How do I achieve the theoretical maximum of 4 FLOPs per cycle?
Using Intels icc Version 11.1 on a 2.4GHz Intel Core 2 Duo I get
Macintosh:~ mackie$ icc -O3 -mssse3 -oaddmul addmul.cc && ./addmul 1000
addmul: 0.105 s, 9.525 Gflops, res=0.000000
Macintosh:~ mackie$ icc -v
Version 11.1
That is very close to the ideal 9.6 Gflops.
EDIT:
Oops, looking at the assembly code it seems that icc not only vectorized the multiplication, but also pulled the additions out of the loop. Forcing a stricter fp semantics the code is no longer vectorized:
Macintosh:~ mackie$ icc -O3 -mssse3 -oaddmul addmul.cc -fp-model precise && ./addmul 1000
addmul: 0.516 s, 1.938 Gflops, res=1.326463
EDIT2:
As requested:
Macintosh:~ mackie$ clang -O3 -mssse3 -oaddmul addmul.cc && ./addmul 1000
addmul: 0.209 s, 4.786 Gflops, res=1.326463
Macintosh:~ mackie$ clang -v
Apple clang version 3.0 (tags/Apple/clang-211.10.1) (based on LLVM 3.0svn)
Target: x86_64-apple-darwin11.2.0
Thread model: posix
The inner loop of clang's code looks like this:
.align 4, 0x90
LBB2_4: ## =>This Inner Loop Header: Depth=1
addsd %xmm2, %xmm3
addsd %xmm2, %xmm14
addsd %xmm2, %xmm5
addsd %xmm2, %xmm1
addsd %xmm2, %xmm4
mulsd %xmm2, %xmm0
mulsd %xmm2, %xmm6
mulsd %xmm2, %xmm7
mulsd %xmm2, %xmm11
mulsd %xmm2, %xmm13
incl %eax
cmpl %r14d, %eax
jl LBB2_4
EDIT3:
Finally, two suggestions: First, if you like this type of benchmarking, consider using the rdtsc instruction istead of gettimeofday(2). It is much more accurate and delivers the time in cycles, which is usually what you are interested in anyway. For gcc and friends you can define it like this:
#include <stdint.h>
static __inline__ uint64_t rdtsc(void)
{
uint64_t rval;
__asm__ volatile ("rdtsc" : "=A" (rval));
return rval;
}
Second, you should run your benchmark program several times and use the best performance only. In modern operating systems many things happen in parallel, the cpu may be in a low frequency power saving mode, etc. Running the program repeatedly gives you a result that is closer to the ideal case.
Branches can definitely keep you from sustaining peak theoretical performance. Do you see a difference if you manually do some loop-unrolling? For example, if you put 5 or 10 times as many ops per loop iteration:
for(int i=0; i<loops/5; i++) {
mul1*=mul; mul2*=mul; mul3*=mul; mul4*=mul; mul5*=mul;
sum1+=add; sum2+=add; sum3+=add; sum4+=add; sum5+=add;
mul1*=mul; mul2*=mul; mul3*=mul; mul4*=mul; mul5*=mul;
sum1+=add; sum2+=add; sum3+=add; sum4+=add; sum5+=add;
mul1*=mul; mul2*=mul; mul3*=mul; mul4*=mul; mul5*=mul;
sum1+=add; sum2+=add; sum3+=add; sum4+=add; sum5+=add;
mul1*=mul; mul2*=mul; mul3*=mul; mul4*=mul; mul5*=mul;
sum1+=add; sum2+=add; sum3+=add; sum4+=add; sum5+=add;
mul1*=mul; mul2*=mul; mul3*=mul; mul4*=mul; mul5*=mul;
sum1+=add; sum2+=add; sum3+=add; sum4+=add; sum5+=add;
}
I've done this exact task before. But it was mainly to measure power consumption and CPU temperatures. The following code (which is fairly long) achieves close to optimal on my Core i7 2600K.
The key thing to note here is the massive amount of manual loop-unrolling as well as interleaving of multiplies and adds...
