Smooth approximations

Some examples of using the logistic function in smooth approximations for ifthen and max/min:

• ifthen(x1>x2, y, z) = z + (y-z)*0.5*(1+tanh(k*(x1-x2)))
• ifthen(x1<x2, y, z) = y + (z-y)*0.5*(1+tanh(k*(x1-x2)))
• max(x,y) = ifthen(x>y,x,y) = y + (x-y)*0.5*(1+tanh(k*(x-y)))
• min(x,y) = ifthen(x<y,x,y) = x + (y-x)*0.5*(1+tanh(k*(x-y)))

Some graphs with k=100:

See also: http://yetanothermathprogrammingconsultant.blogspot.com/2008/08/smooth-approximation-for-ifthen.html.