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TPS Damage Formula

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UVHM is here and we are getting some sort of difficulty and what once seemed like a strong build might now start to feel weak. I’m going to attempt to explain how to get the most out of your skill trees and gear to better min/max and have success in UVHM.
Let’s start with the damage formula.

Output = (NormalHit * SpecMult * CriticalHit + SplashDamage) * ElemBonus * TypeMod

That is from @Sljm math wizardry in his Basic Gun Damage Formula. I’m not going to go into extreme detail how each of these work, read Sljm’s linked thread for that because he already nails it.

How ever I will touch on Cryo since that is not in that thread. From the DDD master thread found here

Cryo DOT gives +200% Type A Critical Damage.
Cryo DOT gives +200% Explosive Damage (multiplier)
Cryo DOT gives +250% Melee Damage (multiplier)

What I want to touch on in this thread more deeply is Multiplicative bonuses vs Additive bonuses. A lot of you know these terms and understand them but a lot of people do not. Even with the people that know these many don’t really understand them that well either. I see a lot of people dismiss additive damage skills because of Multiplicative ones and that is a mistake.
What do these terms mean in relation to Borderlands?

Multiplicative means to multiply other numbers.

Additive means to add to other numbers.

Lets go back to the damage formula again

Output = (NormalHit * SpecMult * CriticalHit + SplashDamage) * ElemBonus * TypeMod

Here is Normal hit from @Sljm thread

NormalHit = (BaseDamage * (1 + GunDamage) + AmpDamage)

Base is your gun damage and .GunDamage is the following these are typically what you call Additive

Most skills, will detail later
Gun damage from Oz kits
Bar gun damage
Gun damage from Com’s

Multiplicative boosts are much more complex

Critical Hits are multiplicative to Normal hit + Amp damage
Splash Damage is multiplicative to Normal hit + Amp damage as well
Spec Multipliers are multiplicative to normal, amp, and crits or multiplicative post crit damage and some can be post splash some are not, more details will be later in the thread.
Elemental Bonuses are multiplicative to everything before them, Noramal + amp, Critical hits, Special Multipliers, and Splash damage
Cryo bonuses are a mixed bag, crits go into the crits, explosive goes at the end of everything.
Some Splash Damage gets grenade buffs and those grenade buffs are Multiplictive to splash damage only.

If critical hits are multiplicative why do people call them additive?

This has caused confusion in the past, there are three types of critical hits
The formula is as follows

Crit Damage = 2 * (1 + .TypeA) * (1 + .TypeB) / (1 + .TypeC)

TypeA is also what is Additive critical hits are since the vast majority of Critical hits are these and they add to other critical hit bonuses. Any Critical hit bonus from Bar, Oz kits, Com’s, Accessories, Luneshine, or your skills are type A,
TypeB is most commonly found on Jakobs AR’s, Pistols, and Shotguns. A few others guns like the Gwen’s head has this. These are considered to be Multiplicative since they will multiply your other critical hit buffs.
TypeC is a divider which lowers your critical hit bonus that is found on Non Jakobs Assault rifles and is a 20% penalty.

For more information on critical hits read @Chuck80 guide, Critical hit bonus sources
Next lets look Splash Damage

Splash Damage is not really part of the gun damage but post gun damage. When you shoot an enemy you will see 2 numbers, the damage from your bullet and the damage from the splash. Now if the bullet kills the enemy outright you will not see the splash damage show up. There have not been any Splash damage testing threads done yet for TPS, mostly because without a test dummy its a massive pain in the ass. So go sign or like this thread for a test dummy.

Splash Damage is based on your Normal hit + Amp damage but can be multiplied by more here is the formula from @Sljm thread linked at the top of this thread.

SplashDamage = NormalHit * SplashAmount * (1 + GrenadeDamage)? * SpecMult?

What is missing here because this is still figured into the total damage formula is they Also get buffed by elemental damage.

There are 2 main types of Splash Damage

Benefits from Grenade Damage. Usually comes attached to a normal bullet impact, such as on most basic Torgue weaponry. Some Uniques/Legendaries break this rule.
Does not benefit from Grenade Damage. Usually occurs on its own without a normal bullet impact, such as with Rocket Launchers or Grenadier Assault Rifles. Some Uniques/Legendaries break this rule.


Some skills also add Splash Damage to your guns

Claptrap’s Coincidental Combustion, @Exotek made a thread about it here
Nisha’s Short Fused
Jacks Sponsored by…, @impala made a thread about it here

Elemental Bonuses are one of the big ones

Many people that played bl2 are familiar with the Bone of the Ancients and for good reason it’s damage bonus was huge because it was elemental.

Here is the formula for elemental bonuses

ElemBonus = (1 + ElementalDamage + [SpecificElement]Damage) * FreezeBonus?

It’s not the formula that makes it so good, its where it comes in the total formula which I will post again

Output = (NormalHit * SpecMult * CriticalHit + SplashDamage) * ElemBonus * TypeMod

You see here that elemental bonuses multiply pretty much everything before it making all of your numbers that much bigger.

Elements that get these bonuses are

:fire: Fire
:zap: Shock
:snowflake: Cryo
:boom: Explosive

Last but not least Special multipliers!!!

So these are the fun part, what separates Borderlands from most shooters and many games in general.

