Just a suggestion - where you discuss the formula you could also mention Jos Leys' algorithm for accurate analytical DE for the formula (and related formulas such as Garth Thornton's variations and Rudy Rucker's formula).

Hi David, That's an idea, at least a mention would be good. I think Paul's done a good job of collecting the different formulas, so I probably won't repeat the math. But in any case, I may first link to some thread posts until I understand some of them better. Out of interest, how much faster is rendering after using Jos Leys' algorithm compared to before?

Cheers for the formulae Paul. I look forward to implementing it, especially the 4th power one. I take it that one can use it in conjunction with Jos Leys' DE algorithm?

Hi all, just did some accurate timing tests (using my core2duo instead of the somewhat unreliable timings on this P4HT) just to see the difference between my Delta DE and Jos Leys' Analytical DE and also to see how much faster the non-trig versions are.

First to anwer Daniel's question - yes you can use the non-trig versions with Jos' analytical DE *but* at the moment I only know how to do that using the trig version of the formula for calculating the derivative (with the "normal" iteration using the non-trig version). There's probably a way to get the derivative without the trig but it's beyond my maths ability

OK now the timings - all rendered within UF at the same (high) detail settings with shadowcasting and all at the same magnification. In all cases the "entire" Mandelbrot was in view (all renders 1024 wide).

z^2+c

Delta DE (trig): 9 mins 12 secs

Analytical DE (trig): 5 mins 25 secs

Delta DE (no trig): 1 min 44 secs

Analytical DE (trig for derivative only): 4 mins 26 secs

z^3+c

Delta DE (trig): 4 mins 53 secs

Analytical DE (trig): 3 mins 07 secs

Delta DE (no trig): 0 min 53 secs

Analytical DE (trig for derivative only): 2 mins 29 secs

z^4+c

Delta DE (trig): 4 mins 04 secs

Analytical DE (trig): 2 mins 30 secs

Delta DE (no trig): 0 min 59 secs

Analytical DE (trig for derivative only): 2 mins 04 secs

Pretty much as you'd expect based on the relative amounts of calculation involved

If anyone's wondering there was essentially zero visible difference between the trig renders and the non-trig renders - just the odd pixel in the "bowl" areas of the z^3 and z^4 analytical DE renders - probably down to greater inaccuracies in the trig versions as compared to the non-trig versions (the way I calculate the normals is especially sensitive to small variations).