How is numpy so fast?

I'm trying to understand how numpy can be so fast, based on my shocking comparison with optimized C/C++ code which is still far from reproducing numpy's speed.

Consider the following example: Given a 2D array with shape=(N, N) and dtype=float32, which represents a list of N vectors of N dimensions, I am computing the pairwise differences between every pair of vectors. Using numpy broadcasting, this simply writes as:

def pairwise_sub_numpy( X ):
    return X - X[:, None, :]

Using timeit I can measure the performance for N=512: it takes 88 ms per call on my laptop.

Now, in C/C++ a naive implementation writes as:

#define X(i, j)     _X[(i)*N + (j)]
#define res(i, j, k)  _res[((i)*N + (j))*N + (k)]

float* pairwise_sub_naive( const float* _X, int N ) 
{
    float* _res = (float*) aligned_alloc( 32, N*N*N*sizeof(float));

    for (int i = 0; i < N; i++) {
        for (int j = 0; j < N; j++) {
            for (int k = 0; k < N; k++)
                res(i,j,k) = X(i,k) - X(j,k);
          }
    }
    return _res;
}

Compiling using gcc 7.3.0 with -O3 flag, I get 195 ms per call for pairwise_sub_naive(X), which is not too bad given the simplicity of the code, but about 2 times slower than numpy.

Now I start getting serious and add some small optimizations, by indexing the row vectors directly:

float* pairwise_sub_better( const float* _X, int N ) 
{
    float* _res = (float*) aligned_alloc( 32, N*N*N*sizeof(float));
    
    for (int i = 0; i < N; i++) {
        const float* xi = & X(i,0);

        for (int j = 0; j < N; j++) {
            const float* xj = & X(j,0);
            
            float* r = &res(i,j,0);
            for (int k = 0; k < N; k++)
                 r[k] = xi[k] - xj[k];
        }
    }
    return _res;
}

The speed stays the same at 195 ms, which means that the compiler was able to figure that much. Let's now use SIMD vector instructions:

float* pairwise_sub_simd( const float* _X, int N ) 
{
    float* _res = (float*) aligned_alloc( 32, N*N*N*sizeof(float));

    // create caches for row vectors which are memory-aligned
    float* xi = (float*)aligned_alloc(32, N * sizeof(float));
    float* xj = (float*)aligned_alloc(32, N * sizeof(float));
    
    for (int i = 0; i < N; i++) {
        memcpy(xi, & X(i,0), N*sizeof(float));
        
        for (int j = 0; j < N; j++) {
            memcpy(xj, & X(j,0), N*sizeof(float));
            
            float* r = &res(i,j,0);
            for (int k = 0; k < N; k += 256/sizeof(float)) {
                const __m256 A = _mm256_load_ps(xi+k);
                const __m256 B = _mm256_load_ps(xj+k);
                _mm256_store_ps(r+k, _mm256_sub_ps( A, B ));
            }
        }
    }
    free(xi); 
    free(xj);
    return _res;
}

This only yields a small boost (178 ms instead of 194 ms per function call).

Then I was wondering if a "block-wise" approach, like what is used to optimize dot-products, could be beneficials:

float* pairwise_sub_blocks( const float* _X, int N ) 
{
    float* _res = (float*) aligned_alloc( 32, N*N*N*sizeof(float));

    #define B 8
    float cache1[B*B], cache2[B*B];

    for (int bi = 0; bi < N; bi+=B)
      for (int bj = 0; bj < N; bj+=B)
        for (int bk = 0; bk < N; bk+=B) {
        
            // load first 8x8 block in the cache
            for (int i = 0; i < B; i++)
              for (int k = 0; k < B; k++)
                cache1[B*i + k] = X(bi+i, bk+k);

            // load second 8x8 block in the cache
            for (int j = 0; j < B; j++)
              for (int k = 0; k < B; k++)
                cache2[B*j + k] = X(bj+j, bk+k);

            // compute local operations on the caches
            for (int i = 0; i < B; i++)
             for (int j = 0; j < B; j++)
              for (int k = 0; k < B; k++)
                 res(bi+i,bj+j,bk+k) = cache1[B*i + k] - cache2[B*j + k];
         }
    return _res;
}

And surprisingly, this is the slowest method so far (258 ms per function call).