I like to separate these into 2 classes

Total multiplicative, these come at the end of the formula and multiply everything, much like elemental but separate from them.

Athena’s Omega Senshu and Clear! are both examples of this

Claptrap’s One Last thing is also Total @Exotek made a thread about it here

Partial multiplicative are ones that come before splash damage but after critical hit damage
-Athena’s Tear is an example of this

Jacks skills still need to be tested to know if he has any of these
So how to put all this together?

Finally after all that I’m back to my starting point. Now that we understand all that (hopefully) how do you get the most out of your gear. One reason I decided to take the time to write all this up is I see people making the mistake of ignoring Additive skills and just picking Multiplicative skills. Yeah Multi are your Michael Jordans’, they are the stars but Jordan is not Jordan with out the players around him. Let me show you what I mean using Athena.

For simplicity I’m going to use a level 9 build 57 and an elemental gun with 100 damage and lets say 300 stacks

100 x (1 + 120%) = 220 damage or x 2.2 to your gun damage on card

Now change Invictus to Ephodos in this level 9 build 30

100 x (1 + 20%) x (1 + 120%) = 264 damage or x 2.64 to your gun damage on card

So by adding 20% to your base damage you received a 44% buff instead.

Let’s do it again but this time on a critical hit, no crit bonuses

Build 1

100 x 2 x (1 + 120%) = 440 damage or x 4.4 to your gun damage on card

Build 2

100 x (1 + 20%) x 2 x (1 + 120%) = 528 or x 5.28 to your gun damage on card

Now that 20% has turned into difference of 88%. Now the pattern is clear the more multiplicative buffs you add to base damage buffs your numbers start to sky rocket.

I hope this is clear enough to not underestimate Additive gun damage skills, 20% additive might not sound like a lot on paper but it is. Multiplicative skills are not enough, they need role players or a base to build off of. Don’t skip the base skills for the flashy skills.
Getting the most out of Critical hits

Damage can be much more important than Critical hit buffs when it come to crits. A base rule that even “experts” of this game get wrong is that a Damage acc on a sniper adds more damage than a Critical hit acc does. The critical hit acc adds 20% crit damage to a sniper while a damage adds 15% base damage. So lets look and understand this one

Crit acc on a Jakobs sniper

100 x 2 x (1 + 160% + 20%) = 560 damage or x 5.6 to your damage on card

Damage acc on a Jakobs sniper

100 x (1 + 15%) x 2 x (1 + 160%) = 598 or x 5.98 to your damage on card

So that 15% buff is more than the 20% buff because it is multiplied twice by 2 other buffs, while the 20% is not.

This leads to a larger point, if you have a crit build Base damage skills once again are very important. Going back to bl2 again a lot of people would think to make the lady fist better adding critical hit skills would be important but that was a bad idea since it had a tiny base damage and a massive crit bonus. The same rules apply to this game because the formula is the same.
The bigger the critical hit bonus is on a gun, the more important base damage is to it.

But not only skills, adding base damage with BAR, Oz kits, Com’s, etc… But also don’t over stack your damage and ignore other skills that keep you alive, there is such a thing as a “Damage ceiling” where you are wasting points with overkill damage and sacrificing survival. That is a topic for a another thread…
Spreading out your buffs and avoiding diminishing returns

When you are building a character some get too involved with stacking a buff to that theme instead of spreading out those buffs.

For an example lets say you have 5 buffs that are all 20% the best way to use those would all be seperate.

100 x (1 + 20%) x (1 + 20%) x (1 + 20%) x (1 +20%) x (1 + 20%) = 248

compared with

100 x (1 + 20% + 20% + 20% + 20% + 20%) = 200

Now I know the first doesn’t exactly fit into the damage formula but it’s as an example but its not as far off as it seems. To use Athena as an example.

Ephodos, Prepare for Glory, and Gun Kata are all Normal hit skills
Tear is a Partial Multiplicative Special Multi skill
Maelstrom is a Elemental Buff skill
Omega – Senshu is a Total Multiplicative Special Multi skill

So that is 4 separate buff categories you can put buffs into, then you can use an Oz kit to expand that to 5 to maybe a grenade buff if you are using a gun with splash damage that is buffed by grenade damage.

Now not saying every time you should try and spread it out to the max like that, each case is different and some guns have more limitations. Non Elemental guns like Jakobs you can only really buff Normal hit and Critical hit. But even with those the theory remains true to spread your buffs around.

More on the Diminishing returns side of it lets go back to this example

100 x (1 + 20% + 20% + 20% + 20% + 20%) = 200

So here you have 100% worth of buffs and you got a 100% buff from those skills right? Wrong.

100 x (1 + 20%) = 120

That is a 20% damage increase on a 20% damage buff.

100 x (1 + 20% + 20%) = 140

Now sure that is 40% buff from the card damage but from the 120 you had above its only a 16.6% increase when you keep doing this each buff is a lesser % increase do your damage.

100 x (1 + 20% + 20% + 20% + 20% + 20%) = 200

So doing all 5 like that the 5th 20% is only adding 11% to your damage.

By spreading them out you avoid the diminishing returns and max out your damage.

Again to emphasize its not always best to spread them out in a certain way, it always depends on your gun stats, the size of each buff and more.

Please give me some feed back if this isn’t clear, if you want more sections, or better examples, etc…

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