Add concise LaTeX pseudocode for FP32 and INT32 ExpCast8

expcast8-vs-i32/expcast8_single_score_pseudocode.tex

@@ -0,0 +1,130 @@
+\documentclass[10pt]{article}
+
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage[margin=1in]{geometry}
+
+\newcommand{\sat}[1]{\operatorname{sat}_{#1}}
+\newcommand{\bits}{\operatorname{bits}}
+\newcommand{\bytezero}{\operatorname{byte}_0}
+\newcommand{\lowthirtytwo}{\operatorname{low32}}
+\newcommand{\RNE}{\operatorname{RNE}}
+\newcommand{\RNfp}{\operatorname{RN}_{\mathrm{fp32}}}
+
+\begin{document}
+
+\section*{ExpCast8 for One Score}
+
+For both algorithms, let
+\[
+z=cS-m+\Omega,
+\qquad
+q^{\star}
+=
+\RNE\!\left(
+\operatorname{clamp}(8z+56+\beta,0,126)
+\right),
+\]
+where $S$ is one attention score, $c>0$ is the combined QK and
+softmax scale, $m$ is the row maximum, $\Omega$ is a shared offset,
+and $\beta$ is the ExpCast8 correction bias.
+
+\subsection*{Pseudocode 1: $S$ is FP32}
+
+Define
+\[
+A_f=\frac{8c}{126},
+\qquad
+B_f=\frac{-8m+8\Omega+56+\beta}{126},
+\qquad
+M=1.5\cdot 2^{23}.
+\]
+
+The GPU applies the following operations to one FP32 score:
+\[
+\begin{array}{rcll}
+1. & u &\leftarrow
+   \sat{[0,1]}\!\left(\RNfp(SA_f+B_f)\right)
+   & \text{\rm [\texttt{FFMA.SAT}]}\\[4pt]
+2. & y &\leftarrow
+   \RNfp(126u+M)
+   & \text{\rm [\texttt{FFMA2}, two scores per instruction]}\\[4pt]
+3. & q &\leftarrow
+   \bytezero\!\left(\bits(y)\right)
+   & \text{\rm [byte extraction; \texttt{PRMT} when packed]}\\[4pt]
+4. &   &\mathbf{return}\ q. &
+\end{array}
+\]
+
+Up to the FP32 rounding in the first line,
+\[
+126u
+\approx
+\operatorname{clamp}(8z+56+\beta,0,126).
+\]
+At $M=1.5\cdot2^{23}$, one FP32 ULP is exactly $1$.  Therefore the
+second line rounds $126u$ to an integer and stores that integer in the
+low bits of $y$:
+\[
+\bits(y)=\bits(M)+q.
+\]
+Hence the lowest byte of $y$ is the returned E4M3 code $q$, which
+approximates $q^{\star}$.
+
+\subsection*{Pseudocode 2: $S$ is INT32}
+
+Choose $F=20$ fractional bits.  Once per tile, compute
+\[
+A=\RNE\!\left(8c\,2^F\right),
+\]
+\[
+B=
+\RNE\!\left(
+\left[8(-m+\Omega)+56+\beta\right]2^F
+\right)
++2^{F-1}.
+\]
+
+In the production configuration, $z\leq 8$, so the largest positive
+code is below $126$.  Therefore unsigned-byte saturation is sufficient;
+no separate upper clamp is required in the per-score path.
+
+The GPU then applies the following operations to one INT32 score:
+\[
+\begin{array}{rcll}
+1. & t &\leftarrow
+   \lowthirtytwo(SA+B)
+   & \text{\rm [\texttt{IMAD}]}\\[4pt]
+2. & r &\leftarrow
+   t\gg_{\mathrm{arith}}F
+   & \text{\rm [\texttt{SHF.R.S32.HI}]}\\[4pt]
+3. & q &\leftarrow
+   \sat{\mathrm{u8}}(r)
+   & \text{\rm [\texttt{I2IP.U8.S32.SAT}]}\\[4pt]
+4. &   &\mathbf{return}\ q. &
+\end{array}
+\]
+
+Ignoring fixed-point coefficient error and assuming that the 32-bit
+multiply-add does not wrap,
+\[
+\frac{SA+B-2^{F-1}}{2^F}
+\approx
+8z+56+\beta.
+\]
+Thus the half-unit $2^{F-1}$ followed by the arithmetic shift performs
+integer rounding:
+\[
+r
+=
+\left\lfloor
+\frac{SA+B}{2^F}
+\right\rfloor
+\approx
+\RNE(8z+56+\beta),
+\]
+for non-tie values.  The exact shift implementation uses half-up
+rounding at ties.  Finally, \texttt{I2IP} directly saturates the result
+to one unsigned byte.
+
+\end{document}