Which Special should I use on X hero? (Math Edition)
Hi guys,
With numbers crunching going strong atm while everyone is trying to figure out the best SI builds to each hero, I have decided to do a little calculation to help everyone out in figuring out which Special will be the best on their waifu/husbando.
Below is a giant wall of algebra comparing the numbers need to choose one Special over the other. I have compared all offensive Specials within their own cooldown group, and listed down the condition to take one over another. Skip to the QED part if you want a quick answer, but I posted the workings so that someone else can cross check it.
#Notations
A = Attacker's total attack (For Blazing/Glowing *** attack calculations)
a = Attacker's attack after weapon triangle modifications
d = Defender's relevant defense stat(Def/Res)
r = Attacker's relevant defense stat(For Bonfire/Ignis and Iceburg/Glacies)
m = Attacker's max hp
h = Attacker's current hp
#Abilities Definition
##2 Turn
| Name | Definition |
|---|---|
| Moonbow | a-0.7d |
| Reprisal | (a-d)+0.3(m-h) |
##3 Turn
| Name | Definition |
|---|---|
| Glimmer | 1.5(a-d) |
| Draconic Aura | 1.3a - d |
| Luna | a-0.5d |
| Vengence | (a-d) + 0.5(m-h) |
| Bonfire/Iceburg | a-d+0.5r |
##4 Turn
| Name | Definition |
|---|---|
| Night Sky | 1.5(a-d) |
| Draconic Fang | 1.5a - d |
| Ignis/Glacies | a-d+0.8r |
##5 Turn
| Name | Definition |
|---|---|
| Astra | 2.5(a-d) |
| Aether | a-0.5d |
| Rising AOE | A+a-2d |
| Blazing AOE | 1.5A+a-2d |
#Quick Reference
Read: Row is stronger than Column when Cell Condition is fulfilled.
##2 Turn
| Ability | Moonbow | Reprisal |
|---|---|---|
| Moonbow | NA | d > m-h |
| Reprisal | m-h >d | NA |
##3 Turn
| Ability | Glimmer | Draconic Aura | Luna | Vengence | Bonfire/Iceburg |
|---|---|---|---|---|---|
| Glimmer | NA | a > 2.5d | a > 2d | a > d+(m-h) | a > d+r |
| Draconic Aura | a < 2.5d | NA | a > 1.67d | a > 1.67(m-h) | 0.6a > r |
| Luna | a < 2d | a < 1.67d | NA | d > m-h | d > r |
| Vengence | (m-h) > a-d | (m-h) > 0.6a | m-h > d | NA | (m-h) > r |
| Bonfire/Iceburg | r > a-d | r > 0.6a | r > d | r > (m-h) | NA |
##4 Turn
| Ability | Night Sky | Draconic Fang | Ignis/Glacies |
|---|---|---|---|
| Night Sky | NA | d < 0 | a > d+1.6r |
| Draconic Fang | d > 0 | NA | a > 1.6r |
| Ignis/Glacies | 1.6r > a-d | r > 0.625a | NA |
##5 Turn
| Ability | Astra | Aether | Rising AOE | Blazing AOE |
|---|---|---|---|---|
| Astra | NA | a > 1.33d | 3a-d > 2A * | 3a-d > 3A * |
| Aether | a < 0.75d | NA | 1.5d > A | d > A |
| Rising AOE | 3a-d < 2A * | A > 1.5d | NA | NA |
| Blazing AOE | 3a-d < 3A * | A > d | NA | NA |
| * See Special | ||||
##Special
Astra vs Rising/Blazing AOE
Because Rising/Blazing AOE have a unmodified attack component in its calculations, I have decided to break it down to its specific matchups factoring in neutral, WTA, WTD, and triangle boosted versions to give a clearer picture. If you want to know how these values are derived, you can check the math below.
Read: Astra is stronger than Row when Cell Condition is fulfilled.
