I messed with the f_mandelbulb internal function, squeezed about 10-15% better
performance out of it, and posted what I have so far below.  (Note the handy
sincos() function in the standard library saving a couple of trig calls.)

Interestingly, I also found an additional 10-15% improvement in render time by
seriously overloading the processors with one to two hundred POV-Ray worker
threads on eight cores and using smaller render block size (16 for 640*360
images).  I'm guessing the extra switching cost is more than offset by keeping
the floating-point pipelines full.

~David Wagner

DBL f_mandelbulb(FPUContext *ctx, DBL *ptr, unsigned int) // 79
{  // Coded by David Wagner, and based on
   // Daniel White's original squaring formula,
   // Paul Nylander's generalization and phi phase shift formulation,
   // the Normalized Iteration Count Algorithm,
   // and Abram Hindle's POV-Ray internal function patch.
   DBL cx=PARAM_X, cy=PARAM_Y, cz=PARAM_Z;
   DBL halfp_r   = PARAM(0);// Radial power exponent.
   DBL p_theta   = PARAM(1);// Power of elevation theta.
   DBL p_phi     = PARAM(2);// Power of azimuth phi.
   DBL phase     = PARAM(3);// Phase shift of phi.
   int i_bailout = PARAM(4);// Maximum number of iterations.
   DBL r2_bailout = pow(PARAM(5),2);// Assumed divergence radius.
   DBL x = cx, y = cy, z = cz;
   DBL r2 = x*x + y*y + z*z;
   DBL rp, sinpphi, cospphi, sinptheta, cosptheta;
   int i=1;
   halfp_r*=0.5;
   while(i < i_bailout && r2 < r2_bailout) {
       ++i;
       rp = pow(r2,halfp_r);
       sincos(p_phi * atan2(z,sqrt(x*x + y*y)) + phase, &sinpphi, &cospphi);
       sincos(p_theta * atan2(y,x), &sinptheta, &cosptheta);
       z = cz + rp * sinpphi;
       x = cx + cosptheta * rp*cospphi;
       y = cy + sinptheta * rp*cospphi;
       r2 = x*x + y*y + z*z;
   }
   if (i < i_bailout) {
       return 1/( i + log(log(r2_bailout) / log(  r2))
          / (log(halfp_r)+0.693147180559945) );
   }//          ==log(2*halfp_r)
   return 1/( i + log(log(r2_bailout) / log(1+r2))
      / (log(halfp_r)+0.693147180559945) );
}

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