To summarize, despite some efforts with some optimized C++ code, I can't come anywhere close the 88 ms / call that numpy achieves effortlessly. Any idea why?

Note: By the way, I am disabling numpy multi-threading and anyway, this kind of operation is not multi-threaded.

Edit: Exact code to benchmark the numpy code:

import numpy as np

def pairwise_sub_numpy( X ):
    return X - X[:, None, :]

N = 512
X = np.random.rand(N,N).astype(np.float32)

import timeit
times = timeit.repeat('pairwise_sub_numpy( X )', globals=globals(), number=1, repeat=5)
print(f">> best of 5 = {1000*min(times):.3f} ms")

Full benchmark for C code:

#include <stdio.h>
#include <string.h>
#include <xmmintrin.h>   // compile with -mavx -msse4.1
#include <pmmintrin.h>
#include <immintrin.h>
#include <time.h>


#define X(i, j)     _x[(i)*N + (j)]
#define res(i, j, k)  _res[((i)*N + (j))*N + (k)]

float* pairwise_sub_naive( const float* _x, int N ) 
{
    float* _res = (float*) aligned_alloc( 32, N*N*N*sizeof(float));

    for (int i = 0; i < N; i++) {
        for (int j = 0; j < N; j++) {
            for (int k = 0; k < N; k++)
                res(i,j,k) = X(i,k) - X(j,k);
          }
    }
    return _res;
}

float* pairwise_sub_better( const float* _x, int N ) 
{
    float* _res = (float*) aligned_alloc( 32, N*N*N*sizeof(float));
    
    for (int i = 0; i < N; i++) {
        const float* xi = & X(i,0);

        for (int j = 0; j < N; j++) {
            const float* xj = & X(j,0);
            
            float* r = &res(i,j,0);
            for (int k = 0; k < N; k++)
                 r[k] = xi[k] - xj[k];
        }
    }
    return _res;
}

float* pairwise_sub_simd( const float* _x, int N ) 
{
    float* _res = (float*) aligned_alloc( 32, N*N*N*sizeof(float));

    // create caches for row vectors which are memory-aligned
    float* xi = (float*)aligned_alloc(32, N * sizeof(float));
    float* xj = (float*)aligned_alloc(32, N * sizeof(float));
    
    for (int i = 0; i < N; i++) {
        memcpy(xi, & X(i,0), N*sizeof(float));
        
        for (int j = 0; j < N; j++) {
            memcpy(xj, & X(j,0), N*sizeof(float));
            
            float* r = &res(i,j,0);
            for (int k = 0; k < N; k += 256/sizeof(float)) {
                const __m256 A = _mm256_load_ps(xi+k);
                const __m256 B = _mm256_load_ps(xj+k);
                _mm256_store_ps(r+k, _mm256_sub_ps( A, B ));
            }
        }
    }
    free(xi); 
    free(xj);
    return _res;
}


float* pairwise_sub_blocks( const float* _x, int N ) 
{
    float* _res = (float*) aligned_alloc( 32, N*N*N*sizeof(float));

    #define B 8
    float cache1[B*B], cache2[B*B];

    for (int bi = 0; bi < N; bi+=B)
      for (int bj = 0; bj < N; bj+=B)
        for (int bk = 0; bk < N; bk+=B) {
        
            // load first 8x8 block in the cache
            for (int i = 0; i < B; i++)
              for (int k = 0; k < B; k++)
                cache1[B*i + k] = X(bi+i, bk+k);

            // load second 8x8 block in the cache
            for (int j = 0; j < B; j++)
              for (int k = 0; k < B; k++)
                cache2[B*j + k] = X(bj+j, bk+k);

            // compute local operations on the caches
            for (int i = 0; i < B; i++)
             for (int j = 0; j < B; j++)
              for (int k = 0; k < B; k++)
                 res(bi+i,bj+j,bk+k) = cache1[B*i + k] - cache2[B*j + k];
         }
    return _res;
}

int main() 
{
    const int N = 512;
    float* _x = (float*) malloc( N * N * sizeof(float) );
    for( int i = 0; i < N; i++)
      for( int j = 0; j < N; j++)
        X(i,j) = ((i+j*j+17*i+101) % N) / float(N);

    double best = 9e9;
    for( int i = 0; i < 5; i++)
    {
        struct timespec start, stop;
        clock_gettime(CLOCK_THREAD_CPUTIME_ID, &start);
        
