IIRC both Grisu and Errol has a fairly specific class of pathological inputs, referred as to pathological midpoints in the paper. I have implemented Grisu3 myse

IIRC both Grisu and Errol has a fairly specific class of pathological inputs, referred as to pathological midpoints in the paper. I have implemented Grisu3 myself so I would use that as an example. (I don't know if this example is also problematic in Errol1.) 10^23 is not exactly representable in IEEE 754 binary64 format (aka double). 10^23 - 3 * 2^23, 10^23 - 2^23 and 10^23 + 2^23 are exactly representable however, so any number from 10^23 - 2 * 2^24 to 10^23 (inclusive, because IEEE 754 has a specific rounding mode here) is round to 10^23 - 2^23. Clearly the shortest correct output in the range would be 10^23, but 10^23 is the midpoint between two FP values! Both Grisu and Errol use a fixed size mantissa (64 bits and 106 bits respectively) and have a conservative error margin as a result, and that margin would always contain such midpoints. Errol uses more precise error analysis to show that pathological midpoints are always integral and in the specific range. Since the specific range is <= 2^131, it simply tries to calculate the midpoint as a 128-bit integer and works around the conservative error margin. This removes a major source of pathological inputs, but the conservative error margin still exists and numbers may still fall in that hole. The final enumeration shows that there are only 45 such numbers.