Understanding gsl::narrow implementation

This is checking for overflow. Lets look at

auto foo = narrow<int>(std::numeric_limits<unsigned int>::max())

T will be int and U will be unsigned int. So

T t = narrow_cast<T>(u);

will give store -1 in t. When you cast that back in

if (static_cast<U>(t) != u)

the -1 will convert back to std::numeric_limits<unsigned int>::max() so the check will pass. This isn't a valid cast though as std::numeric_limits<unsigned int>::max() overflows an int and is undefined behavior. So then we move on to

if (!details::is_same_signedness<T, U>::value && ((t < T{}) != (u < U{})))

and since the signs aren't the same we evaluate

(t < T{}) != (u < U{})

which is

(-1 < 0) != (really_big_number < 0)
==  true != false
==  true

So we throw an exception. If we go even farther and wrap back around using so that t becomes a positive number then the second check will pass but the first one will fail since t would be positive and that cast back to the source type is still the same positive value which isn't equal to its original value.