        //float* res = pairwise_sub_naive( _x, N );
        //float* res = pairwise_sub_better( _x, N );
        //float* res = pairwise_sub_simd( _x, N );
        float* res = pairwise_sub_blocks( _x, N );

        clock_gettime(CLOCK_THREAD_CPUTIME_ID, &stop);

        double t = (stop.tv_sec - start.tv_sec) * 1e6 + (stop.tv_nsec - start.tv_nsec) / 1e3;    // in microseconds
        if (t < best) best = t;
        free( res );
    }
    printf("Best of 5 = %f ms\n", best / 1000);

    free( _x );
    return 0;
}

Compiled using gcc 7.3.0 gcc -Wall -O3 -mavx -msse4.1 -o test_simd test_simd.c

Summary of timings on my machine:

Implementation	Time
numpy	88 ms
C++ naive	194 ms
C++ better	195 ms
C++ SIMD	178 ms
C++ blocked	258 ms
C++ blocked (gcc 8.3.1)	217 ms

As pointed out by some of the comments numpy uses SIMD in its implementation and it does not allocate memory at the point of computation. If I eliminate the memory allocation from your implementation, pre-allocating all the buffers ahead of the computation then I get a better time compared to numpy even with the scaler version(that is the one without any optimizations).

Also in terms of SIMD and why your implementation does not perform much better than the scaler is because your memory access patterns are not ideal for SIMD usage - you do memcopy and you load into SIMD registers from locations that are far apart from each other - e.g. you fill vectors from line 0 and line 511, which might not play well with the cache or with the SIMD prefetcher.

There is also a mistake in how you load the SIMD registers(if I understood correctly what you're trying to compute): a 256 bit SIMD register can load 8 single-precision floating-point numbers 8 * 32 = 256, but in your loop you jump k by "256/sizeof(float)" which is 256/4 = 64; _x and _res are float pointers and the SIMD intrinsics expect also float pointers as arguments so instead of reading all elements from those lines every 8 floats you read them every 64 floats.

The computation can be optimized further by changing the access patterns but also by observing that you repeat some computations: e.g. when iterating with line0 as a base you compute line0 - line1 but at some future time, when iterating with line1 as a base, you need to compute line1 - line0 which is basically -(line0 - line1), that is for each line after line0 a lot of results could be reused from previous computations. A lot of times SIMD usage or parallelization requires one to change how data is accessed or reasoned about in order to provide meaningful improvements.

Here is what I have done as a first step based on your initial implementation and it is faster than the numpy(don't mind the OpenMP stuff as it's not how its supposed to be done, I just wanted to see how it behaves trying the naive way).

C++
Time scaler version: 55 ms
Time SIMD version: 53 ms
**Time SIMD 2 version: 33 ms**
Time SIMD 3 version: 168 ms
Time OpenMP version: 59 ms

Python numpy
>> best of 5 = 88.794 ms


#include <cstdlib>
#include <xmmintrin.h>   // compile with -mavx -msse4.1
#include <pmmintrin.h>
#include <immintrin.h>

#include <numeric>
#include <algorithm>
#include <chrono>
#include <iostream>
#include <cstring>

using namespace std;

float* pairwise_sub_naive (const float* input, float* output, int n) 
{
    for (int i = 0; i < n; i++) {
        for (int j = 0; j < n; j++) {
            for (int k = 0; k < n; k++)
                output[(i * n + j) * n + k] = input[i * n + k] - input[j * n + k];
          }
    }
    return output;
}

float* pairwise_sub_simd (const float* input, float* output, int n) 
{    
    for (int i = 0; i < n; i++) 
    {
        const int idxi = i * n;
        for (int j = 0; j < n; j++)
        {
            const int idxj = j * n;
            const int outidx = idxi + j;
            for (int k = 0; k < n; k += 8) 
            {
                __m256 A = _mm256_load_ps(input + idxi + k);
                __m256 B = _mm256_load_ps(input + idxj + k);
                _mm256_store_ps(output + outidx * n + k, _mm256_sub_ps( A, B ));
            }
        }
    }
    
