Answer by Nemo for Why do I need 17 significant digits (and not 16) to represent a double?

I think the guy on that thread is wrong, and 16 base-10 digits are always enough to represent an IEEE double.

My attempt at a proof would go something like this:

Suppose otherwise. Then, necessarily, two distinct double-precision numbers must be represented by the same 16-significant-digit base-10 number.

But two distinct double-precision numbers must differ by at least one part in 2⁵³, which is greater than one part in 10¹⁶. And no two numbers differing by more than one part in 10¹⁶ could possibly round to the same 16-significant-digit base-10 number.

This is not completely rigorous and could be wrong. :-)