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Intro to Bl3 Math


[WHOOPS! This article is unedited – give us some time to clean it up! 🧹]

 

The Borderlands series is choc full of mathematics, mathematics that for the most part the community has been left to work out for themselves. Scouring these forums will turn up a lot of maths breakdowns for specific Vault Hunters or Weapons. If you dig deeper in the comments there are references to obscure formulas for calculating common mechanics like Action Skill cooldown.

This community knows a lot about the mechanics behind Borderlands, however the maths isn’t very accessible to those starting fresh. I aim to change that with this guide, this guide will present very little new information. My intention is to compile a basic guide that explains the terminology used in Borderlands Math and to give a brief overview of the maths behinds mechanics such as Gun Damage, Health and Critical Hits.

I would deem this guide successful if by the end you feel you have the toolkit to understand in broad terms what the game is doing with all the numbers it shows you. I aim to provide the proverbial toolkit that you can then use to take apart mechanics you’re interested in.

So if you are new to Borderlands mathematics and want to learn, welcome to the rabbit hole! I warn you, grab a hot drink and buckle in because this is not going to be a quick read.

This guide is standing on the shoulders of giants. Community testing for Borderlands 3 is happening at a rate of knots however a lot of it is made possible by the work done on Borderlands 2. The early research done for Borderlands 2 is unfortunately lost to the internet archives, however there is still a wealth of knowledge in the Borderlands 2 and Pre-Sequel sub forums. I’ll do my best to point out where Borderlands 3 has overhauled mechanics, however for the most part if you read a guide on Borderlands 2 mechanics chances are you can extrapolate near seamlessly to Borderlands 3.

# <h1 id=”heading–Toc”>[color=darkcyan]Table of Contents[/color]</h1>
* [Explaining Terminology](https://forums.gearboxsoftware.com/t/guide-an-introduction-to-borderlands-math/4268831#heading–Terminology)
* [Borderlands Scaling](https://forums.gearboxsoftware.com/t/guide-an-introduction-to-borderlands-math/4268831#heading–Scaling)
* [Calculating Health](https://forums.gearboxsoftware.com/t/guide-an-introduction-to-borderlands-math/4268831#heading–health)
* [A Brief Look at Calculating Gun Damage](https://forums.gearboxsoftware.com/t/guide-an-introduction-to-borderlands-math/4268831#heading–CalcGD)
* [Stacks and Skills that Scale](https://forums.gearboxsoftware.com/t/guide-an-introduction-to-borderlands-math/4268831#heading–StacksAndScaling)
* [Elemental Damage](https://forums.gearboxsoftware.com/t/guide-an-introduction-to-borderlands-math/4268831#heading–EleDmg)
* [The Value of Multiplicative Damage and Comparing Bonuses](https://forums.gearboxsoftware.com/t/guide-an-introduction-to-borderlands-math/4268831#heading–multValue)
* [Double Dipping Bonuses](https://forums.gearboxsoftware.com/t/guide-an-introduction-to-borderlands-math/4268831#heading–doubleDip)
* [Introducing Inverses & Why they Look Weird](https://forums.gearboxsoftware.com/t/guide-an-introduction-to-borderlands-math/4268831#heading–Inverses)
* [Deriving Cooldown Rate](https://forums.gearboxsoftware.com/t/guide-an-introduction-to-borderlands-math/4268831#heading–derivingCDR)
* [Cooldown Rate Examples](https://forums.gearboxsoftware.com/t/guide-an-introduction-to-borderlands-math/4268831#heading–CDR)
* [Closing Words](https://forums.gearboxsoftware.com/t/guide-an-introduction-to-borderlands-math/4268831#heading–conclusion)

# <h1 id=”heading–Terminology”>[color=darkcyan]Terminology and Maths Mumbo Jumbo[/color] <h1>

Before it is possible to begin any discussion of numbers I want to explain some basic terminology and what it means.

## <h2 id=”heading–AddMult”>[color=darkcyan]Additive & Multiplicative[/color]<h2>

When discussing damage boosts terms such as additive and multiplicative are thrown around loosely. Forum folk and mathematicians have been using the terms for years so the assumption is that everyone knows what they mean, you know what they mean right?

When 2 terms are said to be additive it means the bonuses are added together. So if my first bonus is [color=orchid]Purple[/color], my second bonus is [color=skyblue]Blue[/color] and they are said to be additive then when we do the calculation it will appear as so:

> Total = [color=orchid]Purple[/color] + [color=skyblue]Blue[/color]

Now lets add a third variable [color=red]Red[/color] and we’ll say this is multiplicative to our friends [color=orchid]Purple[/color] and [color=skyblue]Blue[/color]. The calculation thus becomes:

> Total = ([color=orchid]Purple[/color] + [color=skyblue]Blue[/color]) x [color=red]Red[/color]

Simple enough right? Well lets up it another notch. A 4th variable [color=gold]yellow[/color] is introduced and this is additive to [color=red]Red[/color].
*Well hold on Prismatic, isn’t Red multiplicative?*
Glad you caught that, yes [color=red]Red[/color] is multiplicative to [color=orchid]Purple[/color] and [color=skyblue]Blue[/color], however that doesn’t stop it being additive to [color=gold]yellow[/color]. Additive and Multiplicative are relative terms, they only make sense when comparing terms or groupings of terms. When I say [color=gold]yellow[/color] is Additive to [color=red]Red[/color] I make no mention of its relation to [color=orchid]Purple[/color] and [color=skyblue]Blue[/color]. However as we know how [color=red]Red[/color] interacts with [color=orchid]Purple[/color] and [color=skyblue]Blue[/color] we can interpret how [color=gold]yellow[/color] fits into our overall formula.

