Smooth spectrum for Mandelbrot Set rendering

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This is the smooth color algorithm:

Lets say you start with the complex number z0 and iterate n times until it escapes. Let the end point be zn.

A smooth value would be 

nsmooth := n + 1 - Math.log(Math.log(zn.abs()))/Math.log(2)

This only works for mandelbrot, if you want to compute a smooth function for julia sets, then use

Complex z = new Complex(x,y);
double smoothcolor = Math.exp(-z.abs());

for(i=0;i<max_iter && z.abs() < 30;i++) {
    z = f(z);
    smoothcolor += Math.exp(-z.abs());
}

Then smoothcolor is in the interval (0,max_iter).

Divide smoothcolor with max_iter to get a value between 0 and 1.

To get a smooth color from the value:

This can be called, for example (in Java):

Color.HSBtoRGB(0.95f + 10 * smoothcolor ,0.6f,1.0f);

since the first value in HSB color parameters is used to define the color from the color circle.

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