Formulae

Damages are calculated as follow^[1]:

${\displaystyle {\begin{aligned}BaseDamage=&~{\big (}Atk~\lfloor \pm Ans\times AdvantageMod\rfloor \lfloor +Ans\times EffectiveMod\rfloor -Def~\lfloor -Def\times TerrainMod\rfloor {\big )}^{+}\\BaseDamage^{*}=&~{\big (}Atk~\lfloor \pm Ans\times AdvantageMod\rfloor \lfloor +Ans\times EffectiveMod\rfloor -Def~\lfloor -Def\times TerrainMod\rfloor \lfloor +SpecialDamage\rfloor {\big )}^{+}\\Damage=&~{\big (}{\big \lfloor }(BaseDamage^{*}~\lfloor +FixedDamage\rfloor )\times StaffMod{\big \rfloor }\lfloor -Ans\times PercentReduction\rfloor \lfloor -FixedReduction\rfloor {\big )}^{+}\end{aligned}}}$

The Combat Forecast's^[2] damages use a shortened formula:

${\displaystyle ForecastDamage={\big \lfloor }{\big (}BaseDamage~\lfloor +ConstantFixedDamage\rfloor {\big )}\times StaffMod{\big \rfloor }}$

Notation

• ${\displaystyle Ans}$ is used as a placeholder for "the result so far".
• ${\displaystyle \lfloor ~\rfloor }$ is used to represent the Floor functions, that is rounding the result down. For example, ${\displaystyle \lfloor 2.4\rfloor = 2}$.
• ${\displaystyle \square ^{+}}$ is used to mean the Positive parts, which set negative values to 0.

A detailed explanation of the remaining terms is provided below.

Atk

The ${\displaystyle Atk}$ is the result of the attacker's visible Atk stat before entering the combat, plus all the Temporary buffs and debuffs affecting it. This second value is visible through the Combat Forecast (any temporary buffs and debuffs greater than 99 are displayed as 99, but are used properly).

Def

The ${\displaystyle Def}$ is the result of the defender's visible Def or Res stat before entering the combat, plus all the Temporary buffs and debuffs affecting it. The Def is used when the attacker has a  physical weapon type, while the Res is used for  magical weapon type.

AdvantageMod

The ${\displaystyle AdvantageMod}$ is the result of the Weapon-triangle mechanism of the attacker's weapon type over the defender's weapon type. It is equal to 0.2 (20%) when there is a dis/advantage, 0 otherwise.
Note the ± sign in front of it. The actual operator depends on whether there is an advantage or a disadvantage: + in case of advantage, - in case of disadvantage.

EffectiveMod

The ${\displaystyle EffectiveMod}$ is the result of the effectiveness of the attacker over the defender's weapon type or move type. It is equal to 0.5 (50%) in case of effectiveness, 0 otherwise. Being effective against both weapon type and move type does not increase this modifier any more.

TerrainMod

The ${\displaystyle TerrainMod}$ is the effect of Defensive terrains applied on the defender. It is equal to 0.3 (30%) when the defender is on a defensive terrain, 0 otherwise.

StaffMod

The ${\displaystyle StaffMod}$ is the particular effect that unconditionally affects all attackers that use a  Staff, halving their damage output. It is defined as 0.5 (50%) for  Staff users, 1 otherwise.

SpecialDamage

The ${\displaystyle SpecialDamage}$ is the result of the additional damages granted during the activation of an Offensive Special.
Specials that "Boosts damage by percentage" use ${\displaystyle BaseDamage}$ as the source of boosting.

FixedDamage

The ${\displaystyle FixedDamage}$ is the result of all effects that "Deals damage during combat". It is effectively the sum of all such effects targeted on the defender.
Note that the rounding operation is applied after the summation, therefore the decimal demages of individual effect are also summed.

${\displaystyle FixedDamage}$ can actually be divided into two categories: ${\displaystyle ConstantFixedDamage}$ and ${\displaystyle SituationalFixedDamage}$.
${\displaystyle SituationalFixedDamage}$ includes all effects that depends on factors that can change during a combat. This concerns the following effects:

• Effects that depends on turn order, i.e. "During first attack", "After foe's attack", etc.
• Effects that depends on Specials and its current cooldown count, i.e. "When Special triggers", "When cooldown count is 0", etc.
• Effects that depends on the current HP, i.e. "Deals damage = 10% foe's HP", "When unit's HP > 25%", etc.

All other effects are categorized as ${\displaystyle ConstantFixedDamage}$. Note that effects that depends on the maximum value (of Special cooldown count and HP) or the start of combat's value are categorized as ${\displaystyle ConstantFixedDamage}$.

