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In many areas in computational science (e.g. neural networks, fuzzy logic ... ) there is special interest in function like sigmoid ( erf, arctan, tanh ... ) which are kind of blured version of Heaviside step function, and it's derivative in shape of smooth peak ( Gaussian, Lorenz ... ).

I'm interested also in it's integral: Smooth monotoneously growing function which has smooth transition from $f'(x<<0) \approx 0$ to $f'(x>>0)\approx1$

one of such function is: $f(x) = (x + \sqrt{1+x^2})/2$

I would like some function of similar character which is fast to evaluate numericaly on computer.

  • This means that it preferably should not use any transcedental functions like $\sqrt , \log, \exp, \sin, \sinh ... $.
  • It should be preferably also as smooth as possible ( it should not contain terms like $|x|$ or if () then {} else {} )
  • It would be also nice, that not only the function $f(x)$, but also it's first and second derivative ( $f(x)$, $f'(x)$ (sigmoid), $f''(x)$ ( peak ) ) has this properties (fast to evaluate and smooth)

For example, Fast Sigmoid $f'(x) = x/(1+|x|)$ is not so good because it's integral $f(x)$ constains term $\log (1 \pm x)$ which is slow to evaluate, and it's derivative $f''(x) = 1/(1+|x|)^2 $ has sharp kink (derivative discontinuity) at $x=0$

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