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I programmed c in Visual studio c++.
In visual studio c++, I operate complex number with <complex> at c code grammar, not c++.
a[] is 4001 array, so using b[4001] store operated value and finally return a[].
NXm is defined 4001 from main.
when I compared with the result of matlab's fft, The difference occurs from the 169th to 4000th value.
Would you see if there is any error? or what is the cause?
Thank you for reading the question.
I've tried reversely progressing "for", from NXm to 0.
I've tried changing double ak = (double)k * (double)n * (2.0 * M_PI / (double)NXm); to double ak = k * n * (2.0 * M_PI / (double)NXm);
When I've tried operating short length of field, function worked well.
This is the code.
void fft(complex<double> a[], int NXm)
{
complex<double> sum = 0.0 + 0.0*I; ;
complex<double> c = 1.0*I;
complex<double> b[4001] = { 0 };
for (int k = 0; k < NXm; k++)
{
sum = 0.0 + 0.0*I;
for (int n = 0; n < NXm; n++)
{
double ak = (double)k * (double)n * (2.0 * M_PI / (double)NXm);
sum = sum + a[n] * exp(-c * ak);
}
b[k] = sum;
}
for (int i = 0; i < NXm; i++)
{
a[i] = b[i];
}
}
I expect almost same result as fft of matlab.
Slight error of epsilon level is ok
Complex exponential is:
exp(x + I*y) = exp(x) ( cos(y) + I*sin(y)) = exp(x) * cis(y)
problem may be caused by a flaw in FPU part of processor, if y is rather small, built-in a intrinsic function for sin(y) got precision worse than |y - sin(y)|. Naive implementations of Euler formula may err with small arguments, sloppy ones may assume that sin(y) is zero.
It's better to replace sin() by cosecant intrinsic, which is a reciprocal sinus (1/sin(x)).
Another, less common issue is when trig functions are supplied by values which are greater than 2*PI*n, where n is a natural greater than 1. Precision drops with increase of n and sin (x) != sin(2*PI*n + x)
Third, depending on compiler, there might be optimizing flags which regulate how compiler treats use of math functions and floating point math in general. They may default to imprecise but fast calculations. This may supercede with use of C header in C++ code.
MATLAB used FFT (Fast Fourier Transform) algorithm for computing DFT. It is both much faster (O(n log n) as opposed to O(n^2) with the direct method) and more stable numerically. So don't expect the same results as MATLAB makes.
I need to calculate the value of a high dimensional integral in C++. I have found numerous libraries capable of solving this task for fixed limit integrals,
\int_{0}^{L} \int_{0}^{L} dx dy f(x,y) .
However the integrals which I am looking at have variable limits,
\int_{0}^{L} \int_{x}^{L} dx dy f(x,y) .
To clarify what i mean, here is a naive 2D Riemann sum implementation in 2D, which returns the desired result,
int steps = 100;
double integral = 0;
double dl = L/((double) steps);
double x[2] = {0};
for(int i = 0; i < steps; i ++){
x[0] = dl*i;
for(int j = i; j < steps; j ++){
x[1] = dl*j;
double val = f(x);
integral += val*val*dl*dl;
}
}
where f is some arbitrary function and L the common upper integration limit. While this implementation works, it's slow and thus impractical for higher dimensions.
Effective algorithms for higher dimensions exist, but to my knowledge, library implementations (e.g. Cuba) take a fixed value vector as the limit argument which renders them useless for my problem.
Is there any reason for this and/or is there any trick to circumvent the problem?
Your integration order is wrong, should be dy dx.
You are integrating over the triangle
0 <= x <= y <= L
inside the square [0,L]x[0,L]. This can be simulated by integrating over the full square where the integrand f is defined as 0 outside of the triangle. In many cases, when f is defined on the full square, this can be accomplished by taking the product of f with the indicator function of the triangle as new integrand.
When integrating over a triangular region such as 0<=x<=y<=L one can take advantage of symmetry: integrate f(min(x,y),max(x,y)) over the square 0<=x,y<=L and divide the result by 2. This has an advantage over extending f by zero (the method mentioned by LutzL) in that the extended function is continuous, which improves the performance of the integration routine.
I compared these on the example of the integral of 2x+y over 0<=x<=y<=1. The true value of the integral is 2/3. Let's compare the performance; for demonstration purpose I use Matlab routine, but this is not specific to language or library used.
Extending by zero
f = #(x,y) (2*x+y).*(x<=y);
result = integral2(f, 0, 1, 0, 1);
fprintf('%.9f\n',result);
Output:
Warning: Reached the maximum number of function evaluations
(10000). The result fails the global error test.
