{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "*This post was entirely written using the IPython notebook. Its content is BSD-licensed. You can see a static view or download this notebook with the help of nbviewer at [20170731_CythonAndComplexNumbers.ipynb](http://nbviewer.ipython.org/urls/raw.github.com/flothesof/posts/master/20170731_CythonAndComplexNumbers.ipynb).*" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Last week, I had the pleasure to dive deep into the Cython world in order to solve a physics problem involving complex numbers.\n", "This post describes some of the things I've learnt concerning Cython, complex numbers and parallelization." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# What problem am I trying to solve? " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The problem I was trying to solve was the following: given a vector of complex numbers ($w_i$, \"weights\"), I wanted to compute its inverse Fourier transform given a certain angular frequency, $\\omega$.\n", "\n", "So what I want to compute is this: \n", "\n", "$$\n", "\\mathcal{Re}(\\sum_{i=1}^n w_i \\exp(i \\omega t))\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Beware that $t$ is actually a vector." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Reference implementation with NumPy " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "First, let's see our reference implementation. We make use of NumPy broadcasting to effectively perform the summation part in a single operation over all time steps." ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": true }, "outputs": [], "source": [ "import numpy as np" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def synthesis(weights, omega, time_vector):\n", " \"\"\"Sums weighted complex exponentials.\"\"\"\n", " return (weights[:, np.newaxis] * np.exp(1j * omega * time_vector[np.newaxis, :])).sum(axis=0).real" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's apply this on some sample data." ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "weights = (np.random.randn(1000) + 1j * np.random.randn(1000)) * np.exp(-np.arange(1000))\n", "omega = 2 * np.pi * 1e-3\n", "time_vector = np.arange(2000, dtype=np.float)" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [], "source": [ "ref_result = synthesis(weights, omega, time_vector)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's inspect the shape and the dtype of the result :" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "(2000,)" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "ref_result.shape" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "dtype('float64')" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "ref_result.dtype" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This is the expected shape since time step vector is of size 2000. Also, our output is floats, meaning the dtype is as expected. Let's plot the result to have some fun." ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": true }, "outputs": [], "source": [ "import matplotlib.pyplot as plt\n", "plt.style.use('bmh')\n", "%matplotlib inline" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[]" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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6cr1aqHrY1WKqBtT1C/JsNotg51i+N6lKb15Vz/TierVQmruryQnJytXpfpWr\nY4K8VmNqtQh2Fub/PqJAb77Ta3JjqSxng3GsIj/zZKtS1S+4yqalDFS5W1TVM724tNJJLVXWqjrd\nr3LpEuSFEO8QQpwSQpwVQjxWYfsvCCGOCiFeE0I8L4TYpcdxS1VaY6r9xX9VgZ5Wp9fkHvNHyUm4\n0dvc1MLlUtUvuMqm3bGo0pNX1TM9uGKpLGdm41h06kRAZ/tVSS0HeSGEFfgT4J3ADuAjQogdZbu9\nAbxVSnkL8DvAn7d63HKVlh9pF6GWRjBTnV6udbTg8a063Uqr6hdcZbthyEWfw4o/ksIfMWahh0ak\nqmd6cB2f1joRvfQ6Wu9EQGf7VUl69OTvBs5KKc9LKVPAV4BHS3eQUj4vpdTqg14A1ulw3GtUuhBA\n6UU4HTE3/6baAgWa9OI66tc3yKvqF1xls5YsWKFCR0JVz/Tg0lKueqVqoLP9qiQ9gvxa4FLJ68uF\n31XTPuB7Ohz3GoVCoeLP116E5t5Sl3KpJD24tFtpvfLxoK5fcC2bNrX1qwrMeqqqZ3pwadevXoOu\n0Nl+VVLr9UgNSAjxEPkg/5ZK22dmZti3bx82m41sNsvevXvZv38/fr8ft9uN1WolHA7j9XqZm5tD\nSonX62V6ehq73U4wGCQajTI2NsZGV4YDwMFLC9zYE2VgYIBsNkssFmN8fBy/34/dbsfj8TA7O4vH\n4yGVSpFIJIrbHQ4H/f39BINBBgcHSSQSJJPJ4nan04nL5WJ+fp7h4WEikQipVKq43eVy0dPTg8/n\nY2RkhFAoRDqdLm6v1aa+vvwXW2tTIBBACMHQ0BCBQKClNqXTacLhcFNtcjgchEIhfCkXOQlbPFYs\nuQw+3+WW2+RwOJidnW37edLatNx5SqfTxONx5ubmGLfkyydfubTA7OwsQghDzlM9bert7WVqaqqp\nNhn53bNYLPh8vqbPk38uxNnZBDYL9KfmmJlJ6tKmdDpNMpls+3mq9d1zu91MTk421aZl426ra1YK\nId4E/LaU8u2F158BkFJ+rmy/W4F/At4ppTxd6bMOHDggJyYmmuKYnJxkzZo1xdfngwk++U+v43Xb\n+dsP39zSfCqtqJxLFenB9cWXrvDVozN8aNcY++7Sp42q+gXXsuWk5EN/d4xQMsNffeAm1nrMmw9F\nVc9a5TrgC/FbT5zn5jE3//M925XhMkqtcB06dOjgnj17dlfapke65mVgmxBisxDCAXwY+HbpDkKI\nDcA3gI9h4/H3AAAgAElEQVRVC/CtKp2+9sGUTUNOBnqsBGJppkzMy5dzqSI9uIr18TreSqvqF1zL\nZhGi2O4jJuflVfWsVS4jUjXQuX5VU8tBXkqZAT4F/AA4CXxVSnlcCPFJIcQnC7v9JjAM/KkQ4ogQ\n4pVWj1uu8hpTixDcqj2CbmLetFNrcuMlpW07RvXJx4O6fsFStmKQNzkvr6pnrXId1fEhu1J1ql/V\npEudvJTycSnldinlDVLK3y387gtSyi8Ufv5XUspBKeVthX8VbytaUaUa0+JDKyb2tDq1Jvf4dIyc\nLDxhrFNpG6jrFyxlK51Co9W0ZytS1bNWuMLJDOeCCewWoWsnAjrTr+XUMU+8ut1Lvwi7SiogzLoI\nK3GpoFa5jhp0K62qX7CUbZ2nh6FeGwvJDL4F85aUU9WzVrhe80eRwMSomx6bvmGqE/1aTh0T5K3W\npb3JDaucDLpszCUyXAqZ89BKJS4V1CrXq8WHoPS9lVbVL1jKJoTgtmJK0Ly7RVU9a4XLiPEeTZ3o\n13LqmCAfDoeX/E4IUXxIx6x65kpcKqgVrkQ6y2mdHzXXpKpfUJlNyxcfNTElqKpnrXBpd4p6TC1c\nrk70azl1TJD3er0Vf689KWdWXr4al9lqhUvLx2/TOR8P6voFldluLUyG95rfvLy8qp41yxVKZjg/\nl8RhFUx49U9hdJpftdQxQX5ubq7i70sHX824CKtxma1WuIy8lVbVL6jMtmbAwXCvnZCJeXlVPWuW\nS7srumnUjUPnfDx0nl+11DFBvloAXzvQY+pFaGbVxXJqhUu7ldZrvppSqeoXVGYrTQmalbJR1bNm\nuYr18TqXTmrqNL9qqWOCfLVbHbMvwk67NUyks5wOaPl4/YO8qn5BdbZbtJRNNyV4jZrl0u4UbzOg\nEwGd51ctdUyQn56errptl4kzBi7HZaaa5To+HSNbyMe7dc7Hg7p+QXW2YifCpLy8qp41wzWfSOOb\nT9Jjs3Cjt9cAqs7yqx51TJBfbqIebfD1qAl5+XomEDJDzXIV548fN6ZdqvoF1dnWe3oYdNmYT2S4\nbEKprqqeNcOl3Q3tGHVjtxoTnjrJr3rUMUF+Oa0ZcDBi8uBYp0jvRUI6QUKIYspGm1+/q+ak+WfE\noP5KVccE+Wi0+sVlZl5+OS4z1QxXIp3lVCCGRVBcR1dvqeoXLM9m5riPqp41w9WOTkQn+VWPOibI\nj42NLbvdrLx8LS6z1AzXiUI+fuuwMfl4UNcvWJ5NC0qvmZASVNWzRrlCyQwX5vP18dsNysdD5/hV\nrzomyAcCgWW331qSl8+18SKsxWWWmuFqRy9LVb9gebaNq5x4nDZm4+2f2lpVzxrles1fUh9vUD4e\nOsevetUxQb7WoiDX5OXn25eXN2uxklpqhkvv9VwrSVW/YHm2fF4+/3Rmu1M2qnrWKFe7xns6xa96\n1TFBfmhoaNntZuXla3GZpUa58vn4fH38Tp3nqymVqn5BbTazBl9V9axRLqMrtzR1il/1qmOCfD23\nOtoTdO3My3fKreHJmRiZnGTLkIu+HuOWBlbVL6gnJah1Ito7GZ6qnjXCFVnM8MZcfv74CZ3njy9X\nJ/jViDomyA8MDNTcRxt8fc3fvrx8PVxmqFEuI+erKZWqfkFtts1DLvp7rMxE0/gj7auXV9WzRriO\n+WNI4MbRXt3njy9XJ/jViDomyGez2Zr7rO53MOJub16+Hi4z1CjXawbNH18uVf2C2mwWIdg51v6U\noKqeNcJVnA/J4FQNdIZfjUiXIC+EeIcQ4pQQ4qwQ4rEK24UQ4o8K248KIe7Q47ilisVi9XC2vZSy\nHi4z1AhXMpPj9UAcAcXBRaOkql9QH9stJXeL7ZKqnjXC1Y5BfU2d4FcjajnICyGswJ8A7wR2AB8R\nQuwo2+2dwLbCv18C/nerxy1XvYvgXi2lbE/etBMWDT45nc/H3zBsbD4e1PUL6mMzY3BfVc/q5Yql\nspwLJrBZBDcZnI+H69+vRqVHT/5u4KyU8ryUMgV8BXi0bJ9Hgb+Reb0ArBJCrNbh2EXVuwjurpKL\nsB15+U5YNLidvSxV/YL62G4YctFrtzAVSRGItadeXlXP6uU6Ph0tLgrvshu/NN/17lej0iPIrwUu\nlby+XPhdo/u0JLvdXtd+q/sdeN12wovZtuTl6+VqtxrhKs7vbXA+HtT1C+pjs1pK5rFpU29eVc/q\n5Wr3fEjXu1+Nyth77wY1MzPDvn37sNlsZLNZ9u7dy/79+/H7/bjdbqxWK+FwGK/Xy9zcHFJKvF4v\n09PT2O12gsEg0WiUsbExAoEAQgiGhoYIBAIMDAyQzWaJxWLsHOvlqfMhfvL6FUZ3rWZ2dhaPx0Mq\nlSKRSDA+Po7f78fhcNDf308wGGRwcJBEIkEymSxudzqduFwu5ufnGR4eJhKJkEqlittdLhcWiwWf\nz8fIyAihUIh0Ol3cXqtN2qx09bRJ+0y73Y7H46nZpng8Tjgcrtmm2YUwr0/HEIAnPcfMzCIOh4NQ\nKGRImxwOB7Ozs021qZXzVE+b4vE48Xi8Zps2uDK8CBy6tMBWe6Sl81RPm3p7e5mammqqTUZ+97LZ\nLD6fr2abXp1MALDemSYcDrd8nmq1KR6Pk0wmdb2e9Pjuud1uJicnm2rTchKtzrMhhHgT8NtSyrcX\nXn8GQEr5uZJ9/gx4Wkr55cLrU8CDUsqp0s86cOCAnJiYaIrD5/OxcePGuvb9/qkgf/jsRd6yycNv\nPrylqeMZwdVO1ct1eDLCpx8/yw3DLv73+5o7N0ZwmaF62V6fifEr3z7NOk8Pf/mB8uEp87jarXq4\n4qkse//vUQC+8bFbdV8zuFkuM9QK16FDhw7u2bNnd6VteqRrXga2CSE2CyEcwIeBb5ft823g44Uq\nm3uBUHmAb1Uej6fufUsrbIzOyzfC1U7Vy/Vam2+lVfUL6mfbNtKLy27hcmiRYDxtMJW6ntXDdWLG\nuEXhq+l69qsZtRzkpZQZ4FPAD4CTwFellMeFEJ8UQnyysNvjwHngLPB/gH/b6nHLlUrVP8g1XsjL\nRxazXJgzNi/fCFc7VS9Xux6C0qSqX1A/m9UiuHmsffPYqOpZPVztmsqgVNezX81Ilzp5KeXjUsrt\nUsobpJS/W/jdF6SUXyj8LKWU+wvbb5FSvqLHcUuVSCTq3vfaenljSykb4Wqn6uFazOR4fSafj99p\nwHqulaSqX9AYWzvXfVXVs3q4zFiE5nr2qxl1zBOvjdaYlk49bKSu55rc12dipHOSzUMuBpztGaNX\n1S9ojK103VejpapntbiSmRynZ+OGLkJTSderX82qY4J8ozWmu9ZcvQiNzMtfzzW57U7VgLp+QWNs\n20d66bEKLi4kmU8Ym5dX1bNaXNpDdluGXIYtQlNJ16tfzapjgrzD4Who//E+B6N9xuflG+Vql+rh\n0u5ybmljkFfVL2iMzW61sKOQlzd6igNVPavF1c6H7Ep1vfrVrDomyPf3N/agTn5+eW3qYePy8o1y\ntUu1uFKZHCcD+bk02jkopqpf0DjbLYXvl9F5eVU9q8V1tHin2F7+69WvZtUxQT4YDDb8nnZMVtYM\nVztUi+v1QIx0VrJ50Nm2fDyo6xc0zrarTfPYqOrZclypTI7XA4VBfYMnvSvX9ehXK+qYID84ONjw\ne25tw/zyzXC1Q7W4jkwWellr2tvrUdUvaJztRm8vDqvgjfkk4WTGICp1PVuOq9iJGHLRb/Ckd+W6\nHv1qRR0T5JspPyrNy78xZ0z50vVarmVGaRuo6xc0zuawWoqzKhqZl1fVs+W4XjXp+wXXp1+tqGOC\nfDLZ+OBpvl7e2FLKZrjaoeW4tHy8oL35eFDXL2iOrR2Tlanq2XJcZjwEpel69KsVdUyQb7bG1Oi8\n/PVYk3ty5uqtdDvz8aCuX9AcWzvGfVT1rBpXKpvj5Ex+UL+dlVuarje/WlXHBPlma0yNzstfjzW5\nZtTHa1LVL2iO7aZRN3ar4PxcwrC8vKqeVeM6HYiTyko2DjrxtLkTAdefX62qY4K80+ls6n3j/T2M\n9TkMy8s3y2W0luMyKx8P6voFzbE5bBZ2FPLyRj39qqpn1bjMTNXA9edXq+qYIO9yuZp+760G3lK3\nwmWkqnGV5uNvMeEiVNUvaJ6tmLKZNCbIq+pZNS6zHoLSdL351ao6JsjPz883/V4j86atcBmpalxm\n5uNBXb+geTatDNWoh+5U9awSVyYnOTFdyMeb1JO/nvzSQx0T5IeHh5t+r9ajOGZAXr4VLiNVjcvM\nfDyo6xc0z6bVy1+YT7JgwDw2qnpWievMbJxkJsc6Tw9DveYsw3c9+aWHOibIRyLN95KMzMu3wmWk\nqnGZWb8M6voFzbM5rBZD55dX1bNKXEcmtfWCzfl+wfXllx7qmCDf6oT72pfuiM550+tpgYLS+ePN\nupVW1S9ojW1XcZ4k/YO8qp5V4tLaf1ubn6Qu1fXklx7qmCDfao3prQbNM3I91eSeLMwfv2XYnHw8\nqOsXtMZm5LiPqp6Vc6WyOY6bPOgK149feqljgnyrNaal9fLZnH55+eupJtfM0klNqvoFrbFt9/bS\nY7NwcSHJnM7rvqrqWTnXqUCcxaxk06CTQZc5+Xi4fvzSSy0FeSHEkBDiCSHEmcL/S2bYEUKsF0I8\nJYQ4IYQ4LoT41VaOWU2tlh9pefloSt+8/PVUrmX2oCuo6xe0xma3WthZyMvr3ZtX1bNyrqv5eHOn\n+r1e/NJLrfbkHwOelFJuA54svC5XBvg1KeUO4F5gvxBiR4vHXSI9Jtw34pb6elmgwIz1XCtJVb+g\ndbbiamQ6l1Kq6lk5l/acwG1rzPt+wfXjl15qNcg/Cnyp8POXgPeW7yClnJJSHir8HAFOAmtbPO4S\nhUKhlj/j6kWoX5DXg8sIlXOpkI8Hdf2C1tmMGnxV1bNSrsVMfr4agbnpQLg+/NJTrV7NY1LKqcLP\nfmBsuZ2FEJuA24EXK22fmZlh37592Gw2stkse/fuZf/+/fj9ftxuN1arlXA4jNfrZW5uDiklXq+X\n6elp7HY7wWCQaDTK2NgYgUAAIQRDQ0MEAgEGBgbIZrPEYjHGx8fx+/3Y7XY8Hg+zs7N4PB7W2hcB\nOOqP8MaFCzh7eujv7ycYDDI4OEgikSCZTBbf73Q6cblczM/PMzw8TCQSIZVKFbe7XC56enrw+XyM\njIwQCoVIp9PF7bXa1NeXvxhaaVMqlSKRSBS3OxwO+vv7SafThMPhYpsO+vN/77cOWAiHw8u2yeFw\nEAqFDGmTw+Fgdna2qTa1cp7qaVM6nSYejzd9nhyxKC6bhcuhRY6fv8SmsSFd2tTb28vU1FRbz1M9\n3z2LxYLP52N8fJynjl/KLwo/2MOc/wrSwPNUq03pdJpkMqnr9aTHd8/tdjM5OdlUm5aNu7LGwz9C\niB8BlYZ9Pwt8SUq5qmTfeSllxZnvhRB9wDPA70opv1FpnwMHDsiJiYma0JU0OTnJmjVrmnpvqT7+\nD8fxR1L8yXtvZNtIb8ufpxeX3irn+vffOc0xf4zffmQzb964apl3tpdLJenB9tnvn+Ply2E+/eBG\n9mwdUobLCJVy/dXLk3z51Wnef8sov3SP7jfyTXOppFa4Dh06dHDPnj27K22rma6RUj4spdxZ4d+3\ngGkhxGqAwv8zlT5DCGEHvg78XbUA36rSaX0qFvTOy+vFpbdKuZKZHKdm4qbWx2tS1S/Qh82IeWxU\n9ayUS4VBfU3Xg196qtWc/LeBTxR+/gTwrfIdhBAC+AvgpJTyD1s8XlXpVWN6tV5en8Gx66Em95g/\nSjon2TrS/qXYyqWqX6APW3Hcx6/f4Kuqnmlc8VSWU4EYFgE7Te5EgPp+6a1Wg/zvAY8IIc4ADxde\nI4RYI4R4vLDPfcDHgJ8RQhwp/HtXi8ddIr1qTLXBsdf8MV3q5a+HmlyttO12E59C1KSqX6AP29bh\nXnrtFibDKWai+jzhqKpnGtex6ShZCdtHenE7rCZTqe+X3mqp2yalDAJ7Kvx+EnhX4eefAqKV49Qj\nt1ufFd9H+xys7ncwFUlxfi7Rcl5eLy69Vcp16Eo+yJv5qLkmVf0CfdisFsEt4328eCnMq1MRHtnW\n+qRUqnqmcV0tnTT/+wXq+6W3OuaJV6tVvx6CnvPL68mlpzSucDLDuWACu0UocSutql+gH5veeXlV\nPdO4jkyZPylZqVT3S291TJAPh8O6fVaxnnmy9bypnlx6SuM6MhVBAjvG3Dht5n8dVPUL9GPTerRH\npiLUqm6rR6p6Fg6HiSzmOxE2i+BmBToRoLZfRsj8q1oneb1e3T5Lz3ls9OTSUxrXkSv53qQK+XhQ\n1y/Qj23LsIuBHisz0TRXwostf56qnnm93sLayTAx2qtEJwLU9ssIqeG6Dpqbm9Pts7S8fDyd41yL\n89joyaWnNK5D2qDrWjWCvKp+gX5sFiGKf1S18ZBWpKpnc3NzHNbGe0yer6ZUKvtlhDomyOtx21uq\nYilliykbvbn0kpSSmWiKyfAivXYL23V48EsPqeoX6Mt2R+GP6kEdgryqnkkpi+27U5FOBKjtlxHq\nmCCv962OXvOMqHxreLhkVkCrxfACqLqkql+gL9sdaweA/LhPp6YE6V3F5VC+E3HjqDoVLar61U3X\n1ND09LSun6dXXl5vLr00PT1dTBWokqoBdf0CfdnG+h2sHeghns7xeiDW0mep6tkzp/LTWu1a049N\nkU4EqOuXUVwdE+TrmainEY32OVgzkM/Lnw3Gm/4cvbn0ktvtLnkISh1GVf0C/dm0lE2reXlVPTsT\nyneOVErVgLp+GcXVMUHeCGmDY69cVnPh31Z0OZJmPpFhqNfGhlVOs3FWpPQK8ioqJyXHAkngaju7\nMkcdE+SjUf3XzrxzXT5vevBy8/WrRnDpIa1Nt6/pJz+9kBpS1S/Qn+22Nf1YRH4u/1gq2/TnqOjZ\nuWCCSCrHaJ+dtQM9ZuNcIxX9AuO4OibIj40tO5V9U7q9cBGeaOEiNIJLD52L5k+9KvXxmlT1C/Rn\nczusTHjd5GRrC9Wo6NnBK/lOxJ1rB5TqRICafoFxXB0T5AOBgO6f6XZY2TGavwgPN1lKaQRXq0pl\nchz154OKdreiilT0S5MRbHqkbFT0TGuPiqkaFf0C47g6Jsgb1VtoNWWjWi8G8rMCprKwZcjFcK/d\nbJxrpKJfmoxguxrkm08JquZZMpPjuD+/1J9qd4qgnl+ajOLqmCA/NKTPKjvl2r3u6uBrMw8rGMXV\nirSB5LvWqXcBquiXJiPYJkbd9NotXAotEog1N/Wwap69NpVfn+CGIaep6wVXk2p+aTKKq2OCvFG3\nOluHexnosTIdTTU1z4iKt4YvF+5KdiuWqgE1/dJkBJvNIorPZBxssopLNc+0u5Jt6n29APX80tRN\n19TQwIAx3yirRRRvqZsppTSKq1nNRFP45pM4bYIdY+o8hahJNb9KZRSb9sf2lSZTgqp5pk1lsHu9\nx2SSylLNL01GcXVMkM9mmy9Bq6XdLeTljeRqRlobbh5xYreqd/pV86tURrHdpX2/rkTINPF0tUqe\nzURTXJhP0mu3sG1QrfEeTSr5VSqjuFq6yoUQQ0KIJ4QQZwr/Dy6zr1UIcVgI8Z1WjllNsVhrj4Yv\npzsL84wcmYqSyuYaeq+RXM3o5cLdyMQqNQefVPOrVEaxrR7oYb2nh1gqy4npxkspVfLspUv5TsQd\na/tZTDT/pLiRUsmvUhnF1WpX7jHgSSnlNuDJwutq+lXgZIvHqyojF+cddtvZPOhkMZPj+HRjJ0Kl\nRYMzOVnMlz44scZkmspSya9yGcl2z4Z8akMLko1IJc9evBgC4K71HqW4SrXSuFoN8o8CXyr8/CXg\nvZV2EkKsA94NfLHF41WV0YvzaqWULzd4Eaq0aPDrMzHi6RzrPD0QU3NObZX8KpeRbHetz3+/mgny\nqniWyuSK8yHdvW5AGa5yrTSuVoP8mJRyqvCzH6j2yNbngd8AGst1NCC73dj8392Fi1DrqdQro7ka\nkVZVc9e6AaW4SqUqFxjLtnPMjctu4cJ8kploY6WUqnh21B9lMSvZOuxi2G1XhqtcK42rZhGrEOJH\nQKX7iM+WvpBSSiHEklEjIcS/AGaklAeFEA8ud6yZmRn27duHzWYjm82yd+9e9u/fj9/vx+12Y7Va\nCYfDeL1e5ubmkFLi9XqZnp7GbrcTDAaJRqOMjY0RCAQQQjA0NEQgEGBgYIBsNkssFmN8fBy/34/d\nbsfj8TA7O4vH4yGVSpFIJIrbHQ4H/f39BINBNnhW0WsTXAot8kYggiU+h9PpxOVyMT8/z/DwMJFI\nhFQqVXy/y+XCYrHg8/kYGRkhFAqRTqeL22u1SZuVTq82/fTcLAA3j9iJxyOEw2ESiQTJZLL4/nra\n5HA4CIVChrTJ4XAwOzvb9HkaHBw0rE3