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report (roofline) begonnen, kernels implementiert, komische ergebnisse - irgendwo wird noch immer was wegoptimiert glaub ich

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Armin Friedl 2016-06-19 20:30:49 +02:00
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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.
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\subsection{Theoretical Peak Performance}
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
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 \\
\bottomrule
\end{tabular}
\caption{Intel\textregistered{} Core\texttrademark{} i5-4210U processor specifications~\cite{ark4210}}
\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.
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.
At max frequency the processor is therefore capable of a theoretical peak performance of $32*2.7 = 86.4$ GFLOP/s.
\printbibliography
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[0] Config.pm:318> INFO - This is Biber 1.8
[0] Config.pm:321> INFO - Logfile is 'report.blg'
[58] biber:275> INFO - === Sun Jun 19, 2016, 09:41:00
[58] Biber.pm:333> INFO - Reading 'report.bcf'
[134] Biber.pm:630> INFO - Found 3 citekeys in bib section 0
[147] Biber.pm:3053> INFO - Processing section 0
[164] Biber.pm:3190> INFO - Looking for bibtex format file 'roofline.bib' for section 0
[165] bibtex.pm:937> INFO - Decoding LaTeX character macros into UTF-8
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\documentclass[a4paper, DIV=12]{scrartcl}
\usepackage[english]{babel}
\usepackage[utf8]{inputenc}
\usepackage[dvipsnames]{xcolor}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{stmaryrd}
\usepackage{graphicx}
\usepackage{pdflscape}
\usepackage{listingsutf8}
\usepackage{spverbatim}
\usepackage{placeins}
\usepackage{lmodern}
%\usepackage{helvet}
\usepackage{booktabs}
\usepackage[T1]{fontenc}
\usepackage{microtype}
\usepackage{framed}
\usepackage[colorlinks=true,
linkcolor=blue,
urlcolor=blue,
breaklinks=true,
citecolor=blue]{hyperref}
\usepackage{prettyref}
\usepackage{lastpage}
\usepackage{subcaption}
\usepackage{tabularx}
\usepackage{adjustbox}
\usepackage{pdfpages}
\usepackage{xspace}
\usepackage[inline]{enumitem}
\usepackage[abbreviate=false,maxbibnames=99,backend=biber]{biblatex}
\usepackage{textcomp}
\bibliography{roofline}
\setkomafont{disposition}{\normalfont\bfseries}
\setlist[itemize]{itemsep=0.1em}
\setlist[enumerate]{itemsep=0.1em}
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\renewcommand{\arraystretch}{1.2}
\newcommand\bigforall{\mbox{\Large $\mathsurround0pt\forall$}}
\everymath{\displaystyle}
\lstset{ %
backgroundcolor=\color{white}, % choose the background color; you must add \usepackage{color} or
basicstyle=\ttfamily, % the size of the fonts that are used for the code
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tabsize=2, % sets default tabsize to 2 spaces
title=\lstname, % show the filename of files included with \lstinputlisting; also try caption instead of title
emph=[3]{int:,array,set,of,int,if,then,else,constraint,var,union,endif,function,where,in,div,predicate,let,opt,full,format,def,for,True,False,return,or},
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\begin{document}
\subject{High Performance Computing}
\title{Roofline}
\subtitle{Project 3}
\author{Johannes Winklehner\\1226104 \and Armin Friedl\\1053597}
\date{\today}
\maketitle
\begin{abstract}
A \emph{roofline model} for a multicore-processor is obtained by calcuating the theoretical peak performance of the processor and benchmarking the peak memory bandwith. Two artificial computational kernels with arithmetic intensities of $\frac{1}{16}$ GFLOPs/Byte and $8$ GFLOPs/Byte are devised. The performance of the two kernels is then compared to the theoretical calculations in the roofline model.
\end{abstract}
\tableofcontents
\newpage
\section{Introduction}
\label{sec:introduction}
\input{inputs/introduction.tex}
\FloatBarrier
\section{Roofline Model}
\label{sec:roofline}
\input{inputs/roofline.tex}
\FloatBarrier
\section{Kernels}
\label{sec:kernels}
\input{inputs/kernels.tex}
\end{document}
%%% Local Variables:
%%% mode: latex
%%% TeX-master: t
%%% End:

