diff --git a/plot/plot.py b/plot/plot.py new file mode 100644 index 0000000..850ec95 --- /dev/null +++ b/plot/plot.py @@ -0,0 +1,67 @@ +import pandas as pd +import numpy as np +import matplotlib.pyplot as plt +import matplotlib +from matplotlib.ticker import ScalarFormatter, FormatStrFormatter + + +matplotlib.style.use('ggplot') + +# Roofline Plot + +#log x +i = 32 +xlbl = [] +while i > 1: + xlbl.append("1/"+repr(i)) + i //= 2 +xlbl.append(repr(i)) #1 +i += 1 +while i<=64: + xlbl.append(repr(i)) + i *= 2 + +# memory +values = [] +bandwidth = 10.6 +peak = 86.4 +ymem = [] +ypeak = [] + +for i in np.arange(0,64,0.1): + if bandwidth*i < peak: + values.append(bandwidth*i) + else: + values.append(peak) + +i=1/32 +while i<=64: + if bandwidth*i < peak and bandwidth*i*2 < peak: + ymem.append(bandwidth*i) + ypeak.append(None) + elif bandwidth*i < peak and bandwidth*i*2 > peak: + ypeak.append(peak) + ymem.append(bandwidth*i) + else: + ypeak.append(peak) + ymem.append(None) + i*=2 + +#plot data +#data = pd.Series(data=values, name='Peak Memory Bandwidth', index=np.arange(0,64,0.1)) + +data = {'Peak Memory Bandwidth': pd.Series(ymem, index=xlbl), 'Peak Floating-Point Performance': pd.Series(ypeak, index=xlbl)} +df = pd.DataFrame(data) + +ax = df.plot() +ax.set_xlabel("Operational Itensity (FLOP/Byte)") +ax.set_ylabel("Attainable GFLOP/s") +ax.set_yscale('log', basey=2) +ax.yaxis.set_major_formatter(FormatStrFormatter('%.0f')) + +print(repr(data)) + + + + +plt.show() diff --git a/roofline/report/inputs/discussion.tex b/roofline/report/inputs/discussion.tex index 916b95c..0a19393 100644 --- a/roofline/report/inputs/discussion.tex +++ b/roofline/report/inputs/discussion.tex @@ -1,5 +1,7 @@ According to the definition used the arithmetic intensity is measured by operations per byte. This might not be adequat for haswell processors (and later). Due to the fused multiply-add\footnote{although called multiply-add there are 36 different slightly instructions} extension two floating point operations can be performed with a single instruction. +- worse results for 4 threads @ NUMA-STREAM not necessarily expected + %%% Local Variables: %%% mode: latex %%% TeX-master: "../report" diff --git a/roofline/report/inputs/roofline.tex b/roofline/report/inputs/roofline.tex index 59e0222..b94d152 100644 --- a/roofline/report/inputs/roofline.tex +++ b/roofline/report/inputs/roofline.tex @@ -1,32 +1,78 @@ +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} -The first 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}. +\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 \\ + Specification & Value \\ \midrule - Instruction Set Extension & SSE4.1/4.2, AVX 2.0 \\ - \# of Cores & 2 \\ - Processor Base Frequency & 1.7 GHz \\ - Max Turbo Frequency & 2.7 GHz \\ - Microarchitecture & Haswell \\ + \# 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{Intel\textregistered{} Core\texttrademark{} i5-4210U processor specifications~\cite{ark4210}} + \caption{Relevant processor specifications} \label{tbl:spec-4210} \end{table} -The 4th generation Intel Core processors provide FMA\footnote{Fused Multiply Add} and AVX\footnote{Advanced Vector Extension} extensions~\cite[5-2 Vol.1]{intel2016}. An FMA unit is capable of ``[...] 