The full project can be found on my GitHub: https://github.com/Mysticial/Flops
Warning:
If you decide to compile and run this, pay attention to your CPU temperatures!!!
Make sure you don't overheat it. And make sure CPU-throttling doesn't affect your results!
Furthermore, I take no responsibility for whatever damage that may result from running this code.
Notes:
- This code is optimized for x64. x86 doesn't have enough registers for this to compile well.
- This code has been tested to work well on Visual Studio 2010/2012 and GCC 4.6.
ICC 11 (Intel Compiler 11) surprisingly has trouble compiling it well. - These are for pre-FMA processors. In order to achieve peak FLOPS on Intel Haswell and AMD Bulldozer processors (and later), FMA (Fused Multiply Add) instructions will be needed. These are beyond the scope of this benchmark.
#include <emmintrin.h>
#include <omp.h>
#include <iostream>
using namespace std;
typedef unsigned long long uint64;
double test_dp_mac_SSE(double x,double y,uint64 iterations){
register __m128d r0,r1,r2,r3,r4,r5,r6,r7,r8,r9,rA,rB,rC,rD,rE,rF;
// Generate starting data.
r0 = _mm_set1_pd(x);
r1 = _mm_set1_pd(y);
r8 = _mm_set1_pd(-0.0);
r2 = _mm_xor_pd(r0,r8);
r3 = _mm_or_pd(r0,r8);
r4 = _mm_andnot_pd(r8,r0);
r5 = _mm_mul_pd(r1,_mm_set1_pd(0.37796447300922722721));
r6 = _mm_mul_pd(r1,_mm_set1_pd(0.24253562503633297352));
r7 = _mm_mul_pd(r1,_mm_set1_pd(4.1231056256176605498));
r8 = _mm_add_pd(r0,_mm_set1_pd(0.37796447300922722721));
r9 = _mm_add_pd(r1,_mm_set1_pd(0.24253562503633297352));
rA = _mm_sub_pd(r0,_mm_set1_pd(4.1231056256176605498));
rB = _mm_sub_pd(r1,_mm_set1_pd(4.1231056256176605498));
rC = _mm_set1_pd(1.4142135623730950488);
rD = _mm_set1_pd(1.7320508075688772935);
rE = _mm_set1_pd(0.57735026918962576451);
rF = _mm_set1_pd(0.70710678118654752440);
uint64 iMASK = 0x800fffffffffffffull;
__m128d MASK = _mm_set1_pd(*(double*)&iMASK);
__m128d vONE = _mm_set1_pd(1.0);
uint64 c = 0;
while (c < iterations){
size_t i = 0;
while (i < 1000){
// Here's the meat - the part that really matters.
r0 = _mm_mul_pd(r0,rC);
r1 = _mm_add_pd(r1,rD);
r2 = _mm_mul_pd(r2,rE);
r3 = _mm_sub_pd(r3,rF);
r4 = _mm_mul_pd(r4,rC);
r5 = _mm_add_pd(r5,rD);
r6 = _mm_mul_pd(r6,rE);
r7 = _mm_sub_pd(r7,rF);
r8 = _mm_mul_pd(r8,rC);
r9 = _mm_add_pd(r9,rD);
rA = _mm_mul_pd(rA,rE);
rB = _mm_sub_pd(rB,rF);
r0 = _mm_add_pd(r0,rF);
r1 = _mm_mul_pd(r1,rE);
r2 = _mm_sub_pd(r2,rD);
r3 = _mm_mul_pd(r3,rC);
r4 = _mm_add_pd(r4,rF);
r5 = _mm_mul_pd(r5,rE);
r6 = _mm_sub_pd(r6,rD);
r7 = _mm_mul_pd(r7,rC);
r8 = _mm_add_pd(r8,rF);
r9 = _mm_mul_pd(r9,rE);
rA = _mm_sub_pd(rA,rD);
rB = _mm_mul_pd(rB,rC);
r0 = _mm_mul_pd(r0,rC);
r1 = _mm_add_pd(r1,rD);
r2 = _mm_mul_pd(r2,rE);
r3 = _mm_sub_pd(r3,rF);
r4 = _mm_mul_pd(r4,rC);
r5 = _mm_add_pd(r5,rD);
r6 = _mm_mul_pd(r6,rE);
r7 = _mm_sub_pd(r7,rF);
r8 = _mm_mul_pd(r8,rC);
r9 = _mm_add_pd(r9,rD);
rA = _mm_mul_pd(rA,rE);
rB = _mm_sub_pd(rB,rF);
r0 = _mm_add_pd(r0,rF);
r1 = _mm_mul_pd(r1,rE);
r2 = _mm_sub_pd(r2,rD);
r3 = _mm_mul_pd(r3,rC);
r4 = _mm_add_pd(r4,rF);
r5 = _mm_mul_pd(r5,rE);
r6 = _mm_sub_pd(r6,rD);
r7 = _mm_mul_pd(r7,rC);
r8 = _mm_add_pd(r8,rF);
r9 = _mm_mul_pd(r9,rE);
rA = _mm_sub_pd(rA,rD);
rB = _mm_mul_pd(rB,rC);
i++;
}
// Need to renormalize to prevent denormal/overflow.
