I was thinking about this expression, and I wondered if it holds true when the clock is slow. I can imagine a slow clock which is not right at all in the span of 12 hours—imagine a clock which ticks 5 minutes every 12 hours, which points to 11:59 at exactly 12:00. 12 hours later it is 12:00 and the clock is pointing to 12:04. The clock won't be right for another 5 minutes or so, and will have gone more than 12 hours without being right.
Is there an upper bound on the amount of time that a clock which moves at a slow but constant rate will spend being wrong before it is right? What would be a good way to model this? How should I tag this question?