The relative floating-point model excludes overflow, underflow, and invalid operations

Overflow. The representable numbers $x=(3\mathrm{/}2)\cdot 2^{1023}$ and $y=2$ have product $x\cdot y=3\cdot 2^{1023}$, which exceeds $N_{\max }=(2-2^{-52})2^{1023}$ because $3>2-2^{-52}$; the computed result is $+\mathrm{\infty }$, which is not a real number of the form $(x\cdot y)(1+\delta )$ for any real $\delta$.

Underflow. For the smallest positive subnormal $x=2^{-1074}$, the exact quotient $x\mathrm{/}2=2^{-1075}$ lies strictly between $0$ and the smallest positive subnormal, and round-to-nearest returns $0$; but $(x\mathrm{/}2)(1+\delta )=2^{-1075}(1+\delta )$ is positive for every real $\delta$ with $\mathrm{\mid }\delta \mathrm{\mid }\le u$, since $1+\delta \ge 1-u>0$, so the computed value $0$ is not of the required form.

Invalid operation. The exact value of $0\mathrm{/}0$ is not a real number, so no real $\delta$ can satisfy $\mathrm{fl}(0\mathrm{/}0)=(0\mathrm{/}0)(1+\delta )$; the computed result is $\mathrm{NaN}$, which is not a real number at all.

Each of the three exhibits produces a value outside the model of [L1]: infinity, the lost subnormal result, and not-a-number respectively.

Step 2.1 refutes the false statement: the exclusions in [L1] are not decorative, and any statement proved under the relative model must carry the hypothesis that its operations stay in the normal range.