0.666727294
Extending by symmetry
g = #(x,y) (2*min(x,y)+max(x,y));
result2 = integral2(g, 0, 1, 0, 1)/2;
fprintf('%.9f\n',result2);
Output:
0.666666776
The second result is 500 times more accurate than the first.
Unfortunately, this symmetry trick is not available for general domains; but integration over a triangle comes up often enough so it's useful to keep it in mind.
I was a bit confused by your integral definition but from your code i see it like this:
just did some testing so here is your code:
//---------------------------------------------------------------------------
double f(double *x) { return (x[0]+x[1]); }
void integral0()
{
double L=10.0;
int steps = 10000;
double integral = 0;
double dl = L/((double) steps);
double x[2] = {0};
for(int i = 0; i < steps; i ++){
x[0] = dl*i;
for(int j = i; j < steps; j ++){
x[1] = dl*j;
double val = f(x);
integral += val*val*dl*dl;
}
}
}
//---------------------------------------------------------------------------
Here is optimized code:
//---------------------------------------------------------------------------
void integral1()
{
double L=10.0;
int i0,i1,steps = 10000;
double x[2]={0.0,0.0};
double integral,val,dl=L/((double)steps);
#define f(x) (x[0]+x[1])
integral=0.0;
for(x[0]= 0.0,i0= 0;i0<steps;i0++,x[0]+=dl)
for(x[1]=x[0],i1=i0;i1<steps;i1++,x[1]+=dl)
{
val=f(x);
integral+=val*val;
}
integral*=dl*dl;
#undef f
}
//---------------------------------------------------------------------------
results:
[ 452.639 ms] integral0
[ 336.268 ms] integral1
so the increase in speed is ~ 1.3 times (on 32bit app on WOW64 AMD 3.2GHz)
for higher dimensions it will multiply
but still I think this approach is slow
The only thing to reduce complexity I can think of is algebraically simplify things
either by integration tables or by Laplace or Z transforms
but for that the f(*x) must be know ...
constant time reduction can of course be done
by the use of multi-threading
and or GPU ussage
this can give you N times speed increase
because this is all directly parallelisable
I can see with the CPU profiler, that the compute_variances() is the bottleneck of my project.
% cumulative self self total
time seconds seconds calls ms/call ms/call name
75.63 5.43 5.43 40 135.75 135.75 compute_variances(unsigned int, std::vector<Point, std::allocator<Point> > const&, float*, float*, unsigned int*)
19.08 6.80 1.37 readDivisionSpace(Division_Euclidean_space&, char*)
...
Here is the body of the function:
void compute_variances(size_t t, const std::vector<Point>& points, float* avg,
float* var, size_t* split_dims) {
for (size_t d = 0; d < points[0].dim(); d++) {
avg[d] = 0.0;
var[d] = 0.0;
}
float delta, n;
for (size_t i = 0; i < points.size(); ++i) {
n = 1.0 + i;
for (size_t d = 0; d < points[0].dim(); ++d) {
delta = (points[i][d]) - avg[d];
avg[d] += delta / n;
var[d] += delta * ((points[i][d]) - avg[d]);
}
}
/* Find t dimensions with largest scaled variance. */
kthLargest(var, points[0].dim(), t, split_dims);
}
where kthLargest() doesn't seem to be a problem, since I see that:
0.00 7.18 0.00 40 0.00 0.00 kthLargest(float*, int, int, unsigned int*)
The compute_variances() takes a vector of vectors of floats (i.e. a vector of Points, where Points is a class I have implemented) and computes the variance of them, in each dimension (with regard to the algorithm of Knuth).
Here is how I call the function:
float avg[(*points)[0].dim()];
float var[(*points)[0].dim()];
size_t split_dims[t];
compute_variances(t, *points, avg, var, split_dims);
The question is, can I do better? I would really happy to pay the trade-off between speed and approximate computation of variances. Or maybe I could make the code more cache friendly or something?
I compiled like this:
g++ main_noTime.cpp -std=c++0x -p -pg -O3 -o eg
Notice, that before edit, I had used -o3, not with a capital 'o'. Thanks to ypnos, I compiled now with the optimization flag -O3. I am sure that there was a difference between them, since I performed time measurements with one of these methods in my pseudo-site.
Note that now, compute_variances is dominating the overall project's time!
[EDIT]
copute_variances() is called 40 times.
Per 10 calls, the following hold true:
points.size() = 1000 and points[0].dim = 10000
points.size() = 10000 and points[0].dim = 100
points.size() = 10000 and points[0].dim = 10000
points.size() = 100000 and points[0].dim = 100
Each call handles different data.
Q: How fast is access to points[i][d]?
