Yield, chiplets and weights-in-silicon

Three tabs:

Yield & cost per die — die area (and aspect ratio), defect density D₀ and a yield model give the fraction of good dies. Dies are counted whole on a grid inside the wafer’s usable circle (the classic formula is shown beside the count), and wafer cost ÷ (dies × yield) is the cost of each good die. A chart plots yield against area for all three models with the 858 mm² reticle marked, and a wafer map shows good and defective dies.
Chiplets — one design built as a single die or split into n chiplets (plus any die-to-die interface overhead). With known-good-die testing only working chiplets are bonded, so the package yield is just the bond yield; without it, untested dies go in and the package works only if all of them do. It shows cost and silicon per good part for all three, and flags a design too big for one exposure. Wafer cost and yield model come from the first tab.
Weights in silicon — the die area a model of N parameters needs at a mask-ROM density, in mm², cm² and reticles: Taalas’s HC1 density (Llama 3.1 8B in 815 mm² on TSMC N6), the conventional compute-in-ROM estimate from the BitROM paper (7B in more than 1,000 cm² at 14 nm), or your own.

The models: Poisson Y = e^−AD; Murphy Y = ((1 − e^−AD) / AD)²; negative binomial Y = (1 + AD/α)^−α, with A in cm² and D₀ in defects per cm².

Where the book’s figures differ

"More than 1,000 cm²" for a 7B model is 100,000 mm² ÷ 858 = 116.6 reticles — at least 117 dies, slightly more than the book’s "roughly 116×".
At a mature defect density (0.1/cm²) a full-reticle die yields about 0.45 against 0.81 for a quarter-size chiplet (Murphy): about half, not "a small fraction". The gap becomes large only at higher defect densities (0.05 against 0.38 at 0.5/cm²).

Limits

Yield here is defect-limited (random particles). Parametric and systematic yield loss, redundancy and repair (spare cores, fused-off units — how big GPUs really ship) are not modelled; real large dies yield better than these formulas because of them.
Wafer costs are published estimates (CSET 2020 for 7 nm and 5 nm; press reports for 3 nm), not foundry prices. Test, packaging and interposer costs are a single input.
The wafer map marks a share of 1 − yield of the dies as defective for illustration; real defects cluster.
The mask-ROM densities are derived from two published figures (one whole product die, one paper estimate), not from a cell library.

Sources

Murphy, "Cost-size optima of monolithic integrated circuits", Proc. IEEE 52 (1964); Stapper, "On yield, fault distributions, and clustering of particles", IBM J. Res. Dev. 30 (1986).
CSET, "AI Chips: What They Are and Why They Matter" (2020) — wafer-cost estimates.
BitROM: Weight Reload-Free CiROM Architecture Towards Billion-Parameter 1.58-bit LLM Inference (ASP-DAC 2026) — "LLaMA-7B … exceeding 1,000 cm² in 14nm CMOS".
Taalas HC1: Llama 3.1 8B in an 815 mm² TSMC N6 die (company statements and press, 2026), as cited in the book.

Related: AI hardware calculator (what those dies compute and how much memory a model needs), AI cost and energy economics (from the cost of a chip to the cost of a token).

Companion to Chiplets and Packaging, From Design to Silicon and Weights-in-Silicon of _AI Acceleration: From Transistors to Trillion-Parameter Models.

Linking to an example

Links can be written by hand. Add the tab after #: yield, chiplets or rom. Settings not in the link take their defaults.