> Total = ([color=orchid]Purple[/color] + [color=skyblue]Blue[/color]) x ([color=red]Red[/color] + [color=gold]yellow[/color])

## <h2 id=”heading–WhyMult”>[color=darkcyan]Why Do We Like Multiplicative Damage so Much?[/color]</h2>

Simply put because Multiplying Numbers gives us bigger bonuses than adding them. If we add 2 bonuses of 30% we’d get a 60% boost, however if the bonuses where multiplicative we’d get a bonus of 69%. Why do we get this boost? Multiplicative damage boosts effectively work in stages, we first boost by 30% so we’re at 130% of our original value, If we multiplicatively boost this by another 30% then we’re increasing the 130% by a further 30% whereas the additive boost would only be boosting the smaller 100% by 30%.

## <h2 id=”heading–LinearScaling”>[color=darkcyan]Linear Scaling[/color]</h2>

Linear is a term that rears its head quite consistently. Most Skills that have a variable damage output use a Linear scaling function. So I hear your question:

*”WTF IS A LINEAR FUNCTION?!?”*

Linear functions refer to functions that move in a straight line. So for instance with skills that have stacks, if it scales linearly then 25 stacks will give half the bonus that 50 stacks give and a quarter the bonus of 100 stacks. Half the stacks = half the bonus, simple right?

So what doesn’t count as a linear function?

Things like the amount of health you have at each level is not on a linear function. At level 25 you didn’t have half the health you have at level 50, this is because health is not based on a linear function.

## <h2 id=”heading–Rounded8″>[color=darkcyan]The Rounded 8[/color]</h3>

This is more a warning for your own testing than being integral to understanding how any of the maths work. There are many values in Borderlands that are rounded. Ever wondered why boosting the Mag Size of your Butcher by 40% takes it from 10 to 13 and not 14? This is due to the actual mag size value being less than 10, my guess is that it’s probably 9.5 . So a 40% boost only takes it to 13.3 . The game rounds the final value, so while it may work in your favour when the mag is 9.5 it works against you for a mag size of 13.3 .

I wouldn’t bring rounding up if the game was consistent and rounded all values like we were taught to in primary school. There are exceptions to this rule, common values like health and damage numbers over 10k are rounded down. When you where level 16 your base health before bonuses would of shown up as 317, in truth your max health was 317.6 .

Damage numbers over 10k are another example of this 18 900 will display in game as 18k not the 19k that you would expect. I’m unaware of any hard and fast rule for which way Borderlands will round numbers so take it with a grain of salt and be aware that just because the game has presented you a nice number doesn’t mean that there isn’t funny rounding going on in the background. The game engine will work with the unrounded number for all calculations and only round off at the end.

When testing it’s wise to be aware of the variation that rounding can have on your calculations. It you have a card damage of 500 for instance and you’re predicting the formula is 500 x 3 and the game spits out 1501 you’re probably correct as 500.4 would round down to 500 on the item card but the in game calculation would be 500.4 x 3 = 1501.2.

You ought to be worried however if the game spit out 1502 as 500.5 x 3 would be required to get the game to round to 1502, however 500.5 would show up on the item card as 501 so it’s likely you’ve made a calculation or logic error somewhere and you need to go back to the drawing board.

## <h2 id=”heading–ExpressingPercent”>[color=darkcyan]Expressing Percentages[/color]</h2>

When working with Borderlands formulas the default is to express all percentages in their decimal form, eg 10% = 0.1 . Throughout this guide I will refer to percentage boosts but when I actually write the maths for it you’ll notice I immediately switch to representing the percentages as decimals.

[Back to Table of Contents](https://forums.gearboxsoftware.com/t/guide-an-introduction-to-borderlands-math/4268831#heading–Toc)

*I am now transitioning away from the introduction and high level discussions of general Borderlands maths. So you’ve been warned. The kid gloves are coming off. Tighten up the big-boy pants, because the numbers are coming.*

# <h1 id=”heading–Scaling”>[color=darkcyan]Borderlands Scaling[/color]

Have you ever wondered how the game can generate numbers for damage, player health, experience and so on but no matter how many times you level up the values seem to remain relatively constant? Gearbox hasn’t randomly pulled values out of a hat or magically selected values they think fit.

Everything in Borderlands is based off a scaling formula. In Borderlands 3 the magic formula is

> Scale = 1.09<sup>Level</sup>

This generates a number for each level. The game then applies this value to base values for health, melee and so on.

So for instance the community has found that the base value for health is 80 and the base value for melee is 18. So let’s have some fun and work out the base health and melee values for our level 50 characters and for a bit of fun we’ll see what the values would be at level 99.

**Level 50:**
> Health = 80 x 1.09<sup>50</sup>= **5948.6**
Melee = 18 x 1.09<sup>50</sup> = **1338.4**

**Level 99:**
> Health = 80 x 1.09<sup>99</sup> = **405801.15**
Melee = 18 x 1.09<sup>99</sup> = **91305.2**

This magic formula is useful for more than just calculated health and melee values. Another use for this magic formula is to work out how long gear lasts. Say I wanted to know how many levels it takes for my guns damage to be half what an on level version would deal. I can use the scale formula to find out how many levels it will take.

There are 2 ways, you can be fancy and use logarithms to solve
>1.09<sup>Level</sup> = 2

Or you can be lazy and plug in numbers until you’re close enough. I’m proudly lazy about this stuff so I plugged in a few numbers and found that

> 1.09<sup>**8**</sup> = 1.99256

This is close enough for any practical purposes in Borderlands 3. We have just used the scaling formula to work that it takes 8 levels for our health to double in value or for our guns damage to double.

