Possible to have ``a < b and not(a - b < 0)`` with floats

floating-point, python

Solution

It depends on the floating point format, and specifically whether it has gradual underflow. IEEE 754 binary floating point does. That means the gap between two distinct floats is always non-zero, even if it can only be represented with one significant bit in the extreme case.

The smallest possible absolute difference between two distinct floats comes with equal sign (or one of the numbers zero), zero exponent, and significands that differ by 1. In that case, subtracting the larger from the smaller would result in the smallest magnitude negative number, with negative sign, zero exponent, and significand 1. That number compares less than zero.

There are other calculations that do underflow to zero. For example, dividing the smallest positive number by a number greater than or equal to two results in zero in the normal rounding mode.

Problem

Original source