slides.tex

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\usepackage{subcaption}
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\usepackage[font=scriptsize,labelfont=bf]{caption}

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% Definitions

\def\R{\mathbb{R}}

\def\doublehalf{double-fp16}
\newcommand{\half}{fp16}
\newcommand{\bfloat}{bf16}
\newcommand{\tfloat}{tf32}
\newcommand{\single}{fp32}
\newcommand{\double}{fp64}
\newcommand{\tflops}{T\flops}

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% Theme configuration

\usetheme{Madrid}
\usecolortheme{whale}

\setbeamertemplate{navigation symbols}{}

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% Title page

\title[Shortened title of the talk]{Title of the Talk}

\subtitle{}

\author[Author's surname] % (optional, for multiple authors)
{Author's full name}

\institute[Univ. Leeds]
{
  {\color{blue}
    School of Computer Science,
    University of Leeds,
    Leeds, UK
  }
}

\date[March 2025]{Event name\\
  {\color{Blue} Location, Mar. 27, 2025}}

\titlegraphic{
  \begin{tikzpicture}[overlay,remember picture]
\node[left=0.2cm] at (current page.34){
    \includegraphics[width=1in]{images/uollogo}
};
\end{tikzpicture}
\includegraphics[width=1.2in]{images/bragg}
}

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% Main slides

\begin{document}
\frame{\titlepage}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Example diagram in PGFPlots with text underneath}

  \input{diagrams/top500-statistics}

  Devices counted: P100, V100, A100, H100, MI210, MI250X, MI300X, Intel Data Center GPU, from \url{https://www.top500.org}.\\~\\

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Example slide with a citation and example box}
  In the level-index representation {\color{blue} [Clenshaw and Olver, 1984]} $x \in \mathbb{R}$ is represented as
  $$
  x = \pm e^{e^{.^{.^{.^{e^{f}}}}}}
  $$
  where the process of exponentiation is done $l$ times. Here $f \in [0,1)$.\\~\\

  When $l=0$ we set $x=f$.\\~\\

  \begin{example}
    $0.5=f=0.5$ (level 0) \\
    $1=e^0$      (level 1) \\
    $1.5 \approx e^{0.4}$ (level 1) \\
    $2 \approx e^{0.69}$ (level 1) \\
    $3 \approx e^{e^{0.09}}$ (level is now 2)
  \end{example}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Example slide with aligned equations}
  Level 0 in LI is a special case that was removed in symmetric level-index system {\color{blue} [Clenshaw and Turner, 1988]}.\\~\\

  In SLI a number $x \in \mathbb{R}$ has the form
  $$
  x=s(x)\phi (\zeta)^{r(x)},
  $$
  where $s(x)=\pm 1$ is the sign, $r(x)=\pm 1$ is the reciprocal sign defined by
  \begin{align*}
    r(x)=
    \begin{cases}
      +1, & \text{if } |x| \geq 1, \\
      -1, & \text{if } |x| < 1,
    \end{cases}
  \end{align*}
  and $\phi(\zeta)$ is the standard LI generalized exponential with $\zeta=l+f$:
  \begin{align*}
  \phi(\zeta)=
  \begin{cases}
    \zeta, & \text{if } 0 \leq \zeta < 1, \\
    e^{\phi(\zeta-1)}, & \text{if } \zeta \geq 1.
  \end{cases}
  \end{align*}

  Allows to preserve relative precision of numbers below $1$.

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Example slide with alertblock box and nested bullet points}

  \begin{alertblock}{Notation}
    With sli-$k$.$p$ we refer to a SLI encoding with a $k$-bit level and a $p$-bit index. Total width of the encoding is $k+p+2$.
  \end{alertblock}

  \begin{itemize}
  \item {\color{blue} [Olver, ARITH'87]} proves \emph{closure} properties of SLI and shows that 3 bits for the level is enough for practical purposes.
  \item {\color{blue}[Olver and Turner, ARITH'87]} show table-based parallel algorithms for basic operations. Fast but big circuits.
  \item {\color{blue} [Turner, ARITH'89]}
    \begin{itemize}
    \item Simulation of sli-3.27 in Turbo Pascal;
    \item implements scalar, vector, polynomial ops;
    \item sli-integer mixed-format ops.
    \end{itemize}
  \item {\color{blue} [Kwak and Swartzlander, 1998]} show how to implement LI in hardware by using CORDIC. From $>$11K Full Adders down to $\sim$3800.
  \end{itemize}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Example slide with a picture drawn in TikZ}

