// the mandelbrot set, basically, is a set of points on the complex plane -
// those points are found using the following equation:
//
// z = z^2 + c
//
// c is an initial point on this complex plane. if the value of z is bigger than
// 2 (which means that it's distance from the origin is larger than 2), it means
// the value is out of the set.
//
// but what's going on with all the shapes and colors? this is a recursive
// function.
//
// we start with z as 0, and then with the equation we get a new z, and then put
// that new z onto the equation and we do that over and over. if z doesn't grow
// over 2, then c is part of the set. if it does pass 2, we know for how many
// iterations we have checked and we color based on that

int mandelbrot_iterations(double cr, double ci, int max_iter) {
    double zr = 0.0, zi = 0.0;
    int iter = 0;
    while (iter < max_iter) {
        double zr2 = zr * zr;
        double zi2 = zi * zi;
        // avoids checking for sqrt
        if (zr2 + zi2 > 4.0) {
            break;
        }
        zi = 2.0 * zr * zi + ci;
        zr = zr2 - zi2 + cr;
        iter++;
    }
    return iter;
}