*To check you understand what I’ve done above see if you can work out how many levels it takes to triple your health. If you’re lost on how to do this please ask below and I’ll explain in better detail. If you tried to solve it and got [spoiler]13[/spoiler] then Congratulations!*

[Back to Table of Contents](https://forums.gearboxsoftware.com/t/guide-an-introduction-to-borderlands-math/4268831#heading–Toc)

# <h1 id=”heading–health”>[color=darkcyan]Health[/color]</h1>

I’m going to hold off on damage for a bit in favour of starting with the simpler health calculations. I’ve already mentioned that we can calculate the base health before boosts for a character of any level with:

> [color=lightcoral]Base Health[/color] = 80 x 1.09<sup>Level</sup>

*Going forward I’m going to reference this value as [color=lightcoral]BH[/color] for purposes of brevity in our formulas.*

## <h2 id=”heading–incHealth”>[color=darkcyan]Increasing Health[/color]</h2>

There 2 methods for boosting health in Borderlands 3. We have percentage based boosts, such as Guardian Ranks, and we have additive boosts from gear. The percentage boosts come first and behave nicely among each other. If you have 3 boosts, say 10%, 20% and 50% the resultant boost is just an addition of the 3, in this case 80%. Additive boosts are then added last. The formula is thus:

> [color=#f22]Max Health[/color] = [ [color=lightcoral]BH[/color] x ( 1 + boost1 + boost2 +… ) ] + additive1 + additive2 + …

I’ve expanded the formula here to be explicit in how multiple boosts of the same type work. For future I’m going to state it as:

> [color=#f22]Max Health[/color] = [ [color=lightcoral]BH[/color] x boosts ] + additives

This form is merely shorter to write, you can always expand the groupings to the form shown in the first equation. I should mention adaptive parts briefly, on the card they state an 8% health increase, the actual boost is 7.5% max health per adaptive part.

## <h2 id=”heading–ReserveOrReduce”>[color=darkcyan]Reserving and Reducing[/color]</h2>

Borderlands 3 has 2 main methods that diminish your health. Reducing and reserving health have the same overall effect of giving you less health to work with, however the way they give you less health is different and there are times when one method is desirable while the other is detrimental.

Reducing max health is the more common method and has been seen in the franchise before. There is nothing fancy to reducing max health, you simply have less. You can do this through a variety of methods, the [color=orange]Loaded Dice[/color] or shields with turtle parts are examples of this.

Reservers on the other hand are a more complex mechanic. If you reserve health you aren’t actually changing your max health. Reserving health caps how much health you can heal, but it doesn’t change your max health. If you reserve any amount of health you can’t trigger effects that require you be at full health, furthermore if you reserve more than 50% you lose the safety net provided by health gating.

[details=”What is Health Gating”]
Health Gating refers to a Borderlands mechanic that stops you from being 1 shot if your health is over 50%.
[/details]

### <h3 id=”heading–WhyReduceHealth”>[color=darkcyan]But why have less Health?[/color]</h3>

It sounds entirely counter intuitive to intentionally want less health, so why do builds purposefully use these mechanics?

#### Reducing

Reducing Health is generally desirable when the goal is to take max advantage of Health Gating. The second key component to this is that your healing method is not based on Max Health. Life steal skills and guns are the popular choice but set ups can also be based around passive health regen from gear. Health is reduced in an effort to minimize how much health you need to recover to exceed 50% max health.

#### Reserving

The general aim when reserving health is to exchange it for large gains in shield capacity. Moze is the character most likely to Reserve health as her [color=indianred]Shield of Retribution[/color] tree synergizes really well with low health and big shields.

## <h2 id=”heading–ReduceHealthCalc”>[color=darkcyan]The Full Health Calculation[/color]</h2>

Earlier we looked at the math of increasing health, now I want to expand the formula to include decreasing health.

> [color=#f22]Max Health[/color] = [ [color=lightcoral]BH[/color] x boosts ] + additives

Percentage based reductions in health from turtle shield parts and [color=orange]Loaded Dice[/color] Artifacts are multiplicative to our boosts, but come before the additives. This means a +x max health stat roll on your class mod or artifact are not affected by the health reducers.

> [color=#f22]Max Health[/color] = [ [color=lightcoral]BH[/color] x boosts x Reducers] + additives

Reducers however don’t play nicely between themselves. The [color=orange]Loaded Dice[/color] works how many would expect. To reduce health by 75% we would times by ( 1 – 0.75 ), and the [color=orange]Loaded Dice[/color] does indeed function this way. Shields with turtle parts however play differently. The turtle penalty is added to our denominator.

> Reducers = [color=orange]Loaded Dice[/color] x ( 1 / ( 1 + Turtle Penalty) )

[details=”A small example calculation”]
> Suppose we have a turtle shield with -10% health and we equip a [color=orange]Loaded Dice[/color]. If we plug this into our formula for Reducers we get:
>> Reducers = ( 1 – 0.75 ) x ( 1 / ( 1 +0.1 ) ) = 0.2273
[/details]

Reservers fall at the end of the formula and have an multiply all previous calculations. The Reservers are 1 minus the sum ( addition ) of all percentages you reserved.

> [color=#f44]Final Health[/color] = ( [ [color=lightcoral]BH[/color] x boosts x Reducers] + additives ) x Reservers
Reservers = 1 – (Reserver1 + Reserver2 +…)

So a Moze example with a [color=orange]Front Loader[/color] (60%) and 1 point in [color=indianred]Thin Red Line[/color] (20%) would produce:

> Reservers = ( 1 – ( 0.6 + 0.2)) = 0.2

In cases where 100% or more is reserved, your final health is set to 1 and you receive no additianl penalties for having 120% or more reserved.

### <h2 id=”heading–maxvsfinalHealth”>Max Health and Final Health</h3>

It’s important to note the distinction in my formulas between [color=#f22]Max Health[/color] which I’ve used the majority of the time and [color=#f44]Final Health[/color] which I introduced at the end when talking about Reducers.

I’ve done my best to make [color=#f22]Max Health[/color] consistent with in game terminology. If skills or gear say “regens x% max health per second” then you can base your calculations off what I have marked as [color=#f22]Max Health[/color].

[color=#f44]Final Health[/color] is my own term for health after you’ve reserved a portion of your max. In cases where you have not reserved [color=#f22]Max Health[/color], both terms will be equal.

