How to make the graph have high detail?

I’m trying to create the mandelbrot set but I haven’t found a way to force the graphing to do the extra work even if it takes more time. I tried to reach out to YouTube and Google, but I didn’t find anything useful.
My functions:

  1. f(z) = (z.x^2 - z.y^2 x,2z.xz.y + y)
  2. D(z) = sqrt(z.x^2 + z.y^2)
    graph) D(f(f(f(f(f(f(f(f((x,y)))))))))) ≥ 2

I would also like to know how to make a function repeat itself without having to retype it billions of times.