r0 = _mm_and_pd(r0,MASK);
r1 = _mm_and_pd(r1,MASK);
r2 = _mm_and_pd(r2,MASK);
r3 = _mm_and_pd(r3,MASK);
r4 = _mm_and_pd(r4,MASK);
r5 = _mm_and_pd(r5,MASK);
r6 = _mm_and_pd(r6,MASK);
r7 = _mm_and_pd(r7,MASK);
r8 = _mm_and_pd(r8,MASK);
r9 = _mm_and_pd(r9,MASK);
rA = _mm_and_pd(rA,MASK);
rB = _mm_and_pd(rB,MASK);
r0 = _mm_or_pd(r0,vONE);
r1 = _mm_or_pd(r1,vONE);
r2 = _mm_or_pd(r2,vONE);
r3 = _mm_or_pd(r3,vONE);
r4 = _mm_or_pd(r4,vONE);
r5 = _mm_or_pd(r5,vONE);
r6 = _mm_or_pd(r6,vONE);
r7 = _mm_or_pd(r7,vONE);
r8 = _mm_or_pd(r8,vONE);
r9 = _mm_or_pd(r9,vONE);
rA = _mm_or_pd(rA,vONE);
rB = _mm_or_pd(rB,vONE);
c++;
}
r0 = _mm_add_pd(r0,r1);
r2 = _mm_add_pd(r2,r3);
r4 = _mm_add_pd(r4,r5);
r6 = _mm_add_pd(r6,r7);
r8 = _mm_add_pd(r8,r9);
rA = _mm_add_pd(rA,rB);
r0 = _mm_add_pd(r0,r2);
r4 = _mm_add_pd(r4,r6);
r8 = _mm_add_pd(r8,rA);
r0 = _mm_add_pd(r0,r4);
r0 = _mm_add_pd(r0,r8);
// Prevent Dead Code Elimination
double out = 0;
__m128d temp = r0;
out += ((double*)&temp)[0];
out += ((double*)&temp)[1];
return out;
}
void test_dp_mac_SSE(int tds,uint64 iterations){
double *sum = (double*)malloc(tds * sizeof(double));
double start = omp_get_wtime();
#pragma omp parallel num_threads(tds)
{
double ret = test_dp_mac_SSE(1.1,2.1,iterations);
sum[omp_get_thread_num()] = ret;
}
double secs = omp_get_wtime() - start;
uint64 ops = 48 * 1000 * iterations * tds * 2;
cout << "Seconds = " << secs << endl;
cout << "FP Ops = " << ops << endl;
cout << "FLOPs = " << ops / secs << endl;
double out = 0;
int c = 0;
while (c < tds){
out += sum[c++];
}
cout << "sum = " << out << endl;
cout << endl;
free(sum);
}
int main(){
// (threads, iterations)
test_dp_mac_SSE(8,10000000);
system("pause");
}
Output (1 thread, 10000000 iterations) - Compiled with Visual Studio 2010 SP1 - x64 Release:
Seconds = 55.5104
FP Ops = 960000000000
FLOPs = 1.7294e+010
sum = 2.22652
The machine is a Core i7 2600K @ 4.4 GHz. Theoretical SSE peak is 4 flops * 4.4 GHz = 17.6 GFlops. This code achieves 17.3 GFlops - not bad.
