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Author Topic: Relationship between bailout and accuracy of analytical DE  (Read 992 times)
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David Makin
Global Moderator
Fractal Senior
Posts: 2279

Makin' Magic Fractals
« on: April 05, 2010, 04:27:31 PM »

Hi all,

Since <see subject> is pretty fundamental to most 3D rendering I decided to investigate the relationship in more detail and wrote the following in UF:

  z = @start
  complex dz = (0,0)
  float d = 0.0
  float dold = 0.0
  float m = 0.0
  bool bail = true
  dold = d
  dz = @power*dz*z^(@power-1.0) + 1.0
  z = z^@power + #pixel
  m = |z|
  d = 0.25*log(m)*sqrt(m)/cabs(dz)
  if |(d-dold)/d|<@smallbail
    bail = false

And then created a normal layer of the standard Mandelbrot and adjusted the smallbail on this layer and then adjusted the bailout on the normal layer until the iteration bands matched as closely as possible.
I found the result very significant, at least for 2D complex fractals - I'm hoping the result extends to 3D+ smiley

Basically if your bailout is 10000 i.e. you're testing magnitude squared against 10000 then your DE estimates are accurate to just two decimal figures i.e. if your DE is 0.01 then it's only accurate to +/-0.0001, if you're using a bailout of 1000000 then your DE is accurate to 3 decimal figures etc.
This result was confirmed for all powers I tried on the standard z^p+c Mandelbrot.

The meaning and purpose of life is to give life purpose and meaning.

"Makin' Magic Music" on Jango
« Reply #1 on: April 08, 2010, 02:56:03 AM »

Oh, that's interesting. It makes a lot of sense, though. I'd think it would hold for the Mandelbulb and similar, too.
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