I'm trying to create an idea fitness function using an inverse parabola but I'm struggling with the math.
Basically what I have is:
score = -1 * (actual-ideal)^2 + weight
Or 0 if the score is negative. This works great except that it's not flat enough, and it gives no score if the actual isn't close enough to the idea (at least not a positive value).
What I would ideally like is some kind of minimum score, so that f(0)=0, f(1)>0, etc. and anything above that in terms of x has some kind of value peaking at the ideal value.
Even better would be if I could have the score slowly decrease when the value goes past the ideal, but I'm ok with say 2*ideal being 0, so it's an inverse parabola. A more complex function would be awesome but right now I'm just looking for something simpler.
How would you adjust the equation above so that the score always starts at 0 and peaks out at the ideal?
So you want to map the difference between actual and ideal to a score, where $f(0)$ is the max?
$f(x) = -(x-c)(x+c)$
Where $x$ is actual minus ideal and $c$ is just how big the difference needs to be to give $0$.
If you want a function directly of the actual:
$g(x) = -((x-I)-c)((x-I)+c)$
Where $I$ is the ideal
Put $g(0) = 0$ and see that $c$ can be $I$ or $-I$ (clearly doesn't matter which).
$g(x) = -x(x-2I)$
Is this what you're looking for?