>> This of course is thwarted by functions that are extremely steep at the
>> edges, yet quite shallow near where the solutions are. [*cough*
>> Mandelbrot set *cough*] You end up needing an insanely high
>> max_gradient, yet in the vicinity of the solutions you could actually
>> afford to take much bigger steps.
>
> One thing that helps is to use log log smoothing for the exterior. It helps, but
> it's slow. And I could never find a successful way to multiply or divide a
> fractal's gradient in order to speed things up.

Yeah, simply dividing the function by a constant makes the maximum 
gradient smaller, but also makes all the sample values smaller as well, 
resulting in identical performance. Whereas taking the log (or even log 
log) makes the steep parts less steep, while not affecting the shallow 
parts very much. But, as you say, the log function itself is quite slow. 
I wonder if a piece-wise linear curve would be any faster... (Not that 
POV-Ray can do that.)

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