    return output;
}

float* pairwise_sub_simd_2 (const float* input, float* output, int n) 
{
    float* line_buffer = (float*) aligned_alloc(32, n * sizeof(float));

    for (int i = 0; i < n; i++) 
    {
        const int idxi = i * n;
        for (int j = 0; j < n; j++)
        {
            const int idxj = j * n;
            const int outidx = idxi + j;
            for (int k = 0; k < n; k += 8) 
            {
                __m256 A = _mm256_load_ps(input + idxi + k);
                __m256 B = _mm256_load_ps(input + idxj + k);
                _mm256_store_ps(line_buffer + k, _mm256_sub_ps( A, B ));
            }
            memcpy(output + outidx * n, line_buffer, n);
        }
    }
    
    return output;
}

float* pairwise_sub_simd_3 (const float* input, float* output, int n) 
{    
    for (int i = 0; i < n; i++) 
    {
        const int idxi = i * n;
        for (int k = 0; k < n; k += 8) 
        {
            __m256 A = _mm256_load_ps(input + idxi + k);
            for (int j = 0; j < n; j++)
            {
                const int idxj = j * n;
                const int outidx = (idxi + j) * n;
                __m256 B = _mm256_load_ps(input + idxj + k);
                _mm256_store_ps(output + outidx + k, _mm256_sub_ps( A, B     ));
             }
        }
    }

    return output;
}

float* pairwise_sub_openmp (const float* input, float* output, int n)
{
    int i, j;
    #pragma omp parallel for private(j)
    for (i = 0; i < n; i++) 
    {
        for (j = 0; j < n; j++)
        {
            const int idxi = i * n; 
            const int idxj = j * n;
            const int outidx = idxi + j;
            for (int k = 0; k < n; k += 8) 
            {
                __m256 A = _mm256_load_ps(input + idxi + k);
                __m256 B = _mm256_load_ps(input + idxj + k);
                _mm256_store_ps(output + outidx * n + k, _mm256_sub_ps( A, B ));
            }
        }
    }
    /*for (i = 0; i < n; i++) 
    {
        for (j = 0; j < n; j++) 
        {
            for (int k = 0; k < n; k++)
            {
                output[(i * n + j) * n + k] = input[i * n + k] - input[j * n + k];
            }
        }
    }*/
    
    return output;
}

int main ()
{
    constexpr size_t n = 512;
    constexpr size_t input_size = n * n;
    constexpr size_t output_size = n * n * n;

    float* input = (float*) aligned_alloc(32, input_size * sizeof(float));
    float* output = (float*) aligned_alloc(32, output_size * sizeof(float));

    float* input_simd = (float*) aligned_alloc(32, input_size * sizeof(float));
    float* output_simd = (float*) aligned_alloc(32, output_size * sizeof(float));

    float* input_par = (float*) aligned_alloc(32, input_size * sizeof(float));
    float* output_par = (float*) aligned_alloc(32, output_size * sizeof(float));

    iota(input, input + input_size, float(0.0));
    fill(output, output + output_size, float(0.0));

    iota(input_simd, input_simd + input_size, float(0.0));
    fill(output_simd, output_simd + output_size, float(0.0));
    
    iota(input_par, input_par + input_size, float(0.0));
    fill(output_par, output_par + output_size, float(0.0));

    std::chrono::milliseconds best_scaler{100000};
    for (int i = 0; i < 5; ++i)
    {
        auto start = chrono::high_resolution_clock::now();
        pairwise_sub_naive(input, output, n);
        auto stop = chrono::high_resolution_clock::now();

        auto duration = chrono::duration_cast<chrono::milliseconds>(stop - start);
        if (duration < best_scaler)
        {
            best_scaler = duration;
        }
    }
    cout << "Time scaler version: " << best_scaler.count() << " ms\n";

