"correctly rounded" is implementation defined is the problem. You cannot do it portably, and you cannot query it portably. As such, different platforms, compilers, etc do it differently. Thus using formatting for rounding is inconsistent.

Here's [1] where you can query the current floating-point environment in C: "Specifics about this type depend on the library implementation".

Here's [2] where you can set some rounding modes in C++: "Additional rounding modes may be supported by an implementation.". Note this does not have by default bankers rounding which is used to make many scientific calculations more stable (lowers errors and drift in accumulated calculations). Many platforms do this by default, but it's not in the standard.

You can chase down this rabbit hole. I (and several others) did during the issue on the last project, and got to where it was well-known in numerics circles that this is not a well-defined process in C/C++. If it were, printing and parsing should round-trip, and it does not before a recent C++ addition, and now it only is guaranteed in a special case.

[1] http://www.cplusplus.com/reference/cfenv/fenv_t/

[2] https://en.cppreference.com/w/cpp/numeric/fenv/FE_round