xeJx4PG7YeZoYtHJ4Jsf3jrzB224YqLtNvb29TE1NtfU8\nVWrTj0/nV1HbPiBZWFggm83i8/nafp5qtSkej5NMJnWPEa22ye12Mzk52VSblo3hraxGIoQ4BTwo\npZwSQqwGnpZS3li2z+eAjwEZwAkMAN+QUn60/PMOHDggJyYmmmLx+Xxs3LixqffWq//64zd4+vwC\nn7x3LXt3jirDVY9moik++pXjOG0WvvbRW5i6ckkJrnKp4lclGc32vVNB/uezF7l3wwD/5W03KMNV\nj6SU/OJXTzAVSfH592xnx5hbCa5K6kSuQ4cOHdyzZ8/uSttaTdd8G/hE4edPAN8q30FK+Rkp5Top\n5Sbgw8CPKwX4VuXxGF+Te29hcOyFBlI27eCqRxrz7nX9OGwWZbjKpSoXGM92d2Hc5/BklFSm/sym\nCp5dDi0yFUnR32PlRm9+KUkVuCpppXG1GuR/D3hECHEGeLjwGiHEGiHE463CNaJUqrlHwhvRXesH\nsIj8Y9v1zkrZDq56pAV57Q+VKlzlUpULjGcbdtu5YdjFYskEcvVIBc+0AePd6waKS0mqwFVJK42r\npSAvpQxKKfdIKbdJKR+WUs4Vfj8ppXxXhf2fllL+i1aOWU2JRMKIj71G/T02do71kZX1V9m0g6uW\nYqksRyajWMTVAWQVuCpJVS5oD5vWm29kgF8Fz7ROxD3rrz61qQJXJa00LvUeeWxS7ap9vXdD/ktc\nb8pGhZrcg1fCZHKSm0bdrHLlR/BV4KokVbmgPWxv3pS/03rOF6p7QjyzPQsnM7zmj2IVV0tBwXyu\nalppXB0T5NtV+3rvxvxF+PLlcF0LfKtQk/vCxfxdx5s2XM35qcBVSapyQXvYto30MtJrZzaW5vRs\nfU+Mmu3ZCxdD5GR+we7+nqsFe2ZzVdNK4+qYIO9wONpynHUeJ+s8PUQWs3U9/dourmrK5iQvafn4\njVeDvNlc1aQqF7SHzSLE1d78hfruFs327DlfnvPNG68dODSbq5pWGlfHBPn+/vbNja4NXj7nW6i5\nbzu5KunETIzwYpa1hflRNJnNVU2qckH72O7buAqA5y7U/n6BuZ4l0tnipHflQV7Vc7nSuDomyAeD\nwbYd6y2b8hfhT99YqJk3bSdXJWmB4t4N1661aTZXNanKBe1ju2V1H/09Vi6FFrm4kKy5v5meHbwS\nIZWVTHh7GXFf2xNV9VyuNK6OCfKDg1UnwNRdE6O9jLjtBGJpTgWWz5u2k6tcOSl59o18kL9/87Uc\nZnItJ1W5oH1sNosoTlhWT2/eTM+eL/BpKaZSqXouVxpXxwT5dpZFWYTg/kJv/idvLH8RmlmudSoQ\nJxBLM+K2MzHae822lVZGpofayXZfIfXxvK92Xt4szzI5WRzU11JMpVL1XK40ro4J8slk7dtaPXX/\n5vyX+tkaKZt2c5XqJ+fngTyrpWyRYDO5lpOqXNBetjvXDdBjFYU/1Ms/JGOWZ69ORoimsmxY5WT9\nKueS7aqey5XG1TFBvt21rzvG3Az12piOpjgTrP4X2KyaXCklzxZupR/YvLSXtdJqhfVQO9mcNkux\n5vwn55e/WzTLs6cLnYhK3y9Q91yuNK6OCfLtrn21CFEcgH12mZSNWTW5rwfizETTjPTauWl06Vqu\nK61WWA+1m+3BLfkcrRZMq8kMz1LZXLHE861bKgd5Vc/lSuPqmCDvdC69XTRaDxRTNvNVUzZmcAEl\nA65LUzVgHlctqcoF7We7Z4MHl93CqUCcK6Hqt/JmeHbwcj5Vs2XIycZBV8V9VD2XK42rY4K8y1X5\ni2akbh7rY9BlYzKcqvp0ohlcOSn5yRvL30qbwVWPVOWC9rP12CzF2vOnlknZmOGZdnfx4A3VK0JU\nPZcrjatjgvz8/PK3tEbIahHFL/mPzlQ+vhlcr01FmYmmGe2zc9PY0lQNmMNVj1TlAnPYHip8v54+\nV/1usd1cyUyOAz4tVVM9yKt6LlcaV8cE+eHhYVOO+/DWISDfs8lUmMvGDK4fnc2v37pn61DFVA2Y\n51ctqcoF5rDdsXaAgR4rFxeSnJ+rPMDfbq4XL4ZIZnJMeHtZ3d9TdT9Vz+VK4+qYIB+JREw57tZh\nFxtXOQklM7xyeen0w+3mSmZyxXy89geokszyq5ZU5QJz2GwWwQOFB9meOle5p9durh+dyXcilkvV\ngLrncqVxdUyQN2shACEEP7M1/2V/svDlL1W7uQ74QsTTOW709lasXda00hZO0ENmsT1U+H796Oxc\nxZlP28kVjKd5+XIYq4CfqRHkVT2XK42rY4K8mbWvewo95ucvhpasGNVuLq2X9ci26r14WHm1wnrI\nLLadY27WDvQwF8/wcoW7xXZyPXlmjpzMT9KnrU1QTaqey5XG1TFB3sza19E+B7tW95HOyiU1ze3k\nmounOXgl38tabkAMVl6tsB4yi00IwTtuzOdrv39q6SRW7eKSUvL90/njv/3G2vljVc/lSuNqKcgL\nIYaEEE8IIc4U/q8YWYQQq4QQXxNCvC6EOCmEeFMrx60ks8uitIvwuydnr6mCaCfX908Fi70sj9O2\n7DPI77cAABIwSURBVL5m+1VNqnKBuWyPbBvCIvKDnnPx9DXb2sV1cibO5dAiQy4bd60bqLm/qudy\npXG12pN/DHhSSrkNeLLwupL+F/B9KeUEsAs42eJxl8jshQDu37SKgR4rZ4OJa2rm28WVzUkePzUL\nwLtvGqm5v9l+VZOqXGAu21CvnXvWe8jKqyk5Te3i0u4iHt42VFysezmpei5XGlerQf5R4EuFn78E\nvLd8ByGEB3gA+AsAKWVKSlnfaggNKBSqf+FjI+SwWXjb9nxv/jsnZ4u/bxfXK5fDzETTrBlwcMfa\n2osPmO1XNanKBeazFVM2p4PkSu4W28EVTmZ46lz+j8vbt9dX6me2X9W00rhaDfJjUsqpws9+YKzC\nPpuBAPBXQojDQogvCiEqP6HTgkZGavdejda7J/Jf/qfPLxBdzADt49L+sLxrYqRqbXypVPCrklTl\nAvPZ7l4/wEivncuhRQ5duVpu1w6uH5wOspiV3Lm2f9mqrVKZ7Vc1rTSu5RO3gBDiR0ClYd/Plr6Q\nUkohRKVH8mzAHcAvSylfFEL8L/Jpnf9UvuPMzAz79u3DZrORzWbZu3cv+/fvx+/343a7sVqthMNh\nvF4vc3NzSCnxer1MT0+TSqUYGhoiGo0yNjZGIBBACMHQ0BCBQICBgQGy2SyxWIzx8XH8fj92ux2P\nx8Ps7Cwej4dUKkUikShudzgc9Pf3EwwGGRwcJJFIkEwmi9udTicul4v5+XmGh4exxCPcNGTj5FyG\nf3jpHO+5cZBIJILNZmNkZIRQKEQ6nS6+v1ab+vr6AGq2yTcb5uXLYWwCbu1PE4vFarbpwoULbNiw\noWabIpEIqVSquN3lcuFwOAiFQoa0KZ1Os2rVKkPPU7NtunLlClu2bGn6POnRpnfdOMjfHJ7hKwcv\ns31gHfPz8+RyOZxOp2HnacQ7yj+9lh8UfMfWAXw+X11tmpmZoaenp+3nqVabpqam2Lp1qykxYrk2\nSSmbbtOyMbzW8nXLvlmIU8CDUsopIcRq4Gkp5Y1l+4wDL0gpNxVe3w88JqV8d/nnHThwQE5MTDTF\n4vP52LhxY1Pv1VPPXVjgP//oDcb6HPz1B3dw+dJFw7n+9MBlvnk8wMNbB/mNBzfV9R5V/CqXqlyg\nBls4meEXvnyMxazki++/iQ2rnIZzHfCF+K0nzrO638FffmBHXfl4UMOvSupErkOHDh3cs2fP7krb\nWk3XfBv4ROHnTwDfKt9BSukHLgkhtOC/BzjR4nGXSJXa1zdt9LDO08N0NMVP3lgwnCuczPC9woDY\n+2+plC2rLFX8KpeqXKAG24DTxp7CMxDfPB4AjOf6xrEZAN5z00jdAR7U8KuSVhpXq0H+94BHhBBn\ngIcLrxFCrBFCPF6y3y8DfyeEOArcBvzXFo+7RKrUvlqE4OduGQXgH49OMzU1VeMdrem7r8+ymMmx\ne10/W4brL8FSxa9yqcoF6rC972YvAE+cDjIfTxvKdXw6yqtTUdwOK++caCxnrIpf5VppXC0FeSll\nUEq5R0q5TUr5sJRyrvD7SSnlu0r2OyKl3C2lvFVK+V4ppe7Trbnduo/lNq1Htg6xymnjbDDBG/Ga\nwx5NK5nJFXtz7y/8YalXKvlVKlW5QB22jYMu3rTBw2JW8o+vzRjK9eUj0wA8umMEt8Pa0HtV8atc\nK42rY554tVob+wIaKYfNwvt25ntbX389vOwasK3o2ycCzCcybB/p5fY1tcsmS6WSX6VSlQvUYvuF\nO/K39v98cpZYxphjnJ2N89KlMD02C+/b2VgnAtTyq1Qrjatjgnw4vHRODzP16A4vHqeN03OLvHhJ\nf7ZYKss/vJrvZf3i7tWIOsomS6WaX5pU5QK12LaP9HLP+gEWMzm+dixgyDH++mA+1fjuieGaT1BX\nkkp+lWqlcXVMkPd6vWYjXKNeh5UP78oPhP71K5PXPLyih75xbIbIYpad427urOPhp3Kp5pcmVblA\nPbaPFnrzT11aZDqi7wyGhycjvHQpTK/dwod21T+gXyrV/NK00rg6JsjPzS2d5tdsveemEQadFs7P\nJXmiwjTEzWommuKrR/MVD79455qGe/Ggpl+gLheox3aj181DNwySykq++PIV3T43JyVffCn/eR+8\ndYzBGrNNVpNqfmlaaVwdE+SNynu3IofNws9t6wXgiy9NEk7qkzz9sxevsJjJcf/mVdy6uvbDEJWk\nol+gLheoybbvrjXYLfDM+QWO+aO6fOb3TgU5M5tguNfO3gYH9Eulol+w8rg6Jsiregv2nlvXcet4\nH6Fkhr98ZbLlz3vlcphn31igx2bh39yztunPUdUvVblATbbRPgd7b86XNn7+p5dIZXItfV4wnuaL\nL+W/p5+8dy1OW/MhQkW/YOVxdUyQn56eNhuhomZ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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "plt.plot(time_vector, ref_result)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's move on and port this synthesis function to Cython. We will then time the execution speed and see if Cython allows us to make this function faster.\n", "\n", "But first, let's make a detour. To port our function to Cython, we need to be able to do two things:\n", "\n", "- use complex number types in Cython\n", "- use the complex exponential \"from the C/C++ world\"" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Prerequesites for Cython & complex numbers " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Declaring complex numbers in Cython" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "So how do you declare a complex number in Cython? Luckily, the Cython tutorial from Scipy 2017 by Kurt Smith comes to the rescue (specifically, [this notebook](https://github.com/kwmsmith/scipy-2017-cython-tutorial/blob/master/03-cython-types.ipynb), but check out [the whole tutorial](https://github.com/kwmsmith/scipy-2017-cython-tutorial), it's a great learning resource)." ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [], "source": [ "%load_ext cython" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "(1+1j) (1+1j) (1+1j)\n", "1.0 1.0 (1-1j)\n" ] }, { "data": { "text/html": [ "\n", "\n", "\n", "\n", " \n", " Cython: _cython_magic_535820ea99a6bba3370e40ba9afac1e3.pyx\n", " \n", " \n", "\n", "\n", "