A: point[i] is just the i-th element of std::vector, where the second [], is implemented as this, in the Point class.
const FT& operator [](const int i) const {
if (i < (int) coords.size() && i >= 0)
return coords.at(i);
else {
std::cout << "Error at Point::[]" << std::endl;
exit(1);
}
return coords[0]; // Clear -Wall warning
}
where coords is a std::vector of float values. This seems a bit heavy, but shouldn't the compiler be smart enough to predict correctly that the branch is always true? (I mean after the cold start). Moreover, the std::vector.at() is supposed to be constant time (as said in the ref). I changed this to have only .at() in the body of the function and the time measurements remained, pretty much, the same.
The division in the compute_variances() is for sure something heavy! However, Knuth's algorithm was a numerical stable one and I was not able to find another algorithm, that would de both numerical stable and without division.
Note that I am not interesting in parallelism right now.
[EDIT.2]
Minimal example of Point class (I think I didn't forget to show something):
class Point {
public:
typedef float FT;
...
/**
* Get dimension of point.
*/
size_t dim() const {
return coords.size();
}
/**
* Operator that returns the coordinate at the given index.
* #param i - index of the coordinate
* #return the coordinate at index i
*/
FT& operator [](const int i) {
return coords.at(i);
//it's the same if I have the commented code below
/*if (i < (int) coords.size() && i >= 0)
return coords.at(i);
else {
std::cout << "Error at Point::[]" << std::endl;
exit(1);
}
return coords[0]; // Clear -Wall warning*/
}
/**
* Operator that returns the coordinate at the given index. (constant)
* #param i - index of the coordinate
* #return the coordinate at index i
*/
const FT& operator [](const int i) const {
return coords.at(i);
/*if (i < (int) coords.size() && i >= 0)
return coords.at(i);
else {
std::cout << "Error at Point::[]" << std::endl;
exit(1);
}
return coords[0]; // Clear -Wall warning*/
}
private:
std::vector<FT> coords;
};
1. SIMD
One easy speedup for this is to use vector instructions (SIMD) for the computation. On x86 that means SSE, AVX instructions. Based on your word length and processor you can get speedups of about x4 or even more. This code here:
for (size_t d = 0; d < points[0].dim(); ++d) {
delta = (points[i][d]) - avg[d];
avg[d] += delta / n;
var[d] += delta * ((points[i][d]) - avg[d]);
}
can be sped-up by doing the computation for four elements at once with SSE. As your code really only processes one single element in each loop iteration, there is no bottleneck. If you go down to 16bit short instead of 32bit float (an approximation then), you can fit eight elements in one instruction. With AVX it would be even more, but you need a recent processor for that.
It is not the solution to your performance problem, but just one of them that can also be combined with others.
2. Micro-parallelizm
The second easy speedup when you have that many loops is to use parallel processing. I typically use Intel TBB, others might suggest OpenMP instead. For this you would probably have to change the loop order. So parallelize over d in the outer loop, not over i.
You can combine both techniques, and if you do it right, on a quadcore with HT you might get a speed-up of 25-30 for the combination without any loss in accuracy.
3. Compiler optimization
First of all maybe it is just a typo here on SO, but it needs to be -O3, not -o3!
As a general note, it might be easier for the compiler to optimize your code if you declare the variables delta, n within the scope where you actually use them. You should also try the -funroll-loops compiler option as well as -march. The option to the latter depends on your CPU, but nowadays typically -march core2 is fine (also for recent AMDs), and includes SSE optimizations (but I would not trust the compiler just yet to do that for your loop).
The big problem with your data structure is that it's essentially a vector<vector<float> >. That's a pointer to an array of pointers to arrays of float with some bells and whistles attached. In particular, accessing consecutive Points in the vector doesn't correspond to accessing consecutive memory locations. I bet you see tons and tons of cache misses when you profile this code.
Fix this before horsing around with anything else.
Lower-order concerns include the floating-point division in the inner loop (compute 1/n in the outer loop instead) and the big load-store chain that is your inner loop. You can compute the means and variances of slices of your array using SIMD and combine them at the end, for instance.
The bounds-checking once per access probably doesn't help, either. Get rid of that too, or at least hoist it out of the inner loop; don't assume the compiler knows how to fix that on its own.
Here's what I would do, in guesstimated order of importance:
Return the floating-point from the Point::operator[] by value, not by reference.
Use coords[i] instead of coords.at(i), since you already assert that it's within bounds. The at member checks the bounds. You only need to check it once.
Replace the home-baked error indication/checking in the Point::operator[] with an assert. That's what asserts are for. They are nominally no-ops in release mode - I doubt that you need to check it in release code.
Replace the repeated division with a single division and repeated multiplication.
