"Nice" tweakable S-function

Although I'm not positive, it looks like the function in the accepted answer here:

https://math.stackexchange.com/questions/121720/ease-in-out-function/121755

Might have the properties you're looking for.

Try this:

1/(1+exp(-s*(-log(1/x-1))))

s=1 is the identity. As s grows, the function converges to the step function, as you wanted. Values of s between 0 and 1 are also possible, and behave reasonably.

EDIT: Oh, I think my function is actually the same as x^s/(x^s+(1-x)^s), which is what Zakwayda pointed to:

http://www.wolframalpha.com/input/?i=simplify+1%2F(1%2Bexp(s*ln(1%2Fx-1)))-x^s%2F(x^s%2B(1-x)^s),+assuming+s>1+and+0

Works perfectly! Thank you!

This might be overkill for your needs, but if you want to also be able to tweak when you switch from accelerating to decelerating I put this together a while back:

https://www.desmos.com/calculator/3zhzwbfrxd

I was looking for some functions like this recently as well. I found these functions very useful:

https://en.wikipedia.org/wiki/Smoothstep

https://blog.demofox.org/2012/09/24/bias-and-gain-are-your-friend/

http://demofox.org/biasgain.html

You could also check out the generalized logistic function.

This thread is a great resource!

At one point in time, I wanted a tweakable S-curve that could have a linear section in the middle and ended up with this mess:

float SCurve( float x, float white = 10, float shoulder = 3, float slope = 1.5; )
{
	float invWhite = 1/((slope/shoulder)*(white-shoulder*2));
	float one_on_shoulder = 1/shoulder;
	float slope_on_shoulder = slope/shoulder;
	float white_minus_two_shoulder = white-shoulder*2;
	float white_minus_shoulder = white-shoulder;
	x = min(x,white);
	float a = pow( abs(x*one_on_shoulder), slope );
	float b = slope_on_shoulder*(x-shoulder)+1;
	float c = slope_on_shoulder*white_minus_two_shoulder + 2 - pow(abs((white-x)*one_on_shoulder), slope);
	return x < shoulder ? a : (x > white_minus_shoulder ? c : b);
}

The maximum x value is called "white" because I was using it for tonemapping HDR images
http://www.wolframalpha.com/input/?i=plot+piecewise[{+{abs(x%2F3)^1.5,+x+<+3},+{(1.5%2F3)*(10-2*3)+%2B+2+-+abs((10-x)%2F3)^1.5,+x+>+10-3},+{(1.5%2F3)*(x-3)%2B1,+3<%3Dx<%3D10-3}+}],+x+%3D+0+to+10
http://www.wolframalpha.com/input/?i=plot+piecewise[{+{abs(x%2F5)^4,+x+<+5},+{(4%2F5)*(10-2*5)+%2B+2+-+abs((10-x)%2F5)^4,+x+>+10-5}+}],+x+%3D+0+to+10
Left: white = 10, slope = 1.5, shoulder = 3 (x<3 is the bottom shoulder, 3 Right: white = 10, slope = 4, shoulder = 5 (x<5 is the bottom shoulder, 5

[edit]

https://www.desmos.com/calculator/p7emj3bmmq

18 hours ago, Hodgman said:

This thread is a great resource! 

Then you will appreciate this one as well. Accidentally stumbled over it yesterday:

https://www.youtube.com/watch?reload=9&v=mr5xkf6zSzk

1 hour ago, Dirk Gregorius said:

Then you will appreciate this one as well. Accidentally stumbled over it yesterday:

https://www.youtube.com/watch?reload=9&v=mr5xkf6zSzk 

Sorry to go off-topic. What is 'reload' in the URL? I've never seen it and I can't make sense out of it by changing values. I don't understand how you got that URL or how it works.

On 10/18/2018 at 7:43 PM, AntiTwister said:

https://www.desmos.com/calculator/3zhzwbfrxd

Indeed great resources. I usually use http://www.iquilezles.org/apps/graphtoy/ to play with functions, but that desmos thingy is nice. Graphtoy on the other hand allows a more C-like/shader syntax.

Sorry, no idea. I just copied it here when I came across it. Maybe I was scrubbing or went backwards in the browser history.