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The formula you cite is a back-of-the-envelope approximation that only works so long as c and d have much smaller magnitudes than a and b. (Try to derive that formula yourself. You'll find it pop up in a limit, but it's certainly not generally true.)

You also need to clear up your own confusion about what exactly you mean by "uncertainty". You seem to go back and forth between several different measures.

Uncertainties aren't negative. But solve the rest of the issues, and what I meant by cases should become more clear.