Remove the need for wasted initialization by unrolling the first two iterations of the outer loop.
To lessen impact of cache misses, run the inner loop alternatively forwards then backwards. This at least gives you a chance at using some cached avg and var. It may in fact remove all cache misses on avg and var if prefetch works on reverse order of iteration, as it well should.
On modern C++ compilers, the std::fill and std::copy can leverage type alignment and have a chance at being faster than the C library memset and memcpy.
The Point::operator[] will have a chance of getting inlined in the release build and can reduce to two machine instructions (effective address computation and floating point load). That's what you want. Of course it must be defined in the header file, otherwise the inlining will only be performed if you enable link-time code generation (a.k.a. LTO).
Note that the Point::operator[]'s body is only equivalent to the single-line
return coords.at(i) in a debug build. In a release build the entire body is equivalent to return coords[i], not return coords.at(i).
FT Point::operator[](int i) const {
assert(i >= 0 && i < (int)coords.size());
return coords[i];
}
const FT * Point::constData() const {
return &coords[0];
}
void compute_variances(size_t t, const std::vector<Point>& points, float* avg,
float* var, size_t* split_dims)
{
assert(points.size() > 0);
const int D = points[0].dim();
// i = 0, i_n = 1
assert(D > 0);
#if __cplusplus >= 201103L
std::copy_n(points[0].constData(), D, avg);
#else
std::copy(points[0].constData(), points[0].constData() + D, avg);
#endif
// i = 1, i_n = 0.5
if (points.size() >= 2) {
assert(points[1].dim() == D);
for (int d = D - 1; d >= 0; --d) {
float const delta = points[1][d] - avg[d];
avg[d] += delta * 0.5f;
var[d] = delta * (points[1][d] - avg[d]);
}
} else {
std::fill_n(var, D, 0.0f);
}
// i = 2, ...
for (size_t i = 2; i < points.size(); ) {
{
const float i_n = 1.0f / (1.0f + i);
assert(points[i].dim() == D);
for (int d = 0; d < D; ++d) {
float const delta = points[i][d] - avg[d];
avg[d] += delta * i_n;
var[d] += delta * (points[i][d] - avg[d]);
}
}
++ i;
if (i >= points.size()) break;
{
const float i_n = 1.0f / (1.0f + i);
assert(points[i].dim() == D);
for (int d = D - 1; d >= 0; --d) {
float const delta = points[i][d] - avg[d];
avg[d] += delta * i_n;
var[d] += delta * (points[i][d] - avg[d]);
}
}
++ i;
}
/* Find t dimensions with largest scaled variance. */
kthLargest(var, D, t, split_dims);
}
for (size_t d = 0; d < points[0].dim(); d++) {
avg[d] = 0.0;
var[d] = 0.0;
}
This code could be optimized by simply using memset. The IEEE754 representation of 0.0 in 32bits is 0x00000000. If the dimension is big, it worth it.
Something like:
memset((void*)avg, 0, points[0].dim() * sizeof(float));
In your code, you have a lot of calls to points[0].dim(). It would be better to call once at the beginning of the function and store in a variable. Likely, the compiler already does this (since you are using -O3).
The division operations are a lot more expensive (from clock-cycle POV) than other operations (addition, subtraction).
avg[d] += delta / n;
It could make sense, to try to reduce the number of divisions: use partial non-cumulative average calculation, that would result in Dim division operation for N elements (instead of N x Dim); N < points.size()
Huge speedup could be achieved, using Cuda or OpenCL, since the calculation of avg and var could be done simultaneously for each dimension (consider using a GPU).
Another optimization is cache optimization including both data cache and instruction cache.
High level optimization techniques
Data Cache optimizations
Example of data cache optimization & unrolling
for (size_t d = 0; d < points[0].dim(); d += 4)
{
// Perform loading all at once.
register const float p1 = points[i][d + 0];
register const float p2 = points[i][d + 1];
register const float p3 = points[i][d + 2];
register const float p4 = points[i][d + 3];
register const float delta1 = p1 - avg[d+0];
register const float delta2 = p2 - avg[d+1];
register const float delta3 = p3 - avg[d+2];
register const float delta4 = p4 - avg[d+3];
// Perform calculations
avg[d + 0] += delta1 / n;
var[d + 0] += delta1 * ((p1) - avg[d + 0]);
avg[d + 1] += delta2 / n;
var[d + 1] += delta2 * ((p2) - avg[d + 1]);
avg[d + 2] += delta3 / n;
var[d + 2] += delta3 * ((p3) - avg[d + 2]);
avg[d + 3] += delta4 / n;
var[d + 3] += delta4 * ((p4) - avg[d + 3]);
}
This differs from classic loop unrolling in that loading from the matrix is performed as a group at the top of the loop.
