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 Post subject: Formulas
PostPosted: Sun Dec 16, 2012 7:11 pm 
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Note: The topic is not finished yet. This post will be updated with new formulas.
If you want anything else added or have any questions, feel free to post here.

Formulas for Crits
1) How much crit chance (c) does my tower need to have the highest possible chance on at least a (m)-times multicrit:
Quote:
c >= 0.8^(2-m)

Note: With c > 0.8^(2-m) the chance to get exactly a (m)-times multicrit decreases because the chance to get a higher one increases (i.e. m+1-times multicrit or more).

2) Given a crit chance (c), for which multicrits (m) does my tower have a maximum chance (opposite of 1)):
Quote:
m <= [2 - log(c)/log(0.8)] // Rounded down!

Note: The base of the logarithm you use doesn't matter.

3) What's the chance (p) of a (m)-times multicrit if the chance for that multicrit is maximal:
Quote:
p(m) = 0.8^m


4) What's the chance (p) to do at least a (m)-times multicrit with a crit chance of (c) (no restrictions here):
Quote:
l := [2 - log(c)/log(0.8)] // Rounded down! Note: If l < 0 then set l = 0, if l > m then set l = m

n := m - l
p(m, c) = 0.8^((n^2 + n*(2l - 1) + 2l ) / 2) * c^n
OR
p(m, c) = 0.8^((m^2 - m + 3l - l^2) / 2) * c^(m - l)


5)What is the average multicrit my tower does on each attack (assuming that the crit chance for each crit is maximal):
Quote:
m := multicrit count of your tower
averageMulticrit = 4*(1-0.8^m)

As m approaches infinity, the averageMulticrit approaches 4. So at a certain multicrit value, every further multcrit is pretty much negligible.

Note: It is not possible to get an easy formula which also considers a tower's crit chance. However, it is still true that 4 is the highest average multicrit. If you want to have an approximation for certain crit chances, you can calculate the maximum multicrit for which the tower has still a chance of 80% using formula 2) and put that value into this formula. The real average is then between the calculated value and 4.


Formulas for Damage and Attackspeed
1) Time between attacks:
Quote:
BAT := Base attacktime of your tower
AS := total attackspeed of your tower

TimeBetweenAttacks = BAT/AS


2) Damage Per Second (DPS):
Quote:
DPS = Damage/TimeBetweenAttacks


3) Damage per attack (DPA):
Quote:
ar := armor reduction of enemy
atm := armor type modifier
d := towers attack damage (white + green number)
am := Average multicrit
rm := race modfier
sm := size modfier
cd := crit damage

DPA = d * rm * sm * atm * (1 + am * cd) * (1-ar)


General mathematical formulas
1) Sum formula:
Quote:
sum over i from m to n = (n+1-m)(n+m)/2


2) Geometric series:
Quote:
sum over a*x^i from m to n = a*[x^m + x^(n+1)]/(1-x)
Special case m = 0, n = infinity, |x| < 1
sum over a*x^i from 0 to infinity = a/(1-x)


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 Post subject: Re: Formulas: Crit
PostPosted: Sun Dec 16, 2012 7:18 pm 
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add the formula that determines creep hp

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 Post subject: Re: Formulas: Crit
PostPosted: Sun Dec 16, 2012 7:20 pm 
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Shall I expand this thread to contain all formulas? Sadly I don't know how creeps HP is calculated.


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 Post subject: Re: Formulas: Crit
PostPosted: Sun Dec 16, 2012 7:22 pm 
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Yes
der_kleine_Tomy wrote:
Shall I expand this thread to contain all formulas? YESSadly I don't know how creeps HP is calculated.I am sure you will find this out soon enough.

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 Post subject: Re: Formulas: Crit
PostPosted: Sun Dec 16, 2012 7:26 pm 
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mkay. I guess I can add a few formulas for damage calculation. The formulas for diminishing returns would be nice, too, but I don't know how the diminishing return is implemented. Maybe Glow will reveal his secret, then I put it into this thread.


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 Post subject: Re: Formulas
PostPosted: Tue Dec 18, 2012 11:46 am 
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Maybe we could also extend this into the average dps of various mechanic types, to make ability factor calculations easier?

I will add in geometric sequences(badly formatted), not fancy anything yet:

Wikipedia Sayssss:

In mathematics, a geometric series is a series with a constant ratio between successive terms. For example, the series
1/2+1/4+1/8......
This ALSO refers to many other types of effects, such as stacking slows with only 80% return on additonal stacks.
The way you calculate is simple: Find the ratio. Do 1-ratio. Do intial amount/(1-ratio), and you get the sum. Simple example, a stacking slow with initial value of 10%, and each next stack only 80% of the previous. We get 10%+8.0%+6.4%.... Ratio is .8. 1-.8=.2. .1/.2 = .5. So, even with infinite stacks, the maximum is 50%.


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 Post subject: Re: Formulas
PostPosted: Tue Dec 18, 2012 12:54 pm 
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Ok, I will add some mathematical formulas which are not directly related to the mechanics but can be used to calculate the ability factor of some specific towers.


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 Post subject: Re: Formulas
PostPosted: Tue Dec 18, 2012 4:25 pm 
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Awesome.
Now, one I have been trying to figure out.... the average chance of occurrence for Frosty Rock, basically...
Given an initial chance X, with a gain of Y, every time an event does not occur (1-X chance), X+=Y, until the event occurs, at which point X is reset.


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 Post subject: Re: Formulas
PostPosted: Tue Dec 18, 2012 6:32 pm 
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Frosty Rock (first family member) stuns after 8.5 attacks at a chance of 17% in average. Unfortunately, my formula has no explicit form, so I had to calculate it with a program. I also ran 1 million tests (simply simulating frosty rock) to see whether my formula calculates the correct value. If you want, I can tell you the values for the other members as well.


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 Post subject: Re: Formulas
PostPosted: Tue Dec 18, 2012 6:51 pm 
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Naw, its ok. I was trying to figure out some formula for it since I and some other people have towers using that idea.
Hold on, I have some notes on it I will post, maybe they can help...


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 Post subject: Re: Formulas
PostPosted: Wed Dec 19, 2012 5:19 pm 
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I have found a mistake in the average crit formula. With my fix, the max average should be around 16 instead of 4. I will do some further calculations to whether the new fomula is correct so I won't update it yet.


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 Post subject: Re: Formulas
PostPosted: Thu Dec 20, 2012 2:28 pm 
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The first formula was correct -.-. So the max average multicrit is still 4.


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