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\defcounter {refsection}{0}\relax
\contentsline {section}{\numberline {2}Roofline Model}{2}{section.2}
\defcounter {refsection}{0}\relax
\contentsline {subsection}{\numberline {2.1}Theoretical Peak Performance}{2}{subsection.2.1}
\defcounter {refsection}{0}\relax
\contentsline {section}{\numberline {3}Kernels}{2}{section.3}

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% This file was created with JabRef 2.10.
% Encoding: UTF-8
@Electronic{intel2016,
Title = {Intel® 64 and IA-32 Architectures Software Developer’s Manual},
Author = {Intel},
Url = {https://www-ssl.intel.com/content/dam/www/public/us/en/documents/manuals/64-ia-32-architectures-software-developer-manual-325462.pdf},
Year = {2016},
Month = {April},
Organization = {Intel},
Subtitle = {Combined Volumes: 1, 2A, 2B, 2C, 3A, 3B, 3C and 3D},
Owner = {armin},
Timestamp = {2016.06.19}
}
@Online{ark4210,
Title = {Intel® Core™ i5-4210U Processor Specifications},
Author = {{Intel Ark}},
Url = {http://ark.intel.com/products/81016/},
Urldate = {2016-06-19},
Owner = {armin},
Timestamp = {2016.06.19}
}
@Online{shimpi2012,
Title = {Haswell's Wide Execution Engine},
Author = {Anand Lal Shimpi},
Date = {2012-10-05},
Url = {http://www.anandtech.com/show/6355/intels-haswell-architecture/8},
Urldate = {2016-06-19},
Owner = {armin},
Timestamp = {2016.06.19}
}
@Article{williams2009,
Title = {Roofline: an insightful visual performance model for multicore architectures},
Author = {Williams, Samuel and Waterman, Andrew and Patterson, David},
Year = {2009},
Number = {4},
Pages = {65--76},
Volume = {52},
Journal = {Communications of the ACM},
Owner = {armin},
Publisher = {ACM},
Timestamp = {2016.06.17}
}

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all: roofline aikern.a
roofline: roofline.c aikern.a
gcc -Wall -Wextra -O3 -std=c99 -fopenmp $^ -o $@
aikern.a: aikern.c aikern.h
gcc -O3 -c -o aikern.o aikern.c
ar rcs aikern.a aikern.o
clean:
rm -f roofline
rm -f *.o
rm -f *.a
rm -f *.so

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# include <stdlib.h>
void kernel_1_16_simple(double* a, double* b, double* c, size_t size)
{
// volatile to prevent compiler from optimizing this away
// register to advise compiler to put this in register
volatile double tmp = 0.1;
#pragma omp parallel for
for(size_t i=0; i<size; i++){
/* COMM: 2 reads = 16 bytes, COMP: 1 FLOP -> AI = 1/16 */
tmp = a[i] * b[i];
}
}
void kernel_1_16_fuseaware(double* a, double* b, double* c, size_t size)
{
#pragma omp parallel for
for(size_t i=0; i<size; i++){
/* COMM: 3 reads, 1 write = 32 bytes, COMP: 2 FLOP -> AI = 2/32 = 1/16 */
a[i] = a[i] * b[i] + c[i];
}
}
void kernel_8_1_simple(double* a, double* b, double* c, size_t size)
{
// volatile to prevent compiler from optimizing this away
// register to advise compiler to put this in register
volatile register double tmp = 0.1;
#pragma omp parallel for
for(size_t i=0; i<size; i++){
/* COMM: 1 read, COMP: 8 FLOP -> AI = 8 */
tmp = a[i] * a[i] * a[i] * a[i] *
a[i] * a[i] * a[i] * a[i];
}
}
void kernel_8_1_fuseaware(double* a, double* b, double* c, size_t size)
{
// volatile to prevent compiler from optimizing this away
// register to advise compiler to put this in register
register volatile double tmp = 0.1;
#pragma omp parallel for
for(size_t i=0; i<size; i++){
/* COMM: 1 read, COMP: 8 FLOP -> AI = 8 */
tmp = a[i];
tmp = tmp * tmp + tmp;
tmp = tmp * tmp + tmp;
tmp = tmp * tmp + tmp;
tmp = tmp * tmp + tmp;
}
}
void kernel_1_8_vo(double* a, double* b, double* c, size_t size)
{
volatile double tmp=0.0;
for(size_t i=0; i<size; i++) {
tmp = a[i] * a[i];
}
}