Parameter Meaning Example

Parameter	Meaning	Example
`a`	Die area, mm² (Yield tab)	`a=800`
`ar`	Aspect ratio, height ÷ width	`ar=1.27`
`d0`	Defect density, per cm²	`d0=0.1`
`model`	`murphy`, `poisson` or `negbin`	`model=poisson`
`alpha`	Clustering α for the negative binomial	`alpha=2`
`wafer`, `edge`, `scribe`	Wafer diameter, edge exclusion and scribe lane, mm	`wafer=300&edge=3&scribe=0.1`
`node`	Wafer-cost preset: `n7`, `n5`, `n3`, `custom`	`node=n3`
`wc`	Wafer cost, $	`wc=20000`
`ca`	Design area, mm² (Chiplets tab)	`ca=858`
`n`	Number of chiplets	`n=4`
`ovh`	Die-to-die overhead per chiplet, %	`ovh=10`
`cd0`	Defect density for the chiplets tab, per cm²	`cd0=0.5`
`pk`	Packaging $ per package	`pk=500`
`bond`	Bond yield per die, %	`bond=99`
`p`	Parameters, billions (Weights in silicon)	`p=8.03`
`dens`	`taalas`, `cirom` or `custom`	`dens=cirom`
`wpm`	Custom density, weights per mm²	`wpm=1e7`
`rd0`	Defect density for the ROM die’s yield	`rd0=0.1`
`example`	One of the presets below; parameters given with it override the preset’s	`example=p10-murphy-8cm2`

a

Die area, mm² (Yield tab)

a=800

ar

Aspect ratio, height ÷ width

ar=1.27

d0

Defect density, per cm²

d0=0.1

model

murphy, poisson or negbin

model=poisson

alpha

Clustering α for the negative binomial

alpha=2

wafer, edge, scribe

Wafer diameter, edge exclusion and scribe lane, mm

wafer=300&edge=3&scribe=0.1

node

Wafer-cost preset: n7, n5, n3, custom

node=n3

wc

Wafer cost, $

wc=20000

ca

Design area, mm² (Chiplets tab)

ca=858

n

Number of chiplets

n=4

ovh

Die-to-die overhead per chiplet, %

ovh=10

cd0

Defect density for the chiplets tab, per cm²

cd0=0.5

pk

Packaging $ per package

pk=500

bond

Bond yield per die, %

bond=99

p

Parameters, billions (Weights in silicon)

p=8.03

dens

taalas, cirom or custom

dens=cirom

wpm

Custom density, weights per mm²

wpm=1e7

rd0

Defect density for the ROM die’s yield

rd0=0.1

example

One of the presets below; parameters given with it override the preset’s

example=p10-murphy-8cm2

In a URL + means a space: write wpm=1e7 or 1e%2B7; a hand-typed 1e+7 also works.

Presets:

example= Shows

`example=`	Shows
`p10-murphy-8cm2`	An 8 cm² die at D₀ = 0.1/cm²: Murphy yield 0.47 (open)
`p10-murphy-2cm2`	A 2 cm² die at the same density: 0.82 (open)
`p10-poisson-8cm2`	The 8 cm² die under the harsher Poisson model: 0.45 (open)
`p10-reticle-die`	A full 26 × 33 mm reticle-size die: dies per wafer and cost per good die (open)
`p10-reticle-four`	One 858 mm² die against four quarter-size chiplets with known-good-die testing (open)
`p10-b200-two-dies`	Two reticle-size dies (B200-style): ≈ 1,600 mm² that one exposure cannot print (open)
`p07-taalas-hc1`	Taalas HC1: Llama 3.1 8B in 815 mm², just under the reticle (open)
`p07-bitrom-7b`	A 7B model in conventional mask ROM: more than 1,000 cm², at least 117 reticles (open)
`p07-taalas-405b`	A 405B model at Taalas’s density: ≈ 41,000 mm², 48 reticles (open)

p10-murphy-8cm2

An 8 cm² die at D₀ = 0.1/cm²: Murphy yield 0.47 (open)

p10-murphy-2cm2

A 2 cm² die at the same density: 0.82 (open)

p10-poisson-8cm2

The 8 cm² die under the harsher Poisson model: 0.45 (open)

p10-reticle-die

A full 26 × 33 mm reticle-size die: dies per wafer and cost per good die (open)

p10-reticle-four

One 858 mm² die against four quarter-size chiplets with known-good-die testing (open)

p10-b200-two-dies

Two reticle-size dies (B200-style): ≈ 1,600 mm² that one exposure cannot print (open)

p07-taalas-hc1

Taalas HC1: Llama 3.1 8B in 815 mm², just under the reticle (open)

p07-bitrom-7b

A 7B model in conventional mask ROM: more than 1,000 cm², at least 117 reticles (open)

p07-taalas-405b

A 405B model at Taalas’s density: ≈ 41,000 mm², 48 reticles (open)