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feat: simpler equations

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sBubshait
2026-06-13 20:28:32 +01:00
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@@ -213,10 +213,6 @@
all constraints. $\text{Feasible@}K$ is then the fraction of problems where model $m$ produces a feasible
design in $K$ attempts:
</p>
<div class="equation-block">
$$\text{feasible}(d, p) = \mathbf{1}[\text{correct}(d, p)] \cdot \mathbf{1}[\text{synth}(d)] \cdot
\prod_{j=1}^{N_c} \mathbf{1}[d \text{ meets constraint } c_p(j)]$$
</div>
<div class="equation-block">
$$\text{Feasible@}K = \frac{1}{|P|} \sum_{p \in P} \max_{1 \le i \le K} \text{feasible}(d^{(i)}, p)$$
</div>
@@ -228,35 +224,12 @@
<strong>PPA Optimization Index</strong>
</div>
<p>
$\text{Feasible@}K$ measures whether a model can produce a usable design, but not how well that design
optimises the target hardware. To quantify synthesis quality among feasible designs, we define the
<strong>Maximum Objective Quality (MOQ)</strong>.
</p>
<p>
For each problem $p$, let $f_p^*$ denote the objective value achieved by the reference implementation. For a
feasible generated design $d$, define its normalised quality $q_p(d)$:
For a feasible design $d$, the quality ratio $q_p(d)$ measures performance relative to the reference design ($q_p(d) &gt; 1$ improves on the reference, while $q_p(d) &lt; 1$ underperforms). $\text{MOQ@}K$ is the geometric mean of the best quality ratio found within $K$ attempts:
</p>
<div class="equation-block">
$$q_p(d) = \begin{cases} \frac{f(d)}{f_p^*} & \text{if } \text{dir}_p = \;\uparrow (F_{\text{max}}) \\
\frac{f_p^*}{f(d)} & \text{if } \text{dir}_p = \;\downarrow (\text{LUTs}) \end{cases}$$
$$\text{MOQ@}K = \left( \prod_{p \in P} \max_{\substack{1 \le i \le K \\ \text{feasible}(d^{(i)}, p) =
1}} q_p(d^{(i)}) \right)^{\frac{1}{|P|}}$$
</div>
<p>
Here, $q_p(d) = 1$ means the generated design matches the reference, $q_p(d) &gt; 1$ means it improves on the
reference, and $q_p(d) &lt; 1$ indicates underperformance. </p>
<p>
Across a benchmark set $P$, $\text{MOQ@}K$ is defined as the geometric mean, over all problems, of the
best feasible objective ratio found within the first $K$ attempts:
</p>
<div class="equation-block">
$$\text{MOQ@}K = \left( \prod_{p \in P} \max_{\substack{1 \le i \le K \\ \text{feasible}(d^{(i)}, p) =
1}} q_p(d^{(i)}) \right)^{\frac{1}{|P|}}$$
</div>
<p>
The geometric mean is the natural aggregator for ratio-valued scores: a design $2\times$ better and one
$0.5\times$ worse correctly average to $1.0$ ($\sqrt{2 \times 0.5} = 1.0$), whereas the arithmetic mean
would incorrectly report a net gain of $1.25$. $\text{MOQ@}K$ is conditional on feasibility and must be
interpreted alongside $\text{Feasible@}K$.
</p>
</div>
</div>
</section>