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void kernel_1_16_simple(double* a, double* b, double* c, size_t size);
void kernel_1_16_fuseaware(double* a, double* b, double* c, size_t size);
void kernel_8_1_simple(double* a, double* b, double* c, size_t size);
void kernel_8_1_fuseaware(double* a, double* b, double* c, size_t size);
void kernel_1_8_vo(double* a, double* b, double* c, size_t size);

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# include <stdlib.h>
# include <stdio.h>
# include <unistd.h>
# include <ctype.h>
# include <sys/time.h>
# include <errno.h>
# include <string.h>
# include <stdint.h>
# include <getopt.h>
# include <stdarg.h>
# include <limits.h>
# include "aikern.h"
/* === Macros === */
#ifdef ENDEBUG
#define DEBUG(...) do { fprintf(stderr, __VA_ARGS__); fprintf(stderr, "\n"); } while(0)
#else
#define DEBUG(...)
#endif
/* === Constants === */
/* === Global Variables === */
char* prog_name;
/* === Prototypes === */
/**
* @brief print usage message
*/
static void usage(void);
/**
* @brief terminate program on program error
* @param msg additional message to print
* @param ret exit value
*/
static void bail_out(char* fmt, ...);
/**
* @brief converts the argument to size_t if possible.
* bails out on error.
* @param oparg the argument to convert
*/
static size_t get_size(char* oparg);
/**
* @brief converts the argument to int if possible.
* bails out on error.
* @param oparg the argument to convert
*/
static int get_int(char* oparg);
/**
* @brief microseconds since epoch
*/
static double mysecond(void);
/**
* @brief a simple test kernel with ai of 1/16
*/
static void testkern(double* a, double* b, double* c, size_t size);
int main(int argc, char* argv[]) {
prog_name = argv[0];
int opt;
char *size_arg = NULL;
char *runs_arg = NULL;
while((opt = getopt(argc, argv, "s:r:")) != -1)
{
switch(opt)
{
case 's':
size_arg = optarg;
break;
case 'r':
runs_arg = optarg;
break;
case '?':
usage();
default:
usage();
}
}
if(optind < argc)
{
for (int index = optind; index < argc; index++)
bail_out ("Non-option argument %s\n", argv[index]);
usage();
}
if(size_arg == NULL || runs_arg == NULL)
usage();
size_t size = get_size(size_arg);
int runs = get_int(runs_arg);
printf("Will run with array sizes of %zu\n", size);
printf("Will calculate min, max, avg for %d runs\n", runs);
/* Make this volatile so that nothing is optimized away here */
double* a = malloc(sizeof(double)*(size));
double* b = malloc(sizeof(double)*(size));
double* c = malloc(sizeof(double)*(size));
if(a==NULL || b==NULL || c == NULL)
bail_out("One of the mallocs failed\n. a = %p, b=%p, c=%p", a, b, c);
printf("Allocated 3 arrays\n");
printf("Filling arrays with dummy values\n");
#pragma omp parallel for
for (size_t j=0; j<size; j++)
{
a[j] = 1.0;
b[j] = 2.0;
c[j] = 3.0;
}