    std::chrono::milliseconds best_simd{100000};
for (int i = 0; i < 5; ++i)
{
    auto start = chrono::high_resolution_clock::now();
    pairwise_sub_simd(input_simd, output_simd, n);
    auto stop = chrono::high_resolution_clock::now();

    auto duration = chrono::duration_cast<chrono::milliseconds>(stop - start);
     if (duration < best_simd)
    {
        best_simd = duration;
    }
}
cout << "Time SIMD version: " << best_simd.count() << " ms\n";

std::chrono::milliseconds best_simd_2{100000};
for (int i = 0; i < 5; ++i)
{
    auto start = chrono::high_resolution_clock::now();
    pairwise_sub_simd_2(input_simd, output_simd, n);
    auto stop = chrono::high_resolution_clock::now();

    auto duration = chrono::duration_cast<chrono::milliseconds>(stop - start);
     if (duration < best_simd_2)
    {
        best_simd_2 = duration;
    }
}
cout << "Time SIMD 2 version: " << best_simd_2.count() << " ms\n";

std::chrono::milliseconds best_simd_3{100000};
for (int i = 0; i < 5; ++i)
{
    auto start = chrono::high_resolution_clock::now();
    pairwise_sub_simd_3(input_simd, output_simd, n);
    auto stop = chrono::high_resolution_clock::now();

    auto duration = chrono::duration_cast<chrono::milliseconds>(stop - start);
     if (duration < best_simd_3)
    {
        best_simd_3 = duration;
    }
}
cout << "Time SIMD 3 version: " << best_simd_3.count() << " ms\n";

    std::chrono::milliseconds best_par{100000};
    for (int i = 0; i < 5; ++i)
    {
        auto start = chrono::high_resolution_clock::now();
        pairwise_sub_openmp(input_par, output_par, n);
        auto stop = chrono::high_resolution_clock::now();

        auto duration = chrono::duration_cast<chrono::milliseconds>(stop - start);
         if (duration < best_par)
        {
            best_par = duration;
        }
    }
    cout << "Time OpenMP version: " << best_par.count() << " ms\n";

    cout << "Verification\n";
    if (equal(output, output + output_size, output_simd))
    {
        cout << "PASSED\n";
    }
    else
    {
        cout << "FAILED\n";
    }

    return 0;
}

Edit: Small correction as there was a wrong call related to the second version of SIMD implementation.

As you can see now, the second implementation is the fastest as it behaves the best from the point of view of the locality of reference of the cache. Examples 2 and 3 of SIMD implementations are there to illustrate for you how changing memory access patterns to influence the performance of your SIMD optimizations. To summarize(knowing that I'm far from being complete in my advice) be mindful of your memory access patterns and of the loads and stores to\from the SIMD unit; the SIMD is a different hardware unit inside the processor's core so there is a penalty in shuffling data back and forth, hence when you load a register from memory try to do as many operations as possible with that data and do not be too eager to store it back(of course, in your example that might be all you need to do with the data). Be mindful also that there is a limited number of SIMD registers available and if you load too many then they will "spill", that is they will be stored back to temporary locations in main memory behind the scenes killing all your gains. SIMD optimization, it's a true balance act!

There is some effort to put a cross-platform intrinsics wrapper into the standard(I developed myself a closed source one in my glorious past) and even it's far from being complete, it's worth taking a look at(read the accompanying papers if you're truly interested to learn how SIMD works). https://github.com/VcDevel/std-simd

This is a complement to the answer posted by @celakev . I think I finally got to understand what exactly was the issue. The issue was not about allocating the memory in the main function that does the computation.

What was actually taking time is to access new (fresh) memory. I believe that the malloc call returns pages of memory which are virtual, i.e. that does not corresponds to actual physical memory -- until it is explicitly accessed. What actually takes time is the process of allocating physical memory on the fly (which I think is OS-level) when it is accessed in the function code.