[Back to Table of Contents](https://forums.gearboxsoftware.com/t/guide-an-introduction-to-borderlands-math/4268831#heading–Toc)

# <h1 id=”heading–CalcGD”>[color=darkcyan]A Brief Look at Calculating Gun Damage[/color]</h1>

The maths everyone is really interested in the first time they decided to do maths in their gaming time. I unfortunately won’t be going into depth here as the formula is very different for every vault hunter, if you’re interested in a specific Vault hunters formula I recommend you check their community resource to see if a write up has been done. I’ll be following [Sljm’s example and providing an overview of the formula](https://forums.gearboxsoftware.com/t/guide-basic-gun-damage-formula/45401). If you want proper depth on the damage formula I recommend looking [here](https://forums.gearboxsoftware.com/t/guide-borderlands-3-damage-formulas/4385216) and within the resources linked from the thread.

At a high level the formula for gun damage has barely changed from Borderlands 2. The separate categories are multiplied together to produce the final damage. We’re yet to dig into any of the following terms so it’s likely the following terms will be gibberish, however I believe it’s good to get a look at them now, I’ll then unpack a bit of what the terms mean and are doing. We can then revisit the formula later on once we’ve reached a better understanding.

> Gun Damage = [color=steelblue]Normal Hit[/color] x [color=lightpink]Splash[/color] x [color=wheat]Special Multipliers[/color] x [color=#f44]Critical Damage[/color] x [color=beige]Guardian Rank[/color] x [color=turquoise]Elemental Multiplier[/color]

Categories such as [color=steelblue]Normal Hit[/color] and [color=turquoise]Elemental Multiplier[/color] come up in nearly all damage calculations, be they normal gun damage, bonus elemental or even Tediore reloads. [color=steelblue]Normal Hit[/color] is the weapon card damage times by the Gun Damage boosts.

> [color=steelblue]Normal Hit[/color] = Gun Card Damage x Gun Damage Boosts

I’ve labelled these terms as Gun Damage Boosts as the in game terminology tends to use Gun Damage as the descriptive word to indicate these boosts. In discussions about damage these boosts are commonly referred to as additive gun damage. Again I encourage you to look at your individual characters damage formula to understand exactly which bonuses fit in here. Gear bonuses like +x% Weapon Damage or +y% Manufacturer Damage fit in here.

[color=turquoise]Elemental Multipliers[/color] are your typical fire vs flesh, shock vs armour multipliers. If you have boosts to an element damage then it’s multiplicative to the base modifier. If you have multiple boosts to 1 element then the boosts are generally additive between each other.

[details=”Table of Elemental Bonuses”]
![image|690×454, 75%](upload://wuHSsSBBYqVSp8q54CrPcZ952xN.jpeg)
[/details]

[color=wheat]Special Multipliers[/color] as the name suggests are a unique bunch. Amara for instance has 2 of them and they are multiplicative to each other, Moze on the other hand doesn’t have any special multipliers. A noteworthy gear bonus that fits in here is class mod boosts to specific weapon types. So if you have a class mod boosting something like pistol damage then it fits in here, I believe this bonus is multiplicative to everything but Amara’s [color=limegreen]Personal Space[/color].

[color=beige]Guardian Ranks[/color] include both the gun damage bonus and the boost from C-C-Combo, these are multiplicative to each other. C-C-Combo follows a simplistic stacking mechanism, 1 stack is 2% bonus, 2 stacks is 4%, 5 stacks is 10% etc..

[color=#f44]Critical Damage[/color] has been reworked slightly in how it’s calculated from Borderlands 2. I’ll avoid going into any depth here as @DocStrangelove has done an excellent job in his [Critical Hit Calculation](https://forums.gearboxsoftware.com/t/critical-hit-calculation/3761404) guide.

The important numbers to know even if you don’t know the whole calculation is that all critical hits apply a 2x multiplier to your gun damage.

Sniper’s have a hidden 20% bonus critical hit multiplier that multiplies with the base crit bonus for a 2.4x modifier. Jakobs and Hyperion also have unique manufacturer specific crit bonuses on all their guns. These multiply with everything, Hyperion’s bonus is 5% and Jakob’s bonus is 10%.

> [color=#f44]Critical Damage Multiplier[/color] = 2 x ( 1 + Listed Crit Bonus on Weapon) x ( 1 + Skills and Gear) x ( 1 + Manufacturer Specific Bonus) x ( 1 + Sniper Bonus )

Guns with [color=lightpink]Splash[/color] damage are easily identified by shooting at your feet, it it didn’t harm you the gun isn’t splash and you can ignore the modifier. Otherwise this modifier will include all gear bonuses to both splash and AOE damage. All bonuses I’ve tested have been additive to each other within splash damage.

If you would like a breakdown that goes into better detail on gun damage; https://forums.gearboxsoftware.com/t/guide-borderlands-3-damage-formulas/4385216

[Back to Table of Contents](https://forums.gearboxsoftware.com/t/guide-an-introduction-to-borderlands-math/4268831#heading–Toc)

# <h1 id=”heading–StacksAndScaling”>[color=darkcyan]Stacks and Skills that Scale[/color]</h1>

Borderlands has become progressively more fond of stack based skills as the series has worn on. In Borderlands 3 it is an active effort to create a build that doesn’t include a skill that has a stacking effect. Stack based skills scale very simply as:

> Total bonus = bonus per stack x number of stacks.

The only exception I am aware of to this rule is the [color=limegreen]Samsara[/color] healing. It’s convoluted and doesn’t generalize well to anything else I’ve tested, so if you’re interested ask and I’ll show it, otherwise just know it’s an exception.