For examples,  Wo Dao+,  Godlike Reflexes and  Domain of Flame are ${\displaystyle SituationalFixedDamage}$, as they respectively depends on Special activation, Special cooldown count value, and turn order.  Shining Bow+,  Exposure and  Gambit 4 on the other hands are considered ${\displaystyle ConstantFixedDamage}$, as they respectively are dependent of the Def stat, unconditional, and dependent on the maximum cooldown count value.

PercentReduction

The ${\displaystyle PercentReduction}$ is the result of all effects that "Reduce damage by percentage" on the defender. The individual effects should not be summed, but rather multiplied. It is defined as ${\displaystyle PercentReduction=1-\prod {\frac {100-percent}{100}}}$ when at least one effect exists, 0 otherwise. In this expression, ${\displaystyle \prod}$ is the Product function applied on all such effects, while ${\displaystyle percent}$ is the individual reduction percentage.

By example, if there is one effect that reduces damage by 30% on the defender, ${\displaystyle PercentReduction}$ will be ${\displaystyle 1-{\frac {100-30}{100}}=0.3}$. If there are two such effects, ${\displaystyle PercentReduction}$ will be ${\displaystyle 1-{\frac {100-30}{100}}\times {\frac {100-30}{100}}=0.51}$, which corresponds to a 51% of damage reduction.

Effects that "Reduces damage by percentage" are usually written as is, or as "Reduces damage by X%", followed by a period, comma, or a conjunction. A few examples are  Aegis,  Reduces damage when initiating,  Domain of Ice and  Close Call 4.

FixedReduction

The ${\displaystyle FixedReduction}$ is the result of all other effects that "Reduces damage". It is defined as the sum of all such effects on the defender.
Note that the rounding operation is applied after the summation, therefore the decimal part of individual effect are also summed.

Effects that "Reduces damage" are usually written as "Reduces damage by X", where X can be a number, and "Reduces damage by X% of ...". A few examples are  Shield Pulse 3,  Flow Flight 3 and  Wyvern Ono.

Procedural formula

The formula can also be understood has a serie of operations, which are listed below:

${\displaystyle {\begin{aligned}Ans=&AttackersVisibleAtk+AttackersTemporaryAtkBonus&(Atk)\\Ans=&{\begin{cases}\lfloor Ans\times 1.2\rfloor ,&{\mbox{attacker has an advantage}}\\\lceil Ans\times 0.8\rceil ,&{\mbox{attacker has a disadvantage}}\\Ans,&{\mbox{otherwise}}\end{cases}}&(AdvantageMod)\\Ans=&{\begin{cases}\lfloor Ans\times 1.5\rfloor ,&{\mbox{attacker is effective}}\\Ans,&{\mbox{otherwise}}\end{cases}}&(EffectiveMod)\\Def=&{\begin{cases}DefendersVisibleDef+DefendersTemporaryDefBonus,&{\mbox{attacker has physical weapon}}\\DefendersVisibleRes+DefendersTemporaryResBonus,&{\mbox{attacker has magical weapon}}\end{cases}}\\Ans=&Ans-{\begin{cases}\lfloor Def\times 1.3\rfloor ,&{\mbox{defender is on a defensive terrain}}\\Def,&{\mbox{otherwise}}\end{cases}}&(Def+TerrainMod)\\BaseDamage=&Ans^{+}\\Ans=&{\begin{cases}Ans+\lfloor SpecialDamage\rfloor ,&{\mbox{Special triggers}}\\Ans,&{\mbox{otherwise}}\end{cases}}&(SpecialDamage)\\Ans=&Ans^{+}+\lfloor FixedDamage\rfloor &(Bounding+FixedDamage)\\Ans=&{\begin{cases}\lfloor Ans\times 0.5\rfloor ,&{\mbox{attacker uses Staff}}\\Ans,&{\mbox{otherwise}}\end{cases}}&(StaffMod)\\Ans=&{\begin{cases}\left\lceil Ans\times \prod {\frac {100-PercentReduction}{100}}\right\rceil ,&{\mbox{defender has percent-based damage reduction}}\\Ans,&{\mbox{otherwise}}\end{cases}}&(PercentReduction)\\Ans=&Ans-\lfloor FixedReduction\rfloor &(FixedReduction)\\Damage=&Ans^{+}&(Bounding)\end{aligned}}}$

Note that the formulae have been factorized whenever possible, which required the introduction of the ceil function ${\displaystyle \lceil ~\rceil }$, which rounds the result up.