Generated by Cython 0.25.2

\n", "

\n", " Yellow lines hint at Python interaction.
\n", " Click on a line that starts with a \"+\" to see the C code that Cython generated for it.\n", "

\n", "
 01:
\n", "
 02: # `double complex` is preferred for compatibility with Python's `complex` type:
\n", "
 03: # https://docs.python.org/3/c-api/complex.html
\n", "
 04:
\n", "
 05: cdef:
\n", "
+06:     float complex fc = 1+1j
\n", "
  __pyx_t_1 = __Pyx_c_sum_double(__pyx_t_double_complex_from_parts(1, 0), __pyx_t_double_complex_from_parts(0, 1.0));\n",
       "  __pyx_v_46_cython_magic_535820ea99a6bba3370e40ba9afac1e3_fc = __pyx_t_float_complex_from_parts(__Pyx_CREAL(__pyx_t_1), __Pyx_CIMAG(__pyx_t_1));\n",
       "
+07:     double complex dc = 1+1j
\n", "
  __pyx_v_46_cython_magic_535820ea99a6bba3370e40ba9afac1e3_dc = __Pyx_c_sum_double(__pyx_t_double_complex_from_parts(1, 0), __pyx_t_double_complex_from_parts(0, 1.0));\n",
       "
+08:     long double complex ldc = 1+1j
\n", "
  __pyx_t_1 = __Pyx_c_sum_double(__pyx_t_double_complex_from_parts(1, 0), __pyx_t_double_complex_from_parts(0, 1.0));\n",
       "  __pyx_v_46_cython_magic_535820ea99a6bba3370e40ba9afac1e3_ldc = __pyx_t_long_double_complex_from_parts(__Pyx_CREAL(__pyx_t_1), __Pyx_CIMAG(__pyx_t_1));\n",
       "
 09:
\n", "
+10: print(fc, dc, ldc)
\n", "
  __pyx_t_2 = __pyx_PyComplex_FromComplex(__pyx_v_46_cython_magic_535820ea99a6bba3370e40ba9afac1e3_fc); if (unlikely(!__pyx_t_2)) __PYX_ERR(0, 10, __pyx_L1_error)\n",
       "  __Pyx_GOTREF(__pyx_t_2);\n",
       "  __pyx_t_3 = __pyx_PyComplex_FromComplex(__pyx_v_46_cython_magic_535820ea99a6bba3370e40ba9afac1e3_dc); if (unlikely(!__pyx_t_3)) __PYX_ERR(0, 10, __pyx_L1_error)\n",
       "  __Pyx_GOTREF(__pyx_t_3);\n",
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" ], "text/plain": [ "" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "%%cython -a\n", "cimport numpy as np\n", "\n", "def identity(np.ndarray [np.complex128_t, ndim=1] weights):\n", " return weights" ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "True" ] }, "execution_count": 14, "metadata": {}, "output_type": "execute_result" } ], "source": [ "np.allclose(weights, identity(weights))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We will stick to the above syntax (*np.ndarray*) in what follows." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## The complex exponential" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Another thing we need is to have access to the complex exponential in the Cython code. A naive approach is to use the NumPy exponetial: " ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n", "\n", "\n", " \n", " Cython: _cython_magic_e6c0f8824314565e8d5c73d53395537a.pyx\n", " \n", " \n", "\n", "\n", "

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 03: import cython
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 04:
\n", "
 05: @cython.boundscheck(False)
\n", "
 06: @cython.wraparound(False)
\n", "
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" ], "text/plain": [ "" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ "%%cython -a\n", "cimport numpy as np\n", "import numpy as np\n", "import cython\n", "\n", "@cython.boundscheck(False)\n", "@cython.wraparound(False)\n", "def apply_complex_exp(np.ndarray [np.complex128_t, ndim=1] weights, np.ndarray [np.complex128_t, ndim=1] out):\n", " cdef int i\n", " for i in range(weights.shape[0]):\n", " out[i] = np.exp(1j * weights[i])" ] }, { "cell_type": "code", "execution_count": 16, "metadata": { "collapsed": true }, "outputs": [], "source": [ "out = np.empty_like(weights)" ] }, { "cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "dtype('complex128')" ] }, "execution_count": 17, "metadata": {}, "output_type": "execute_result" } ], "source": [ "out.dtype" ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [], "source": [ "apply_complex_exp(weights, out)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's check the result:" ] }, { "cell_type": "code", "execution_count": 19, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "True" ] }, "execution_count": 19, "metadata": {}, "output_type": "execute_result" } ], "source": [ "np.allclose(np.exp(1j * weights), out)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "However, the problem is that we don't get pure C speed on this operation due to the call to a numpy function. There must be a better way. Searching the internet, I found [this StackOverflow thread](https://stackoverflow.com/questions/27906862/complex-valued-calculations-using-cython), which suggests to do the following." ] }, { "cell_type": "code", "execution_count": 20, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n", "\n", "\n", " \n", " Cython: _cython_magic_d867f5e67c8f54a1604814be92f853ab.pyx\n", " \n", " \n", "\n", "\n", "

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\n", "
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\n", "
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\n", "
 07:
\n", "
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\n", "
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" ], "text/plain": [ "" ] }, "execution_count": 20, "metadata": {}, "output_type": "execute_result" } ], "source": [ "%%cython -a --cplus\n", "cimport numpy as np\n", "import numpy as np\n", "import cython\n", "\n", "cdef extern from \"complex.h\":\n", " double complex exp(double complex)\n", "\n", "@cython.boundscheck(False)\n", "@cython.wraparound(False)\n", "def apply_complex_exp2(np.ndarray [np.complex128_t, ndim=1] weights, np.ndarray [np.complex128_t, ndim=1] out):\n", " cdef int i\n", " for i in range(weights.shape[0]):\n", " out[i] = exp(1j * weights[i])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This allows us translating to pure C... wait-for-it... ++! If you look closely at the above code cell, I had to add the `--cplus` flag to the compiler since the extern definition we are referring to is the built-in header for the a file that on my computer can be found in `Anaconda\\Lib\\site-packages\\Cython\\Includes\\libcpp`! If I don't add this, I get an error from the compiler..." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's check that this returns the right result." ] }, { "cell_type": "code", "execution_count": 21, "metadata": {}, "outputs": [], "source": [ "apply_complex_exp2(weights, out)" ] }, { "cell_type": "code", "execution_count": 22, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "True" ] }, "execution_count": 22, "metadata": {}, "output_type": "execute_result" } ], "source": [ "np.allclose(np.exp(1j * weights), out)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's also check that this is faster:" ] }, { "cell_type": "code", "execution_count": 23, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "1.24 ms ± 100 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)\n" ] } ], "source": [ "%timeit apply_complex_exp(weights, out)" ] }, { "cell_type": "code", "execution_count": 24, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "40.8 µs ± 3.87 µs per loop (mean ± std. dev. of 7 runs, 10000 loops each)\n" ] } ], "source": [ "%timeit apply_complex_exp2(weights, out)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Indeed, the second approach that uses C++ is much faster!" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "With these things in mind, let's now write the Cython version of our reference implementation." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Cython optimization of the synthesis function " ] }, { "cell_type": "code", "execution_count": 25, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n", "\n", "\n", " \n", " Cython: _cython_magic_180265c64445be1e3c6d6a7fae5a6faf.pyx\n", " \n", " \n", "\n", "\n", "

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\n", "

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\n", " Click on a line that starts with a \"+\" to see the C code that Cython generated for it.\n", "