Output (8 threads, 10000000 iterations) - Compiled with Visual Studio 2010 SP1 - x64 Release:
Seconds = 117.202
FP Ops = 7680000000000
FLOPs = 6.55279e+010
sum = 17.8122
Theoretical SSE peak is 4 flops * 4 cores * 4.4 GHz = 70.4 GFlops. Actual is 65.5 GFlops.
Let's take this one step further. AVX...
#include <immintrin.h>
#include <omp.h>
#include <iostream>
using namespace std;
typedef unsigned long long uint64;
double test_dp_mac_AVX(double x,double y,uint64 iterations){
register __m256d r0,r1,r2,r3,r4,r5,r6,r7,r8,r9,rA,rB,rC,rD,rE,rF;
// Generate starting data.
r0 = _mm256_set1_pd(x);
r1 = _mm256_set1_pd(y);
r8 = _mm256_set1_pd(-0.0);
r2 = _mm256_xor_pd(r0,r8);
r3 = _mm256_or_pd(r0,r8);
r4 = _mm256_andnot_pd(r8,r0);
r5 = _mm256_mul_pd(r1,_mm256_set1_pd(0.37796447300922722721));
r6 = _mm256_mul_pd(r1,_mm256_set1_pd(0.24253562503633297352));
r7 = _mm256_mul_pd(r1,_mm256_set1_pd(4.1231056256176605498));
r8 = _mm256_add_pd(r0,_mm256_set1_pd(0.37796447300922722721));
r9 = _mm256_add_pd(r1,_mm256_set1_pd(0.24253562503633297352));
rA = _mm256_sub_pd(r0,_mm256_set1_pd(4.1231056256176605498));
rB = _mm256_sub_pd(r1,_mm256_set1_pd(4.1231056256176605498));
rC = _mm256_set1_pd(1.4142135623730950488);
rD = _mm256_set1_pd(1.7320508075688772935);
rE = _mm256_set1_pd(0.57735026918962576451);
rF = _mm256_set1_pd(0.70710678118654752440);
uint64 iMASK = 0x800fffffffffffffull;
__m256d MASK = _mm256_set1_pd(*(double*)&iMASK);
__m256d vONE = _mm256_set1_pd(1.0);
uint64 c = 0;
while (c < iterations){
size_t i = 0;
while (i < 1000){
// Here's the meat - the part that really matters.
r0 = _mm256_mul_pd(r0,rC);
r1 = _mm256_add_pd(r1,rD);
r2 = _mm256_mul_pd(r2,rE);
r3 = _mm256_sub_pd(r3,rF);
r4 = _mm256_mul_pd(r4,rC);
r5 = _mm256_add_pd(r5,rD);
r6 = _mm256_mul_pd(r6,rE);
r7 = _mm256_sub_pd(r7,rF);
r8 = _mm256_mul_pd(r8,rC);
r9 = _mm256_add_pd(r9,rD);
rA = _mm256_mul_pd(rA,rE);
rB = _mm256_sub_pd(rB,rF);
r0 = _mm256_add_pd(r0,rF);
r1 = _mm256_mul_pd(r1,rE);
r2 = _mm256_sub_pd(r2,rD);
r3 = _mm256_mul_pd(r3,rC);
r4 = _mm256_add_pd(r4,rF);
r5 = _mm256_mul_pd(r5,rE);
r6 = _mm256_sub_pd(r6,rD);
r7 = _mm256_mul_pd(r7,rC);
r8 = _mm256_add_pd(r8,rF);
r9 = _mm256_mul_pd(r9,rE);
rA = _mm256_sub_pd(rA,rD);
rB = _mm256_mul_pd(rB,rC);
r0 = _mm256_mul_pd(r0,rC);
r1 = _mm256_add_pd(r1,rD);
r2 = _mm256_mul_pd(r2,rE);
r3 = _mm256_sub_pd(r3,rF);
r4 = _mm256_mul_pd(r4,rC);
r5 = _mm256_add_pd(r5,rD);
r6 = _mm256_mul_pd(r6,rE);
r7 = _mm256_sub_pd(r7,rF);
r8 = _mm256_mul_pd(r8,rC);
r9 = _mm256_add_pd(r9,rD);
rA = _mm256_mul_pd(rA,rE);
