# Damage System - math

This contains in-depth mathematical analysis of the damage dealing system and functions, and breaks down the steps of the damage dealing process

This is sub-layout for documentation pages

## Contents

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#### 1 Phases

The damage is calculated in several phases, all of them are going to be explained below, but before we do so, let's start with some definitions

• Let $A$, $D$ be characters in the game.
• Let $A$ be an attacker, and $D$ be the defender.
• Let $d$ be final damage that is dealt from attacker $A$ to the defender $D$
• Let $rand(a,b)$ be a function, which generates a random number from interval $<a, b>$, using uniform probability distribution. $a \in \mathbb{R}$

##### 1.1 Roll phase

Rolling phase is such a phase, where attack rating is determined for the attacker, and defence rating be determined for the defender.
Attacker has some set of attack stats, from which the relevant attack stat is chosen which respect to the attack style(magic / melee / blood / fire / ice / slash / stab etc.)
Let such a chosen number be $Aa = 5$

The same process is done for the defender, where defence stat $Dd$ is chosen, with respect to the attacker's attack style Let $Aa = 8$
Then, we calculate roll for both participants.

• Let $u$ be the chosen stat number. (eighter $Aa$ or $Dd$)
Then, the roll function $f(u)$ is defined, as follows:
$(u \lt 0) \rightarrow f(u) = rand(u, 0)$
$(u \geq 0) \rightarrow f(u) = rand(0, u)$

Then, we calculate rolls $r_a$ for attacker and $r_b$ for defender
$r_a = f(Aa)$
$r_d = f(Dd)$

Then, damage is evaded completelly, if $r_a < r_d$
And damage is evaded with 50% chance, if $r_a = r_d$

##### 1.1.1 Example

Here is an example to support the theory above.

Let's imagine, that attacker is using ice barrage spell to attack. Because of that, we choose attack - ice stat as $Aa$. Let
$Aa = 12$

For defender, we choose defence - ice stat as $Dd$. Let
$Dd = -3$

We calculate
$r_a = rand(0, 12) = 3$
$r_d = rand(-3, 0) = -1$
Because $r_a \gt r_d$, damage will be dealt, and system proceeds to the next phase,

##### 1.2 Damage calculation phase

Damage calculation phase is such a phase, which takes place after roll phase, if attacker was successfull in a roll. Here, final damage $d$ is calculated.
This phase takes previously calculated rolls $r_a$ and $r_d$ as an additional input.

The formula for final damage $d$ is:
$d = rand(1, d_{max})$
Where $d_{max}$ is the maximum possible damasge that can be dealt by the attacker. It applies:

$d_{max} = (1.7 * (rankMainW + dmgBonusForSpeed) + A_l * 0.3) * (1 + 0.8 * rollDifference)$

Where:

• $rankMainW$ is the rank of the main weapon currently wielded by the attacker, or 0 if no weapon is wielded
• $dmgBonusForSpeed = \frac{S_w - S_d}{1000} * 0.8$
Where
• $S_w$ is the main weapon attack speed
• $S_d$ is the default attack speed, defined by system
• $A_l$ is the level of the attacker
• $rollDifference = min(1 - r_d / r_a, 1)$

Then, that damage number is altered through series of modifier phases.

##### 1.3 Damage modifier phases

Damage modification phase, takes place always after damage calculation phase.

Input - output specification
Input: the damage $d$
Output: the damage $d1$, which can be whatever

##### 1.3.1 Worn item sets modifier phase

Input - output specification
Input: the damage structure $s$ containing damage $d$
Output: the damage structure $s$ with modified damage or any values

This phase takes in consideration attacker's worn item sets, and it's special effects.
For example, The unholy robe set has a chance of ignoring target's protection prayers. That is calculated here, and the structure $s$ is changed in appropriate way for future phase.

##### 1.3.2 Prayer damage modifier phase

Input - output specification
Input: the damage structure $s$ containing damage $d$
Output: the damage structure $s$ with modified damage or any values

This phase takes in consideration all attacker's activated prayers, and also, all defender's activated prayers.
This sorely depends on how the prayers are defined, but for example, melee / magic protection prayer can nullyfy the damage, if the attack style matches

If, however, input damage structure $s$ contains flag to skip protection prayers, they are ommited

##### 1.4 After effects modifier phase

After effect phase is the last phase, and takes place after damage all modifier phases were completed (and if any damage was dealt at all.)
It is responsible for buff / debuff modifiers, such as poison / freeze / teleblock or healing (blood spells).