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 03: import cython
\n", "
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\n", "
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\n", "
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\n", "
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\n", "
 08:
\n", "
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 18:     cdef double complex temp_sum
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\n", "
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       "      __pyx_v_temp_sum = __Pyx_c_sum_double(__pyx_v_temp_sum, __Pyx_c_prod_double((*__Pyx_BufPtrStrided1d(__pyx_t_double_complex *, __pyx_pybuffernd_weights.rcbuffer->pybuffer.buf, __pyx_t_11, __pyx_pybuffernd_weights.diminfo[0].strides)), std::exp(__pyx_t_double_complex_from_parts(__Pyx_CREAL(__pyx_t_13), __Pyx_CIMAG(__pyx_t_13)))));\n",
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\n", "
    __pyx_t_14 = __pyx_v_i;\n",
       "    *__Pyx_BufPtrStrided1d(__pyx_t_5numpy_float64_t *, __pyx_pybuffernd_out.rcbuffer->pybuffer.buf, __pyx_t_14, __pyx_pybuffernd_out.diminfo[0].strides) = std::real(__pyx_v_temp_sum);\n",
       "  }\n",
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+24:     return out
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       "  __Pyx_INCREF(((PyObject *)__pyx_v_out));\n",
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       "
" ], "text/plain": [ "" ] }, "execution_count": 25, "metadata": {}, "output_type": "execute_result" } ], "source": [ "%%cython -a --cplus\n", "cimport numpy as np\n", "import numpy as np\n", "import cython\n", "\n", "cdef extern from \"\" namespace \"std\" nogil:\n", " double complex exp(double complex z)\n", " double real(double complex z)\n", "\n", "cdef double complex I = 1j \n", " \n", "@cython.boundscheck(False)\n", "@cython.wraparound(False)\n", "def synthesis_cython1(np.ndarray [np.complex128_t, ndim=1] weights,\n", " double omega,\n", " np.ndarray [np.float64_t, ndim=1] time_vector):\n", " cdef int i, j\n", " cdef np.ndarray [np.float64_t, ndim=1] out = np.empty_like(time_vector, dtype=np.float)\n", " cdef double complex temp_sum \n", " for i in range(time_vector.shape[0]):\n", " temp_sum = 0\n", " for j in range(weights.shape[0]):\n", " temp_sum += weights[j] * exp(I * time_vector[i] * omega )\n", " out[i] = real(temp_sum)\n", " return out" ] }, { "cell_type": "code", "execution_count": 26, "metadata": { "collapsed": true }, "outputs": [], "source": [ "out = synthesis_cython1(weights, omega, time_vector)" ] }, { "cell_type": "code", "execution_count": 27, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "True" ] }, "execution_count": 27, "metadata": {}, "output_type": "execute_result" } ], "source": [ "np.allclose(ref_result, out)" ] }, { "cell_type": "code", "execution_count": 28, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[]" ] }, "execution_count": 28, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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Xqp7pzeXbpv1xLbs5/YsIGD1+9StngnxTUxNioTZv4lmFPoSq1eT2S2+uY9sO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\n3162LFtcx3+q7Tey6lbsDnvC8aqey9HGlWmQfwD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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "plt.plot(ref_result)\n", "plt.plot(out)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's now do some timings." ] }, { "cell_type": "code", "execution_count": 29, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "30.1 ms ± 625 µs per loop (mean ± std. dev. of 7 runs, 10 loops each)\n" ] } ], "source": [ "%timeit synthesis(weights, omega, time_vector)" ] }, { "cell_type": "code", "execution_count": 30, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "127 ms ± 5.81 ms per loop (mean ± std. dev. of 7 runs, 10 loops each)\n" ] } ], "source": [ "%timeit synthesis_cython1(weights, omega, time_vector)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "As we can see, our Cython version is almost ten times slower than our vectorized implementation. Let's try tweaking it." ] }, { "cell_type": "code", "execution_count": 31, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n", "\n", "\n", " \n", " Cython: _cython_magic_59850e01233c87321a5c3910c7be7295.pyx\n", " \n", " \n", "\n", "\n", "

Generated by Cython 0.25.2

\n", "

\n", " Yellow lines hint at Python interaction.
\n", " Click on a line that starts with a \"+\" to see the C code that Cython generated for it.\n", "

\n", "
+01: cimport numpy as np
\n", "
  __pyx_t_1 = PyDict_New(); if (unlikely(!__pyx_t_1)) __PYX_ERR(0, 1, __pyx_L1_error)\n",
       "  __Pyx_GOTREF(__pyx_t_1);\n",
       "  if (PyDict_SetItem(__pyx_d, __pyx_n_s_test, __pyx_t_1) < 0) __PYX_ERR(0, 1, __pyx_L1_error)\n",
       "  __Pyx_DECREF(__pyx_t_1); __pyx_t_1 = 0;\n",
       "
+02: import numpy as np
\n", "
  __pyx_t_1 = __Pyx_Import(__pyx_n_s_numpy, 0, 0); if (unlikely(!__pyx_t_1)) __PYX_ERR(0, 2, __pyx_L1_error)\n",
       "  __Pyx_GOTREF(__pyx_t_1);\n",
       "  if (PyDict_SetItem(__pyx_d, __pyx_n_s_np, __pyx_t_1) < 0) __PYX_ERR(0, 2, __pyx_L1_error)\n",
       "  __Pyx_DECREF(__pyx_t_1); __pyx_t_1 = 0;\n",
       "
 03: import cython
\n", "
 04:
\n", "
 05: cdef extern from "<complex.h>" namespace "std" nogil:
\n", "
 06:     double complex exp(double complex z)
\n", "
 07:     double real(double complex z)
\n", "
 08:
\n", "
+09: cdef double complex I = 1j
\n", "
  __pyx_v_46_cython_magic_59850e01233c87321a5c3910c7be7295_I = __pyx_t_double_complex_from_parts(0, 1.0);\n",
       "
 10:
\n", "
 11: @cython.boundscheck(False)
\n", "
 12: @cython.wraparound(False)
\n", "
+13: def synthesis_cython2(np.ndarray [np.complex128_t, ndim=1] weights,
\n", "
/* Python wrapper */\n",
       "static PyObject *__pyx_pw_46_cython_magic_59850e01233c87321a5c3910c7be7295_1synthesis_cython2(PyObject *__pyx_self, PyObject *__pyx_args, PyObject *__pyx_kwds); /*proto*/\n",
       "static PyMethodDef __pyx_mdef_46_cython_magic_59850e01233c87321a5c3910c7be7295_1synthesis_cython2 = {\"synthesis_cython2\", (PyCFunction)__pyx_pw_46_cython_magic_59850e01233c87321a5c3910c7be7295_1synthesis_cython2, METH_VARARGS|METH_KEYWORDS, 0};\n",
       "static PyObject *__pyx_pw_46_cython_magic_59850e01233c87321a5c3910c7be7295_1synthesis_cython2(PyObject *__pyx_self, PyObject *__pyx_args, PyObject *__pyx_kwds) {\n",
       "  PyArrayObject *__pyx_v_weights = 0;\n",
       "  double __pyx_v_omega;\n",
       "  PyArrayObject *__pyx_v_time_vector = 0;\n",
       "  PyObject *__pyx_r = 0;\n",
       "  __Pyx_RefNannyDeclarations\n",
       "  __Pyx_RefNannySetupContext(\"synthesis_cython2 (wrapper)\", 0);\n",
       "  {\n",
       "    static PyObject **__pyx_pyargnames[] = {&__pyx_n_s_weights,&__pyx_n_s_omega,&__pyx_n_s_time_vector,0};\n",
       "    PyObject* values[3] = {0,0,0};\n",
       "    if (unlikely(__pyx_kwds)) {\n",
       "      Py_ssize_t kw_args;\n",
       "      const Py_ssize_t pos_args = PyTuple_GET_SIZE(__pyx_args);\n",
       "      switch (pos_args) {\n",
       "        case  3: values[2] = PyTuple_GET_ITEM(__pyx_args, 2);\n",
       "        case  2: values[1] = PyTuple_GET_ITEM(__pyx_args, 1);\n",
       "        case  1: values[0] = PyTuple_GET_ITEM(__pyx_args, 0);\n",
       "        case  0: break;\n",
       "        default: goto __pyx_L5_argtuple_error;\n",
       "      }\n",
       "      kw_args = PyDict_Size(__pyx_kwds);\n",
       "      switch (pos_args) {\n",
       "        case  0:\n",
       "        if (likely((values[0] = PyDict_GetItem(__pyx_kwds, __pyx_n_s_weights)) != 0)) kw_args--;\n",
       "        else goto __pyx_L5_argtuple_error;\n",
       "        case  1:\n",
       "        if (likely((values[1] = PyDict_GetItem(__pyx_kwds, __pyx_n_s_omega)) != 0)) kw_args--;\n",
       "        else {\n",
       "          __Pyx_RaiseArgtupleInvalid(\"synthesis_cython2\", 1, 3, 3, 1); __PYX_ERR(0, 13, __pyx_L3_error)\n",
       "        }\n",
       "        case  2:\n",
       "        if (likely((values[2] = PyDict_GetItem(__pyx_kwds, __pyx_n_s_time_vector)) != 0)) kw_args--;\n",
       "        else {\n",
       "          __Pyx_RaiseArgtupleInvalid(\"synthesis_cython2\", 1, 3, 3, 2); __PYX_ERR(0, 13, __pyx_L3_error)\n",
       "        }\n",
       "      }\n",
       "      if (unlikely(kw_args > 0)) {\n",
       "        if (unlikely(__Pyx_ParseOptionalKeywords(__pyx_kwds, __pyx_pyargnames, 0, values, pos_args, \"synthesis_cython2\") < 0)) __PYX_ERR(0, 13, __pyx_L3_error)\n",
       "      }\n",
       "    } else if (PyTuple_GET_SIZE(__pyx_args) != 3) {\n",
       "      goto __pyx_L5_argtuple_error;\n",
       "    } else {\n",
       "      values[0] = PyTuple_GET_ITEM(__pyx_args, 0);\n",
       "      values[1] = PyTuple_GET_ITEM(__pyx_args, 1);\n",
       "      values[2] = PyTuple_GET_ITEM(__pyx_args, 2);\n",
       "    }\n",
       "    __pyx_v_weights = ((PyArrayObject *)values[0]);\n",
       "    __pyx_v_omega = __pyx_PyFloat_AsDouble(values[1]); if (unlikely((__pyx_v_omega == (double)-1) && PyErr_Occurred())) __PYX_ERR(0, 14, __pyx_L3_error)\n",
       "    __pyx_v_time_vector = ((PyArrayObject *)values[2]);\n",
       "  }\n",
       "  goto __pyx_L4_argument_unpacking_done;\n",
       "  __pyx_L5_argtuple_error:;\n",
       "  __Pyx_RaiseArgtupleInvalid(\"synthesis_cython2\", 1, 3, 3, PyTuple_GET_SIZE(__pyx_args)); __PYX_ERR(0, 13, __pyx_L3_error)\n",
       "  __pyx_L3_error:;\n",
       "  __Pyx_AddTraceback(\"_cython_magic_59850e01233c87321a5c3910c7be7295.synthesis_cython2\", __pyx_clineno, __pyx_lineno, __pyx_filename);\n",
       "  __Pyx_RefNannyFinishContext();\n",
       "  return NULL;\n",
       "  __pyx_L4_argument_unpacking_done:;\n",
       "  if (unlikely(!__Pyx_ArgTypeTest(((PyObject *)__pyx_v_weights), __pyx_ptype_5numpy_ndarray, 1, \"weights\", 0))) __PYX_ERR(0, 13, __pyx_L1_error)\n",
       "  if (unlikely(!__Pyx_ArgTypeTest(((PyObject *)__pyx_v_time_vector), __pyx_ptype_5numpy_ndarray, 1, \"time_vector\", 0))) __PYX_ERR(0, 15, __pyx_L1_error)\n",
       "  __pyx_r = __pyx_pf_46_cython_magic_59850e01233c87321a5c3910c7be7295_synthesis_cython2(__pyx_self, __pyx_v_weights, __pyx_v_omega, __pyx_v_time_vector);\n",
       "\n",
       "  /* function exit code */\n",
       "  goto __pyx_L0;\n",
       "  __pyx_L1_error:;\n",
       "  __pyx_r = NULL;\n",
       "  __pyx_L0:;\n",
       "  __Pyx_RefNannyFinishContext();\n",
       "  return __pyx_r;\n",
       "}\n",
       "\n",
       "static PyObject *__pyx_pf_46_cython_magic_59850e01233c87321a5c3910c7be7295_synthesis_cython2(CYTHON_UNUSED PyObject *__pyx_self, PyArrayObject *__pyx_v_weights, double __pyx_v_omega, PyArrayObject *__pyx_v_time_vector) {\n",
       "  int __pyx_v_i;\n",
       "  int __pyx_v_j;\n",
       "  PyArrayObject *__pyx_v_out = 0;\n",
       "  __pyx_t_double_complex __pyx_v_temp_sum;\n",
       "  __pyx_t_double_complex __pyx_v_temp_mult;\n",
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" ], "text/plain": [ "" ] }, "execution_count": 31, "metadata": {}, "output_type": "execute_result" } ], "source": [ "%%cython -a --cplus\n", "cimport numpy as np\n", "import numpy as np\n", "import cython\n", "\n", "cdef extern from \"\" namespace \"std\" nogil:\n", " double complex exp(double complex z)\n", " double real(double complex z)\n", "\n", "cdef double complex I = 1j \n", " \n", "@cython.boundscheck(False)\n", "@cython.wraparound(False)\n", "def synthesis_cython2(np.ndarray [np.complex128_t, ndim=1] weights,\n", " double omega,\n", " np.ndarray [np.float64_t, ndim=1] time_vector):\n", " cdef int i, j\n", " cdef np.ndarray [np.float64_t, ndim=1] out = np.empty_like(time_vector, dtype=np.float)\n", " cdef double complex temp_sum\n", " cdef double complex temp_mult\n", " for i in range(time_vector.shape[0]):\n", " temp_sum = 0\n", " temp_mult = exp(I * time_vector[i] * omega)\n", " for j in range(weights.shape[0]):\n", " temp_sum += weights[j] * temp_mult\n", " out[i] = real(temp_sum)\n", " return out" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's check it's correct again:" ] }, { "cell_type": "code", "execution_count": 32, "metadata": {}, "outputs": [], "source": [ "out = synthesis_cython2(weights, omega, time_vector)" ] }, { "cell_type": "code", "execution_count": 33, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "True" ] }, "execution_count": 33, "metadata": {}, "output_type": "execute_result" } ], "source": [ "np.allclose(ref_result, out)" ] }, { "cell_type": "code", "execution_count": 34, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "29.7 ms ± 573 µs per loop (mean ± std. dev. of 7 runs, 10 loops each)\n" ] } ], "source": [ "%timeit synthesis(weights, omega, time_vector)" ] }, { "cell_type": "code", "execution_count": 35, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "27.6 ms ± 392 µs per loop (mean ± std. dev. of 7 runs, 10 loops each)\n" ] } ], "source": [ "%timeit synthesis_cython2(weights, omega, time_vector)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Nice! Moving the computation of the complex exponent outside of the loop helped us match NumPy. As you can see above however, line 22 is still yellow, meaning there is some Python interaction going on. Can we eliminate that?" ] }, { "cell_type": "code", "execution_count": 36, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n", "\n", "\n", " \n", " Cython: _cython_magic_9686ecbb26e36aaeeb31076454f5407d.pyx\n", " \n", " \n", "\n", "\n", "

