# Why does this distance estimator method render The Mandelbrot set incorrectly (non-divergent regions as divergent)?

I am using the following algorithm to render the Mandelbrot set and the exterior:

• for each test point, calculate $$c$$

• initialise $$z_{0}=(0+0i)$$

• also initialise the gradient $$dz_{0}=(0+0i)$$

• iterate the following for the maximum allowed iterations

– calculate next $$z$$ using $$z\mapsto z^{2}+c$$

– calculate next gradient $$dz$$ using $$dz\mapsto2\cdot z\cdot dz+1$$

– break out of iteration loop if $$|z|>4$$ escape condition met

• calculate distance estimate as $$d=\left(|z|\cdot\log|z|\right)/|dz|$$

• colour the pixel based on distance estimate, eg $$255\times\tanh(d\times \text{resolution}/\text{size})$$ using a grey scale, 0-black, 255-white

The following shows a square viewport that has a bottom left at $$(-0.7416363282638+0.1804439806419i)$$, and a width of $$0.008304417869$$.

It seems to me that the red-circled mini-Mandelbrot shapes are part of the fully-connected Mandelbrot set and should be coloured black.

Question: Why are they white?

Thoughts: Even with non-zoomed views, I have seem some of the bulbs of The Mandelbrot coloured white.

The source of my algorithm is here, and this comment.