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Hi everyone.
If our iteration formula is z->z^2+c, then d(z^2+c)/dc=2*z*(dz/dc)+1 so we have dz->2*z*dz+1 in our iteration loop.
Then we can compute the distance to the set thanks to |z|*ln|z|/|dz| (see http://www.iquilezles.org/www/articles/distancefractals/distancefractals.htm).

The fact is that the derivative grows very fast, and from the beginning (not just after |z|>2) for a point out of the set.

Does anyone know a trick so that with only a complex number represented by two double numbers, we can compute dz through more iterations ?

Thanks .

It is easy to construct the poly for skipping iterations. You have already done this for one iteration.

z1 = z^2 + c

2 iterations

z2 = z1^2 +c = z^4 + 2*z^2*c + c^2 +c

The order of the poly grows as 2^iteration which quickly becomes intractable. Also I think you have to renormalize the DE for higher powered polys.

You mean this: "To put it another way, if you take a point at distance 1 unit from the set and you use a bailout of b then for z^p+c the maximum magnitude of the final value of dz will be (approx.) |b^p|*ln(|b^p|) where p is the power in z^p+c." ?

That's pretty simple, though I did miss something

Assuming that the standard formula for analytical DE is (at least approximately) correct then we have:

DE = |z|*ln|z|/|dz|

So if the DE is 1 then |dz| = |z|*ln|z| and if DE is d then |dz| = |z|*ln|z|/d so although I did inadvertently turn this upside-down in my head my mistake wasn't that bad....in fact the scale of |dz| does increase as you approach the set relative to |z|*ln|z| but only in a simple inverse relationship with the DE

So if your bailout is m (if testing cabs(z)) then for z^p+c the maximum possible value of |z| "outside" is approx. m^p so the max. poss value of |dz| is m^p*ln(m^p)/d where d is the DE value  i.e. as I said in the previous post the value at 1 unit distance and a bailout of m then the max storage required for dz is that required for m^p*ln(m^p) and if your min. distance (i.e. surface solid threshold) is say 1e-6 then if you require space for 1e100 for the components of |z|*ln(|z|) then you just require 1e106 for the components of dz, although allowing 1e107 would be better

Apologies for the previous mistake !! I sometimes rush in without working things out clearly enough (especially when tired) !!

Of course the closer you want to get to the set then not only will this require more iterations but also higher bailout values since the %age accuracy of the DE does tend to reduce as you get closer to the boundary even if using a "perfect" analytical method for the DE.

With respect to the derivative.....

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=1151

If you are going to do extra iterations then you may as well simply use a larger bailout value in the first place - this increases the accuracy of the DE everywhere (analytical or delta).

Only if you want the ridiculously deep zooms that some folks do in 2D

In practice if the DE is reasonably accurate you should be able to get pretty close to the true boundary....

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=1043

i.e. in 2D "double" gets you to around 1e16 magnification so here this is reduced to around 1e12 - though this is an old render and more accurate DE would do better.......

Edit: Note the render time includes full hard-light tracing i.e. full ray-stepping back towards light source for every visible surface pixel....