5th Order ABS Mandelbrot Variations

5th Order Mandelbrot Variations. These formula share remarkable similarities to my previous 3rd order formulas. The 5th Order polynomials each have six terms. Because i^2 is in the real domain, every even term is real and every odd term is imaginary. When Zi and Zr are factored out, that leaves all forth power terms remaining. To optimize the code for computational efficiency, additional variables are created for (Zr^2)^2, (Zi^2)^2 and (Zi^2)*(Zr^2). Each of these terms appears twice in the equation, reducing the number of necessary bignum multiplication operations significantly. It may be possible to further optimize code in software but this is beyond my scope.

Mandelbrot 5th:
        zi = zi * (5 * zrsqrsqr - 10 * zrzisqr + zisqrsqr) +JuliaI;
        zr = zr * (zrsqrsqr - 10 * zrzisqr + 5 * zisqrsqr) +JuliaR;
        zisqr = zi * zi;
        zrsqr = zr * zr;
        zisqrsqr = zisqr * zisqr;
        zrsqrsqr = zrsqr * zrsqr;
        zrzisqr = zrsqr * zisqr;


Mandelbar 5th (Vertical):
        zi = -zi * (5 * zrsqrsqr - 10 * zrzisqr + zisqrsqr) +JuliaI;
        zr = zr * (zrsqrsqr - 10 * zrzisqr + 5 * zisqrsqr) +JuliaR;
        zisqr = zi * zi;
        zrsqr = zr * zr;
        zisqrsqr = zisqr * zisqr;
        zrsqrsqr = zrsqr * zrsqr;
        zrzisqr = zrsqr * zisqr;


Mandelbar 5th (horizontal):
        zi = zi * (5 * zrsqrsqr - 10 * zrzisqr + zisqrsqr) +JuliaI;
        zr = -zr * (zrsqrsqr - 10 * zrzisqr + 5 * zisqrsqr) +JuliaR;
        zisqr = zi * zi;
        zrsqr = zr * zr;
        zisqrsqr = zisqr * zisqr;
        zrsqrsqr = zrsqr * zrsqr;
        zrzisqr = zrsqr * zisqr;


Burning Ship 5th:
        zi = abs(zi) * (5 * zrsqrsqr - 10 * zrzisqr + zisqrsqr) +JuliaI;
        zr = abs(zr) * (zrsqrsqr - 10 * zrzisqr + 5 * zisqrsqr) +JuliaR;
        zisqr = zi * zi;
        zrsqr = zr * zr;
        zisqrsqr = zisqr * zisqr;
        zrsqrsqr = zrsqr * zrsqr;
        zrzisqr = zrsqr * zisqr;


Buffalo 5th:
        zi = abs(zi * (5 * zrsqrsqr - 10 * zrzisqr + zisqrsqr)) +JuliaI;
        zr = abs(zr * (zrsqrsqr - 10 * zrzisqr + 5 * zisqrsqr)) +JuliaR;
        zisqr = zi * zi;
        zrsqr = zr * zr;
        zisqrsqr = zisqr * zisqr;
        zrsqrsqr = zrsqr * zrsqr;
        zrzisqr = zrsqr * zisqr;


Burning Ship 5th Partial:
        zi = zi * (5 * zrsqrsqr - 10 * zrzisqr + zisqrsqr) +JuliaI;
        zr = abs(zr) * (zrsqrsqr - 10 * zrzisqr + 5 * zisqrsqr) +JuliaR;
        zisqr = zi * zi;
        zrsqr = zr * zr;
        zisqrsqr = zisqr * zisqr;
        zrsqrsqr = zrsqr * zrsqr;
        zrzisqr = zrsqr * zisqr;


Burning Ship 5th Partial Mbar:
        zi = -zi * (5 * zrsqrsqr - 10 * zrzisqr + zisqrsqr) +JuliaI;
        zr = abs(zr) * (zrsqrsqr - 10 * zrzisqr + 5 * zisqrsqr) +JuliaR;
        zisqr = zi * zi;
        zrsqr = zr * zr;
        zisqrsqr = zisqr * zisqr;
        zrsqrsqr = zrsqr * zrsqr;
        zrzisqr = zrsqr * zisqr;


Celtic 5th (Buffalo 5th Partial):
        zi = zi * (5 * zrsqrsqr - 10 * zrzisqr + zisqrsqr) +JuliaI;
        zr = abs(zr * (zrsqrsqr - 10 * zrzisqr + 5 * zisqrsqr)) +JuliaR;
        zisqr = zi * zi;
        zrsqr = zr * zr;
        zisqrsqr = zisqr * zisqr;
        zrsqrsqr = zrsqr * zrsqr;
        zrzisqr = zrsqr * zisqr;


Celtic 5th Mbar:
        zi = -zi * (5 * zrsqrsqr - 10 * zrzisqr + zisqrsqr) +JuliaI;
        zr = abs(zr * (zrsqrsqr - 10 * zrzisqr + 5 * zisqrsqr)) +JuliaR;
        zisqr = zi * zi;
        zrsqr = zr * zr;
        zisqrsqr = zisqr * zisqr;
        zrsqrsqr = zrsqr * zrsqr;
        zrzisqr = zrsqr * zisqr;


Quazi Burning Ship 5th (BS/Buffalo Hybrid):
        zi = -abs(zi * (5 * zrsqrsqr - 10 * zrzisqr + zisqrsqr)) +JuliaI;
        zr = abs(zr) * (zrsqrsqr - 10 * zrzisqr + 5 * zisqrsqr) +JuliaR;
        zisqr = zi * zi;
        zrsqr = zr * zr;
        zisqrsqr = zisqr * zisqr;
        zrsqrsqr = zrsqr * zrsqr;
        zrzisqr = zrsqr * zisqr;


quazi Perpendicular 5th:
        zi = -zi * abs(5 * zrsqrsqr - 10 * zrzisqr + zisqrsqr) +JuliaI;
        zr = abs(zr) * (zrsqrsqr - 10 * zrzisqr + 5 * zisqrsqr) +JuliaR;
        zisqr = zi * zi;
        zrsqr = zr * zr;
        zisqrsqr = zisqr * zisqr;
        zrsqrsqr = zrsqr * zrsqr;
        zrzisqr = zrsqr * zisqr;


Quazi Heart 5th:
        zi = zi * abs(5 * zrsqrsqr - 10 * zrzisqr + zisqrsqr) +JuliaI;
        zr = abs(zr) * (zrsqrsqr - 10 * zrzisqr + 5 * zisqrsqr) +JuliaR;
        zisqr = zi * zi;
        zrsqr = zr * zr;
        zisqrsqr = zisqr * zisqr;
        zrsqrsqr = zrsqr * zrsqr;
        zrzisqr = zrsqr * zisqr;


Special thanks to Kalles Fraktaler and Command Line Cowboy (Panzerboy) for adding my 2nd and 3rd order fractals to their software plugins. Feel free to use any of the above formulas in any software program, image, or video render. Credit is appreciated but not mandatory. Thanks for viewing...

See Also:

2nd Order ABS Formula:
stardust4ever.deviantart.com/a…

3rd Order ABS Formula:
stardust4ever.deviantart.com/a…

4th Order ABS Formula:
stardust4ever.deviantart.com/a…

5th Order ABS Formula:
stardust4ever.deviantart.com/a…