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Increase the bailout value ?

Apologies - I missed the video which shows it's obviously not uniform but based on viewpoint location, I'd guess marius is correct ?

Another question:

Is there anything like  a distance estimation formula for hybrid fractals (for a mandelbox with rotation)?

Whatever formula combinations you use each should simply take the current DE value and apply it's own modification i.e. every formula applied in a hybrid needs to also apply it's own (correct) modification to the running DE value.
Of course if using the brute-force smooth iteration method (Buddhi's delta DE) then no running DE is needed and the more complex the overall iteration becomes then the more optimum using the delta DE becomes basically because errors in the combined analytical DE are cumulative in the worst places whereas the accuracy of the delta DE is more independent of the overall complexity (assuming of course that an accurate smooth iteration value is possible for the overall formula) and the calculations involved in delta DE are essentially independent of the complexity whereas analytical DE gets slower as more formulas are added to a hybrid.
In either case (analytical or delta) the worst errors occur in fractals that have areas with discontinuities in the smooth iteration value e.g. a space of 5 iterations touching a space of 7 with no space of 6 in between - typically this is formulas with conditionals or fractional powers.

You don't need to worry about the trig too much, in the case of polynomials (even using the odd Mandelbulb triplex maths) it turns out that the running DE is exactly the same as that for plain complex or quaternions etc.

i.e. If your current running DE value is d and you're applying z^p+c on a given iteration then the new running DE value is

 d = p*d*z^(p-1)

(That may need a +1 on the end for either Mandy's or Julias I forget which)
Of course the z^(p-1) needs to be calculated using the correct number form i.e. using triplex math for the Mandelbulb matching the triplex math used for z^p+c.

This is "off the top of my head" and it's a while since I messed with it and my memory isn't what it was but...
My delta DE is simply done by calculating the smooth iteration value for the current point on the ray, for a point deltax up the x axis, a point deltay (magnitude same as deltax) up the y axis and a point deltaz (same magnitude) up the z axis - the delta being a suitably small value according to the current required distance threshold for "solid".
Of course this is an overhead of 3 full iteration loops per step along the ray compared to using analytical DE *but* of course the analytical DE calculation involves calculations done on every iteration whereas the smooth iteration value only involves calculation on bailout.
You then take the differences in the smooth iteration value i.e. (value for point offset in x - value for base point (at step on ray), value for point offset in y - value for base point, value for point offset in z - value for base point) and get the magnitude of this (as a 3D vector) and then adjust appropriately to give a DE value.
Buddhi's method I think is similar except if I remember correctly he iterates at the delta offsets always to the same iteration count as the base point (at current step on the ray)  and uses the bailout values rather than the smooth iteration values.

With respect to the iteration count - I was talking about points along the ray before you reach solid - here the ray will pass through space of different iteration levels at bailout and for many fractals (like the Mandelbulb) although the shape of the boundaries from one iteration depth to the next are fractal and irregular they are always consistent i.e. there will never be a point of 5 iterations at bailout directly next (I mean in the infinitessimal limit) to a point of 7 iterations at bailout - there's always a point between of 6 iterations at bailout.
To a certain degree even analytical DE relies on this but some fractals, notably some fractals with conditionals in the formula and fractals with fractional powers, have adjacent points where the iteration count at bailout is discontinuous and such formulas always present rendering issues due to "incorrect" calculation of the distance estimate.

Final iter is indeed the iteration count at bailout and in log(log(|z|)/log(|zold|)) |z| can be either the final magnitude or its square as long as |zold| matches as either the penultimate magnitude or its square.
Obviously if avoiding the square root in the bailout testing (e.g. in a shader using dot(z,z) instead of length(z)) i.e. testing against the square of the bailout then you have to ensure you use the correct values in  (log(log(bailout))-log(log(final magnitude))).
i.e. again either both the non-squared or the squared, obviously it's simplest to match which you use in the bailout test.


The original smooth iteration this is based on was outlined (I believe) by Linas Vepstas and an online search should find info on that.
The key to it being successful as a deltaDE method is ensuring that it remains as smooth as possible everywhere even for more esoteric formulas i.e. such that no hard-breaks are introduced at iteration boundaries and the technique of using the actual divergence at bailout i.e. log(log(|z|)/log(|zold|)) was my idea - it came suddenly to me when trying to sleep after thinking about possible ways of getting correct smoothing everywhere on the standard complex "lambda" fractal - I suddenly realised that the reason for variation across the fractal in that case was due to varying divergence at bailout based on position (because of "c" being a scaling factor in that case) and the idea of using the actual divergence on bailout was a sudden inspiration and I went to sleep happy convinced it would work on the lambda mandy which it did, the surprise was that it also worked in many other cases where smooth iteration colouring was problematic - even exp(z)+c or sin(z)+c and vastly improving the Barnsely's - though for the Barnsley's Ron Barnett's cumulative exp smoothing is better though it doesn't quite follow the contours of plain banded iteration.
Of course for delta DE you can avoid the final iteration value altogether and just work with the rest of the values provided that you iterate at all 4 points to the same iteration count (i.e. whatever iteration count the base point on the ray bails out at, then perform that number of iterations at each of the "delta" offset points).
Also in some formulas using the plain magnitude differences for deltaDE as Buddhi outlined somewhere works better but for other formulas the smooth iteration differences work better.

Apologies, I forgot that Jos Leys showed that you could just use the magnitude rather than the full triplex - here's some code directly from my UF formula:
                if @useDE==0 ;(i.e. type 0)
                  dr = @mpwr*dr*magn^(@mpwr-1.0)
                  if @fractaltype==1
                    dr = dr + 1.0
                  endif
                endif

Done at the start of each iteration i.e. using magn from the previous one - note that this should be the true magnitude not the square.

Yes, you are correct that to render the surface based solely on the distance estimate then ideally the max iteration value should be set so that no points tested on the ray actually reach max iterations - though this possibility must be allowed for due to possible over-stepping as you say.
The simple solution is to assume "solid" is either the point at the correct distance threshold or the nearest point to the viewer that reaches the maximum iteration count.
In fact using DE works quite well even if you want to view the true iteration iso-surface (i.e. have "solid" at a given iteration count) at least at lower iterations (typically <20), all a user need to do here (if programmed correctly) is set the threshold distance to (near) zero but set the max iterations value to set the iso-surface you want to be solid.