Hmm...
I was trying to work out a similar idea for deep zooms.
When you get up to doing 256 bit calculations what you're really doing at the processor level is lots of 32 bit calcs added together like long multiplication.
But the area of the image will be completely within the least significant 32 bit word, ie even 24000 (pixels) is a number smaller than 2^32.
So it would make sense to save every calculation result of every iteration step because all that really needed to be recalculated was the least significant word.
Of course the least significant still needs to multiply by the most significant through to the middle significant so your saving still having to do maybe 1/4 of the calculations.
Then theres the effect of carries possibly altering the 2nd least significant, numbers like .999999 that might trigger a cascade of carries and it all became too much
for my mental scratchpad.
Power series, okay thats getting a little more familiar to my high school maths
, at least I remember what the sigma summation symbol does, & I remember the teacher saying the word polynomial, but that was 30 years ago now
.