The other common form of bonus that varies is skills that scale by percent full or empty. For instance Moze has [color=limegreen]Click, Click[/color] and [color=indianred]Desperate Measures[/color] while Zane has [color=limegreen]Confident Competence[/color] and [color=limegreen]Adrenaline[/color]. These skill scale simply with percent full or empty.
So in the case of [color=limegreen]Confident Competence[/color] if your shields max capacity is 600 and and you currently have 300 shields then you’ll get 300/600 or half full then you’ll receive half the full bonus. Putting this into a formula:

> [color=limegreen]Confident Competence bonus[/color] = ([color=limegreen]Confident Competence max bonus[/color]) x (Current Shield)/(Max Shield)

Moze’s skills scale on percent empty not full so we have to rejig the formula slightly. [color=indianred]Desperate Measures[/color] scales on how empty Moze’s health is. We can use the old statisticians trick that 1-A is the same as the opposite of A. So if

> ( Current Health) / ( Max Health )

Is how full our health is then:

> 1 – ( Current Health ) / ( Max Health )

Is how empty our health is. So we can use this to calculate the Desperate Measures Bonus

> [color=indianred]Desperate Measures Bonus[/color] = ([color=indianred]Desperate Measures max bonus[/color]) x [ 1 – (Current Health)/(Max Health) ]

[color=limegreen]Click Click[/color] could be calculated with the same method as [color=indianred]Desperate Measures[/color] by merely swapping Health for current and max Magazine size.

[Back to Table of Contents](https://forums.gearboxsoftware.com/t/guide-an-introduction-to-borderlands-math/4268831#heading–Toc)

# <h1 id=”heading–EleDmg”>[color=darkcyan]Elemental Damage[/color]</h1>

Elemental Damage is the bread and butter that has sustained everyone since the latter half of normal mode. There’s not much to say about the modifiers except that they’re powerful multiplicative bonuses that don’t take much effort to maintain and can boost all damage you deal.

Elemental Damage boosting skills and gear are so valuable because they are multiplicative bonuses to these Elemental Modifiers. In TVHM with Fire on Flesh the default modifier is 1.75x, with a 30% boost from an artifact or skill we get 1.75 x 1.3 = 2.275. The Elemental Projector takes this to another level as 1.75 x 1.9 = 3.325. Elemental boosts however are additive to each other, so combining our previous two examples would produce 1.75 x ( 1 + 0.9 + 0.3) = 3.85 .

> [color=turquoise]Elemental Multiplier[/color] = [color=darkcyan]Elemental Modifier[/color] x ( 1 + Elemental boost 1 + Elemental Boost 2 + ..)

## [color=darkcyan]Bonus Elemental Damage[/color]

The Annointments introduced with the Maliwan Takedown introduced a wave of gear that adds elemental bonuses on action skill end making this calculation more relevant than ever. Theses Bonuses are extremely powerful as they essentially act as a second bullet with a percentage of the damage. They get all the bonuses from Normal Hit, Splash, Special Multipliers, Critical Damage and Guardian Rank then they get their own elemental modifier at the end.

Recall I mentioned the general formula for gun damage is:

> Gun Damage = [color=steelblue]Normal Hit[/color] x [color=lightpink]Splash[/color] x [color=wheat]Special Multipliers[/color] x [color=#f44]Critical Damage[/color] x [color=beige]Guardian Rank[/color] x [color=turquoise]Elemental Multiplier[/color]

The Bonus Elemental Modifier Receives most of the same boosts and would be calculated as

> Bonus Elemental Damage = Bonus Damage Percent x [color=steelblue]Normal Hit[/color] x [color=wheat]Special Multipliers[/color] x [color=#f44]Critical Damage[/color] x [color=beige]Guardian Rank[/color] x [color=turquoise]Elemental Multiplier 1[/color]

The bonus damage percent is to account for the percentage bonus damage given by the annointment. Secondly I’ve labelled the element as [color=turquoise]Elemental Multiplier 1[/color] as it is not necessarily the same element as your gun and might receive a different multiplier.

Gun Damage and Bonus Elemental Damage are then added together to give your final damage output.

While it is possible to have multiple bonus elemental damage annointments active at 1 time giving you multiple Bonus Elemental Damage equations to include in your final damage it is not possible to stack the grenade and shield annointments to give +100% of a single element. You may however have 2 different elements each adding bonus damage.

The grenade and shield ASE annointments have a caveat that they don’t receive gun type buffs. So having +smg damage on your com and using an smg doesn’t transfer this damage to the bonus elements you are getting from annointments on your grenade and shield. Annointments on the gun however still get this buff.

[Back to Table of Contents](https://forums.gearboxsoftware.com/t/guide-an-introduction-to-borderlands-math/4268831#heading–Toc)

# <h1 id=”heading–multValue”>[color=darkcyan]The Value of Multiplicative Damage and Comparing Bonuses[/color]</h1>

I’ve now briefly mentioned the different types of multipliers that fit into the damage formula. I suspect the following formula will look more digestible now than it did at first.

> Gun Damage = [color=steelblue]Normal Hit[/color] x [color=lightpink]Splash[/color] x [color=wheat]Special Multipliers[/color] x [color=#f44]Critical Damage[/color] x [color=beige]Guardian Rank[/color] x [color=turquoise]Elemental Multiplier[/color]

I’ve briefly discussed what each of these modifiers mean. I don’t however want to leave you without an example of how to apply this formula. Crunching pure numbers gets boring fast so I am going to use this opportunity to demonstrate to you the value of multiplicative bonuses.

*Anyone who’s already realized that my pretty formula a few lines up is only listing multiplicative bonuses gets brownie points.*

[color=lightpink]Splash[/color] multiplies [color=steelblue]Normal Hit[/color] and all the rest, therefore [color=lightpink]Splash[/color] is a multiplicative bonus. We could say the same about any of the above modifiers, therefore they are all multiplicative.

So lets try an example and see what we can manage. We’ll use Amara for this as she has a nice spread of damage skills. For reason’s of simplicity I’ll assume our gun is a splash pistol and has 100 damage on the card and we don’t have any guardian ranks.

We’ll start by adding 3 points in [color=limegreen]Samsara[/color] and accumulating 5 stacks, which is a 25% Gun Damage Boost. This falls under [color=steelblue]Normal Hit[/color].

