM) sand TYPE B (Phalanx Doctrine and Drowning in Brass), since they all work the same and are additive within a group (at least until gearbox gets fucky again 
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With that in mind, IB damage looks like (Base Damage * TYPE A ( + Scorching RPMs + Stainless Steel Bear + Desperate Measures ) * TYPE B ( + Drowning in Brass + Phalanx Doctrine) * Specialist Bear * (Splash Damage Bonuses when applicable) * Critical Hit (2 * 1+( Scorching RPM’s) * Elemental Multipliers (+Stoke the Embers if fire) + (Bonus fire damage (Fire in the Skag Den, Cloud of Lead, Experimental Munitions).
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Don’t be too desperate now…