  \begin{figure}
  \centering
  \footnotesize
  \begin{tikzpicture}
    \node (s) [rectangle,draw, minimum height = 1cm, minimum width = 0.5cm] {$s$};
    \node (r) [rectangle,draw, minimum height = 1cm, minimum width = 0.5cm, right=-\the\pgflinewidth of s.east] {$r$};
    \node (l) [rectangle,draw, minimum height = 1cm, minimum width = 1.5cm, right=-\the\pgflinewidth of r.east] {$l$};
    \node (f) [rectangle,draw, minimum height = 1cm, minimum width = 5.5cm, right=-\the\pgflinewidth of l.east] {$f$};
    \draw[decorate,decoration={brace}, rotate=180] ($(s)+(-0.2,0.6)$) -- ($(s)+(0.2,0.6)$)
    node[below=2pt, pos=0.5]{$1$};
    \draw[decorate,decoration={brace}, rotate=180] ($(r)+(-0.2,0.6)$) -- ($(r)+(0.2,0.6)$)
    node[below=2pt, pos=0.5]{$1$};
    \draw[decorate,decoration={brace}, rotate=180] ($(l)+(-0.6,0.6)$) -- ($(l)+(0.6,0.6)$)
    node[below=2pt, pos=0.5]{$3$};
    \draw[decorate,decoration={brace}, rotate=180] ($(f)+(-2.7,0.6)$) -- ($(f)+(2.7,0.6)$)
    node[below=2pt, pos=0.5]{$11$ bits};
  \end{tikzpicture}
\end{figure}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Example slide with three pictures drawn in TikZ}
  \begin{figure}
  \centering
  \footnotesize
  \begin{tabular}{c}
  \begin{tikzpicture}
    \node (e) [rectangle,draw, minimum height = 1cm, minimum width = 1.5cm] {$E$};
    \node (m) [rectangle,draw, minimum height = 1cm, minimum width = 1cm, right=-\the\pgflinewidth of e.east] {$M$};
    \draw[decorate,decoration={brace}, rotate=180] ($(e)+(-0.6,0.6)$) -- ($(e)+(0.6,0.6)$)
    node[below=2pt, pos=0.5]{$3$};
    \draw[decorate,decoration={brace}, rotate=180] ($(m)+(-0.4,0.6)$) -- ($(m)+(0.4,0.6)$)
    node[below=2pt, pos=0.5]{$2$ bits};
  \end{tikzpicture} \\
  \end{tabular}

  \begin{tabular}{cc}
  \begin{tikzpicture}
    \node (r) [rectangle,draw, minimum height = 1cm, minimum width = 0.5cm] {$r$};
    \node (l) [rectangle,draw, minimum height = 1cm, minimum width = 0.5cm, right=-\the\pgflinewidth of r.east] {$l$};
    \node (f) [rectangle,draw, minimum height = 1cm, minimum width = 2cm, right=-\the\pgflinewidth of l.east] {$f$};
    \draw[decorate,decoration={brace}, rotate=180] ($(r)+(-0.2,0.6)$) -- ($(r)+(0.2,0.6)$)
    node[below=2pt, pos=0.5]{$1$};
    \draw[decorate,decoration={brace}, rotate=180] ($(l)+(-0.2,0.6)$) -- ($(l)+(0.2,0.6)$)
    node[below=2pt, pos=0.5]{$1$};
    \draw[decorate,decoration={brace}, rotate=180] ($(f)+(-0.8,0.6)$) -- ($(f)+(0.8,0.6)$)
    node[below=2pt, pos=0.5]{$3$ bits};
  \end{tikzpicture} &
  \begin{tikzpicture}
    \node (r) [rectangle,draw, minimum height = 1cm, minimum width = 0.5cm] {$r$};
    \node (l) [rectangle,draw, minimum height = 1cm, minimum width = 1cm, right=-\the\pgflinewidth of r.east] {$l$};
    \node (f) [rectangle,draw, minimum height = 1cm, minimum width = 1.5cm, right=-\the\pgflinewidth of l.east] {$f$};
    \draw[decorate,decoration={brace}, rotate=180] ($(r)+(-0.2,0.6)$) -- ($(r)+(0.2,0.6)$)
    node[below=2pt, pos=0.5]{$1$};
    \draw[decorate,decoration={brace}, rotate=180] ($(l)+(-0.4,0.6)$) -- ($(l)+(0.4,0.6)$)
    node[below=2pt, pos=0.5]{$2$};
    \draw[decorate,decoration={brace}, rotate=180] ($(f)+(-0.6,0.6)$) -- ($(f)+(0.6,0.6)$)
    node[below=2pt, pos=0.5]{$2$ bits};
  \end{tikzpicture}
  \end{tabular}
\end{figure}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Example slide with a table}