###Astra vs ...
| Ability | Neutral | WTA | WTD | Triangle Adapt, WTA | Triangle Adapt, WTD |
|---|---|---|---|---|---|
| Rising AOE | A > d | 1.6A > d | 0.4A > d | 2.2A > d | d < -0.2A |
| Blazing AOE | d < 0 | 0.6A > d | d < -0.6A | 1.2A > d | d < -1.2A |
#Wall of Math Below
2 turn
Moonbow (a-0.7d) vs Reprisal ((a-d)+0.3(m-h))
a-0.7d > (a-d)+0.3(m-h)
a-a > 0.7d-d+0.3(m-h)
0.3d > 0.3(m-h)
d > m-h
QED: Moonbow is stronger when the Defender's defense is greater than Attacker's missing health
3 turn
Glimmer (1.5(a-d)) vs Draconic Aura (1.3a-d)
1.5(a-d) > 1.3a-d
1.5a-1.5d > 1.3a-d
0.2a > 0.5d
a > 2.5d
QED: Glimmer is stronger when Attacker's attack is greater than 2.5 times the Defender's defense
Glimmer (1.5(a-d)) vs Luna (a-0.5d)
1.5(a-d) > a-0.5d
1.5a-1.5d > a-0.5d
0.5a > d
a > 2d
QED: Glimmer is stronger when Attacker's attack is greater than 2 times of Defender's defense
Glimmer (1.5(a-d)) vs Vengence ((a-d) + 0.5(m-h))
1.5(a-d) > a-d+0.5(m-h)
1.5a-1.5d > a-d+0.5(m-h)
0.5a > 0.5d+0.5(m-h)
a > d+(m-h)
QED: Glimmer is stronger when Attacker's attack is greater than Defender's defense plus Attacker's missing health
Glimmer (1.5(a-d)) vs Bonfire/Iceburg (a-d+0.5r)
1.5(a-d) > a-d+0.5r
1.5a-1.5d > a-d+0.5r
0.5a > 0.5d+0.5r
a > d+r
QED: Glimmer is stronger when Attacker's attack is greater than Defender's defense plus Attacker's relevant defense
Draconic Aura (1.3a-d) vs Luna (a-0.5d)
1.3a-d > a-0.5d
0.3a > 0.5d
a > 1.667d
QED: Draconic Aura is stronger when Attacker's attack is greater than 1.667 times of Defender's defense
Draconic Aura (1.3a-d) vs Vengence ((a-d)+0.5(m-h))
1.3a-d > (a-d)+0.5(m-h)
1.3a-d > a-d+0.5(m-h)
0.3a > 0.5(m-h)
a > 1.667(m-h)
QED: Draconic Aura is stronger when Attacker's attack is greater than 1.667 times of Attacker's missing heath
Draconic Aura (1.3a-d) vs Bonfire/Iceburg (a-d+0.5r)
1.3a-d > a-d+0.5r
0.3a > 0.5r
0.6a > r
QED: Draconic Aura is stronger when Attacker's relevant defense is lesser than 0.6 times of Attacker's attack
Luna (a-0.5d) vs Vengence (a-d+0.5(m-h))
a-0.5d > a-d+0.5(m-h)
0.5d > 0.5(m-h)
d > m-h
QED: Same as Moonbow vs Reprisal
Luna (a-0.5d) vs Bonfire/Iceburg (a-d+0.5r)
a-0.5d > a-d+0.5r
0.5d > 0.5r
d > r
QED: Luna is stronger when Defender's defense is greater than Attacker's relevant defense stat
Vengence (a-d+0.5(m-h)) vs Bonfire/Glacies (a-d+0.5r)
a-d+0.5(m-h) > a-d+0.5r
0.5(m-h) > 0.5r
m-h > r
QED: Vengence is stronger when Attacker's missing health is greater than Attacker's relevant stat
4 turn
Night Sky (1.5(a-d)) vs Draconic Fang (1.5a-d)
1.5(a-d) > 1.5a-d
1.5a-1.5d > 1.5a-d
1.5a-1.5a >1.5d - d
d < 0
QED: Night Sky will never be stronger than Draonic Fang (Requires Defender to have negetive defense)
Night Sky (1.5(a-d)) vs Chilling Wind/Ignis (a-d+0.8r)
1.5(a-d) > a-d+0.8r
1.5a-1.5d > a-d+0.8r
1.5a-a > 1.5d-d+0.8r