rB = _mm256_sub_pd(rB,rF);
r0 = _mm256_add_pd(r0,rF);
r1 = _mm256_mul_pd(r1,rE);
r2 = _mm256_sub_pd(r2,rD);
r3 = _mm256_mul_pd(r3,rC);
r4 = _mm256_add_pd(r4,rF);
r5 = _mm256_mul_pd(r5,rE);
r6 = _mm256_sub_pd(r6,rD);
r7 = _mm256_mul_pd(r7,rC);
r8 = _mm256_add_pd(r8,rF);
r9 = _mm256_mul_pd(r9,rE);
rA = _mm256_sub_pd(rA,rD);
rB = _mm256_mul_pd(rB,rC);
i++;
}
// Need to renormalize to prevent denormal/overflow.
r0 = _mm256_and_pd(r0,MASK);
r1 = _mm256_and_pd(r1,MASK);
r2 = _mm256_and_pd(r2,MASK);
r3 = _mm256_and_pd(r3,MASK);
r4 = _mm256_and_pd(r4,MASK);
r5 = _mm256_and_pd(r5,MASK);
r6 = _mm256_and_pd(r6,MASK);
r7 = _mm256_and_pd(r7,MASK);
r8 = _mm256_and_pd(r8,MASK);
r9 = _mm256_and_pd(r9,MASK);
rA = _mm256_and_pd(rA,MASK);
rB = _mm256_and_pd(rB,MASK);
r0 = _mm256_or_pd(r0,vONE);
r1 = _mm256_or_pd(r1,vONE);
r2 = _mm256_or_pd(r2,vONE);
r3 = _mm256_or_pd(r3,vONE);
r4 = _mm256_or_pd(r4,vONE);
r5 = _mm256_or_pd(r5,vONE);
r6 = _mm256_or_pd(r6,vONE);
r7 = _mm256_or_pd(r7,vONE);
r8 = _mm256_or_pd(r8,vONE);
r9 = _mm256_or_pd(r9,vONE);
rA = _mm256_or_pd(rA,vONE);
rB = _mm256_or_pd(rB,vONE);
c++;
}
r0 = _mm256_add_pd(r0,r1);
r2 = _mm256_add_pd(r2,r3);
r4 = _mm256_add_pd(r4,r5);
r6 = _mm256_add_pd(r6,r7);
r8 = _mm256_add_pd(r8,r9);
rA = _mm256_add_pd(rA,rB);
r0 = _mm256_add_pd(r0,r2);
r4 = _mm256_add_pd(r4,r6);
r8 = _mm256_add_pd(r8,rA);
r0 = _mm256_add_pd(r0,r4);
r0 = _mm256_add_pd(r0,r8);
// Prevent Dead Code Elimination
double out = 0;
__m256d temp = r0;
out += ((double*)&temp)[0];
out += ((double*)&temp)[1];
out += ((double*)&temp)[2];
out += ((double*)&temp)[3];
return out;
}
void test_dp_mac_AVX(int tds,uint64 iterations){
double *sum = (double*)malloc(tds * sizeof(double));
double start = omp_get_wtime();
#pragma omp parallel num_threads(tds)
{
double ret = test_dp_mac_AVX(1.1,2.1,iterations);
sum[omp_get_thread_num()] = ret;
}
double secs = omp_get_wtime() - start;
uint64 ops = 48 * 1000 * iterations * tds * 4;
cout << "Seconds = " << secs << endl;
cout << "FP Ops = " << ops << endl;
cout << "FLOPs = " << ops / secs << endl;
double out = 0;
int c = 0;
while (c < tds){
out += sum[c++];
}
cout << "sum = " << out << endl;
cout << endl;
free(sum);
}
int main(){
// (threads, iterations)
test_dp_mac_AVX(8,10000000);
system("pause");
}
Output (1 thread, 10000000 iterations) - Compiled with Visual Studio 2010 SP1 - x64 Release:
Seconds = 57.4679
FP Ops = 1920000000000
FLOPs = 3.34099e+010
sum = 4.45305
Theoretical AVX peak is 8 flops * 4.4 GHz = 35.2 GFlops. Actual is 33.4 GFlops.