Edit 1:
A subtle data optimization is to place the avg and var into a structure. This will ensure that the two arrays are next to each other in memory, sans padding. The data fetching mechanism in processors like datums that are very close to each other. Less chance for data cache miss and better chance to load all of the data into the cache.
You could use Fixed Point math instead of floating point math as an optimization.
Optimization via Fixed Point
Processors love to manipulate integers (signed or unsigned). Floating point may take extra computing power due to the extraction of the parts, performing the math, then reassemblying the parts. One mitigation is to use Fixed Point math.
Simple Example: meters
Given the unit of meters, one could express lengths smaller than a meter by using floating point, such as 3.14159 m. However, the same length can be expressed in a unit of finer detail like millimeters, e.g. 3141.59 mm. For finer resolution, a smaller unit is chosen and the value multiplied, e.g. 3,141,590 um (micrometers). The point is choosing a small enough unit to represent the floating point accuracy as an integer.
The floating point value is converted at input into Fixed Point. All data processing occurs in Fixed Point. The Fixed Point value is convert to Floating Point before outputting.
Power of 2 Fixed Point Base
As with converting from floating point meters to fixed point millimeters, using 1000, one could use a power of 2 instead of 1000. Selecting a power of 2 allows the processor to use bit shifting instead of multiplication or division. Bit shifting by a power of 2 is usually faster than multiplication or division.
Keeping with the theme and accuracy of millimeters, we could use 1024 as the base instead of 1000. Similarly, for higher accuracy, use 65536 or 131072.
Summary
Changing the design or implementation to used Fixed Point math allows the processor to use more integral data processing instructions than floating point. Floating point operations consume more processing power than integral operations in all but specialized processors. Using powers of 2 as the base (or denominator) allows code to use bit shifting instead of multiplication or division. Division and multiplication take more operations than shifting and thus shifting is faster. So rather than optimizing code for execution (such as loop unrolling), one could try using Fixed Point notation rather than floating point.
Point 1.
You're computing the average and the variance at the same time.
Is that right?
Don't you have to calculate the average first, then once you know it, calculate the sum of squared differences from the average?
In addition to being right, it's more likely to help performance than hurt it.
Trying to do two things in one loop is not necessarily faster than two consecutive simple loops.
Point 2.
Are you aware that there is a way to calculate average and variance at the same time, like this:
double sumsq = 0, sum = 0;
for (i = 0; i < n; i++){
double xi = x[i];
sum += xi;
sumsq += xi * xi;
}
double avg = sum / n;
double avgsq = sumsq / n
double variance = avgsq - avg*avg;
Point 3.
The inner loops are doing repetitive indexing.
The compiler might be able to optimize that to something minimal, but I wouldn't bet my socks on it.
Point 4.
You're using gprof or something like it.
The only reasonably reliable number to come out of it is self-time by function.
It won't tell you very well how time is spent inside the function.
I and many others rely on this method, which takes you straight to the heart of what takes time.
I know this is a recurring question, but I haven't really found a useful answer yet. I'm basically looking for a fast approximation of the function acos in C++, I'd like to know if I can significantly beat the standard one.
But some of you might have insights on my specific problem: I'm writing a scientific program which I need to be very fast. The complexity of the main algorithm boils down to computing the following expression (many times with different parameters):
sin( acos(t_1) + acos(t_2) + ... + acos(t_n) )
where the t_i are known real (double) numbers, and n is very small (like smaller than 6). I need a precision of at least 1e-10. I'm currently using the standard sin and acos C++ functions.
Do you think I can significantly gain speed somehow? For those of you who know some maths, do you think it would be smart to expand that sine in order to get an algebraic expression in terms of the t_i (only involving square roots)?
Thank you your your answers.
The code below provides simple implementations of sin() and acos() that should satisfy your accuracy requirements and that you might want to try. Please note that the math library implementation on your platform is very likely highly tuned for the specific hardware capabilities of that platform and is probably also coded in assembly for maximum efficiency, so simple compiled C code not catering to specifics of the hardware is unlikely to provide higher performance, even when the accuracy requirements are somewhat relaxed from full double precision. As Viktor Latypov points out, it may also be worthwhile to search for algorithmic alternatives that do not require expensive calls to transcendental math functions.
In the code below I have tried to stick to simple, portable constructs. If your compiler supports the rint() function [specified by C99 and C++11] you might want to use that instead of my_rint(). On some platforms, the call to floor() can be expensive since it requires dynamic changing of machine state. The functions my_rint(), sin_core(), cos_core(), and asin_core() would want to be inlined for best performance. Your compiler may do that automatically at high optimization levels (e.g. when compiling with -O3), or you could add an appropriate inlining attribute to these functions, e.g. inline or __inline depending on your toolchain.