{\footnotesize
\begin{table}
  \centering
  \begin{tabular}{llrr}
    \toprule
     & FP & sli-1.3 & sli-2.2 \\
    \midrule
    $00000$ & $0$ & $Zero$ & $Zero$ \\
    $00001$ & $0.0625$ & $(e^{0.125})^{-1}\approx 0.8825$ & $(e^{0.25})^{-1}\approx 0.7788$ \\
    $00010$ & $0.125$ & $(e^{0.25})^{-1}\approx 0.7788$ & $(e^{0.5})^{-1}\approx 0.6065$ \\
    $00011$ & $0.1875$ & $\sim 0.6873$ & $\sim 0.4724$ \\
    $00100$ & $0.25$ & $\sim 0.6065$ & $(e^{e^{0}})^{-1}\approx 0.3679$ \\
    $00101$ & $0.3125$ & $\sim 0.5353$ & $\sim 0.2769$ \\
    $00110$ & $0.375$ & $\sim 0.4724$ & $\sim 0.1923$ \\
    $00111$ & $0.4375$ & $\sim 0.4169$ & $\sim 0.1204$ \\
    $01000$ & $0.5$ & $(e^{e^{0}})^{-1}\approx 0.3679$ & $(e^{e^{e^{0}}})^{-1}\approx 0.06599$ \\
    $01001$ & $0.625$ & $(e^{e^{0.125}})^{-1}\approx 0.322$ &  $\sim 0.02702$ \\
    $01010$ & $0.75$ & $(e^{e^{0.25}})^{-1}\approx 0.2769$ &  $\sim 0.0055$ \\
    $01011$ & $0.875$ & $\sim 0.2334$ &  $\sim 2.4 \times 10^{-4}$ \\
    $01100$ & $1$ & $\sim 0.1923$ & $(e^{e^{e^{e^{0}}}})^{-1}\approx 2.6 \times 10^{-7}$ \\
    $01101$ & $1.25$ & $\sim 0.1544$ & $\sim 8.4 \times 10^{-17}$ \\
    $01110$ & $1.5$ & $\sim 0.1204$ & $\sim 1.7 \times 10^{-79}$\\
    $01111$ & $1.75$ & $\sim 0.0908$ & $\sim 10^{-1758}$\\
    \bottomrule
  \end{tabular}
  \label{table:toy-system-values}
\end{table}}

\end{frame}

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\begin{frame}[fragile]{Example slide with MATLAB code}

\begin{lstlisting}
>> x = sli.zeros(2,3,2,12);
>> [rows, cols] = size(x);
>> for i = 1:rows
for j = 1:cols
x(i,j) = x(i,j).set_val(2);
end
end
>> y = x*x';
>> y(1,1)
ans = 
  sli with properties:
    level_bits: 2
    index_bits: 12
          sign: 0
    reciprocal: 1
         level: 2
         index: 0.910400390625000
         value: 12.004930399103673
\end{lstlisting}

\end{frame}

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\begin{frame}{Example slide with PGFPlots diagrams}

Round $1000$ binary64 values around 0 with steps of $10^{-5}$.

\input{diagrams/representation-binary16.tex}

Take bfloat16 arithmetic and steps of $10^{-42}$.

\input{diagrams/representation-bfloat16.tex}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Example slide with equations and a diagram}

Compute the backward error of $y=Ax$ with  $x \in (0,1)^n$: $\max_i\frac{
    |\widehat{y}-y|_i}{(|A||x|)_i}$.\\~\\

  \input{diagrams/matvec-binary16.tex}

\end{frame}

\appendix
\begin{frame}[allowframebreaks]{References}

\def\newblock{}

\begin{thebibliography}{10}

\bibitem{clol84}
  {C.~W.~Clenshaw and F.~W.~Olver}
  \newblock Beyond floating point
  \newblock J. ACM., 31:2. Apr., 1984.

\bibitem{cltu88}
  {C.~W.~Clenshaw and P.~R.~Turner}
  \newblock The symmetric level-index system
  \newblock IMA J. Numer. Anal., 8:4. Oct., 1988.

\bibitem{olve87}
  {F.~W.~Olver}
  \newblock A closed computer arithmetic
  \newblock IEEE 8th Symp. Comp. Arith., May 1987.

\end{thebibliography}

\end{frame}

\end{document}