I don't know very much at all about DE (especially its computation) but I thought that I would add to the discussion of this 0.521 constant that this DE is not accurate - it can be off by a factor as much as 4 or 1/4 at times. This will make it significantly harder to find an analytical value for 0.521, I predict. If someone uses Distance Calculation (completely ridiculous for high-speed renders), which involves truly checking each point in the vicinity until one is found sufficiently close to the set, and that also seems affected by 0.521.... that would probably give could clues to determining it.
Well yes, analytical DE is an estimate after all, but until a better way of rendering 3D+ objects is found I'll stick with it - using "solid on iteration" is nowhere near as nice since you'll always end up either with areas without any detail and/or areas with significant aliasing (unless you use a volumetric or pseudo-volumetric method which would be much slower anyway, though it produces nice results).
I was just hoping/wondering if a method based on the apparent universality of the "0.521" (and the fact that the min distance to origin trap value can match the variation in the smooth iteration colouring) could be arrived at that produces results comparable with the current analytical DE but faster
Also I think you'll find that generally DE is actually pretty accurate, the problem with the "stars" on the Mandelbulbs is something new - I can't think of any similar problems on hypercomplex fractals (in the widest sense) - the problem with Julia Sets is admittedly pretty universal.
Utilising the 0.521 result directly and applying it as a "delta" method yields:
DE = 0.521*delta/abs(smoothiter1-smoothiter0)
Under experimentation this actually needs adjusting to match the analytical method and I found that using 0.2 instead of 0.521 works fine.
This is essentially one of the delta methods I tried previously - arrived at based on applying Newton's method, at the time I settled on this instead:
DE = 0.4/(distadjust + abs(smoothiter1-smoothiter0)/delta)
Where the default value of distadjust is 1.0.
The problem with the delta methods is that they are directional so one has to program around the problem of points where the rays are parallel or near parallel to the fractal surface.
Also the "stars" on the Mandelbulbs present a different problem for the delta methods