So

> [color=steelblue]Normal Hit[/color] = Gun Card Damage x Gun Damage Boosts
[color=steelblue]Normal Hit[/color] = 100 x [1+[color=limegreen]Samsara[/color]]
[color=steelblue]Normal Hit[/color] = 100 x [1+0.25] =125

We haven’t added any other bonuses, so everything else is 1. Therefore:

> Gun Damage = [color=steelblue]Normal Hit[/color] x [color=lightpink]Splash[/color] x [color=wheat]Special Multipliers[/color] x [color=#f44]Critical Damage[/color] x [color=beige]Guardian Rank[/color] x [color=turquoise]Elemental Multiplier[/color]
&nbsp;
Gun Damage = [color=steelblue]125[/color] x [color=lightpink]1[/color] x [color=wheat]1[/color] x [color=#f44]1[/color] x [color=beige]1[/color] x [color=turquoise]1[/color] = 125

Now lets spec another damage skill. We’ll take 5/5 [color=limegreen]Arms Deal[/color], which boosts splash damage by 20%.

> [color=lightpink]Splash[/color] = 1 + [color=limegreen]Arms Deal[/color]
[color=lightpink]Splash[/color] = 1 + 0.2 = 1.2

This leads to our damage formula currently looking as such.

> Gun Damage = [color=steelblue]125[/color] x [color=lightpink]1.2[/color] x [color=wheat]1[/color] x [color=#f44]1[/color] x [color=beige]1[/color] x [color=turquoise]1[/color] = 150

So far we’ve avoided any diminishing returns and only picked up bonuses that are multiplicative. You may ask have we gained anything over them being additive? The answer is yes, if the two boosts we got where additive we would have

> Gun Damage = 100 x [ 1 + 0.25 + 0.2 ] = 145

So by making sure our boosts are multiplicative we’ve managed to find 5 extra damage for free.

Okay it’s all very good to know that multiplying bonuses is better than adding them, but what about when we can’t find more multiplicative bonuses? Do we take the bigger number?

The answer to this has a bit of nuance. Lets reconsider our Amara example with [color=limegreen]Arms Deal[/color] and [color=limegreen]Samsara[/color]. Say you are picking between 2 new bonuses, you can only take one of them. A 25% bonus to [color=steelblue]Normal Hit[/color] or a 25% bonus to [color=lightpink]Splash[/color]. Which is better?

> 25% bonus to [color=steelblue]Normal Hit[/color]:
>> Gun Damage = [color=steelblue]100 x [ 1 + 0.25 + 0.25 ][/color] x [color=lightpink][ 1 + 0.2 ][/color] x [color=wheat]1[/color] x [color=#f44]1[/color] x [color=beige]1[/color] x [color=turquoise]1[/color] = 180
>
>25% bonus to [color=lightpink]Splash[/color]:
> > Gun Damage = [color=steelblue]100 x [ 1 + 0.25 ][/color] x [color=lightpink][ 1 + 0.2 + 0.25 ][/color] x [color=wheat]1[/color] x [color=#f44]1[/color] x [color=beige]1[/color] x [color=turquoise]1[/color] = 181.25

By crunching the numbers we see that the [color=lightpink]Splash[/color] boost is better here. The reason being that we’ve added the bonus to a smaller multiplier. We’ve not boosted splash much yet, so any bonus we do give it has a large effect.

The next question that will arise for many is what happens when the bonuses are not equal? If instead of comparing a 25% bonus for both what if we compared a 25% bonus to [color=steelblue]Normal Hit[/color] to a 20% bonus to [color=lightpink]Splash[/color]. Now which is better?

> 25% bonus to [color=steelblue]Normal Hit[/color]:
>> Gun Damage = [color=steelblue]100 x [ 1 + 0.25 + 0.25 ][/color] x [color=lightpink][ 1 + 0.2 ][/color] x [color=wheat]1[/color] x [color=#f44]1[/color] x [color=beige]1[/color] x [color=turquoise]1[/color] = 180
>
>20% bonus to [color=lightpink]Splash[/color]:
> > Gun Damage = [color=steelblue]100 x [ 1 + 0.25 ][/color] x [color=lightpink][ 1 + 0.2 + 0.2 ][/color] x [color=wheat]1[/color] x [color=#f44]1[/color] x [color=beige]1[/color] x [color=turquoise]1[/color] = 175

This time the [color=steelblue]Normal Hit[/color] is better, okay so how do we know when one is better than the other? This requires a bit of algebraic math trickery, which I’ll spare you the details on – you can ask below if you really want to know.

We can always compare two bonuses [color=orange]bonus1[/color] and [color=orange]bonus2[/color] as long as we know what other modifiers are additive to [color=orange]bonus1[/color] and [color=orange]bonus2[/color], I’ll call these [color=darkorange]additive1[/color] and [color=darkorange]additive2[/color]. So in our above comparison, [color=orange]bonus1[/color] and [color=orange]bonus2[/color] are the boosts to [color=steelblue]Normal Hit[/color] and [color=lightpink]Splash[/color], and [color=darkorange]additive1[/color] and [color=darkorange]additive2[/color] are the boosts provided by [color=limegreen]Samsara[/color] and [color=limegreen]Arms Deal[/color].

I say we’re going to compare two modifiers, in truth we’re going to find out how much of [color=orange]bonus2[/color] we need to get the same benefit as [color=orange]bonus1[/color]. Then we can compare our actual bonus to the number we get for [color=orange]bonus2[/color], if our actual bonus is bigger, then we want it!

So onto finding the comparison, First I just want to show the general formula of how the 2 bonuses apply.

> When applying [color=orange]bonus1[/color] our formula is as such:
>> Comparative Bonus = [ 1 + [color=orange]bonus1[/color] + [color=orange]additive1[/color] ] x [ 1 + [color=orange]additive2[/color] ] x Other Bonuses
>
> When we instead apply [color=orange]bonus2[/color] our formula is as such:
> > Comparative Bonus = [ 1 + [color=orange]additive1[/color] ] x [ 1 + [color=orange]bonus2[/color] + [color=orange]additive2[/color] ] x Other Bonuses

The damage offered by the two modifers will be equal when:

> [color=orange]bonus2[/color] = [color=orange]bonus1[/color] x ( [ 1 + [color=orange]additive2[/color] ] / [ 1 + [color=orange]additive1[/color] ] )

Note: it’s entirely possible 1 of the two additives is equal to zero. If this is the case the method still works fine. It also won’t be rare for 1 of the additives to actually be made up of multiple bonuses, again this is fine.

