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Hi all:

Just testing the delta DE method to it's limits - this is a magnified view of a quaternion Julia Set with magnification *701,518.21 and some ray-points took over 400 iterations.
The "solid threshold" (i.e. distance estimate limit) for solid found was 1e-11 and the delta distance was 1e-14, going any closer starts running out of accuracy without using extended precision.
Rendered at high accuracy (default step distances reduced by a factor of 10) @3840*2880 in just under 5 hours on a 2GHz core2duo system - note that to get rid of the aliasing centre-left would require a higher resolution render.

(http://fc02.deviantart.com/fs51/f/2009/293/a/c/Fractal_Flyover_by_MakinMagic.jpg)

If no image above then go here:

http://makinmagic.deviantart.com/art/Fractal-Flyover-140921878 (http://makinmagic.deviantart.com/art/Fractal-Flyover-140921878)

I decided to test the delta DE method even further and figured that the most "regular" object that would show errors most obviously would be a high magnification of the quaternionic Mandy for z^2+c - so here it is rendered at 640*480 exactly as you see it (it's a png).

Magnification: *2.1582312 e12
DE "solid" value: 1 e-13
delta distance: 1.25 e-14

As you can see it's obviously stretching the limits of "double" accuracy - in particular the real axis is scalloped and the rim of the "horn" is markedly uneven.

Anyone with a fast PC and some spare time is invited to have a go at rendering this one with UF's extended precision set to "always" just to compare the result with this render - if you're interested let me know and I'll send you the formula and the parameter file (though be warned on my 3GHz P4HT it took over an hour with extended precision set to "never").

(http://www.fractalforums.com/gallery/1/141_22_10_09_1_17_45.png)

http://www.fractalforums.com/gallery/?sa=view;id=1015 (http://www.fractalforums.com/gallery/?sa=view;id=1015)

I confess the previous render took so long because I had an issue with the shadow-casting and to fix it I ended up increasing the "accuracy" setting to 10 - thus reducing all step sizes to one tenth of the default, hence it took so long.
However I've fixed that and also discovered that the other errors (the scalloping and edge inaccuracy) where scaleable erors such that going closer to the "solid" also reduced the size of the errors, making them almost unnoticeable:

(http://www.fractalforums.com/gallery/1/141_22_10_09_4_20_01.png)

This render took under 11 minutes at 640*480 and used the following settings:

Magnification: 2.1582312 e12
Solid threshold: 1 e-14
Delta: 1.25 e-15

http://www.fractalforums.com/gallery/?sa=view;id=1016 (http://www.fractalforums.com/gallery/?sa=view;id=1016)

Here's the analytical DE version:

(http://www.fractalforums.com/gallery/1/141_24_10_09_2_05_14.png)

http://www.fractalforums.com/gallery/?sa=view;id=1019 (http://www.fractalforums.com/gallery/?sa=view;id=1019)

As you can see the analytical method smooths the surface detail so aliasing is reduced but the surface detail (rings) is less obvious.
Edit: For instance in the Delta DE version you can see that the "horn" opens out slightly close to the edge, a detail that's missing from the analytical version.
This took just under 8 minutes @640*480.

After a bit more testing I found that the delta DE method starts breaking down at a distance threshold of around 1e-15 whereas the analytical method hangs on to a threshold of around 1e-16 - though I'm still testing ;)

For those interested, in the above renders my maxiter parameter was set high at 400 to ensure "solid" was based purely on the distance threshold rather than on the maxiter value - in actual fact the maximum iteration count attained in the delta DE version was 62 and the maximum attained in the analytical DE version was 58.