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\n", "

\n", " Yellow lines hint at Python interaction.
\n", " Click on a line that starts with a \"+\" to see the C code that Cython generated for it.\n", "

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 03: import cython
\n", "
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\n", "
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\n", "
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\n", "
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" ], "text/plain": [ "" ] }, "execution_count": 36, "metadata": {}, "output_type": "execute_result" } ], "source": [ "%%cython -a --cplus\n", "cimport numpy as np\n", "import numpy as np\n", "import cython\n", "\n", "cdef extern from \"\" namespace \"std\" nogil:\n", " double complex exp(double complex z)\n", " double real(double complex z)\n", "\n", "cdef double complex I = 1j \n", " \n", "@cython.boundscheck(False)\n", "@cython.wraparound(False)\n", "def synthesis_cython3(np.ndarray [complex, ndim=1] weights,\n", " double omega,\n", " np.ndarray [double, ndim=1] time_vector):\n", " cdef int i, j\n", " cdef np.ndarray [np.float64_t, ndim=1] out = np.empty_like(time_vector, dtype=np.float)\n", " cdef double complex temp_sum\n", " cdef double complex temp_mult\n", " for i in range(time_vector.shape[0]):\n", " temp_sum = 0\n", " temp_mult = exp(I * time_vector[i] * omega)\n", " for j in range(weights.shape[0]):\n", " temp_sum += weights[j] * temp_mult\n", " out[i] = real(temp_sum)\n", " return out" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Yes we can! All we did was changing the dtype of the input array. Let's check our function produces correct results again:" ] }, { "cell_type": "code", "execution_count": 37, "metadata": { "collapsed": true }, "outputs": [], "source": [ "out = synthesis_cython3(weights, omega, time_vector)" ] }, { "cell_type": "code", "execution_count": 38, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "True" ] }, "execution_count": 38, "metadata": {}, "output_type": "execute_result" } ], "source": [ "np.allclose(ref_result, out)" ] }, { "cell_type": "code", "execution_count": 39, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "29.9 ms ± 1.36 ms per loop (mean ± std. dev. of 7 runs, 10 loops each)\n" ] } ], "source": [ "%timeit synthesis(weights, omega, time_vector)" ] }, { "cell_type": "code", "execution_count": 40, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "28.1 ms ± 1.36 ms per loop (mean ± std. dev. of 7 runs, 10 loops each)\n" ] } ], "source": [ "%timeit synthesis_cython3(weights, omega, time_vector)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Interestingly, even though our function is now \"completely white\", meaning that Cython has translated it to pure C++, it is not actually faster than our previous version. The last thing we can do in our quest for performance is to make the loop parallel." ] }, { "cell_type": "code", "execution_count": 41, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n", "\n", "\n", " \n", " Cython: _cython_magic_e7cc766498993d3947c8c5bbc875e6a8.pyx\n", " \n", " \n", "\n", "\n", "

Generated by Cython 0.25.2

\n", "

\n", " Yellow lines hint at Python interaction.
\n", " Click on a line that starts with a \"+\" to see the C code that Cython generated for it.\n", "

