Why is C++ much faster than python with boost?
My goal is to write a small library for spectral finite elements in Python and to that purpose I tried extending python with a C++ library using Boost, with the hope that it would make my code faster.
class Quad {
public:
Quad(int, int);
double integrate(boost::function<double(std::vector<double> const&)> const&);
double integrate_wrapper(boost::python::object const&);
std::vector< std::vector<double> > nodes;
std::vector<double> weights;
};
...
namespace std {
typedef std::vector< std::vector< std::vector<double> > > cube;
typedef std::vector< std::vector<double> > mat;
typedef std::vector<double> vec;
}
...
double Quad::integrate(boost::function<double(vec const&)> const& func) {
double result = 0.;
for (unsigned int i = 0; i < nodes.size(); ++i) {
result += func(nodes[i]) * weights[i];
}
return result;
}
// ---- PYTHON WRAPPER ----
double Quad::integrate_wrapper(boost::python::object const& func) {
std::function<double(vec const&)> lambda;
switch (this->nodes[0].size()) {
case 1: lambda = [&func](vec const& v) -> double { return boost::python::extract<double>(func (v[0])); }; break;
case 2: lambda = [&func](vec const& v) -> double { return boost::python::extract<double>(func(v[0], v[1])); }; break;
case 3: lambda = [&func](vec const& v) -> double { return boost::python::extract<double>(func(v[0], v[1], v[2])); }; break;
default: cout << "Dimension must be 1, 2, or 3" << endl; exit(0);
}
return integrate(lambda);
}
// ---- EXPOSE TO PYTHON ----
BOOST_PYTHON_MODULE(hermite)
{
using namespace boost::python;
class_<std::vec>("double_vector")
.def(vector_indexing_suite<std::vec>())
;
class_<std::mat>("double_mat")
.def(vector_indexing_suite<std::mat>())
;
class_<Quad>("Quad", init<int,int>())
.def("integrate", &Quad::integrate_wrapper)
.def_readonly("nodes", &Quad::nodes)
.def_readonly("weights", &Quad::weights)
;
}
I compared the performance of three different methods to calculate the integral of two functions. The two functions are:
f1(x,y,z) = x*x
f2(x,y,z) = np.cos(2*x+2*y+2*z) + x*y + np.exp(-z*z) +np.cos(2*x+2*y+2*z) + x*y + np.exp(-z*z) +np.cos(2*x+2*y+2*z) + x*y + np.exp(-z*z) +np.cos(2*x+2*y+2*z) + x*y + np.exp(-z*z) +np.cos(2*x+2*y+2*z) + x*y + np.exp(-z*z)
The methods used are:
Call the library from a C++ program:
double func(vector<double> v) {
return F1_OR_F2;
}
int main() {
hermite::Quad quadrature(100, 3);
double result = quadrature.integrate(func);
cout << "Result = " << result << endl;
}
Call the library from a Python script:
import hermite
def function(x, y, z): return F1_OR_F2
my_quad = hermite.Quad(100, 3)
result = my_quad.integrate(function)
Use a for
loop in Python:
import hermite
def function(x, y, z): return F1_OR_F2
my_quad = hermite.Quad(100, 3)
weights = my_quad.weights
nodes = my_quad.nodes
result = 0.
for i in range(len(weights)):
result += weights[i] * function(nodes[i][0], nodes[i][1], nodes[i][2])
Here are the execution times of each of the method (The time was measured using the time
command for method 1, and the python module time
for methods 2 and 3, and the C++ code was compiled using Cmake and set (CMAKE_BUILD_TYPE Release)
)
For f1
:
0.07s user 0.01s system 99% cpu 0.083 total
For f2
:
0.28s user 0.01s system 99% cpu 0.289 total
Based on these results, my questions are the following:
Why is the first method so much faster than the second?
Could the python wrapper be improved to reach comparable performance between methods 1 and 2?
Why is method 2 more sensitive than method 3 to the difficulty of the function to integrate?
EDIT : I also tried to define a function that accepts a string as argument, writes it to a file, and proceeds to compile the file and dynamically load the resulting .so
file:
double Quad::integrate_from_string(string const& function_body) {
// Write function to file
ofstream helper_file;
helper_file.open("/tmp/helper_function.cpp");
helper_file << "#include <vector>n#include <cmath>n";
helper_file << "extern "C" double toIntegrate(std::vector<double> v) {n";
helper_file << " return " << function_body << ";n}";
helper_file.close();
// Compile file
system("c++ /tmp/helper_function.cpp -o /tmp/helper_function.so -shared -fPIC");
// Load function dynamically
typedef double (*vec_func)(vec);
void *function_so = dlopen("/tmp/helper_function.so", RTLD_NOW);
vec_func func = (vec_func) dlsym(function_so, "toIntegrate");
double result = integrate(func);
dlclose(function_so);
return result;
}
It's quite dirty and probably not very portable, so I'd be happy to find a better solution, but it works well and plays nicely with the ccode
function of sympy
.
