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BotrK Math! (diminishing returns)

Creator: willeke March 30, 2013 8:34am
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willeke
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Hi,
i got the following problem. I wanna calculate how effectiv the Blade of the ruined King really is, but its effected by diminishing returns (due to the multiple percentual effect).

So i wanna calcualte the damage it deals over a short time - and not by a number of autoattacks.
So the Basic Formula for the number of autoattacks in 5sec goes (i hope it goes) like this:

(Attack Speed at lvl18 + (Basic Attack Speed x (AS Buff from BotrK/100)) x 5) [So i calculate this for Udyr)
(0,957 + (0,658 x(40/100))x5) = 2,273 Attacks in 5sec

So for the unique i got:
(HP_Enemy - (HP_Enemy x (15/100))) = X (HP_Enemy after one auto-attack)
(X- (X x (15/100))) = Y (HP_Enemy after two auto-attacks)

now i have to do this process 2,273 times. So got someone an idea for a formula for this?
DarkPercy
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wtf lol?

It's: Base attack Speed x (1 + literally any freaking source of bonus % attack speed you can get in the game added together) x 5

For Udyr: 0.658 x (100% + (2.67% x 17) + 40%) x 5 = 0.658 x 185.39% x 5 = 1.2198662 x 5 = 6.099331 attacks in the 5 seconds.

So, Blade of the Ruined King would hit the enemy 6.099331 times, dealing 5% of their CURRENT HEALTH for every hit. It is very very hard to calculate how damage that is ._.

You really have to know how much HP your enemy has after every attack to calculate exactly how much damage your Blade of the Ruined King will do over these seconds. Sorry but you need to break it down lol.

Your second formula has ''(HP_Enemy x (15/100))'' in it and that really troubles me cause that would mean each autoattack would do 15% of their HP lol. The main problem is that the 5% of current Health damage thing isn't the only that's gonna lower the enemy's HP. Your entire autoattack will do damage too. So it's not that easy.
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willeke
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I noticed i messed up the braces, it should go like this: (0,957 + (0,658 x(40/100)))x5 = 6,101

So yes , my formula is for the CURRENT HP which is changing. So i just add the 25ADx6,101 and go like this:

I go for an 2000HP Enemey.
1st Auto-Attack: (2000-25) x(15/100) = 296 (Damage)
2nd Auto-Attack: (2000-25x2-296) x (15/100) = 248
3rd Auto Attack: (2000-25x3-296-248) x (15/100) = 207
4th Auto Attack: (2000-25x4-296-248-207) x (15/100) = 172
5th Auto Attack: (2000-25x5-296-248-207-172) x (15/100) = 142
6th Auto Attack: (2000-25x6-296-248-207-172-142) x (15/100) = 117

This is not exactly the 6,1 Attacks but it comes close. It does not care of regeneration effect, armor etc.. And it does not (but i will add this) calculate the damage from other sources (Phoenix, Tiger etc.) in, which will lower the damage output dramaticaly.

I found a formula for this btw:
HP after overall Dmg = Current HP * ( 1- 15/100)^n (n=auto-attacks)
HP after overall Dmg (only from BotrK passiv) starting with 2000 should be: 742HP left. Means 1258 DMG

I got an question for the Liandrys Tourment. If i active Phoenix and it will it pulse 5 times over 5 seconds, but Liandrys got an 0,5sec proc.
( Mechanic:

Liandry's Torment procs every half a second. Liandry will tick 6 times, and each tick does 1% of enemy current health as magic damage. Damage is doubled if enemy movement is impaired, dealing 2% magic damage every tick. Damage from each tick is calculated individually.

The Liandry's Torment checks on every half a second if an enemy is under specified Crowd Control and applies an amplified tick if found and usual if not.

Reapplying Liandry's Torment merely refreshes the duration it has no influence on the passive's damage rate. )

So there should be 10 procs for active phoenix + 1 proc for each auto-attack with udyr while in phoenix stance (with no AS buff 5 attacks in 5sec) means in 5 sec: 15 procs = 95 ticks each for 1% (diminishing returns). Right?
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Broo. Blade of the Ruined King does FIVE percent of the enemy's current health per autoattack. Not 15%. Other than that, you are right.

Yeap that's exactly why I was saying it'd be extremely hard to calculate how effective Botrk is. You have to take soooo much other stuff in consideration.

Your formula ''HP after overall Dmg = Current HP * ( 1- 15/100)^n (n=auto-attacks)''
would work IF:

1- you changed the 15/100 for 5/100
2- The ONLY thing doing damage to the enemy would be Blade of the Ruined King
3- your enemy would have no armor, no HP regen and no flat damage reduction whatsoever.

AKA, your formula does not work :(.

For the Liandry's Torment thing, there is one very important thing that you need to know. You cannot stack Liandry's Torment procs/ticks.

There can only be one liandry's ''burn'' at a time. For example, if you activate Phoenix Stance near an enemy, it will start burning the enemy. On the first Phoenix pulse, the enemy starts taking damage from Liandry's. If you just leave the enemy and stop doing any damage to him after that, the burn will last 3 seconds, dealing 1% of the enemy's current HP every half a second.

However, if you stay near the enemy and deal damage with a Phoenix Stance pulse once again, a second later, what it will do is REFRESH the Liandry's Torment burning duration. It will not make the enemy ''burn'' twice. So Liandry's will keep doing only 1% of the enemy's current HP every half a second but will still have 3 seconds of duration remaining instead of only 2.