\n", "
+01: cimport numpy as np
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  __pyx_t_1 = PyDict_New(); if (unlikely(!__pyx_t_1)) __PYX_ERR(0, 1, __pyx_L1_error)\n",
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       "  __Pyx_DECREF(__pyx_t_1); __pyx_t_1 = 0;\n",
       "
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 03: import cython
\n", "
 04: from cython.parallel import prange
\n", "
 05:
\n", "
 06: cdef extern from "<complex.h>" namespace "std" nogil:
\n", "
 07:     double complex exp(double complex z)
\n", "
 08:     double real(double complex z)
\n", "
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\n", "
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\n", "
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\n", "
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       "static PyObject *__pyx_pw_46_cython_magic_e7cc766498993d3947c8c5bbc875e6a8_1synthesis_cython4(PyObject *__pyx_self, PyObject *__pyx_args, PyObject *__pyx_kwds) {\n",
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       "                            for (__pyx_t_12 = 0; __pyx_t_12 < __pyx_t_11; __pyx_t_12+=1) {\n",
       "                              __pyx_v_j = __pyx_t_12;\n",
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                              __pyx_t_13 = __pyx_v_j;\n",
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       "                            }\n",
       "
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                            __pyx_t_14 = __pyx_v_i;\n",
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+27:     return out
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       "  __Pyx_INCREF(((PyObject *)__pyx_v_out));\n",
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       "
" ], "text/plain": [ "" ] }, "execution_count": 41, "metadata": {}, "output_type": "execute_result" } ], "source": [ "%%cython -a --cplus --compile-args=/openmp\n", "cimport numpy as np\n", "import numpy as np\n", "import cython\n", "from cython.parallel import prange\n", "\n", "cdef extern from \"\" namespace \"std\" nogil:\n", " double complex exp(double complex z)\n", " double real(double complex z)\n", "\n", "cdef double complex I = 1j \n", " \n", "@cython.boundscheck(False)\n", "@cython.wraparound(False)\n", "def synthesis_cython4(np.ndarray [complex, ndim=1] weights,\n", " double omega,\n", " np.ndarray [double, ndim=1] time_vector):\n", " cdef int i, j\n", " cdef np.ndarray [np.float64_t, ndim=1] out = np.empty_like(time_vector, dtype=np.float)\n", " cdef double complex temp_sum\n", " cdef double complex temp_mult\n", " for i in prange(time_vector.shape[0], nogil=True):\n", " temp_sum = 0\n", " temp_mult = exp(I * time_vector[i] * omega)\n", " for j in range(weights.shape[0]):\n", " temp_sum = temp_sum + weights[j] * temp_mult\n", " out[i] = real(temp_sum)\n", " return out" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Note that I had to give up my `+=` term to compile this since I ran into this error message \"Cannot read reduction variable in loop body\" (fix found [here](https://stackoverflow.com/questions/40451203/cython-parallel-loop-problems)).\n", "\n", "Also, note that I supplied the compiler with the *--/openmp* flag, which is valid under Windows only (for Linux it's *--fopenmp*)." ] }, { "cell_type": "code", "execution_count": 42, "metadata": {}, "outputs": [], "source": [ "out = synthesis_cython4(weights, omega, time_vector)" ] }, { "cell_type": "code", "execution_count": 43, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "True" ] }, "execution_count": 43, "metadata": {}, "output_type": "execute_result" } ], "source": [ "np.allclose(ref_result, out)" ] }, { "cell_type": "code", "execution_count": 44, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "29.1 ms ± 250 µs per loop (mean ± std. dev. of 7 runs, 10 loops each)\n" ] } ], "source": [ "%timeit synthesis(weights, omega, time_vector)" ] }, { "cell_type": "code", "execution_count": 45, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "9.23 ms ± 521 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)\n" ] } ], "source": [ "%timeit synthesis_cython4(weights, omega, time_vector)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We achieve a nice little speedup to parallel execution on four cores." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Timings " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's time the different versions of our functions!" ] }, { "cell_type": "code", "execution_count": 46, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "29.3 ms ± 639 µs per loop (mean ± std. dev. of 7 runs, 10 loops each)\n", "123 ms ± 3.54 ms per loop (mean ± std. dev. of 7 runs, 10 loops each)\n", "27.5 ms ± 352 µs per loop (mean ± std. dev. of 7 runs, 10 loops each)\n", "27.3 ms ± 218 µs per loop (mean ± std. dev. of 7 runs, 10 loops each)\n", "8.76 ms ± 599 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)\n" ] } ], "source": [ "timings = {}\n", "for func, label in zip([synthesis, synthesis_cython1, synthesis_cython2, synthesis_cython3, synthesis_cython4],\n", " ['reference', 'cython1-naive', 'cython2-clever', 'cython3-clever-pure-c++-loop', 'cython-parallel']):\n", " obj = %timeit -o func(weights, omega, time_vector)\n", " timings[label] = obj" ] }, { "cell_type": "code", "execution_count": 47, "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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timings (ms)
cython-parallel8.757900
cython3-clever-pure-c++-loop27.341302
cython2-clever27.519439
reference29.275485
cython1-naive123.381547
\n", "
" ], "text/plain": [ " timings (ms)\n", "cython-parallel 8.757900\n", "cython3-clever-pure-c++-loop 27.341302\n", "cython2-clever 27.519439\n", "reference 29.275485\n", "cython1-naive 123.381547" ] }, "execution_count": 47, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import pandas as pd\n", "s = pd.Series({key: timings[key].average * 1e3 for key in timings}).to_frame(name='timings (ms)').sort_values(by='timings (ms)')\n", "s" ] }, { "cell_type": "code", "execution_count": 48, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 48, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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bdtYxMZXMLN01XODg1hIFc+4cLnB0xyS9O5J5lxVNU9w+VODI9kkm3bhvtJ4j\n2ifZOF5HQx2s7pzg6t5FrO6cZHvJ2LC9nsPaJnlwez1tBaercWrmNUcmjd7xOg5pK/HAWD1djVN0\nNuyqaXDCGCjWcVBLiXtH6lnRPEVbYdf2gWIdg4OD85rTIzH38vpCmNly4Kfuvir9+jkkg6knACe6\ne6+ZrQBucvcn7v7/r1u3zg899NCyXjuChx56iMc97nFZlyFlUHaxKb+4lF1lnXLFbfP6es/uKvLj\ngfm7v+J1rz1m3l4LYP369beuWbPm2D1tK/s0n7v3AQ+Z2fRAaQ3wK+Aa4Nz0sXOBq8t9jcg0XR2X\nsotN+cWl7GK7M8en+eZ6Nd8FwL+Y2S+Bo4EPAx8Bnmtm9wEnp1/nzqZNm7IuQcqk7GJTfnEpu9iO\n7shvn6k5DSPd/XZgT1Nea+byfWvB9PlYiUfZxab84lJ2sU2vrcqj/O65iIiISAVoMFUlo6OjWZcg\nZVJ2sSm/uJRdbCua1GdKKmzZsly216oJyi425ReXsovt9iEtQJcK6+/vz7oEKZOyi035xaXsYjuy\nPb8L0DWYqhIzy7oEKZOyi035xaXsYpv0/OanwVSVLF26NOsSpEzKLjblF5eyi+2+0fzeqFqDqSrR\ndHVcyi425ReXsovtCJ3mk0prb2/PugQpk7KLTfnFpexi2zie3yFFfve8ykqlUtYlSJmUXWzKLy5l\nF1tDjkcUOd716hobG8u6BCmTsotN+cWl7GJbtkh9pqTCli9fnnUJUiZlF5vyi0vZxXbroPpMSYX1\n9fVlXYKUSdnFpvziUnaxre7UAnSpsIaGhqxLkDIpu9iUX1zKLrbtJfWZkgrr6OjIugQpk7KLTfnF\npexi27BdfaakwrZs2ZJ1CVImZReb8otL2cV2WJtO80mF6QgrLmUXm/KLS9nF9qBmpqTSisVi1iVI\nmZRdbMovLmUXW1vBsy4hMxpMVcn4+HjWJUiZlF1syi8uZRdbV6P6TEmFqV9KXMouNuUXl7KLTX2m\npOLULyUuZReb8otL2cWmPlNScY2NjVmXIGVSdrEpv7iUXWwjk+ozJRXW1taWdQlSJmUXm/KLS9nF\n1jue3yFFfve8ygYGBrIuQcqk7GJTfnEpu9gOaStlXUJmNJiqkiVLlmRdgpRJ2cWm/OJSdrE9MKY+\nU1JhusQ3LmUXm/KLS9nFptYIUnE7duzIugQpk7KLTfnFpexi62xQ006pMPVLiUvZxab84lJ2sanP\nlFSc+qUmE1BiAAAgAElEQVTEpexiU35xKbvY1GdKKq6pqSnrEqRMyi425ReXsottcEJ9pqTCmpub\nsy5ByqTsYlN+cSm72AaK+R1S5HfPq2zbtm1ZlyBlUnaxKb+4lF1sB7Woz5RUWFdXV9YlSJmUXWzK\nLy5lF9u9I+ozJRU2MjKSdQlSJmUXm/KLS9nFtqJZfabKZmb1ZnabmV2bfr3UzK43s/vS/+aypW2x\nWMy6BCmTsotN+cWl7GJrK6jP1Fy8Gbhn1tcXATe4+8HADenXuaN+KXEpu9iUX1zKLjb1mSqTmR0I\nnA5cMevhM4Cr0n9fBbxoLq8RlfqlxKXsYlN+cSm72NRnqnyfBN4BzD5Ruszde9N/9wHL5vgaIekS\n37iUXWzKLy5lF1ueWyOUPSdnZi8ANrv7rWZ24p6e4+5uZns8ibp582bOP/98CoUCpVKJs846i7Vr\n19LX10dLSwv19fUMDw/T09PD1q1bcXd6enrYtGkTra2tAIyOjrJs2TL6+/sxM5YuXUp/fz/t7e2U\nSiXGxsZYvnw5fX19NDQ00NHRwZYtW+jo6KBYLDI+Pj6zvbGxkba2NgYGBliyZAnj4+Ps2LFjZntT\nUxPNzc1s27aNrq4uRkZGKBaLM9ubm5tpbGxkaGiI7u5uRkZG/uj718I+DQ0NMTExUfP7tHPnTjZs\n2FBT+1SLOe1tn7Zt20Z9fX1N7VMt5rSnfRocHKRYLNbUPmWZU2thitWdkwwU6xiZNFYtLnHPSIGV\ni0ssrnduHSywunOSTTvrmJiCA5unuGu4wMGtJQrm3Dlc4OiOSXp3JIOkFU1T3D5U4Mj2SSbduG+0\nniPaJ9k4XkdDHfxpyyQPbq9jdeck20vGhu31HNY2yYPb62krOF2NUzOvOTJp9I7XcUhbiQfG6ulq\nnKKzYVdNgxPGQLGOg1pK3DtSz4rmKdoKu7YPFOsYHByc15wecUzkXt6