Floats can represent all integers with magnitude <=2^53. Which is a sight smaller than integers, under the standard representation (magnitude <2^62), but still plenty for most applications—the point is to use this to implement integer math, not float math. That covers multiplication, addition, and subtraction; div/mod is admittedly a bit trickier, though it's a doozy with avx512.
(There's also a cute consequence: you can construct numbers outside the contiguous range of representable integers; so long as you never incur any rounding error, the result will be correct, and the inexact flag will not be set.)