The Phoenix Stance does extra damage every 3 autoattacks by the way :P. And all it would do is refresh the Liandry's Torment burning duration again ;p.

I hope this was clear lol.

So yeah, forget about the 95 ticks thing, that would require you to keep refreshing the Liandry's Torment passive for 95/2 - 3 = 44.5 seconds, which is nearly impossible lol.
My AP Xin Zhao guide is on the way!
willeke
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Its for Comparing Items so i ignore 2) and 3) for all and thats fair. so yes i have to work the 5% in instead of 15% ... misread the activ -_-.

anyhow liandrys should do 10 burns in 5sec. Phonix has changed to 1st and 3rd autoattack btw :P
and each burn goes for 1% (if slowed 2%) - right?

the formula is basically this: http://en.wikipedia.org/wiki/Compound_interest
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Liandry's Torment is over 3 seconds, which means 1.65% of target's current health in magic damage per second for 3 seconds, or 3.3% of target's current health in magic damage per second for 3 seconds if their movement is impaired.

Phoenix Stance only does the 1st autoattack upon activation, afterwards, it is only each 3rd autoattack.

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DarkPercy
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Liandry's Torment is over 3 seconds, which means 1.65% of target's current health in magic damage per second for 3 seconds, or 3.3% of target's current health in magic damage per second for 3 seconds if their movement is impaired.


No, mobafire's tooltip on Liandry's Torment is outdated and incorrect. It does 1% of the enemy's current HP every half a second. 2% if their movement is impaired.


Phoenix Stance only does the 1st autoattack upon activation, afterwards, it is only each 3rd autoattack.


Hey. I was gonna say that.

willeke wrote:

Its for Comparing Items so i ignore 2) and 3) for all and thats fair. so yes i have to work the 5% in instead of 15% ... misread the activ -_-.


Uh okay lol. I still insist that the formula doesn't work.

willeke wrote:

anyhow liandrys should do 10 burns in 5sec. Phonix has changed to 1st and 3rd autoattack btw :P
and each burn goes for 1% (if slowed 2%) - right?


Correct. Liandry's will do 10 ticks over the 5 seconds, doing 1% of the enemy's current HP for every tick, 2% if they are slowed/rooted/stunned/kk. The burning effect would also last the full 3 seconds after your Phoenix Stance would be over. So in total you could do 14 ticks over 7 seconds even though Phoenix Stance only lasts 5 seconds.

willeke wrote:

the formula is basically this: http://en.wikipedia.org/wiki/Compound_interest


That's nothing really special lol. It's just that you can't use this for the situation because this formula doesn't take in account armors that reduce/increase your cash income or sudden cash decreases/increases out of nowhere lol.
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DarkPercy wrote:

No, mobafire's tooltip on Liandry's Torment is outdated and incorrect. It does 1% of the enemy's current HP every half a second. 2% if their movement is impaired.


Ah, thank you. I double checked it, and I've fixed the values.

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willeke
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I tested the formula and it gave out the numbers i shown above - the princip is the same, so in my opinion it should work.

so i give here my complete formula for damage, just to double-check if i made more mistakes. thanks for the advice sofar btw :D. (Once again this is not for the ACTUAL damage in the game, it is more for comparing items, so this is THEORETICAL damage done on paper in 5 sec)

Udyr:
Attack Speed Formula [ASF]: (0,957 + (0,658 x(Bonus AS from Item/100)))x5
Attack Damage Formula [ADF]: (107,31 + Bonus AD from Item)
Phoenix Proc [PPr]: (Bonus AP from Item x 0,45 + 200) + ((0,957 + (0,658 x(Bonus AS from Item/100)))x5):3) x (Bonus AP from Item x 0,45 + 200) [this gets the 1st proc + every 3rd proc]
Poenix AoE [PAoE]: (Bonus AP from Item x 0,45 [Patch 3.05] + 55) x5

Overall: ASF x ADF + PPr + PAoE = Overall Damage with Udyr in 5sec

i got an addition question to crit damage: If i have 10% crit chance and i do 100 Damage from ADF i just add up 10% of 100 Damage, so 10%crit just gives me "theoretically" +10Damage, right?
DarkPercy
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Well, I'm pretty sure that the first attack in phoenix stance doesn't count towards the first 3 autoattacks counter. I'm not 101% sure but... I think it goes like this:
1st attack does +200 damage, the second doesn't, the third doesn't either, the fourth does +200 damage, etc.

In that case, the formula is slightly changed:

Phoenix Proc [PPr]: (Bonus AP from Item x 0,45 + 200) + (0,957 + (0,658 x(Bonus AS from Item/100)-1) x 5/3) x (Bonus AP from Item x 0,45 + 200)

Also, I think the AP ratio on the Phoenix activation's 5 pulses is 25%, not 45%. It doesn't really matter because I doubt there'll be any AP in your build but whatever :P.

AND WTF. I'm french so I'm used to putting commas for decimals and I have make an extra effort to put periods instead of commas because that's how english people write decimals... And nop, you use commas. So much confusionnnnnn GG.

Yes. If your critical damage is 200%, 10% crit chance will make you deal on average AD x 110% damage per autoattack.
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quit flirting with my waifu

or i'll fight you

TO THE DEATH

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