Soft saturation

Type : waveshaper
References : Posted by Bram de Jong
Notes :
This only works for positive values of x. a should be in the range 0..1
Code :
x < a:
  f(x) = x
x > a:
  f(x) = a + (x-a)/(1+((x-a)/(1-a))^2)
x > 1:
  f(x) = (a+1)/2

Comments
from : graue[AT]oceanbase[DOT]org
comment : This is a most excellent waveshaper. I have implemented it as an effect for the music tracker Psycle, and so far I am very pleased with the results. Thanks for sharing this knowledge, Bram!

from : brobinson[AT]toptensoftware[DOT]com
comment : I'm wondering about the >1 condition here. If a is 0.8, values <1 approach 0.85 but values >1 are clamped to 0.9 (there's a gap) If you substitute x=1 to the equation for x>a you get: a+((1-a)/4) not (a+1)/2 Have I missed something or is there a reason for this? (Go easy I'm new to all of this)

from : kibibu[AT]gmail[DOT]com
comment : Substituting x=1 into equation 2 (taking many steps) f(x) = a + (x-a)/(1+ ((x-a)/(1-a))^2) = a + (1-a)/(1+ ((1-a)/(1-a))^2) = a + (1-a)/(1+ 1^2) = a + (1-a)/2 = 2a/2 + (1-a)/2 = (2a + 1 - a) /2 = (a+1) / 2

from : musicdsp[at] Nospam dsparsons[dot]co[dot]uk
comment : You can normalise the output: f(x)'=f(x)*(1/((a+1)/2)) This gives a nice variable shaper with smooth curve upto clipping at 0dBFS