Non-trig expansion for RuckerBrot?

[DarkBeam]

Hi, I would like to ask to the geniuses of this wonderful forum: is there a non-trig expansion of the Rucker's MB3D?

Find the formula here:

http://www.fractalforums.com/index.php?action=dlattach;topic=2081.0;attach=1031;image

The entire topic

http://www.fractalforums.com/theory/summary-of-3d-mandelbrot-set-formulas/

This is A Rucker PDF file with interesting explanations. http://www.rudyrucker.com/blog/notebooks/rucker_mandelbulb_ver7_sept24_2009.pdf

Page 8 shows the non-trig expansion of White/Nylander formula (I hoped that was his formula!... )

newx = ( x*x + y*y - z*z )*( x*x - y*y) / ( x*x + y*y )
newy = 2 * ( x*x + y*y - z*z )*x*y / ( x*x + y*y )
newz = - 2 * z * sqrt( x*x + y*y )

Thanks!

[DarkBeam]

Hm.
Obtained this from WolframAlpha (x,y,z are real numbers so complex parts can be safely skipped), but still confused...

cos(2 atan2(y,x) ) = +(y*y-x*x)/(y*y+x*x)
sin(2 atan2(y,x) ) = -(4 y*z)/(y*y+x*x)

then:

cos(2 atan(z/x) ) = (x*x-z*z)/(z*z+x*x)
sin(2 atan(z/x) ) = +(4 x*z)/(z*z+x*x)

The strange thing is he says "atan(z/x)" and not 2*atan...

cos(atan(z/x) ) = abs(x)/sqrt(z*z+x*x)
sin(atan(z/x) ) = z/sqrt(z*z+x*x) * x/abs(x)

What is correct?