// takes a point on the complex plane and determines if it is in the Set or not

// returns the normalised escape value (how many times it was iterated before escaping), or -1 if it is in the Set

// basically this function does z = z^2 + c, but normalises the result for smooth shading

double smoothsample (COMPLEX c)

{

unsigned int i;

double cardoid, z2s;

COMPLEX z, z2 = {0, 0};

// test if this point is in the cardoid or the bulb of the Set

cardoid = (c.r - 0.25f) * (c.r - 0.25f) + c.i * c.i;

if (

4 * cardoid * (cardoid + (c.r - 0.25f)) / c.i / c.i < 1 ||

16 * ((c.r + 1) * (c.r + 1) + c.i * c.i) < 1

)

return -1;

for (i = 0; i < maxIterations; i++)

{

// cache multiplication

z2.r = z.r * z.r;

z2.i = z.i * z.i;

// do actual iteration

z.i = 2 * z.r * z.i + c.i;

z.r = z2.r - z2.i + c.r;

// cache sum

z2s = z2.r + z2.i;

// not part of the Mandelbrot set; colour the pixel and immediately proceed to next pixel

if (z2s > escapeRadius)

return (double)i - log2(log(z2s) / 2);

}

return -1; // point is in the Set

}