How to imagine Float in the form of Integer without loss of information?

There is a task of transforming float values into int values (one float value into one int value) without loss of information, i.e., so that the reverse transformation can be performed later. How can this be implemented?

Important

It is necessary that order and sum are preserved,
i.e., for any a, b: float the following must hold:

to_float(to_int(a) + to_int(b)) == a + b
to_int(a) < to_int(b) when a < b, etc.

I found on the English-speaking Stack Overflow where the structure of float values is represented, but I can't figure out how to apply it to my task.

Note: nan