Not knowing anything about your platform I opted for simple polynomial approximations, which are evaluated using Estrin's scheme plus Horner's scheme. See Wikipedia for a description of these evaluation schemes:
http://en.wikipedia.org/wiki/Estrin%27s_scheme ,
http://en.wikipedia.org/wiki/Horner_scheme
The approximations themselves are of the minimax type and were custom generated for this answer with the Remez algorithm:
http://en.wikipedia.org/wiki/Minimax_approximation_algorithm ,
http://en.wikipedia.org/wiki/Remez_algorithm
The identities used in the argument reduction for acos() are noted in the comments, for sin() I used a Cody/Waite-style argument reduction, as described in the following book:
W. J. Cody, W. Waite, Software Manual for the Elementary Functions. Prentice-Hall, 1980
The error bounds mentioned in the comments are approximate, and have not been rigorously tested or proven.
/* not quite rint(), i.e. results not properly rounded to nearest-or-even */
double my_rint (double x)
{
double t = floor (fabs(x) + 0.5);
return (x < 0.0) ? -t : t;
}
/* minimax approximation to cos on [-pi/4, pi/4] with rel. err. ~= 7.5e-13 */
double cos_core (double x)
{
double x8, x4, x2;
x2 = x * x;
x4 = x2 * x2;
x8 = x4 * x4;
/* evaluate polynomial using Estrin's scheme */
return (-2.7236370439787708e-7 * x2 + 2.4799852696610628e-5) * x8 +
(-1.3888885054799695e-3 * x2 + 4.1666666636943683e-2) * x4 +
(-4.9999999999963024e-1 * x2 + 1.0000000000000000e+0);
}
/* minimax approximation to sin on [-pi/4, pi/4] with rel. err. ~= 5.5e-12 */
double sin_core (double x)
{
double x4, x2, t;
x2 = x * x;
x4 = x2 * x2;
/* evaluate polynomial using a mix of Estrin's and Horner's scheme */
return ((2.7181216275479732e-6 * x2 - 1.9839312269456257e-4) * x4 +
(8.3333293048425631e-3 * x2 - 1.6666666640797048e-1)) * x2 * x + x;
}
/* minimax approximation to arcsin on [0, 0.5625] with rel. err. ~= 1.5e-11 */
double asin_core (double x)
{
double x8, x4, x2;
x2 = x * x;
x4 = x2 * x2;
x8 = x4 * x4;
/* evaluate polynomial using a mix of Estrin's and Horner's scheme */
return (((4.5334220547132049e-2 * x2 - 1.1226216762576600e-2) * x4 +
(2.6334281471361822e-2 * x2 + 2.0596336163223834e-2)) * x8 +
(3.0582043602875735e-2 * x2 + 4.4630538556294605e-2) * x4 +
(7.5000364034134126e-2 * x2 + 1.6666666300567365e-1)) * x2 * x + x;
}
/* relative error < 7e-12 on [-50000, 50000] */
double my_sin (double x)
{
double q, t;
int quadrant;
/* Cody-Waite style argument reduction */
q = my_rint (x * 6.3661977236758138e-1);
quadrant = (int)q;
t = x - q * 1.5707963267923333e+00;
t = t - q * 2.5633441515945189e-12;
if (quadrant & 1) {
t = cos_core(t);
} else {
t = sin_core(t);
}
return (quadrant & 2) ? -t : t;
}
/* relative error < 2e-11 on [-1, 1] */
double my_acos (double x)
{
double xa, t;
xa = fabs (x);
/* arcsin(x) = pi/2 - 2 * arcsin (sqrt ((1-x) / 2))
* arccos(x) = pi/2 - arcsin(x)
* arccos(x) = 2 * arcsin (sqrt ((1-x) / 2))
*/
if (xa > 0.5625) {
t = 2.0 * asin_core (sqrt (0.5 * (1.0 - xa)));
} else {
t = 1.5707963267948966 - asin_core (xa);
}
/* arccos (-x) = pi - arccos(x) */
return (x < 0.0) ? (3.1415926535897932 - t) : t;
}
sin( acos(t1) + acos(t2) + ... + acos(tn) )
boils down to the calculation of
sin( acos(x) ) and cos(acos(x))=x
because
sin(a+b) = cos(a)sin(b)+sin(a)cos(b).
The first thing is
sin( acos(x) ) = sqrt(1-x*x)
Taylor series expansion for the sqrt reduces the problem to polynomial calculations.
