= Integrated circuit cost per good die
{title2=$C_{\rm good}=C_w/(N_gY)$}
Let a circular <semiconductor wafer> have <diameter> $d$ and processed cost $C_w$. For square <semiconductor dies> of <area> $A$, ignoring all edge losses gives the ideal gross count $N_0=\pi d^2/(4A)$. For illustration adopt the geometric edge approximation
$$
N_g\approx\frac{\pi d^2}{4A}-\frac{\pi d}{\sqrt{2A}},\qquad
\mathbb E[G]\approx N_g e^{-DA},\qquad
C_{\rm good}\approx\frac{C_w}{N_g e^{-DA}}.
$$
Here $G$ is the good-die count. The correction scales as wafer perimeter divided by die side length: only a boundary strip can lose partially fitting squares. Its coefficient is an approximation depending on layout; the expression is not an exact packing theorem and is useful only when $\sqrt A\ll d$. Counts from it need not be <integers>. Actual layouts, edge exclusion and separation lanes determine an integer gross count. The <expected value> of $G$ follows by <linearity of expectation>; independence between <semiconductor dies> is not required for that step if each has the same yield.
For the original example $d=200\,\mathrm{mm}$, $A=100\,\mathrm{mm}^2$, $D=0.005\,\mathrm{mm}^{-2}$ and $C_w=1000$ cost units, $N_0=314.159$ and $N_g\approx269.730$. The predicted good count is $163.600$ per wafer, and the processed cost is $6.11248$ units per good die. This is total processed cost divided by expected output, appropriate to a large repeated batch; it is not the <expected value> of a single-wafer random ratio $C_w/G$, which is undefined when $G=0$. Test, packaging, design and equipment costs are excluded.
A hypothetical test readout makes the inference explicit. Suppose $600$ of $1000$ tested <semiconductor dies>, each of <area> $100\,\mathrm{mm}^2$, pass a specified functional test. These are invented counts for calculation, not reported measurements. The pass fraction is $\widehat Y=0.600$; within the <Poisson defect yield> assumptions it implies
$$
\widehat D=-\frac{\ln\widehat Y}{A}=0.00510826\,\mathrm{mm}^{-2}.
$$
If pass indicators are <independent random variables> with common success probability $Y$, their sum has a <binomial distribution>. The fitted standard deviation of the pass fraction is $\sqrt{0.6(1-0.6)/1000}=0.01549$. A test readout estimates pass probability for its tested functions and conditions; it cannot alone establish spatially independent defects, detect every latent failure or distinguish fabrication losses from a faulty test. The <logarithm> only translates the observed fraction into a defect intensity after those additional assumptions are adopted.
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