Improving computation speed for sine/cosine and large arrays - c++

Question

for signal processing I need to compute relatively large C arrays as shown in the code part below. This is working fine so far, unfortunately, the implementation is slow. The size of "calibdata" is arround 150k and needs to be calculated for different frequencies/phases. Is there a way to improve speed significantly? Doing the same with logical indexing in MATLAB is way faster.
What I tried already:
using taylor approximation of sine: no siginificant improvement.
using std::vector, also no siginificant improvement.
code:
double phase_func(double* calibdata, long size, double* freqscale, double fs, double phase, int currentcarrier){
for (int i = 0; i < size; i++)
result += calibdata[i] * cos((2 * PI*freqscale[currentcarrier] * i / fs) + (phase*(PI / 180) - (PI / 2)));
result = fabs(result / size);
return result;}
Best regards,
Thomas

Answer 1

When optimizing code for speed, step 1 is to enable compiler optimizations. I hope you've done that already.
Step 2 is to profile the code and see exactly how the time is being spent. Without profiling, you're just guessing, and you could end up trying to optimize the wrong thing.
For example, your guess seems to be that the cos function is the bottleneck. But the other possibility is that the calculation of the angle is the bottleneck. Here's how I would refactor the code to reduce the time spent calculating the angle.
double phase_func(double* calibdata, long size, double* freqscale, double fs, double phase, int currentcarrier)
{
double result = 0;
double angle = phase * (PI / 180) - (PI / 2);
double delta = 2 * PI * freqscale[currentcarrier] / fs;
for (int i = 0; i < size; i++)
{
result += calibdata[i] * cos( angle );
angle += delta;
}
return fabs(result / size);
}

Answer 2

Okay, I'm probably going to get flogged for this answer, but I would use the GPU for this. Because your array doesn't appear to be self-referential, the best speedup you're going to get for large arrays is through parallelization... by far. I don't use MATLAB, but I just did a quick search for GPU utilization on the MathWorks site:
http://www.mathworks.com/company/newsletters/articles/gpu-programming-in-matlab.html?requestedDomain=www.mathworks.com
Outside of MATLAB you could use OpenCL or CUDA yourself.

Answer 3

Your enemies in execution time are:
Division
Function calls (including implicit ones in loops)
Accessing data from diffent areas
Operating dissimilar instructions
You should research on Data Driving programming and using the data cache effectively.
Division
Whether with hardware support or software support division takes a long time by its very nature. Eliminate if possibly by changing the numeric base or factoring out of the loop (if possible).
Function Calls
The most efficient method of execution is sequential. Processors are optimized for this. A branch may require the processor perform some additional calculation (branch prediction) or reloading of the instruction cache / pipeline. A waste of time (that could be spent executing data instructions).
The optimization for this is to use techniques like loop unrolling and inlining of small functions. Also reduce the quantity of branches by simplifying expressions and using Boolean algebra.
Accessing data from different areas
Modern processors are optimized to operate on local data (data in one area). One example is loading an internal cache with data. Specifically, loading a cache line with data. For example, if the data from your arrays is in one location and the cosine data in another, this may cause the data cache to be reloaded, again wasting time.
A better solution is to place all data contiguously or to contiguously access all the data. Rather than making many discontiguous accesses to the cosine table, look up a batch of cosine values sequentially (without any other data accesses between).
Dissimilar Instructions
Modern processors are more efficient at processing a batch of similar instructions. For example the pattern load, add, store is more efficient for blocks when all the loading is performed, then all adding, then all storing.
Summary
Here's an example:
register double result = 0.0;
register unsigned int i = 0U;
for (i = 0; i < size; i += 2)
{
register double cos_angle1 = /* ... */;
register double cos_angle2 = /* ... */;
result += calibdata[i + 0] * cos_angle1;
result += calibdata[i + 1] * cos_angle2;
}
The above loop is unrolled and like operations are performed in groups.
Although the keyword register may be deprecated, it is a suggestion to the compiler to use dedicated registers (if possible).