To clarify, here's the expansion to n=2, n=3:
sin( acos(t1) + acos(t2) ) = sin(acos(t1))cos(acos(t2)) + sin(acos(t2))cos(acos(t1) = sqrt(1-t1*t1) * t2 + sqrt(1-t2*t2) * t1
cos( acos(t2) + acos(t3) ) = cos(acos(t2)) cos(acos(t3)) - sin(acos(t2))sin(acos(t3)) = t2*t3 - sqrt(1-t2*t2)*sqrt(1-t3*t3)
sin( acos(t1) + acos(t2) + acos(t3)) =
sin(acos(t1))cos(acos(t2) + acos(t3)) + sin(acos(t2)+acos(t3) )cos(acos(t1)=
sqrt(1-t1*t1) * (t2*t3 - sqrt(1-t2*t2)*sqrt(1-t3*t3)) + (sqrt(1-t2*t2) * t3 + sqrt(1-t3*t3) * t2 ) * t1
and so on.
The sqrt() for x in (-1,1) can be computed using
x_0 is some approximation, say, zero
x_(n+1) = 0.5 * (x_n + S/x_n) where S is the argument.
EDIT: I mean the "Babylonian method", see Wikipedia's article for details. You will need not more than 5-6 iterations to achieve 1e-10 with x in (0,1).
As Jonas Wielicki mentions in the comments, there isn't much precision trade-offs you can make.
Your best bet is to try and use the processor intrinsics for the functions (if your compiler doesn't do this already) and using some math to reduce the amount of calculations necessary.
Also very important is to keep everything in a CPU-friendly format, make sure there are few cache misses, etc.
If you are calculating large amounts of functions like acos perhaps moving to the GPU is an option for you?
You can try to create lookup tables, and use them instead of standard c++ functions, and see if you see any performance boost.
Significant gains can be made by aligning memory and streaming in the data to your kernel. Most often this dwarfs the gains that can be made by recreating the math functions. Think of how you can improve memory access to/from your kernel operator.
Memory access can be improved by using buffering techniques. This depends on your hardware platform. If you are running this on a DSP, you could DMA your data onto an L2 cache and schedule the instructions so that multiplier units are fully occupied.
If you are on general purpose CPU, most you can do is to use aligned data, feed the cache lines by prefetching. If you have nested loops, then the inner most loop should go back and forth (i.e. iterate forward and then iterate backward) so that cache lines are utilised, etc.
You could also think of ways to parallelize the computation using multiple cores. If you can use a GPU this could significantly improve performance (albeit with a lesser precision).
In addition to what others have said, here are some techniques at speed optimization:
Profile
Find out where in the code most of the time is spent.
Only optimize that area to gain the mose benefit.
Unroll Loops
The processors don't like branches or jumps or changes in the execution path. In general, the processor has to reload the instruction pipeline which uses up time that can be spent on calculations. This includes function calls.
The technique is to place more "sets" of operations in your loop and reduce the number of iterations.
Declare Variables as Register
Variables that are used frequently should be declared as register. Although many members of SO have stated compilers ignore this suggestion, I have found out otherwise. Worst case, you wasted some time typing.
Keep Intense Calculations Short & Simple
Many processors have enough room in their instruction pipelines to hold small for loops. This reduces the amount of time spent reloading the instruction pipeline.
Distribute your big calculation loop into many small ones.
Perform Work on Small Sections of Arrays & Matrices
Many processors have a data cache, which is ultra fast memory very close to the processor. The processor likes to load the data cache once from off-processor memory. More loads require time that can be spent making calculations. Search the web for "Data Oriented Design Cache".
Think in Parallel Processor Terms
Change the design of your calculations so they can be easily adaptable to use with multiple processors. Many CPUs have multiple cores that can execute instructions in parallel. Some processors have enough intelligence to automatically delegate instructions to their multiple cores.
Some compilers can optimize code for parallel processing (look up the compiler options for your compiler). Designing your code for parallel processing will make this optimization easier for the compiler.
Analyze Assembly Listing of Functions
Print out the assembly language listing of your function.
Change the design of your function to match that of the assembly language or to help the compiler generate more optimal assembly language.
If you really need more efficiency, optimize the assembly language and put in as inline assembly code or as a separate module. I generally prefer the latter.