Output (8 threads, 10000000 iterations) - Compiled with Visual Studio 2010 SP1 - x64 Release:
Seconds = 111.119
FP Ops = 15360000000000
FLOPs = 1.3823e+011
sum = 35.6244
Theoretical AVX peak is 8 flops * 4 cores * 4.4 GHz = 140.8 GFlops. Actual is 138.2 GFlops.
Now for some explanations:
The performance critical part is obviously the 48 instructions inside the inner loop. You'll notice that it's broken into 4 blocks of 12 instructions each. Each of these 12 instructions blocks are completely independent from each other - and take on average 6 cycles to execute.
So there's 12 instructions and 6 cycles between issue-to-use. The latency of multiplication is 5 cycles, so it's just enough to avoid latency stalls.
The normalization step is needed to keep the data from over/underflowing. This is needed since the do-nothing code will slowly increase/decrease the magnitude of the data.
So it's actually possible to do better than this if you just use all zeros and get rid of the normalization step. However, since I wrote the benchmark to measure power consumption and temperature, I had to make sure the flops were on "real" data, rather than zeros - as the execution units may very well have special case-handling for zeros that use less power and produce less heat.
More Results:
- Intel Core i7 920 @ 3.5 GHz
- Windows 7 Ultimate x64
- Visual Studio 2010 SP1 - x64 Release
Threads: 1
Seconds = 72.1116
FP Ops = 960000000000
FLOPs = 1.33127e+010
sum = 2.22652
Theoretical SSE Peak: 4 flops * 3.5 GHz = 14.0 GFlops. Actual is 13.3 GFlops.
Threads: 8
Seconds = 149.576
FP Ops = 7680000000000
FLOPs = 5.13452e+010
sum = 17.8122
Theoretical SSE Peak: 4 flops * 4 cores * 3.5 GHz = 56.0 GFlops. Actual is 51.3 GFlops.
My processor temps hit 76C on the multi-threaded run! If you runs these, be sure the results aren't affected by CPU throttling.
- 2 x Intel Xeon X5482 Harpertown @ 3.2 GHz
- Ubuntu Linux 10 x64
- GCC 4.5.2 x64 - (-O2 -msse3 -fopenmp)
Threads: 1
Seconds = 78.3357
FP Ops = 960000000000
FLOPs = 1.22549e+10
sum = 2.22652
Theoretical SSE Peak: 4 flops * 3.2 GHz = 12.8 GFlops. Actual is 12.3 GFlops.
Threads: 8
Seconds = 78.4733
FP Ops = 7680000000000
FLOPs = 9.78676e+10
sum = 17.8122
Theoretical SSE Peak: 4 flops * 8 cores * 3.2 GHz = 102.4 GFlops. Actual is 97.9 GFlops.
There's a point in the Intel architecture that people often forget, the dispatch ports are shared between Int and FP/SIMD. This means that you will only get a certain amount of bursts of FP/SIMD before the loop logic will create bubbles in your floating point stream. Mystical got more flops out of his code, because he used longer strides in his unrolled loop.
If you look at the Nehalem/Sandy Bridge architecture here http://www.realworldtech.com/page.cfm?ArticleID=RWT091810191937&p=6 it's quite clear what happens.
In contrast, it should be easier to reach peak performance on AMD (Bulldozer) as the INT and FP/SIMD pipes have separate issue ports with their own scheduler.
This is only theoretical as I have neither of these processors to test.