Answer 4

You can try to use the definition of cosine based on the complex exponential:
where j^2=-1.
Store exp((2 * PI*freqscale[currentcarrier] / fs)*j) and exp(phase*j). Evaluating cos(...) then resumes to a couple of products and additions in the for loops, and sin(), cos() and exp() are only called a couple of times.
Here goes the implementation:
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include <complex.h>
#include <time.h>
#define PI 3.141592653589
typedef struct cos_plan{
double complex* expo;
int size;
}cos_plan;
double phase_func(double* calibdata, long size, double* freqscale, double fs, double phase, int currentcarrier){
double result=0; //initialization
for (int i = 0; i < size; i++){
result += calibdata[i] * cos ( (2 * PI*freqscale[currentcarrier] * i / fs) + (phase*(PI / 180.) - (PI / 2.)) );
//printf("i %d cos %g\n",i,cos ( (2 * PI*freqscale[currentcarrier] * i / fs) + (phase*(PI / 180.) - (PI / 2.)) ));
}
result = fabs(result / size);
return result;
}
double phase_func2(double* calibdata, long size, double* freqscale, double fs, double phase, int currentcarrier, cos_plan* plan){
//first, let's compute the exponentials:
//double complex phaseexp=cos(phase*(PI / 180.) - (PI / 2.))+sin(phase*(PI / 180.) - (PI / 2.))*I;
//double complex phaseexpm=conj(phaseexp);
double phasesin=sin(phase*(PI / 180.) - (PI / 2.));
double phasecos=cos(phase*(PI / 180.) - (PI / 2.));
if (plan->size<size){
double complex *tmp=realloc(plan->expo,size*sizeof(double complex));
if(tmp==NULL){fprintf(stderr,"realloc failed\n");exit(1);}
plan->expo=tmp;
plan->size=size;
}
plan->expo[0]=1;
//plan->expo[1]=exp(2 *I* PI*freqscale[currentcarrier]/fs);
plan->expo[1]=cos(2 * PI*freqscale[currentcarrier]/fs)+sin(2 * PI*freqscale[currentcarrier]/fs)*I;
//printf("%g %g\n",creall(plan->expo[1]),cimagl(plan->expo[1]));
for(int i=2;i<size;i++){
if(i%2==0){
plan->expo[i]=plan->expo[i/2]*plan->expo[i/2];
}else{
plan->expo[i]=plan->expo[i/2]*plan->expo[i/2+1];
}
}
//computing the result
double result=0; //initialization
for(int i=0;i<size;i++){
//double coss=0.5*creall(plan->expo[i]*phaseexp+conj(plan->expo[i])*phaseexpm);
double coss=creall(plan->expo[i])*phasecos-cimagl(plan->expo[i])*phasesin;
//printf("i %d cos %g\n",i,coss);
result+=calibdata[i] *coss;
}
result = fabs(result / size);
return result;
}
int main(){
//the parameters
long n=100000000;
double* calibdata=malloc(n*sizeof(double));
if(calibdata==NULL){fprintf(stderr,"malloc failed\n");exit(1);}
int freqnb=42;
double* freqscale=malloc(freqnb*sizeof(double));
if(freqscale==NULL){fprintf(stderr,"malloc failed\n");exit(1);}
for (int i = 0; i < freqnb; i++){
freqscale[i]=i*i*0.007+i;
}
double fs=n;
double phase=0.05;
//populate calibdata
for (int i = 0; i < n; i++){
calibdata[i]=i/((double)n);
calibdata[i]=calibdata[i]*calibdata[i]-calibdata[i]+0.007/(calibdata[i]+3.0);
}
//call to sample code
clock_t t;
t = clock();
double res=phase_func(calibdata,n, freqscale, fs, phase, 13);
t = clock() - t;
printf("first call got %g in %g seconds.\n",res,((float)t)/CLOCKS_PER_SEC);
//initialize
cos_plan plan;
plan.expo=malloc(n*sizeof(double complex));
plan.size=n;
t = clock();
res=phase_func2(calibdata,n, freqscale, fs, phase, 13,&plan);
t = clock() - t;
printf("second call got %g in %g seconds.\n",res,((float)t)/CLOCKS_PER_SEC);
//cleaning
free(plan.expo);
free(calibdata);
free(freqscale);
return 0;
}
Compile with gcc main.c -o main -std=c99 -lm -Wall -O3. Using the code you provided, it take 8 seconds with size=100000000 on my computer while the execution time of the proposed solution takes 1.5 seconds... It is not so impressive, but it is not negligeable.
The solution that is presented does not involve any call to cos of sin in the for loops. Indeed, there are only multiplications and additions. The bottleneck is either the memory bandwidth or the tests and access to memory in the exponentiation by squaring (most likely first issue, since i add to use an additional array of complex).
For complex number in c, see:
How to work with complex numbers in C?
Computing e^(-j) in C
If the problem is memory bandwidth, then parallelism is required... and directly computing cos would be easier. Additional simplifications coud have be performed if freqscale[currentcarrier] / fs were an integer. Your problem is really close to the computation of Discrete Cosine Transform, the present trick is close to the Discrete Fourier Transform and the FFTW library is really good at computing these transforms.
Notice that the present code can produce innacurate results due to loss of significance : result can be much larger than cos(...)*calibdata[] when size is large. Using partial sums can resolve the issue.