0.5a > 0.5d+0.8r
a > d + 1.6r
QED: Night Sky is stronger when Attacker's attack is greater than Defender's defense plus 1.6 times of Attacker's relevant defense stat
Draconic Fang (1.5a-d) vs Chilling Wind/Ignis (a-d+0.8r)
1.5a-d > a-d+0.8r
1.5a > a+0.8r
0.5a > 0.8r
a > 1.6r
QED: Draconic Fang is stronger when Attacker's attack is 1.6 times greater than the Attacker's relevant defense
5 turn
Astra (2.5(a-d)) vs Aether (a-0.5d)
2.5(a-d) > a-0.5d
2.5a-2.5d > a-0.5d
2.5a-a > 2.5d-0.5d
1.5a > 2d
a > 1.3333d
QED: Astra is stronger when Attacker's attack is 1.333 times or more than the Defender's defense, note that this does not factor in Aether's healing properties
Astra(2.5(a-d) vs Rising AOE(A+a-2d)
2.5(a-d) > A+a-2d
2.5a-2.5d > A+a-2d
2.5a-a > A+2.5d-2d
1.5a > A+0.5d
3a-d > 2A
// same color combat
a = A
3A-d > 2A
A > d
// advantage
a = 1.2A
3.6A-d > 2A
1.6A > d
// disadvantage
a = 0.8A
2.4A-d > 2A
0.4A > d
// t-adapt, advantage
a = 1.4A
4.2A-d > 2A
2.2A > d
// t-adapt, disadvantage
a = 0.6A
1.8A-d < 2A
d < -0.2A // impossible
QED: Astra is stronger against same color and triangle advantage matchup. Against triangle disadvantage matchups, Astra wins as long as Defender's defense is less than 0.4 times of Attacker's unmodified attack. AOE is stronger in Triangle Adapt modified disadvantage matchups. Note that this model ignores any additional damage that may occur due to AOE hitting multiple enemies.
Astra(2.5(a-d)) vs Blazing AOE(1.5A+a-2d)
2.5(a-d) > 1.5A+a-2d
2.5a-2.5d > 1.5A+a-2d
2.5a-a > 1.5A+2.5d-2d
1.5a > 1.5A+0.5d
3a > 3A+d
3a-d > 3A
// same color combat
a = A
3A-d > 3A
d < 0 // impossible
// advantage
a = 1.2A
3.6A-d > 3A
0.6A > d
// disadvantage
a = 0.8A
2.4A-d > 3A
d < -0.6A // impossible
// t-adapt, advantage
a = 1.4A
4.2A-d > 3A
1.2A > d
// t-adapt, disadvantage
a = 0.6A
1.8A-d > 3A
d < -1.2A // impossible
QED: Astra is stronger only in matchups with triangle advantage and as long as the Attacker's unmodified attack is 0.6 or 1.2 times greater than the Defender's defense or normal and Triangle Adapt modified advantage respectively.
Aether(a-0.5d) vs Rising AOE(A+a-2d)
a-0.5d > A+a-2d
2d-0.5d > A
1.5d > A
QED: Aether is stronger only when Attacker's unmodified attack is lesser than 1.5 times of Defender's defense. Note that this does not factor in Aether's healing properties or additional damage that may occur due to AOE hitting multiple enemies.
Aether(a-0.5d) vs Fat Man(1.5A+a-2d)
a-0.5d > 1.5A+a-2d
2d-0.5d > 1.5A
1.5d > 1.5A
d > A
QED: Aether is stronger only when Attacker's unmodified attack is lesser than Defender's defense. Again, this does not take into account Aether's healing properties or additional damage that may occur due to AOE hitting multiple enemies.
Enjoy!
PS: Reddit formatting is still a pain in the ass
Posted Mar 22, 2017, 2:12 PM UTC · edited