Notes

There are some effects that can change the default behaviour listed above. Some examples are listed here:

• The  Triangle Adept status effect and other skills with similar behaviour all change the value of ${\displaystyle AdvantageMod}$. The highest effect applied on the attacker is added to the modifier, therefore changing it from 0.2 to 0.4.
  • When such an effect is active on the defender, and the  Cancel Affinity status effect or another similar effect is on the attacker, the Triangle Adept effect is substracted to the multiplier, instead of added. It therefore changes from 0.2 to 0.
  • The Time Traveler skills does not affect ${\displaystyle AdvantageMod}$, but are considered ${\displaystyle ConstantFixedDamage}$ and ${\displaystyle FixedReduction}$ instead.
  • See also Weapon-triangle for more informations.
• The  Hexblade status effect and other similar effects can change the targeted stat of ${\displaystyle Def}$.
• Skills that "Deals damage" based on "Atk minus Def" or "Atk minus Res", such as  Astra Blade, use the actual stats (with consideration of Temporary buffs and debuffs), not ${\displaystyle BaseDamage}$.
• Skills such as  Wrathful Staff 3 or similar on  Staff users change ${\displaystyle StaffMod}$ to 1.
  • Effects that cancel such skills, like  Mystic Boost 3, reset it to 0.5.
• Skills that "Reduce 'Reduce damage by percentage'" affects the ${\displaystyle percent}$ part of the ${\displaystyle PercentReduction}$ expression.
  •  Hardy Fighter 3 allows Defensive Specials to trigger twice. It is therefore not affected by effects such as  Halves non-Special damage reduction}, which targets non-Special percent based damage reduction.

Examples

Simple combat

A very simple combat. The only particularity is  Marth: Altean Prince being on a Defensive terrain.

Enemy attack

${\displaystyle {\begin{array}{rll}Ans=&52+0=52&(Atk)\\Ans=&52-(36+0)=16&(Def)\\Damage=&16\end{array}}}$

Player attack

${\displaystyle {\begin{array}{rll}Ans=&55+6=61&(Atk)\\Ans=&61-\lfloor (31+0)\times 1.3\rfloor &(Def+TerrainMod)\\=&61-\lfloor 40.3\rfloor =21\\Damage=&21\end{array}}}$

Advantage, Effectiveness and Special

A combat accounting for Weapon-triangle mechanisms and Effectiveness.  Sharena: Princess of Askr is equipped with the  Effective against dragons Status effect, and activates her  Glimmer Special.

Enemy attack

${\displaystyle {\begin{array}{rll}Ans=&49+0=49&(Atk)\\Ans=&\lceil 49\times 0.8\rceil &(AdvantageMod)\\=&\lceil 39.2\rceil =40\\Ans=&40-(26+3)=11&(Def)\\Damage=&11\end{array}}}$

Player Forecast

${\displaystyle {\begin{array}{rll}Ans=&58+3=61&(Atk)\\Ans=&\lfloor 61\times 1.2\rfloor &(AdvantageMod)\\=&\lfloor 73.2\rfloor =73\\Ans=&\lfloor 73\times 1.5\rfloor &(EffectiveMod)\\=&\lfloor 109.5\rfloor =109\\Ans=&109-(37+0)=72&(Def)\\Damage=&72&\Leftarrow BaseDamage\end{array}}}$

Player Attack

${\displaystyle {\begin{array}{rll}Ans=&BaseDamage+\lfloor BasedDamage\times 50\%\rfloor &(SpecialDamage)\\=&72+\lfloor 72\times 50\%\rfloor \\=&72+\lfloor 36\rfloor =108\\Damage=&108\end{array}}}$

Staff user

A combat with a  Staff user and ${\displaystyle ConstantFixedDamage}$.  Marth: Altean Prince is afflicted by the  Exposure Status effect.

Player attack

${\displaystyle {\begin{array}{rll}Ans=&48+10=58&(Atk)\\Ans=&58-(25+0)=33&(Def)\\Ans=&33+\lfloor 10\rfloor =43&(FixedDamage)\\Ans=&\lfloor 43\times 0.5\rfloor &(StaffMod)\\=&\lfloor 21.5\rfloor =21\\Damage=&21\end{array}}}$

Fixed Damage and Reduction

A combat with ${\displaystyle ConstantFixedDamage}$ and ${\displaystyle FixedReduction}$.  Wyvern Ono deals damage equal to 20% of the unit's Def, and reduce damage by the same amount.