Examples
In your situation, take first 10 terms of the Taylor expansion, calculate them separately and place into individual variables:
double term1, term2, term3, term4;
double n, n1, n2, n3, n4;
n = 1.0;
for (i = 0; i < 100; ++i)
{
n1 = n + 2;
n2 = n + 4;
n3 = n + 6;
n4 = n + 8;
term1 = 4.0/n;
term2 = 4.0/n1;
term3 = 4.0/n2;
term4 = 4.0/n3;
Then sum up all of your terms:
result = term1 - term2 + term3 - term4;
// Or try sorting by operation, if possible:
// result = term1 + term3;
// result -= term2 + term4;
n = n4 + 2;
}
Lets consider two terms first:
cos(a+b) = cos(a)*cos(b) - sin(a)*sin(b)
or cos(a+b) = cos(a)*cos(b) - sqrt(1-cos(a)*cos(a))*sqrt(1-cos(b)*cos(b))
Taking cos to the RHS
a+b = acos( cos(a)*cos(b) - sqrt(1-cos(a)*cos(a))*sqrt(1-cos(b)*cos(b)) ) ... 1
Here cos(a) = t_1 and cos(b) = t_2
a = acos(t_1) and b = acos(t_2)
By substituting in equation (1), we get
acos(t_1) + acos(t_2) = acos(t_1*t_2 - sqrt(1 - t_1*t_1) * sqrt(1 - t_2*t_2))
Here you can see that you have combined two acos into one. So you can pair up all the acos recursively and form a binary tree. At the end, you'll be left with an expression of the form sin(acos(x)) which equals sqrt(1 - x*x).
This will improve the time complexity.
However, I'm not sure about the complexity of calculating sqrt().
I have the following code that I use to compute the distance between two vectors:
double dist(vector<double> & vecA, vector<double> & vecB){
double curDist = 0.0;
for (size_t i = 0; i < vecA.size(); i++){
double dif = vecA[i] - vecB[i];
curDist += dif * dif;
}
return curDist;
}
This function is a major bottleneck in my application since it relies on a lot of distance calculations, consuming more than 60% of CPU time on a typical input. Additionally, the following line:
double dif = vecA[i] - vecB[i];
is responsible for more than 77% of CPU time in this function. My question is: is it possible to somehow optimize this function?
Notes:
To profile my application I have used Intel Amplifier XE;
Reducing the number of distance computations is not a feasible solution for
me;
There are two possible issues I can think of right now:
This computation is memory bound.
There is an iteration-to-iteration dependency on curDist.
This computation is memory bound.
Your dataset is larger than your CPU cache. So in this case, no amount of optimization is going to help unless you can restructure your algorithm.
There is an iteration-to-iteration dependency on curDist.
You have a dependency on curDist. This will block vectorization by the compiler. (Also, don't always trust the profiler numbers to the line. They can be inaccurate especially after compiler optimizations.)
Normally, the compiler vectorizer can split up the curDist into multiple partial sums to and unroll/vectorize the loop. But it can't do that under strict-floating-point behavior. You can try relaxing your floating-point mode if you haven't already. Or you can split the sum and unroll it yourself.
For example, this kind of optimization is something the compiler can do with integers, but not necessarily with floating-point:
double curDist0 = 0.0;
double curDist1 = 0.0;
double curDist2 = 0.0;
double curDist3 = 0.0;
for (size_t i = 0; i < vecA.size() - 3; i += 4){
double dif0 = vecA[i + 0] - vecB[i + 0];
double dif1 = vecA[i + 1] - vecB[i + 1];
double dif2 = vecA[i + 2] - vecB[i + 2];
double dif3 = vecA[i + 3] - vecB[i + 3];
curDist0 += dif0 * dif0;
curDist1 += dif1 * dif1;
curDist2 += dif2 * dif2;
curDist3 += dif3 * dif3;
}
// Do some sort of cleanup in case (vecA.size() % 4 != 0)
double curDist = curDist0 + curDist1 + curDist2 + curDist3;
You could eliminate the call to vecA.size() for each iteration of the loop, just call it once before the loop. You could also do loop unrolling to give yourself more computation per loop iteration. What compiler are you using, and what optimization settings? Compiler will often do unrolling for you, but you could manually do it.
If it's feasible (if the range of the numbers isn't huge) you may want to explore using fixed point to store these numbers, rather than doubles.
Fixed point would turn these into int operations rather than double operations.
Another interesting thing is that assuming your profile is correct, the lookups seems to be a significant factor (otherwise the multiplication would likely be more costly than the subtractions).
I'd try using a const vector iterator rather than the random access lookup. It may help in two ways: 1 - it is constant, and 2 - the serial nature of the iterator may let the processor do better caching.
If your platform does not have (or is not using) an ALU that supports floating point math, floating point libraries, by nature, are slow and consume additional non-volatile memory. I suggest instead using 32-bit (long) or 64-bit (long long) fixed-point arithmetic. Then convert the final result to floating point at the end of the algorithm. I did this on a project a couple years ago to improve the performance of an I2T algorithm and it worked wonderfully.