So as an example of using this formula lets take the Amara example from earlier where we had:

> Gun Damage = [color=steelblue]125[/color] x [color=lightpink]1.2[/color] x [color=wheat]1[/color] x [color=#f44]1[/color] x [color=beige]1[/color] x [color=turquoise]1[/color] = 150

Say we had a class mod that boosted weapon damage by 25% and we wanted to know what boost to splash damage would get the same value for our set up. We therefore have:
– [color=orange]bonus1[/color] as the 25% com boost
– [color=orange]additive1[/color] is the gun damage from [color=limegreen]Samsara[/color]
– [color=orange]additive2[/color] is the gun damage from [color=limegreen]Arms Deal[/color]
– And finally [color=orange]bonus2[/color] is the mystery splash damage we want to know.

So plugging into our formula we get:

> [color=orange]bonus2[/color] = [color=orange]bonus1[/color] x ( [ 1 + [color=orange]additive2[/color] ] / [ 1 + [color=orange]additive1[/color] ] )
> Equivalent Splash Bonus = CoM Bonus x ( [ 1 + [color=limegreen]Arms Deal[/color] ] / [ 1 + [color=limegreen]Samsara[/color] ] )
> Equivalent Splash Bonus = 0.25 x ( [ 1 + 0.2 ] / [ 1 + 0.25 ] ) = 24%

So we now know that 24% bonus to splash would give us the same value as our current 25% bonus to Weapon damage, therefore we also know that if we got a +28% splash com it would be even better than our current com.

[Back to Table of Contents](https://forums.gearboxsoftware.com/t/guide-an-introduction-to-borderlands-math/4268831#heading–Toc)

# <h1 id=”heading–doubleDip”>[color=darkcyan]Double Dipping Bonuses[/color]</h1>

Double dipping is a rare idea that turns up from time to time in Borderlands, and every time it does people make a big fuss about it. So what is it and why the fuss?

Double dipping refers to when a an effect is boosted by the same bonus twice. This most commonly occurs when an items damage is based off the total damage of whatever triggers it. So for instance the electric Banjo will double dip on elemental bonuses. The first time will be the elemental bonus applied to the bullet that triggers the banjo and the second time will be on the actual chain damage of the banjo.

Double dipping creates a scenario where a bonus multiplies itself, a 30% boost that double dips will create a 69% improvement to the effects damage.

Lets consider Amara’s Ties that Bind as an example. Ties that Bind according to the card transfers 35% of damage dealt to a target to all linked targets. The actual math of it has slightly more intricacies, however we don’t need exactness here so we’ll just use an approximate of what’s going on here.

> Damage to Linked target = [Damage to original target] x 0.35 x [color=turquoise]Elemental Multiplier of Amara’s AS Element[/color] x [ Action Skill Damage ] x Other

For simplicity I’ll assume we have no action skill damage or other damage modifiers active. So in a scenario where we are dealing fire damage to flesh targets. If we hit the original target with 100 elemental damage then our damage to the linked target is

> Damage to Linked target = 100 x 0.35 x [color=turquoise]1.75[/color] x 1 x 1 = 61.25

Now lets include 5/5 Tempest, Tempest is a 30% boost to all elemental damage. As a boost to elemental damage it also happens to be double dipped by Ties that Bind. So what happens to our example? Well first we would be hitting our original target for 30% more damage, so we now start with 130 base.

> Damage to Linked target = 130 x 0.35 x [color=turquoise]1.75 x 1.3[/color] x 1 x 1 = 103.5

Which you should note is 69% more than 61.25.

So what other examples are there of double dipping? There unfortunately aren’t many, Moze has both Short Fuse and Mind Sweeper grenades that double dip. Overkill technically double dips everything, though that’s an strange case.

[Back to Table of Contents](https://forums.gearboxsoftware.com/t/guide-an-introduction-to-borderlands-math/4268831#heading–Toc)

# <h1 id=”heading–Inverses”>[color=darkcyan]Introducing Inverses and Why They Look Weird[/color]</h1>

Up to this point all the math we’ve discussed and seen demonstrated has followed a very simple format. Each buff or de-buff gets its own multiplier and that multiplier is constructed by adding or subtracting the bonus from 1. This sort of bonus makes a lot of sense.

The bonuses we’re going to be discussing next however, don’t follow this form and aren’t as immediately intuitive as to what they’re doing. I thus wanted to preface this math by discussing what is likely going on behind the scenes and how the bonuses make sense in this context before we transition to discussing the results. I don’t know for certain that this is what the game is doing, but it is a fairly well informed guess. I feel this section actually benefits from seeing how some of the math is derived.

*I’m going to talk about action skill cooldown rate as my example, but for any example that uses similar math you can make a similar sort of comparison. Most time related calculations will use this method.*

[For those only interested in the formula, This is your skip button. ](https://forums.gearboxsoftware.com/t/guide-an-introduction-to-borderlands-math/4268831#heading–CDR)

## <h2 id=”heading–derivingCDR”>[color=darkcyan]Deriving Where Cooldown Rate Calculations Come From[/color]</h2>

Action Skill’s all incorporate a cooldown time, which from our perspective immediately makes us think of pulling out a stop watch and timing it. When the time is up, we give the player their action skill back. If the player buffs their cooldown rate? No problem, we’ll just calculate the new shorter time and use that instead.

This account makes a lot of sense if you’re trying to do this manually, however from a coders perspective this is actually a nightmare to implement. Timers are finicky to code, they require a lot of micro management. If you then require that the timer be capable of handling dynamic cooldown bonuses, eg 30% cooldown rate bonus for 3 seconds after dealing splash damage…

It’s a nightmare.

Without me assuming you have knowledge of coding it’s hard for me to better convey why this is so unappealing to code, so please just believe me when I say a method like this is asking for trouble.