CMTO7DHgVMAk0Ae3At4Gn\nAie6e6+ZrQBucvcn7v7/r1u3zg899NCyXjuCDRs2sHLlyqzLkDIou9iUX1zKrrJOueK2eX29E7qL\n3Lylcd5e77rXHjNvrwWwfv36W9esWXPsnraVPSfn7he7+4Huvgo4G/ihu58DXAOcmz7tXODqcl8j\nsu7u7qxLkDIpu9iUX1zKLrZ7RrQAvZI+AjzXzO4DTk6/zp2hoaGsS5AyKbvYlF9cyi62lYvz2wG9\nIsNId78JuCn99wCwphLfN7KJiYmsS5AyKbvYlF9cyi62xfXqMyUVpn4pcSm72JRfXMouNvWZkopT\nv5S4lF1syi8uZReb+kxJxbW0tGRdgpRJ2cWm/OJSdrFt2pnfIUV+97zKpvvcSDzKLjblF5eyi20i\nv/c51mCqWoaHh7MuQcqk7GJTfnEpu9gObM7vaEqDqSrp6enJugQpk7KLTfnFpexiu2tYC9ClwrZu\n3Zp1CVImZReb8otL2cV2cGt++0xpMFUl5d6mR7Kn7GJTfnEpu9gKe74Vby5oMFUlmq6OS9nFpvzi\nUnax3anTfFJpmzZtyroEKZOyi035xaXsYju6Q32mpMJaW1uzLkHKpOxiU35xKbvYenfkd0iR3z0X\nERERqQANpqpkdHQ06xKkTMouNuUXl7KLbUWT+kxJhS1btizrEqRMyi425ReXsovt9iEtQJcK6+/v\nz7oEKZOyi035xaXsYjuyXQvQpcLMLOsSpEzKLjblF5eyi23S85ufBlNVsnTp0qxLkDIpu9iUX1zK\nLrb7RvN7o2oNpqpE09VxKbvYlF9cyi62I3J8mi+/q8WqrL29PesSpEzKLjblV1mnXHHbvL3WQS2T\nPHD9/N2f77rXHjNvr5UHG8fzOz+T3z2vslIpvzd8jE7Zxab84mrQJ1Joec4vx7teXWNjY1mXIGVS\ndrEpv7iWLcpvn6JakOf8dJqvSpYvX551CVImZVdZ83maCKC1MMXopE4VRXTroD6SIstzfvnd8yrr\n6+tj5cqVWZdRE+b7w/iE7iI3b2mct9fTh3Flre6cnNf8pHKUXWx5zk+n+aqkoaEh6xKkTNtL+e2V\nUguUX1zKLrY856fBVJV0dHRkXYKUacP2/PZKqQXKLy5lF1ue89Ngqkq2bNmSdQlSpsPa8tsrpRYo\nv7iUXWx5zk+DqSrRzFRcD+b46KoWKL+4lF1sec5Pg6kqKRaLWZcgZWoreNYlyBwov7iUXWx5zk+D\nqSoZHx/PugQpU1djfnul1ALlF5eyiy3P+WkwVSXqVRRXnnul1ALlF5eyiy3P+WkwVSV9fX1ZlyBl\nWt2Z30WUtUD5xaXsYstzfhpMVUljYz4bl9WCkcn89kqpBcovLmUXW57z02CqStra2rIuQcrUm+M7\nn9cC5ReXsostz/mVvedm9jgzu9HMfmVmd5vZm9PHl5rZ9WZ2X/rfJZUrN46BgYGsS5AyHdJWyroE\nmQPlF5eyiy3P+c1lGDkJvM3dDweeAaw1s8OBi4Ab3P1g4Ib069xZsiSXY8ia8MBYfnul1ALlF5ey\niy3P+ZU9mHL3Xndfn/57BLgHOAA4A7gqfdpVwIvmWmREao0QV54v760Fyi8uZRdbnvOryAlOM1sF\nHAP8DFjm7r3ppj5gWSVeI5odO3ZkXYKUqbMhv43naoHyi0vZxZbn/ObcFMLMWoFvAW9x92GzXav5\n3d3NbI8/3c2bN3P++edTKBQolUqcddZZrF27lr6+PlpaWqivr2d4eJienh62bt2Ku9PT08OmTZto\nbW0FYHR0lGXLltHf34+ZsXTpUvr7+2lvb6dUKjE2Nsby5cvp6+ujoaGBjo4OtmzZQkdHB8VikfHx\n8ZntjY2NtLW1MTAwwJIlSxgfH2fHjh0z25uammhubmbbtm10dXUxMjJCsVic2d7c3ExjYyNDQ0N0\nd3djZmzYsGFmey3s09DQEBMTE/O+Tyd0F7l1sMDqzklGJo3e8ToOaSvxwFg9XY1TdDb4zPbBCWOg\nWMdBLSXuHalnRfMUbYVd2weKdYxMGqsWl7hnpMDKxSUW1+/avmlnHVuKxgndRe4aLnBwa4mCOXcO\nFzi6Y5LeHcnxx4qmKW4fKnBk+ySTbtw3Ws8R7ZNsHK+joQ6WLZqa+Z7bS8aG7fUc1jbJg9vraSs4\nXY27tvf29tZETnv73TuwuVSVnCam4MDmqYfl1FjnHNI6WfGc9va7t3PnzprIaW+/ewc2l6qS057+\nnurNecbSiarktKffvdHR0ZrJaU+/e62FqarktLe/p8Y6n3nNSua0t9+9wcHBec3pEcdC7uWPJM2s\nAbgW+IG7/1362K+BE92918xWADe5+xN3/3/XrVvnhx56aNmvvdBt2LCBlStXZl1GTTjlitvm9fVO\n6C5y85b5a21x3WuPmbfXyoLyi20+81N2laW/vcpav379rWvWrDl2T9vmcjWfAVcC90wPpFLXAOem\n/z4XuLrc14isqakp6xKkTIMT+e2VUguUX1zKLrY85zeX03zPAl4F3Glmt6ePvQv4CPANMzsf2AC8\nbG4lxtTc3Jx1CVKmgWJ+e6XUAuUXl7KLLc/5lT2YcvcfA3sbhq4p9/vWim3bttHe3p51GVKGg1pK\nbBzP7yW+0Sm/uJRdbHnOL7/DyCrr6urKugQp070j+XwzqBXKLy5lF1ue89NgqkpGRkayLkHKtKI5\nv71SaoHyi0vZxZbn/DSYqpJisZh1CVKmtkJ+e6XUAuUXl7KLLc/5aTBVJcuXL8+6BCnTrYNzbr8m\nGVJ+cSm72PKcnwZTVbKvBl+ycK3unMy6BJkD5ReXsostz/lpMFUlao0QV54v760Fyi8uZRdbnvPL\n755XWWPj/HWBlcoamcxv47laoPziUnax5Tk/DaaqZGhoKOsSpEyrFpeyLkHmQPnFpexiy3N+GkxV\nSXd3d9YlSJnuGcnvIspaoPziUnax5Tk/DaaqRDNTca3M8dFVLVB+cSm72PKcnwZTVTIxMZF1CVKm\nxfX57ZVSC5RfXMoutjznp8FUlajPVFx57pVSC5RfXMoutjznp8FUlajPVFx57pVSC5RfXMoutjzn\np8FUlbS0tGRdgpRp0079WUSm/OJSdrHlOb/87nmV1dfn9+7Z0U3k916dNUH5xaXsYstzfhpMVcnw\n8HDWJUiZDszxnc9rgfKLS9nFluf8NJiqkp6enqxLkDLdNZzfRZS1QPnFpexiy3N+GkxVydatW7Mu\nQcp0cGt+e6XUAuUXl7KLLc/5aTBVJe757bcRXcGUXWTKLy5lF1ue88vNnNwpV9w2r6/X0TDF0MSW\neXu96157zLy9Vq27M8dT1bVA+cWl7GLLc36amaqSozvy228jOmUXm/KLS9nFluf8NJiqkt4d+tFG\npexiU35xKbvY8pxffvdcREREpAI0mKqSFU357bcRnbKLTfnFpexiy3N+GkxVye1D+V2IF52yi035\nxaXsYstzfhpMVcmR7fldiBedsotN+cWl7GLLc34aTFXJpFvWJUiZlF1syi8uZRdbnvPTYKpK7hvV\njY6jUnaxKb+4lF1sec5Pg6kqOSLH053RKbvYlF9cyi62POenwVSVbBzXjzYqZReb8otL2cWW5/zy\nu+dV1qCfbFjKLjblF5eyiy3P+eV416tr2aL89tuITtnFpvziUnax5Tk/Daaq5NbB/PbbiE7Zxab8\n4lJ2seU5v6oNpszsVDP7tZndb2YXVet1FqrVnfldiBedsotN+cWl7GLLc35VGUyZWT3wGeA04HDg\nFWZ2eDVea6H6nxv/M+sSpEzKLjblF5eyiy3P+VVrZuppwP3u/ht3LwJfB86o0mstSLfdlN9fquiU\nXWzKLy5lF1ue86vWYOoA4KFZX29MH8uNZq1GC0vZxab84lJ2seU5P3P3yn9Ts5cAp7r7a9OvXwU8\n3d3fNP2c//iP/xjp7e2d+dG3t7f3L126dEvFi8nI1q1bu2tpf/JE2cWm/OJSdrHlIL+Va9as6dnT\nhmotvf898LhZXx+YPjbj+c9/fluVXltERERk3lRrUu4XwMFm9ngzawTOBq6p0muJiIiIZKYqM1Pu\nPmlmbwJ+ANQDX3D3u6vxWiIiIiJZqsqaKRGRWmVm5nrjFJFZcrz2PhYzO9DMWtJ/W9b1iOSNmR1j\nZo/VQGphmH4f1PthbYuSrwZTcbwHuMHMWtzdo/yCyfzT70blmdlxwKeA9qxrkYfNDv6pmS0ys6ZM\ni5KKmvU+1phpIY+STvMFYmZfAXqAl7j7qE43yN6Y2anAamAR8CF335lxSWGZ2ZHAm4A73P2z+rtb\nOMxsLXAmyUVPTcDF7r4j26pkrqb/xtL3sdcCvwIecvfPZ1zaXmlmKhB3fxWwGfiOmbVqhkr2xMye\nRjKLshF4NvBZMzso3abfl/33BJLbYh1tZss0kFoYzOx04GXAS4CDgCWADhpqQPrZdhLwYeDvgGXA\nWWa2KNvK9k6DqQVs1pqAY83sBWa2yt1fTdJd/ls65Se7S2dRXgd8xt2vcveTSD5gPgrJm1SW9UUw\n6+/uMDM7ALgeeAvQBpxsZl1Z1pdXe3ifM+CfgBcDncDr0/fDY81Mn20B7ZbxAcAbSGYcjwbe4O47\nzezxmRS3D/qFW8DSN4YzgM+RzDBcaWanuft5QB/wg+kZqkwLlYXkcSQzKU81s1UA7v7XwGMW6pvQ\nQpP+3Z0G/CtwHnA7cD/wLeBk4Awz686wxNyZfWrVzJ6a/vx/D3yCZBB1irsXzeyNwPkkp7clmPRv\n79lm9lhgB/BN4EPA8939d2b2XODc6YuxFhINphaY6WlMSywFXg2cSPKG3p7+F3c/F/gdyekHyalZ\nsygHm9ly4GZgLUl/t+eZ2ZPM7AiSafLJ7CqNIz0legnwIpJB1A6gzt2/CXwXOAV9WM+rWQOptwIf\nJ5kl/BXw98BvzexMM3sdu2ZlxzMrVubqVOAdwNXALcAf3H3AzNYA/wD8zN3HsixwT7QAfQExs07g\n+8Cb3f1nZrYE+BjQDxwPnOvuD5jZ84Bb3b2W74Ekj1K6SPPjwP8ATwHOITlQ+gCwlGSd3Rfc/Vot\nnt6z3WY+2oE3Ar8F3g6c4+73pX931wHd7t6fXbX5ZGYnApcBp7n7YPrYAcAzgVeQ/J5/xt3vyqxI\nmTMzexbwKnadWv97koPBAvAJd/9ehuXtlQZTC4SZNabT1O8k+UX6S3f/HzO7GDiX5Hzxzekbyj8C\nL3f3X2ZYsiwA6YfJd4G3uPstZvZK4JPAcUAH8DfAj4HLdUXfIzOzZ5Cc1vsrkoHpSpKB01S67aPA\nee7+QIZl5paZPZvk53+emTVPzz6ZWb27l3SgEIuZ9QCPcfe704tmnuDuX023fZPk6r3/k37dTjJe\nGVqoOes03wKQnv+/0sye4u4fBS4H/tnMjiZZp3EN8C4zez/J+qm3ayAlqRHgbpJLw3H3fyEZTL3K\n3f8H+AqwBniZmVXrxuahzVr0eg/weJI1Z2cDQ8Cl6eX3/wj8nQZS82MvF9WMAI9Lr6icHkj9BfB6\nMyssxA9Y2TNL7tn7UmDczBaTzEBdamYfNLMXAhcAi83scAB3H3b3ofTfCzJnzUwtEGb2SZLFw+9z\n91+a2ZtJrmR4KfAgcBKwGNjg7j9dqKNzqa5Z/VfagVHAgf8E7nb3t6XPeS1wuLu/Nf36VOCX7v6H\nrOpeyMysLp19KgDvBgbc/dNm9kTg9cAW4Bfu/l/6u6u+3U65XggcCmwgOUh4K8lFAFcDzSSLzV/o\n7v+bUblSJjNrJlkH/GbgKpLTtH8OnEbyWbgY+Ad3vyqzIveDBlMZm56iTv99GcmC8vekA6q3kJx2\neK27/zzLOiV7swZSpwLvJBlk/47kapefA7cBdwCvAd7p7t/PqtYo0iPfr5I05fwV8BiSG7S/xN1/\nkWVteWdmzyFZM3o5yaXxjwH+kuTD9nBgOfBZDaRi2W2wfCTJ+9VO4Kvufmf6+FtIFqJ/1t2vyazY\n/aDBVIZmfTi2uftI+tj7gWOAd6cDqreTrOE4BhjRUXH+pKcwJtN/P4Xk1N1bSQZSVwC/JJkW/2uS\nWy/c4e7XaxZlz3b/uZjZ60n+vrqBL5DMhDQBH3X3qWyqzLf09N1rgI+7+3Vmtgx4G8mMxYXu3j89\no5hpobJfZn3mHUIyEzVKsrh8LUnfsGvcfV363MXuvj3K+5gGUxmZ9Uv1fOAskg/Gq9x9g5ldQvLm\n/n53v83MVrr7hkwLlkxY0iDyrcCX0ivKnk6yHupNs57z38AHNBO1b7v93T0PmAA+Q9It/pkkV4s1\nk3TTPmIhXoJdi/YwwD2CZL3ode5+QfrYY0juUdpFcpGOazAVj5m9ALgU+FH60CdJZqbeBLQA33T3\nn2RUXtm0AD0j6Rv6ySTT2J8hub/UJ83sBHd/P8kphw+ZWRvJQEvy6UCSWZI3mtmfAsPAGjNbOes5\nN6XPkX1I/+6eB7wX+BfgSSS9a9zdbwHOIOlx8xoNpObHbqd9npde2TVAsk70uZb0lsLdNwPvJ2kd\nU9JAKh4zO5YkwxcB20kujvkQySDqH4EiSfbhaDA1z6avUjGzVuD5wCtJbl48RbIG5h1m9hx3fzfJ\ndLZO7eWYu99B8qE/BlxIshD308B/m9lplnTIfyFB34Dmg5l1p71rph1FcgppOckC2De5+6SZLXH3\nLe7+X+5+016uKJMKmzWQeivwLpK1Mv9KctrnJcB5ZvZ/0+f2u3p8hWO7bu/TRrIO+Ekka98uJBmH\n/C3QClzi7r/OpMg50qXS82jWKYYXkPSweS/JKYWPAye4+6iZ3Qf8hZn9yt3vz7Bcydj074u7r08/\n188iOYq7CNhKcnTXDVzk7j/a+3fKr/QKvdcAq8yswd1vIrlK6FMkBzB/4cltKv4ceKKZ/YO7T8DC\nvQS7Vuw2I3UIcLK7n2BmnyC5gnKzu280s5cDXzSzz7r71ixrlv0zK+N2YNDdb0wf/xjJDOOPLbl1\nUw/Q7O7FDMudE81MzaN0IPUMkm69d7v7cLqpGzjKzJ5AMjt1ubtrpiHn0t8XS/+9nmQNyTjJndR/\n6O5vAM529+9pFmXP0oX73wF6gReY2TEkM3vLSRbqb7CkEe4ngNumB1JSXbsNpB5HsnZtk5ldSnIB\nwF940sT4LHe/G3i2BlLxzFqf+AMzuyydSIBkcPV/07+900k619+RVZ2VoAXo88jMmoB/Aw5194Nn\nPf56kqnPdpJZhhCXgkr1zeqB1EUyRd5OcruYZpIZqnGtHdmzWT+7RpLZqAtJusL/MzBIMjj9X+Ag\nkos9FuRtKmqZmb2aZDH5aSQtKo4Bnuzu45b0SzuPpI+Ubp0VUHr18TtJPveeRHJ7qxvc/Roz+wLJ\n3+U33P3bGZZZERpMVdkerlJZRXL7j5unr8gys3rgAGCxu/9vlEtBpTL28DsyfTp4ejDwWJJbwrzb\n3b9mZquBYXe/L7Oig0gXm/818BckpxJeTTIg/SJwL8mbebu7P5RZkTllZi8jGSxd6O73prMULwP+\nlORKr5cBr3Tday8kM1tBcqPib7v7O9M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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "fig, ax = plt.subplots(figsize=(10, 5))\n", "s.plot(kind='bar', ax=ax, rot=45)\n", "plt.legend(loc='upper left')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Conclusions " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "So here's the list of the things I learnt while doing this work:\n", "\n", "- declaring complex variables within Cython functions\n", "- using C++ std lib functions using cdef extern in Cython code\n", "- Cython allows one to easily parallelize code\n", "- the Cython error messages are often cryptic\n", "- it is useful to read the generated Cython code to understand what's going on under the hood of the translation" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", 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