That is exactly what I did, I implemented some cycle finding algorithm, a tortoise and hare variant maybe but I can not tell from the top of my head, on the reminder sequence giving up after a user-specified number of steps. But it irks me that this may abort just a hand full of steps before finally finding the cycle.

And unfortunately q is just way to loose as a bound if it even deserves the name bound in this case. I initially started looking into this because I wanted to print something like »0.577 215 664 <901,532 more digits>« but in that case q is essentially useless because you either have to really search the cycle which quickly becomes impractical once q reaches millions or billions or you have to live with »0.577 215 664 <up to 999,999,991 more digits but maybe also just one>«.

> I implemented some cycle finding algorithm, a tortoise and hare variant maybe but I can not tell from the top of my head, on the reminder sequence giving up after a user-specified number of steps.

What's the cycle finding for? Isn't it enough to record the remainder sequence with indices? As soon as any remainder reoccurs, that's the cycle.

You could of course just maintain the set of remainders seen so far and then you would detect the cycle by reaching a remainder that is already in your set of remainders seen so far but that requires space proportional to the number of digits expanded and if you are expanding a lot of digits and have a large denominator, this will add substantially to the amount of memory consumed. The digits themselves require of course a linear amount of space, too, but only about one byte per digit compared to the potentially big remainder.