Here is a proof. Consider the two following trivial functions:

#include <stdio.h>
#include <stdlib.h>
#include <time.h>

float* just_alloc( size_t N ) 
{
    return (float*) aligned_alloc( 32, sizeof(float)*N );
}

void just_fill( float* _arr, size_t N ) 
{
    for (size_t i = 0; i < N; i++)
        _arr[i] = 1;
}

#define Time( code_to_benchmark, cleanup_code ) \
    do { \
        double best = 9e9; \
        for( int i = 0; i < 5; i++) { \
            struct timespec start, stop; \
            clock_gettime(CLOCK_THREAD_CPUTIME_ID, &start); \
            code_to_benchmark; \
            clock_gettime(CLOCK_THREAD_CPUTIME_ID, &stop); \
            double t = (stop.tv_sec - start.tv_sec) * 1e3 + (stop.tv_nsec - start.tv_nsec) / 1e6; \
            printf("Time[%d] = %f ms\n", i, t); \
            if (t < best) best = t; \
            cleanup_code; \
        } \
        printf("Best of 5 for '" #code_to_benchmark "' = %f ms\n\n", best); \
    } while(0)

int main() 
{
    const size_t N = 512;

    Time( float* arr = just_alloc(N*N*N), free(arr) );
    
    float* arr = just_alloc(N*N*N);
    Time( just_fill(arr, N*N*N), ; );
    free(arr);

    return 0;
}

I get the following timings, which I now detail for each of the calls:

Time[0] = 0.000931 ms
Time[1] = 0.000540 ms
Time[2] = 0.000523 ms
Time[3] = 0.000524 ms
Time[4] = 0.000521 ms
Best of 5 for 'float* arr = just_alloc(N*N*N)' = 0.000521 ms

Time[0] = 189.822237 ms
Time[1] = 45.041083 ms
Time[2] = 46.331428 ms
Time[3] = 44.729433 ms
Time[4] = 42.241279 ms
Best of 5 for 'just_fill(arr, N*N*N)' = 42.241279 ms

As you can see, allocating memory is blazingly fast, but the first time that the memory is accessed, it is 5 times slower than the other times. So, basically the reason that my code was slow was because i was each time reallocating fresh memory that had no physical address yet. (Correct me if I'm wrong but I think that's the gist of it!)

A bit late to the party, but I wanted to add a pairwise method with Eigen, which is supposed to give C++ a high-level algebra manipulation capability and use SIMD under the hood. Just like numpy.

Here is the implementation

#include <iostream>
#include <vector>
#include <chrono>
#include <algorithm>
        
#include <Eigen/Dense>

auto pairwise_eigen(const Eigen::MatrixXf &input, std::vector<Eigen::MatrixXf> &output) {
        for (int k = 0; k < input.cols(); ++k)
                output[k] = input
                          // subtract matrix with repeated k-th column
                          - input.col(k) * Eigen::RowVectorXf::Ones(input.cols());

}

int main() {
        constexpr size_t n = 512;
        
        // allocate input and output 
        Eigen::MatrixXf input = Eigen::MatrixXf::Random(n, n);
        std::vector<Eigen::MatrixXf> output(n);

        std::chrono::milliseconds best_eigen{100000};
        for (int i = 0; i < 5; ++i) {
                auto start = std::chrono::high_resolution_clock::now();
                pairwise_eigen(input, output);
                auto end = std::chrono::high_resolution_clock::now();
         
                auto duration = std::chrono::duration_cast<std::chrono::milliseconds>(end-start);      
                if (duration < best_eigen)
                        best_eigen = duration;
        }

        std::cout << "Time Eigen version: " << best_eigen.count() << " ms\n";

        return 0;
}

The full benchmark tests suggested by @celavek on my system are

Time scaler version: 57 ms
Time SIMD version: 58 ms
Time SIMD 2 version: 40 ms
Time SIMD 3 version: 58 ms
Time OpenMP version: 58 ms

Time Eigen version: 76 ms

Numpy >> best of 5 = 118.489 ms

Whit Eigen there is still a noticeable improvement with respect to Numpy, but not so impressive compared to the "raw" implementations (there is certainly some overhead). An extra optimization is to allocate the output vector with copies of the input and then subtract directly from each vector entry, simply replacing the following lines

// inside the pairwise method
        for (int k = 0; k < input.cols(); ++k)
                output[k] -= input.col(k) * Eigen::RowVectorXf::Ones(input.cols());


// at allocation time
        std::vector<Eigen::MatrixXf> output(n, input);

This pushes the best of 5 down to 60 ms.