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 V35, on a technical. So damage is being lowered by 20% based on elemental multipliers.

The initial damage shown @:12s is 1274 (The actual damage for V35 is ~1592, so rounding is being done.)

@: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)

@: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:

Exploding Bullets

Molten Roar

Every Grenade Launcher/ Rocket Pod

Corrosive Sabot Round

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 Cccombo (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 .
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 )
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).