Clever fast Mandelbrot - please explain!

[EdS2]

I refer to Valentin Albillo's remarkable exposition VA040b, calculating the area of the Mandelbrot set, as linked here.

The specific optimisation I'd like to understand is the shortcut which detects the two otherwise very expensive central areas: the main cardioid and the largest circular bud, aka the main disc. Normally these are rendered in black, and take the maximum number of iterations for each point evaluated.

Valentin notes:

Quote:if we can quickly check whether a given z belongs or not to the main cardioid or the main disk we’ll save lots of running time and as it happens, indeed we actually can, using just a few steps for the RPN version or just 2 lines of code for the BASIC version.

All this is clear, but it's the two lines of code, or the few steps, which have me baffled, as to how exactly they determine membership in these two areas.

Any explanations welcome!

Ref:
HP Article VA040b - Boldly Going - Mandelbrot Set Area (HP-71B)

Quote:Line 3: checking whether the point belongs to the main cardioid (thus, to M).
Line 4: checking whether the point belongs to the main disk (thus, to M)

See also
HP Article VA040a - Boldly Going - Mandelbrot Set Area (HP42S)

Quote: Steps 45-53: checking whether the point belongs to the main cardioid (thus, to M). { 9 steps }
Steps 54-59: checking whether the point belongs to the main disk (thus, to M). { 6 steps }