Manuscript Psy04-A@1 · Psy04-A-v1.tex · proposition · d5800e2a

Proposition: Rare-branch normalization cost

Robust model and apparatus allowances › An implementation route to e Q · lines 1127–1151 · proof included or linked

Psy04-A@1#prop:normalization

For positive subnormalized blocks $τ, τ^{'}$ , with traces $a, a^{'} \geq a_{m i n} > 0$ , suppose

\frac{1}{2} \norm τ - τ^{'}_{1} \leq e .

Then

d_{tr} (τ / a, τ^{'} / a^{'}) \leq min {1, 2 e / a_{m i n}} .

Proof. Insert $τ^{'} / a$ . Since $\abs a - a^{'} \leq \norm τ - τ^{'}_{1}$ ,

\norm τ / a - τ^{'} / a^{'}_{1} \leq \frac{\norm τ - τ ^{'} _{1} + \abs a - a ^{'}}{a} \leq \frac{2 \norm τ - τ ^{'} _{1}}{a _{m i n}} .

Divide by two.

Displayed formulas in this passage

unlabeled · line 1131
$\frac{1}{2} \norm τ - τ^{'}_{1} \leq e .$
unlabeled · line 1135
$d_{tr} (τ / a, τ^{'} / a^{'}) \leq min {1, 2 e / a_{m i n}} .$
unlabeled · line 1144
$\norm τ / a - τ^{'} / a^{'}_{1} \leq \frac{\norm τ - τ ^{'} _{1} + \abs a - a ^{'}}{a} \leq \frac{2 \norm τ - τ ^{'} _{1}}{a _{m i n}} .$