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I have modified my monotonicity checking algorithm. First I implemented a monotonicity check for the differential of the values, but combined with the harmonic interpolation it produced quite a few visual glitches. Then I inserted a threshold into the original monotonicity check:

if(Math.max(Math.abs(minDiff),Math.abs(maxDiff))<THRESHOLD) then monotonic=true;

After trying different threshold values, I found that 0.3 has the most savings while still not distorting the visuals. Here are two examples about the effect of this threshold (left: original monotonicity checking; right: original monotonicity checking + threshold).

As you can see, the curves in the middle of monotonic areas have disappeared. This results in savings of up to 30%, depending on the image.

where is the difference to "normal" guessing algorithm ?!
and what the heck are these bands, that do not belong to the set, (mostly blue and curly )

Does this also mean that the actual number of iterations before bailout (and including the fractional part) on a straight line (the boundary of an interpolated area) follows an analytic function?