Unexpected need for Multi-precision

[element90]

I have recently been playing with this formula:

z = (alpha(z^3 - z^4) + beta)^5

alpha = position in the complex plane
beta = 1
initial value of z = 0.75

There are certain areas of the fractal that are blank that look like they shouldn't be, zooming to the area shows the characteristics associated with insufficient precision but at low levels of zoom. I have two example pictures both located at -9.3368996120395776 + 7.309115523072i and the width of the complex plane is just 0.0006036665, the picture using long doubles has few features so the first example is with 96 bit values:



The picture above clearly show the features associated with insufficient precision, increasing the precision to 112 bits and the picture becomes:



Note: the blank patches, increasing the precision will extend the filaments into the patches, the outer colouring method is "absolute log of exponential inverse change sum of angle".

The picture False Mandelbrot Byways No. 2 http://www.fractalforums.com/images-showcase-%28rate-my-fractal%29/false-mandelbrot-byways-no-2/ also features these blank areas and uses the same formula but with beta = 0.999999995 and is unaffected. It seems that the closer the value of beta is to 1 the greater the effect.

This phenomenon is peculiar as it only affects parts of the fractal and, depending on position, the requirement for multi-precision calculation can occur with relatively large values, for example, at -9.48 + 0i and a width in the complex plane complex of 1. I'll post pictures of that area later.