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paper(gemm): inline §3.1 pipeline-stage chain; drop display equation

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The five-stage chain was set as a centered display equation, which
overflowed the column and clipped the last 'DMA_WRITE' label. The
inline prose right below it also repeated the same chain in a
shorter form. Consolidate into one inline parenthetical listing the
five stage names, and reuse 'flows through the chain' for the
self-routing description — no clipping, no repetition.

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
This commit is contained in:
Yangwook Kang
2026-06-17 10:40:23 -07:00
parent b64c43b947
commit 0c6ca0aaed
2 changed files with 10 additions and 13 deletions
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docs/report/1H-codesign-paper/sections/03-gemm.tex
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@@ -19,20 +19,17 @@ in command overhead?
\subsection{Design}
The answer is the \emph{composite command}. A single command carries the
ordered tile pipeline
\[
\textsf{DMA\_READ}\rightarrow\textsf{FETCH}\rightarrow\textsf{GEMM}
\rightarrow\textsf{STORE}\rightarrow\textsf{DMA\_WRITE},
\]
The answer is the \emph{composite command}. A single command carries
the ordered five-stage tile pipeline (\textsf{DMA\_READ},
\textsf{FETCH}, \textsf{GEMM}, \textsf{STORE}, \textsf{DMA\_WRITE}),
and the PE scheduler splits the payload into hardware tiles
(here $32\times64\times32$), emitting one tile token per tile. Subsequent
stages are reached by \emph{token self-routing} between the on-PE engines,
so a tile flows DMA\,$\rightarrow$\,fetch\,$\rightarrow$\,GEMM\,$%
\rightarrow$\,store without returning to the scheduler between stages.
Because the whole pipeline is described by one command, the issue cost is
paid once per GEMM rather than once per tile-stage, and the scheduler is
free to keep every stage busy on different tiles simultaneously---tile
(here $32\times64\times32$), emitting one tile token per tile.
Subsequent stages are reached by \emph{token self-routing} between
the on-PE engines, so a tile flows through the chain without
returning to the scheduler between stages. Because the whole
pipeline is described by one command, the issue cost is paid once
per GEMM rather than once per tile-stage, and the scheduler is free
to keep every stage busy on different tiles simultaneously---tile
$i$'s GEMM overlaps tile $i{+}1$'s DMA read. A multi-operation composite
additionally lets an epilogue (for example a vector-math step) ride the
same tile loop, firing per $K$-tile, per output tile, or once per kernel