Why is C++ much faster than python with boost?

Question

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:

• The function f1(x,y,z) = x*x
• A function that is more difficult to evaluate: 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:

1. 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;
   }
   

2. 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)
   

3. 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:

  • Method 1: 0.07s user 0.01s system 99% cpu 0.083 total
  • Method 2: 0.19s
  • Method 3: 3.06s

• For f2:

  • Method 1: 0.28s user 0.01s system 99% cpu 0.289 total
  • Method 2: 12.47s
  • Method 3: 16.31s

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.

Answer 1

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.

Answer 2

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