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Thank you for these valuable informations

Acutally it's Buddhi's formula. I've just modified it a little bit. I think it can be improved further.
BTW! the statment:
distance = (sqrt(x*x+y*y+z*z)-abs(scale-1))/abs(DEfactor) - abs(scale)^(1-i);
is over complicated. Now i do instead:
distance = (sqrt(x*x+y*y+z*z)-BVR)/abs(DEfactor)
BVR is the radius of the bounding volume of the fractal.

It's possible to use max(max(abs(x),abs(y)),abs(z)) as the norm instead of sqrt(x*x+y*y+z*z). This is not an optimization, just another way to do it.
In case you are interrested, it's also possible to keep the size of the (positive scale) mandelbox constant.link.

I guess that happens with positive scale Mandelboxes. I've finally found a tweek to "improve" their DE estimate:
Replace:
by

This way, it's not necessary (AFAIK) to scale down the DE for negative scales. Unfortunately, this is not true with positive scales. There are areas where it's necessary to downscale the DE by a factor down to 0.45 or so... Something is still missing.