Without underflow and overflow, is there any 2 numbers, which A < B in decimal form, but A > B after converting to floating point?

Question

For example, 0.00000000000000000000000000000000000000000000000000000000000000000000000000000011 is greater than 0.0000000000000000000000000000000000000000000000000000000000000000000000000000001 both in decimal and after converting to floating point numbers:

document.writeln(0.00000000000000000000000000000000000000000000000000000000000000000000000000000011>0.0000000000000000000000000000000000000000000000000000000000000000000000000000001);

And 1.9999999999999999999 is smaller than 2 in decimal, but they becomes equal after converting to floating point numbers:

document.writeln(1.9999999999999999999==2);

My question is, is there any 2 numbers, A and B, which is A < B in decimal form, but it becomes A > B after converting to floating point number?

Answer 1

The usual rounding rules are weakly monotonic, so no.

The rounding rules defined by IEEE 754 round a result either to the nearest representable value regardless of which direction it is in or to the nearest representable value in a chosen direction (such as toward zero). It is not possible for the roundings of two numbers to cross each other (when rounded with the same rule), as this would imply one number is not rounded to the nearest representable value.