SECOND EDIT I have rewritten the function in pure Python Using Numpy.
import numpy as np
import numpy.polynomial.hermite_e as herm
import time
def integrate(function, degrees):
dim = len(degrees)
nodes_multidim = []
weights_multidim = []
for i in range(dim):
nodes_1d, weights_1d = herm.hermegauss(degrees[i])
nodes_multidim.append(nodes_1d)
weights_multidim.append(weights_1d)
grid_nodes = np.meshgrid(*nodes_multidim)
grid_weights = np.meshgrid(*weights_multidim)
nodes_flattened = []
weights_flattened = []
for i in range(dim):
nodes_flattened.append(grid_nodes[i].flatten())
weights_flattened.append(grid_weights[i].flatten())
nodes = np.vstack(nodes_flattened)
weights = np.prod(np.vstack(weights_flattened), axis=0)
return np.dot(function(nodes), weights)
def function(v): return F1_OR_F2
result = integrate(function, [100,100,100])
print("-> Result = " + str(result) + ", Time = " + str(end-start))
Somewhat surprisingly (at least to me), there is no significant difference in performance between this method and the pure C++ implementation. In particular, it takes 0.059s for f1
and 0.36s for f2
.
An alternative way
In a bit less general way your problem can be solved a lot easier. You could write the integration and the function in pure python code and compile it using numba.
First approach (running 0.025s (I7-4771) per integration after the first run)
The funktion is compiled at the first call, this takes about 0.5s
function_2:
@nb.njit(fastmath=True)
def function_to_integrate(x,y,z):
return np.cos(2*x+2*y+2*z) + x*y + np.exp(-z*z) +np.cos(2*x+2*y+2*z) + x*y + np.exp(-z*z) +np.cos(2*x+2*y+2*z) + x*y + np.exp(-z*z) +np.cos(2*x+2*y+2*z) + x*y + np.exp(-z*z) +np.cos(2*x+2*y+2*z) + x*y + np.exp(-z*z)
Integration
@nb.jit(fastmath=True)
def integrate3(num_int_Points):
nodes_1d, weights_1d = herm.hermegauss(num_int_Points)
result=0.
for i in range(num_int_Points):
for j in range(num_int_Points):
result+=np.sum(function_to_integrate(nodes_1d[i],nodes_1d[j],nodes_1d[:])*weights_1d[i]*weights_1d[j]*weights_1d[:])
return result
Testing
import numpy as np
import numpy.polynomial.hermite_e as herm
import numba as nb
import time
t1=time.time()
nodes_1d, weights_1d = herm.hermegauss(num_int_Points)
for i in range(100):
#result = integrate3(nodes_1d,weights_1d,100)
result = integrate3(100)
print(time.time()-t1)
print(result)
Second approach
The function can also run in parallell, when integrating over many elements the gauss points and weights may be calculated only once. This will result in a runtime of about 0.005s .
@nb.njit(fastmath=True,parallel=True)
def integrate3(nodes_1d,weights_1d,num_int_Points):
result=0.
for i in nb.prange(num_int_Points):
for j in range(num_int_Points):
result+=np.sum(function_to_integrate(nodes_1d[i],nodes_1d[j],nodes_1d[:])*weights_1d[i]*weights_1d[j]*weights_1d[:])
return result
Passing a abitrary function
import numpy as np
import numpy.polynomial.hermite_e as herm
import numba as nb
import time
def f(x,y,z):
return np.cos(2*x+2*y+2*z) + x*y + np.exp(-z*z) +np.cos(2*x+2*y+2*z) + x*y + np.exp(-z*z) +np.cos(2*x+2*y+2*z) + x*y + np.exp(-z*z) +np.cos(2*x+2*y+2*z) + x*y + np.exp(-z*z) +np.cos(2*x+2*y+2*z) + x*y + np.exp(-z*z)
def make_integrate3(f):
f_jit=nb.njit(f,fastmath=True)
@nb.njit(fastmath=True,parallel=True)
def integrate_3(nodes_1d,weights_1d,num_int_Points):
result=0.
for i in nb.prange(num_int_Points):
for j in range(num_int_Points):
result+=np.sum(f_jit(nodes_1d[i],nodes_1d[j],nodes_1d[:])*weights_1d[i]*weights_1d[j]*weights_1d[:])
return result
return integrate_3
int_fun=make_integrate3(f)
num_int_Points=100
nodes_1d, weights_1d = herm.hermegauss(num_int_Points)
#Calling it the first time (takes about 1s)
result = int_fun(nodes_1d,weights_1d,100)
t1=time.time()
for i in range(100):
result = int_fun(nodes_1d,weights_1d,100)
print(time.time()-t1)
print(result)
After the first call this takes about 0.002s using Numba 0.38 with Intel SVML
Your functions take vectors by value, which involves copying the vector. integrate_wrapper
does extra copies.
It also makes sense to accept boost::function
by reference and capture func
by reference in those lambdas.
Change these to (note the &
and const&
bits):
double integrate(boost::function<double(std::vector<double> const&)> const&);
double Quad::integrate_wrapper(boost::python::object func) {
std::function<double(vec const&)> lambda;
switch (this->nodes[0].size()) {
case 1: lambda = [&func](vec const& v) -> double { return boost::python::extract<double>(func (v[0])); }; break;
case 2: lambda = [&func](vec const& v) -> double { return boost::python::extract<double>(func(v[0], v[1])); }; break;
case 3: lambda = [&func](vec const& v) -> double { return boost::python::extract<double>(func(v[0], v[1], v[2])); }; break;
default: cout << "Dimension must be 1, 2, or 3" << endl; exit(0);
}
return integrate(lambda);
}
Still though, calling a Python function from C++ is more expensive than calling a C++ function.
People normally use numpy
for fast linear algebra in Python, it uses SIMD for many common operations. You should probably consider using numpy
first before rolling out a C++ implementation. In C++ you would have to use Intel MKL on Eigen to vectorize.
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