**So what’s the alternative to this?**

The alternative is to forego basing the method on time. We instead create a target that we must reach and gradually we move closer to it. We ensure that under normal conditions this will always take the same amount of time. You could think of this like a progress bar that fills up at a set rate. Or if we’re using a physical example, like driving to a town that is 100km away and you must do the entire drive at 10km/h. If you think about it, my conditions ensure that your trip will always be 10 hours long.

This method also makes cooldown rate bonuses simple to calculate, if I boost the cooldown rate the analogue in my example is for you to drive faster. Lets reconsider that nightmare example I gave – 30% cooldown rate bonus for 3 seconds after dealing splash damage. If we adapt this to our example, say: “drive 30% faster for an hour after passing a petrol station.” Suddenly working out the effect of the bonus shouldn’t seem so bad.

Lets start the math with a simple example, we’ll use the 100km trip at 10km/h. I claimed the trip would take 10 hours but I didn’t show the math, let me do that now.

> Time x Speed = Distance
Time x 10 = 100
Time = 100 / 10 = 10

I’ve started with a simple equation for distance and subbed in the values I gave. I intentionally made this simple as I want your attention to be on how the formula changes if we increase the cars speed by 60%. For simplicity we’ll assume you drive 60% faster the whole way. So how does our speed increase change things?

> Time x Speed = Distance
Time x ( 10 x 1.6 ) = 100
Time = 100 / ( 10 x 1.6 ) = 6.25

Notice that our fairly simple change to your speed ended up on the bottom of the fraction when we tried to work out the time taken. I don’t know about you, but I don’t want to go through this whole calculation every time you propose a different speed. So, let’s write time in terms of your first speed and work out what we have to do to that time to get the new time.

> Time = 100 / ( 10 x 1.6 ) = ( 100 / 10 ) x (1 / 1.6) = ( 100 / 10 ) x (1 / [1+0.6])

Notice that 100 / 10 is our initial time. So I now want to remove the numbers and replace them with what they stand for:

> Time = Initial Time x [ 1 / (1+speed increase) ]

See the weird impact the faster speed made on the travel time.

Now to bring this back to Borderlands, I was using this extended example to explain why cooldowns are weird. I said that coders use something similar to your trip to town for cooldowns. So lets use the math we’ve just done on your trip, but we’ll sub in the equivalent Borderlands terms. First we replace your destination town with the result that your cooldown is finished, and if instead of travel speed we have cooldown rate then you’ll see that.

> Time x Speed = Distance
*becomes:*
Time x Cooldown Rate = Base Cooldown

However we’re not really interested in this formula, we want to know the effect that boosting cooldown rate will have. So instead we’ll look at the later formula I used when we looked at the difference that increasing your speed made and sub in the Cooldown equivalents:

> Time = Initial Time x [ 1 / (1+speed increase) ]
*to:*
Time = Base Cooldown x [ 1 / (1+Cooldown Rate boosts) ]

Here we’ve arrived at the strange looking formula that is used to calculate cooldowns. I hope that my long winded example has at least given you a sense of why a boost to cooldown rate looks funny when calculating time.

# <h1 id=”heading–CDR”>[color=darkcyan]Cooldown Rate[/color]</h1>

We’ve now looked at the likely source of the funny calculation, but I wanted to discuss cooldown in the context of an example.

For those who jumped here, this is the formula we’re working with:

> New Cooldown = Base Cooldown x [ 1 / (1+Cooldown Rate boosts) ]

Those that read the previous section will notice I swapped ‘Time’ out for ‘New Cooldown,’ I’ve done this just to clear up terminology further. Anyway onto the worked example.

*You have an Amara phasecast build where you’ve specced 5/5 Restless (+25% Cooldown Rate) and you want to know the new cooldown time for your build. You’ve also however got a few guardian ranks and are getting +10% Cooldown rate from the stats. So what’s the new cooldown?*

Phasecast has a base cooldown of 28 Seconds, and within the formula our cooldown rate boosts are additive. So, the new cooldown is:

> New Cooldown = [Phasecast Base Cooldown] x [ 1 / (1 + Restless + Guardian Rank Cooldown) ]
New Cooldown = [28 Seconds] x [ 1 / (1+0.25+0.1) ] = 28 x 1/1.35 = 20.74 Seconds.

This all good and well but now you want to know how much cooldown rate you’d need to bring the cooldown rate to 15 seconds. So let us work that out too!

For this problem we have slightly different information. We know what we want the new cooldown to be and we know the base cooldown, the only missing puzzle piece is the required cooldown rate boosts. If we plug what we know into the formula we get:

> New Cooldown = [Phasecast Base Cooldown] x [ 1 / (1 + Cooldown Rate boosts) ]
15 Seconds = [28 Seconds] x [ 1 / (1 + Cooldown Rate boosts) ]

If we use a bit of algebra we can rearrange the above a few times with the aim of having ‘Cooldown Rate boosts’ alone

> 1 + Cooldown Rate Boosts = 28/15
Cooldown Rate boosts = 28/15 – 1
Cooldown Rate boosts = 13/15 = 0.8667 = 86.67%

In order to get a 15 second cooldown on Phasecast we’d need all our cooldown rate boost to add up to 86.67%.

[Back to Table of Contents](https://forums.gearboxsoftware.com/t/guide-an-introduction-to-borderlands-math/4268831#heading–Toc)

# <h1 id=”heading–conclusion”>[color=darkcyan]Closing Words[/color] <h1>

Phew, that’s a wrap, if you stuck with me through all that thank you! I really hope at least 1 of my explanations has given you a new understanding. I’d love to hear your feedback, if my examples or thought processes are unclear let me know and I’ll take another crack at it!

I’ve considered how long to make this piece, I nearly included sections on splash damage and damage reduction. However I decided to hold off on those for now as this draft has been weighing on me as is. If there is interest I will add sections on them, and if there is other math you want info on let me know and we can see about adding a section on it. However for now we’ll leave it here.

Once again thank you for reading.

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