Player attack

${\displaystyle {\begin{array}{rll}Ans=&53+0=53&(Atk)\\Ans=&\lceil 53\times 0.8\rceil &(AdvantageMod)\\=&\lceil 42.4\rceil =43\\Ans=&43-(37-7)=13&(Def)\\Ans=&13+\lfloor 57\times 20\%\rfloor &(FixedDamage)\\=&13+\lfloor 11.4\rfloor =24\\Damage=&24\end{array}}}$

Enemy forecast

${\displaystyle {\begin{array}{rll}Ans=&49-7=42&(Atk)\\Ans=&\lfloor 42\times 1.2\rfloor &(AdvantageMod)\\=&\lfloor 50.4\rfloor =50\\Ans=&50-(42+0)=8&(Def)\\Damage=&8&\Leftarrow BaseDamage\end{array}}}$

Enemy attack

${\displaystyle {\begin{array}{rll}Ans=&BaseDamage-\lfloor 57\times 20\%\rfloor &(FixedReduction)\\=&8-\lfloor 11.4\rfloor =-3\\Damage=&-3^{+}=0\end{array}}}$

Situational Damage and Percent Reduction

A combat with ${\displaystyle SituationalFixedDamage}$ and ${\displaystyle PercentReduction}$.  Divine Breath and  Dragon's Wrath 4 both deals damage equal to 25% of unit's Atk minus foe's Res during first attack, and reduce damage of first attack by 25%.

Enemy Forecast

${\displaystyle {\begin{array}{rll}Ans=&52+0=52&(Atk)\\Ans=&\lceil 52\times 0.8\rceil &(AdvantageMod)\\=&\lceil 41.6\rceil =42\\Ans=&\lfloor 42\times 1.5\rfloor &(EffectiveMod)\\=&\lfloor 63\rfloor =63\\Ans=&63-(38+10)=15&(Def)\\Damage=&15&\Leftarrow BaseDamage\end{array}}}$

Enemy Attack

${\displaystyle {\begin{array}{rll}Ans=&\lceil BaseDamage\times {\frac {100-25}{100}}\times {\frac {100-25}{100}}\rceil &(PercentReduction)\\=&\lceil 15\times 0.5625\rceil \\=&\lceil 8.4375\rceil =9\\Damage=&9\end{array}}}$

Player Forecast

${\displaystyle {\begin{array}{rll}Ans=&63+15=78&(Atk)\\Ans=&\lfloor 78\times 1.2\rfloor &(AdvantageMod)\\=&\lfloor 93.6\rfloor =93\\Ans=&93-(25+0)=68&(Def)\\Damage=&68&\Leftarrow BaseDamage\end{array}}}$

Player Attack

${\displaystyle {\begin{array}{rll}Ans=&BaseDamage+\lfloor (63+15-25-0)\times 25\%+(63+15-25-0)\times 25\%\rfloor &(FixedDamage)\\=&68+\lfloor 53\times 25\%+53\times 25\%\rfloor \\=&68+\lfloor 26.5\rfloor =94\\Damage=&94\end{array}}}$

Complete case

A complete case where almost every part are included (except ${\displaystyle StaffMod}$ and ${\displaystyle SpecialDamage}$).  Naga: Dragon Divinity has an advantage and is effective against  Tiki: Dragon Scion. She also possess ${\displaystyle SituationalFixedDamage}$ as in the previous case.  Tiki: Dragon Scion is located on a Defensive terrain, is afflicted by the  Exposure Status effect, therefore sustaining ${\displaystyle ConstantFixedDamage}$, is in the range of  Domain of Ice's ${\displaystyle PercentReduction}$, and activates her Special  Escutcheon, which in turn triggers  Shield Pulse 3's ${\displaystyle FixedReduction}$.

Player Forecast

${\displaystyle {\begin{array}{rll}Ans=&63+12=75&(Atk)\\Ans=&\lfloor 75\times 1.2\rfloor &(AdvantageMod)\\=&\lfloor 90\rfloor =90\\Ans=&\lfloor 90\times 1.5\rfloor &(EffectiveMod)\\=&\lfloor 135\rfloor =135\\Ans=&135-\lfloor (31+4)\times 1.3\rfloor &(Def+TerrainMod)\\=&135-\lfloor 45.5\rfloor =90&\Leftarrow BaseDamage\\Ans=&90+\lfloor 10\rfloor =100&(ConstantFixedDamage)\\Damage=&100{\mbox{ (Reduced at 99 due to display limit)}}\end{array}}}$

Player Attack

${\displaystyle {\begin{array}{rll}Ans=&BaseDamage+\lfloor 10+(75-35)\times 25\%+(75-35)\times 25\%\rfloor &(FixedDamage)\\=&90+\lfloor 10+10+10\rfloor =120\\Ans=&\lceil 120\times {\frac {100-30}{100}}\times {\frac {100-30}{100}})\rceil &(PercentReduction)\\=&\lceil 120\times 0.49\rceil =\lceil 58.8\rceil =59\\Ans=&59-\lfloor 5\rfloor =54&(FixedReduction)\\Damage=&54\end{array}}}$

References