My appologies for the double post. Panzerboy, if you feel like I'm spamming Fractal Extreme Plugin requests, please let me know and I'll stop posting about new formulas. In the other thread, I had mentioned briefly the desire to explore the Perpendicular Mandelbrot set, which is a vertical slice of the 2nd order mandelbulb. Here's a zoomed in view of a minibrot that I found within the fractal which I rendered in UF5:
There exists both trig and polynomial variants of the 3D Mandelbulb fractal, which use an arcane form of triplex algebra.
I had a look for the mandlbulb and the formula I found has square roots and atan functions.
I recall there were two different formulas for the 3D Mandelbulbs. One formula version used trig functions, and the other version used polynomials. The poly functions were much faster for the lower orders than the trig functions (especially 2nd order), but due to the complexity of the higher order polynomial, the trig functions became much faster for bulbs higher than order 8. I'm trying to find some old posts about it here in fractalforums which I remembered reading a long time ago, but the search function is practically broken. I know the polynomial functions did not use trig, but they may have still incorporated a variable "r" where r = sqrt(x^2+y^2+z^2) which would still be problematic. I can't find it, but there is a post somewhere on fractalforums which lists the polynomial equivalents of mandelbulbs for orders 2 through 8. If I can find that post, I can just reduce the set x,y,z by setting y=0 and then reducing the equation to x and z using basic algebra, but I know for a fact there is a polynomial version of the equation which does not use trig functions.
Well, it seems the people over at UF have already done the math homework for me. I copy/pasted the Ultrafractal Code for the Perpendicular Mandelbrot from UF5 and I have good news: While polynomial formula for the 3D Mandelbulb (in terms of x,y,z) does utilize the square-root function, the Pythagorean-like formula r=sqrt(y^2+z^2) (or something similar to that effect) cancels itself out when the original y axis is eliminated, leaving sqrt(z^2) which simplifies to abs(z). That is good news, because the trig and square roots have been entirely eliminated from the equation. Here is the UF5 code. It's pretty basic, so the formula should render fast if ported over to Fractal Extreme:
;This produces slice of the second order Mandelbulb directly perpendicular to the regular Mandelbrot set.
xTemp=x^2 - y^2 + Cx
y=-2*y*Abs(x) + Cy
float param Bailout