In this section a roofline model~\cite{williams2009} will be created for the Intel\textregistered{} Core\texttrademark{} i5-4210U. In \prettyref{sec:peak} the theoretical floating-point peak performance of the CPU is calculated. \prettyref{sec:memory} then shows memory bandwidth measurements gathered with NUMA-STREAM~\cite{berstrom}. These ingredients are put together into the roofline model which is constructed in \prettyref{sec:model}.

\subsection{Theoretical Peak Performance}
\label{sec:peak}
The CPU under test was a Intel\textregistered{} Core\texttrademark{} i5-4210U. \prettyref{tbl:spec-4210} shows the relevant specifications for this processor according to~\textcite{ark4210}.

\begin{table}[h!]
  \centering
  \begin{tabular}{ll}
    \toprule
    Specification             & Value                \\
    \midrule
    \# of Cores               & 2                    \\
    \# of Threads             & 4                    \\
    Microarchitecture         & Haswell              \\
    Max Turbo Frequency       & 2.7 GHz              \\
    Processor Base Frequency  & 1.7 GHz              \\
    Instruction Set Extension & SSE 4.1/4.2, AVX 2.0 \\
    \bottomrule
  \end{tabular}
  \caption{Relevant processor specifications}
  \label{tbl:spec-4210}
\end{table}

According to~\textcite[5-2 Vol.1]{intel2016} the 4th generation Intel Core processors provide FMA (Fused Multiply-Add) units and AVX (Advanced Vector Extension). Whereas AVX can be the main driver for floating-point peak performance, the peak in this case is mainly determined by the FMA unit.

In general an FMA unit is capable of multiple floating-point (FP) operations during a single cycle. This is directly backed by the hardware (operations are \emph{``fused''} together). Specifically the FMA unit of a Haswell processor is capable of ``[...] 256-bit floating-point instructions to perform computation on 256-bit vectors''~\cite[5-28 Vol.1]{intel2016}.

Since even a DP (double-precision) FP element has only 64-bit, 256-bit would be obviously overprovisioned. But the FMA instructions do not just take scalars as arguments. Instead up to 4 DP FP elements can be packed together in a vector and operations are conducted pairwise. An example mulitply-add instruction is given in \cite{intelvfmadd132pd}.

Unfortunately no definite source could be found but according to \textcite{shimpi2012} the Haswell architecture is built with 2 FMA units per core. Taking all together we get:

\begin{enumerate}
\item Two operations are conducted at once (``fused'') and up to four DP FP elements can be packed into the argument vectors. At optimal untilization the FMA unit therefore provides $2*4 = 8 ~\text{DP FLOPs}$ each cycle. 
\item Two cores each with two FMAs can then calculate $2 * 2 * 8 = 32 ~\text{DP FLOPs}$
\end{enumerate}

\noindent{}At maximum turbo frequency the processor therefore has a theoretical peak performance of $32*2.7 = 86.4 ~\text{GFLOP/s}$. At base frequency it is capable of $32*1.7 = 54.4 ~\text{GFLOP/s}$.

\subsection{Memory Bandwidth}
\label{sec:memory}
To benchmark the memory bandwidth NUMA-STREAM~\cite{berstrom} was used. The binary ran on a Fedora 23 system with kernel \verb|4.5.7-200.fc23.x86_64 x86_64| in \verb|multi-user.target| to turn off as many distractors as possible. Compilation was done with gcc and the following options: \texttt{-O3 -std=c99 -fopenmp -lnuma -DN=80000000 -DNTIMES=100}.


Again the details of the processor architecture offer a bit of a challenge. The i5-4210U is hyper threaded meaning it provides 4 hardware threads on 2 physical cores. It is not immediately obvious how many threads NUMA-STREAM should be configured with. For this test both configurations\footnote{plus two configurations with 8 and 1 threads respectively for cross checking} were tested and the best one was chosen. The results for NUMA-STREAM configured with two threads are listed in~\prettyref{lst:numa-stream-results}. Prefixes are given in metric scale, i.e. $M = 10^6$ not $2^{20}$.The highest achieved rate was $10608~\text{MB/s}$ with the triad function. The triad function is the most demanding kernel of NUMA-STREAM defined at \cite{bergstrom2} as \lstinline[language=C]|a[j] = b[j]+scalar*c[j]|. All other tested configurations had worse results for all 4 kernels although with at most $300~\text{MB/s}$ difference. 

\bigskip
\begin{lstlisting}[caption={NUMA-STREAM results for two threads}, label=lst:numa-stream-results, numbers=none]
Function      Rate (MB/s)   Avg time     Min time     Max time
Copy:        9373.3846       0.1368       0.1366       0.1390
Scale:       9414.1304       0.1361       0.1360       0.1381
Add:        10614.6002       0.1812       0.1809       0.1835
Triad:      10607.7910       0.1813       0.1810       0.1834
\end{lstlisting}
\bigskip

\subsection{Graph}
\label{sec:model}

The graph of the roofline model is defined by~\cite{williams2009}:
\begin{verbatim}
  Attainable GFLOP/s = Min(Peak FLOP, 
                           Peak Memory Bandwidth*Operational Intensity)
\end{verbatim}

\noindent{}The resulting graph for the values obtained in~\prettyref{sec:peak} and~\prettyref{sec:memory} can be seen in \prettyref{fig:roofline}.

\begin{figure}
  \begin{adjustbox}{center}
    \includegraphics[width=0.8\linewidth]{res/rooftop}
  \end{adjustbox}
  \caption{Roofline graph from the values obtained in~\prettyref{sec:peak} and~\prettyref{sec:memory}}
  \label{fig:roofline}
\end{figure}


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