printf("Warming up cache\n");
double t = mysecond();
testkern(a,b,c, size);
t = mysecond() - t;
printf("Cache warming took %.4f microseconds = %.4f seconds (with test AI of 1/16 FLOPs/Byte)\n", (t*1.0E6), t);
/*
TESTS!!
*/
printf("1/16 simple\n");
t = mysecond();
kernel_1_16_simple(a,b,c, size);
t = mysecond() - t;
printf("Cache warming took %.4f microseconds = %.4f seconds (with test AI of 1/16 FLOPs/Byte)\n", (t*1.0E6), t);
printf("1/16 fuseaware\n");
t = mysecond();
kernel_1_16_fuseaware(a,b,c, size);
t = mysecond() - t;
printf("Cache warming took %.4f microseconds = %.4f seconds (with test AI of 1/16 FLOPs/Byte)\n", (t*1.0E6), t);
printf("8 simple\n");
t = mysecond();
kernel_8_1_simple(a,b,c, size);
t = mysecond() - t;
printf("Cache warming took %.4f microseconds = %.4f seconds (with test AI of 1/16 FLOPs/Byte)\n", (t*1.0E6), t);
printf("8 fuseaware\n");
t = mysecond();
kernel_8_1_fuseaware(a,b,c, size);
t = mysecond() - t;
printf("Cache warming took %.4f microseconds = %.4f seconds (with test AI of 1/16 FLOPs/Byte)\n", (t*1.0E6), t);
printf("1/8 vo\n");
t = mysecond();
kernel_1_8_vo(a,b,c, size);
t = mysecond() - t;
printf("Cache warming took %.4f microseconds = %.4f seconds (with test AI of 1/16 FLOPs/Byte)\n", (t*1.0E6), t);
exit(EXIT_SUCCESS);
}
static void testkern(double* a, double* b, double* c, size_t size)
{
#pragma omp parallel for
for (size_t j = 0; j < size; j++)
{
/* 3*8 Bytes read + 3*8 Bytes write, 3 FLOPs -> AI = 3/(2*3*8) = 1/16 */
a[j] = 2.0E0 * a[j];
b[j] = 2.0E0 * b[j];
c[j] = 2.0E0 * c[j];
}
}
/* === Helper Functions === */
static double mysecond(void)
{
struct timeval tp;
int i;
i = gettimeofday(&tp,NULL);
if(i != 0)
bail_out("Time measurement impossible. gettimeofday error");
return ( (double) tp.tv_sec + (double) tp.tv_usec * 1.e-6 );
}
static size_t get_size(char *oparg)
{
long long int llsize = strtoll(oparg, NULL, 10);
if(llsize <= 0)
usage();
unsigned long long int u_llsize = (unsigned long long int) llsize;
if(u_llsize > SIZE_MAX)
bail_out("Only size between 1 to %zu allowed.", SIZE_MAX);
return (size_t) llsize;
}
static int get_int(char *oparg)
{
long long int llsize = strtoll(oparg, NULL, 10);
if(llsize <= 0)
usage();
unsigned long long int u_llsize = (unsigned long long int) llsize;
if(u_llsize > INT_MAX)
bail_out("Only size between 1 to %d allowed.", INT_MAX);
return (int) llsize;
}
static void usage()
{
fprintf(stderr, "USAGE: ./roofline -s <size> -s <runs> \n");
bail_out(NULL);
}
static void bail_out(char* fmt, ...)
{
if(fmt != NULL)
{
char msgbuf[150];
va_list vl;
va_start(vl, fmt);
if(vsnprintf(msgbuf, sizeof(msgbuf), fmt, vl) < 0)
msgbuf[0] = '\0';
va_end( vl);
if(strlen(msgbuf) > 0)
(void)fprintf(stderr, "%s: %s \n", prog_name, msgbuf);
}
if(errno != 0)
(void)fprintf(stderr, "%s: %s\n", prog_name, strerror(errno));
exit(EXIT_FAILURE);
}