Answer 5

Simple trig identity to eliminate the - (PI / 2). This is also more accurate than attempting the subtraction which uses machine_PI. This is important when values are near π/2.
cosine(x - π/2) == -sine(x)
Use of const and restrict: Good compilers can perform more optimizations with this knowledge. (See also #user3528438)
// double phase_func(double* calibdata, long size,
// double* freqscale, double fs, double phase, int currentcarrier) {
double phase_func(const double* restrict calibdata, long size,
const double* restrict freqscale, double fs, double phase, int currentcarrier) {
Some platforms perform faster calculations with float vs double with a tolerable loss of precision. YMMV. Profile code both ways.
// result += calibdata[i] * cos(...
result += calibdata[i] * cosf(...
Minimize recalculations.
double angle_delta = ...;
double angle_current = ...;
for (int i = 0; i < size; i++) {
result += calibdata[i] * cos(angle_current);
angle_current += angle_delta;
}
Unclear why code uses long size and and int currentcarrier. I'd expect the same type and to use type size_t. This is idiomatic for array indexing. #Daniel Jour
Reversing loops can allow a compare to 0 rather than compare to variable. Sometimes a modest performance gain.
Insure compiler optimizations are well enabled.
All together
double phase_func2(const double* restrict calibdata, size_t size,
const double* restrict freqscale, double fs, double phase,
size_t currentcarrier) {
double result = 0.0;
double angle_delta = 2.0 * PI * freqscale[currentcarrier] / fs;
double angle_current = angle_delta * (size - 1) + phase * (PI / 180);
size_t i = size;
while (i) {
result -= calibdata[--i] * sinf(angle_current);
angle_current -= angle_delta;
}
result = fabs(result / size);
return result;
}

Answer 6

Leveraging the cores you have, without resorting to the GPU, use OpenMP. Testing with VS2015, the invariants are lifted out of the loop by the optimizer. Enabling AVX2 and OpenMP.
double phase_func3(double* calibdata, const int size, const double* freqscale,
const double fs, const double phase, const size_t currentcarrier)
{
double result{};
constexpr double PI = 3.141592653589;
#pragma omp parallel
#pragma omp for reduction(+: result)
for (int i = 0; i < size; ++i) {
result += calibdata[i] *
cos( (2 * PI*freqscale[currentcarrier] * i / fs) + (phase*(PI / 180.0) - (PI / 2.0)));
}
result = fabs(result / size);
return result;
}
The original version with AVX enabled took: ~1.4 seconds
and adding OpenMP brought it down to: ~0.51 seconds.
Pretty nice return for two pragmas and a compiler switch.