256-bit floating-point instructions to perform computation on -256-bit vectors''~\cite[5-28 Vol.1]{intel2016}. Therefore it can execute 2 (multiply-add) times 4 double-precision floating-point instructions each cycle. This results in 8 DP FLOPs per cycle. +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. -Unfortunately no definite source could be found but according to \textcite{shimpi2012} the Haswell architecture has 2 FMA units, equalling to $2 * 8 = 16$ DP FLOPs per core. Furthermore there are 2 cores in a Core i5 processor. Taken together this results in $16 * 2 = 32$ DP FLOPs per cycle for both cores. +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}. -At max frequency the processor is therefore capable of a theoretical peak performance of $32*2.7 = 86.4$ GFLOP/s. +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{intel2}. + +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}. -\printbibliography +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} + %%% Local Variables: %%% mode: latex diff --git a/roofline/report/report.aux b/roofline/report/report.aux index f9d36a2..e5479fb 100644 --- a/roofline/report/report.aux +++ b/roofline/report/report.aux @@ -25,20 +25,32 @@ \@writefile{toc}{\defcounter {refsection}{0}\relax }\@writefile{toc}{\select@language{english}} \@writefile{lof}{\defcounter {refsection}{0}\relax }\@writefile{lof}{\select@language{english}} \@writefile{lot}{\defcounter {refsection}{0}\relax }\@writefile{lot}{\select@language{english}} +\abx@aux@cite{williams2009} +\abx@aux@cite{berstrom} \abx@aux@cite{ark4210} \abx@aux@cite{intel2016} +\abx@aux@cite{intel2} \abx@aux@cite{shimpi2012} -\abx@aux@cite{dukhan} \@writefile{toc}{\defcounter {refsection}{0}\relax }\@writefile{toc}{\contentsline {section}{\numberline {1}Introduction}{2}{section.1}} \newlabel{sec:introduction}{{1}{2}{Introduction}{section.1}{}} 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+\@writefile{toc}{\defcounter {refsection}{0}\relax }\@writefile{toc}{\contentsline {subsection}{\numberline {2.3}Graph}{3}{subsection.2.3}} +\newlabel{sec:model}{{2.3}{3}{Graph}{subsection.2.3}{}} +\@writefile{lof}{\defcounter {refsection}{0}\relax }\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Roofline graph from the values obtained in~\hyperref [sec:peak]{Section~\ref *{sec:peak}} and~\hyperref [sec:memory]{Section~\ref *{sec:memory}}\relax }}{4}{figure.caption.3}} +\newlabel{fig:roofline}{{1}{4}{Roofline graph from the values obtained in~\prettyref {sec:peak} and~\prettyref {sec:memory}\relax }{figure.caption.3}{}} +\@writefile{toc}{\defcounter {refsection}{0}\relax }\@writefile{toc}{\contentsline {section}{\numberline {3}Kernels}{4}{section.3}} +\newlabel{sec:kernels}{{3}{4}{Kernels}{section.3}{}} +\newlabel{LastPage}{{}{4}{}{page.4}{}} +\xdef\lastpage@lastpage{4} +\xdef\lastpage@lastpageHy{4} diff --git a/roofline/report/report.bbl b/roofline/report/report.bbl index 95a4690..23918dd 100644 --- a/roofline/report/report.bbl +++ b/roofline/report/report.bbl @@ -19,23 +19,42 @@ \refsection{0} \sortlist{entry}{nty} - \entry{dukhan}{online}{} + \entry{berstrom}{online}{} \name{labelname}{1}{}{% - {{hash=ed84b6085046bdd016a6224a8de0961a}{Dukhan}{D\bibinitperiod}{Marat}{M\bibinitperiod}{}{}{}{}}% + {{hash=f138f2b760ef7bf1265cad7357981ef4}{Bergstrom}{B\bibinitperiod}{Lars}{L\bibinitperiod}{}{}{}{}}% } \name{author}{1}{}{% - {{hash=ed84b6085046bdd016a6224a8de0961a}{Dukhan}{D\bibinitperiod}{Marat}{M\bibinitperiod}{}{}{}{}}% + {{hash=f138f2b760ef7bf1265cad7357981ef4}{Bergstrom}{B\bibinitperiod}{Lars}{L\bibinitperiod}{}{}{}{}}% } - \strng{namehash}{ed84b6085046bdd016a6224a8de0961a} - \strng{fullhash}{ed84b6085046bdd016a6224a8de0961a} - \field{sortinit}{D} - \field{labeltitle}{FLOPS per cycle for sandy-bridge and haswell SSE2/AVX/AVX2} - \field{title}{FLOPS per cycle for sandy-bridge and haswell SSE2/AVX/AVX2} + \strng{namehash}{f138f2b760ef7bf1265cad7357981ef4} + \strng{fullhash}{f138f2b760ef7bf1265cad7357981ef4} + \field{sortinit}{B} + \field{labeltitle}{NUMA-STREAM} + \field{title}{NUMA-STREAM} \field{urlday}{20} \field{urlmonth}{06} \field{urlyear}{2016} \verb{url} - \verb http://stackoverflow.com/a/15657772 + \verb https://github.com/larsbergstrom/NUMA-STREAM + \endverb + \endentry + \entry{bergstrom2}{online}{} + \name{labelname}{1}{}{% + {{hash=f138f2b760ef7bf1265cad7357981ef4}{Bergstrom}{B\bibinitperiod}{Lars}{L\bibinitperiod}{}{}{}{}}% + } + \name{author}{1}{}{% + {{hash=f138f2b760ef7bf1265cad7357981ef4}{Bergstrom}{B\bibinitperiod}{Lars}{L\bibinitperiod}{}{}{}{}}% + } + \strng{namehash}{f138f2b760ef7bf1265cad7357981ef4} + \strng{fullhash}{f138f2b760ef7bf1265cad7357981ef4} + \field{sortinit}{B} + \field{labeltitle}{stream.c} + \field{title}{stream.c} + \field{urlday}{20} + \field{urlmonth}{06} + \field{urlyear}{2016} + \verb{url} + \verb https://github.com/larsbergstrom/NUMA-STREAM/blob/e5aa9ca4a77623ff6f1c2d5daa7995565b944506/stream.c#L286 \endverb \endentry \entry{intel2016}{online}{} @@ -60,6 +79,25 @@ \verb https://www-ssl.intel.com/content/dam/www/public/us/en/documents/manuals/64-ia-32-architectures-software-developer-manual-325462.pdf \endverb \endentry + \entry{intel2}{online}{} + \name{labelname}{1}{}{% + {{hash=ff97a9fdede09eaf6e1c8ec9f6a61dd5}{{Intel}}{I\bibinitperiod}{}{}{}{}{}{}}% + } + \name{author}{1}{}{% + {{hash=ff97a9fdede09eaf6e1c8ec9f6a61dd5}{{Intel}}{I\bibinitperiod}{}{}{}{}{}{}}% + } + \strng{namehash}{ff97a9fdede09eaf6e1c8ec9f6a61dd5} + \strng{fullhash}{ff97a9fdede09eaf6e1c8ec9f6a61dd5} + \field{sortinit}{I} + \field{labeltitle}{Intel Intrinsics Guide} + \field{title}{Intel Intrinsics Guide} + \field{urlday}{19} + \field{urlmonth}{06} + \field{urlyear}{2016} + \verb{url} + \verb https://software.intel.com/sites/landingpage/IntrinsicsGuide/#techs=AVX2,FMA&text=madd&expand=2365 + \endverb + \endentry \entry{ark4210}{online}{} \name{labelname}{1}{}{% {{hash=7e5a7615b54a4cc38565e3ac9203a5f7}{{Intel\bibnamedelimb Ark}}{I\bibinitperiod}{}{}{}{}{}{}}% @@ -101,6 +139,31 @@ \verb http://www.anandtech.com/show/6355/intels-haswell-architecture/8 \endverb \endentry + \entry{williams2009}{article}{} + \name{labelname}{3}{}{% + {{hash=4fb8fbc5c4bfbbc94f80321c3336b71b}{Williams}{W\bibinitperiod}{Samuel}{S\bibinitperiod}{}{}{}{}}% + {{hash=1cdbf6b1b2783ec44ba9770f07e6528f}{Waterman}{W\bibinitperiod}{Andrew}{A\bibinitperiod}{}{}{}{}}% + {{hash=64f58317379ddd1a24e0e6cba85ae78c}{Patterson}{P\bibinitperiod}{David}{D\bibinitperiod}{}{}{}{}}% + } + \name{author}{3}{}{% + {{hash=4fb8fbc5c4bfbbc94f80321c3336b71b}{Williams}{W\bibinitperiod}{Samuel}{S\bibinitperiod}{}{}{}{}}% + {{hash=1cdbf6b1b2783ec44ba9770f07e6528f}{Waterman}{W\bibinitperiod}{Andrew}{A\bibinitperiod}{}{}{}{}}% + {{hash=64f58317379ddd1a24e0e6cba85ae78c}{Patterson}{P\bibinitperiod}{David}{D\bibinitperiod}{}{}{}{}}% + } + \list{publisher}{1}{% + {ACM}% + } + \strng{namehash}{8fbf7ed9bc5abefeb2fdf60a347400ae} + \strng{fullhash}{8fbf7ed9bc5abefeb2fdf60a347400ae} + \field{sortinit}{W} + \field{labeltitle}{Roofline: an insightful visual performance model for multicore architectures} + \field{journaltitle}{Communications of the ACM} + \field{number}{4} + \field{title}{Roofline: an insightful visual performance model for multicore architectures} + \field{volume}{52} + \field{year}{2009} + \field{pages}{65\bibrangedash 76} + \endentry \endsortlist \endrefsection \endinput diff --git a/roofline/report/report.bcf b/roofline/report/report.bcf index 7375c1c..c7ff053 100644 --- a/roofline/report/report.bcf +++ b/roofline/report/report.bcf @@ -1964,15 +1964,19 @@ roofline.bib - ark4210 - ark4210 + williams2009 + berstrom ark4210 ark4210 intel2016 intel2016 - shimpi2012 - shimpi2012 - dukhan + intel2016 + intel2 + shimpi2012 + shimpi2012 + berstrom + bergstrom2 + williams2009 diff --git a/roofline/report/report.blg b/roofline/report/report.blg index 7d74509..93cbf64 100644 --- a/roofline/report/report.blg +++ b/roofline/report/report.blg @@ -1,14 +1,14 @@ [0] Config.pm:318> INFO - This is Biber 1.8 [0] Config.pm:321> INFO - Logfile is 'report.blg' -[54] biber:275> INFO - === Mon Jun 20, 2016, 01:10:36 -[54] Biber.pm:333> INFO - Reading 'report.bcf' -[123] Biber.pm:630> INFO - Found 4 citekeys in bib section 0 -[134] Biber.pm:3053> INFO - Processing section 0 -[150] Biber.pm:3190> INFO - Looking for bibtex format file 'roofline.bib' for section 0 -[151] bibtex.pm:937> INFO - Decoding LaTeX character macros into UTF-8 -[152] bibtex.pm:812> INFO - Found BibTeX data source 'roofline.bib' -[180] Biber.pm:2939> INFO - Overriding locale 'en_US.UTF-8' default tailoring 'variable = shifted' with 'variable = non-ignorable' -[180] Biber.pm:2945> INFO - Sorting 'entry' list 'nty' keys -[181] Biber.pm:2949> INFO - No sort tailoring available for locale 'en_US.UTF-8' -[187] bbl.pm:482> INFO - Writing 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