When floating-point addition isn't commutative

In this note, the symbols "+", "x", and "=" have their mathematical meanings; the symbols "⊕", "⊗", and "→" denote floating-point addition, multiplication and assignment respectively. Informally, T → A ⊕ B implies T = round (A+B), T → A ⊗ B implies T = round (AB), and T → AB ⊕ C implies T = round (AxB + C).