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Sympy code gen demo
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| { | |
| "metadata": { | |
| "name": "" | |
| }, | |
| "nbformat": 3, | |
| "nbformat_minor": 0, | |
| "worksheets": [ | |
| { | |
| "cells": [ | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "import sympy as S\n", | |
| "S.var(\"x mu sigma\", real=True)" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "latex": [ | |
| "$$\\begin{pmatrix}x, & \\mu, & \\sigma\\end{pmatrix}$$" | |
| ], | |
| "metadata": {}, | |
| "output_type": "pyout", | |
| "png": "iVBORw0KGgoAAAANSUhEUgAAAGgAAAAaBAMAAACp2ARYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nSUVORK5CYII=\n", | |
| "prompt_number": 1, | |
| "text": [ | |
| "(x, \u03bc, \u03c3)" | |
| ] | |
| } | |
| ], | |
| "prompt_number": 1 | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "Here's the likelihood for a 1-d normal distribution, in terms of two parameters: the mean $\\mu$ and standard deviation $\\sigma$" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "\n", | |
| "f = 1/(sqrt(2*S.pi)*sigma)*S.exp(-(x-mu)**2/(2*sigma**2))\n", | |
| "f\n" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "latex": [ | |
| "$$\\frac{\\sqrt{2}}{2 \\sqrt{\\pi} \\sigma} e^{- \\frac{\\left(- \\mu + x\\right)^{2}}{2 \\sigma^{2}}}$$" | |
| ], | |
| "metadata": {}, | |
| "output_type": "pyout", | |
| "png": "iVBORw0KGgoAAAANSUhEUgAAAIsAAAA0BAMAAABIq9ZsAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ngg==\n", | |
| "prompt_number": 2, | |
| "text": [ | |
| " 2 \n", | |
| " -(-\u03bc + x) \n", | |
| " \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n", | |
| " 2 \n", | |
| " ___ 2\u22c5\u03c3 \n", | |
| "\u2572\u2571 2 \u22c5\u212f \n", | |
| "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n", | |
| " ___ \n", | |
| " 2\u22c5\u2572\u2571 \u03c0 \u22c5\u03c3 " | |
| ] | |
| } | |
| ], | |
| "prompt_number": 2 | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "And here's the log likelihood, which we can tidy up:" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "ll = log(f)\n", | |
| "ll" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "latex": [ | |
| "$$\\log{\\left (\\frac{\\sqrt{2}}{2 \\sqrt{\\pi} \\sigma} e^{- \\frac{\\left(- \\mu + x\\right)^{2}}{2 \\sigma^{2}}} \\right )}$$" | |
| ], | |
| "metadata": {}, | |
| "output_type": "pyout", | |
| "png": "iVBORw0KGgoAAAANSUhEUgAAAMsAAAA/BAMAAAC81iEPAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nfvV4rozgAAAAAElFTkSuQmCC\n", | |
| "prompt_number": 3, | |
| "text": [ | |
| " \u239b 2 \u239e\n", | |
| " \u239c -(-\u03bc + x) \u239f\n", | |
| " \u239c \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\n", | |
| " \u239c 2 \u239f\n", | |
| " \u239c ___ 2\u22c5\u03c3 \u239f\n", | |
| " \u239c\u2572\u2571 2 \u22c5\u212f \u239f\n", | |
| " \u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\n", | |
| " \u239c ___ \u239f\n", | |
| " \u239d 2\u22c5\u2572\u2571 \u03c0 \u22c5\u03c3 \u23a0\n", | |
| "log " | |
| ] | |
| } | |
| ], | |
| "prompt_number": 3 | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "ll = simplify(logcombine(ll))\n", | |
| "ll" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "latex": [ | |
| "$$\\log{\\left (\\frac{1}{\\sigma} \\right )} - \\frac{1}{2} \\log{\\left (\\pi \\right )} - \\frac{1}{2} \\log{\\left (2 \\right )} - \\frac{\\left(\\mu - x\\right)^{2}}{2 \\sigma^{2}}$$" | |
| ], | |
| "metadata": {}, | |
| "output_type": "pyout", | |
| "png": "iVBORw0KGgoAAAANSUhEUgAAAYIAAAA3BAMAAADzpFDIAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n3f4DvSUcrYcgV58AAAAASUVORK5CYII=\n", | |
| "prompt_number": 4, | |
| "text": [ | |
| " 2\n", | |
| " \u239b1\u239e log(\u03c0) log(2) (\u03bc - x) \n", | |
| "log\u239c\u2500\u239f - \u2500\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n", | |
| " \u239d\u03c3\u23a0 2 2 2 \n", | |
| " 2\u22c5\u03c3 " | |
| ] | |
| } | |
| ], | |
| "prompt_number": 4 | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "Let's pull out the derivatives of the log-likelihood: $\\frac{\\partial\\mathcal(l)}{\\partial\\mu}$, $\\frac{\\partial\\mathcal(l)}{\\partial\\sigma}$" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "ll.diff(mu)\n" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "latex": [ | |
| "$$- \\frac{1}{2 \\sigma^{2}} \\left(2 \\mu - 2 x\\right)$$" | |
| ], | |
| "metadata": {}, | |
| "output_type": "pyout", | |
| "png": "iVBORw0KGgoAAAANSUhEUgAAAIwAAAAqBAMAAACTq67+AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nlTUebY/VzG/DEqMiKJVV6wAAAABJRU5ErkJggg==\n", | |
| "prompt_number": 5, | |
| "text": [ | |
| "-(2\u22c5\u03bc - 2\u22c5x) \n", | |
| "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n", | |
| " 2 \n", | |
| " 2\u22c5\u03c3 " | |
| ] | |
| } | |
| ], | |
| "prompt_number": 5 | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "ll.diff(sigma)\n" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "latex": [ | |
| "$$- \\frac{1}{\\sigma} + \\frac{1}{\\sigma^{3}} \\left(\\mu - x\\right)^{2}$$" | |
| ], | |
| "metadata": {}, | |
| "output_type": "pyout", | |
| "png": "iVBORw0KGgoAAAANSUhEUgAAAKMAAAAqBAMAAAAt/fajAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nrGs1Y5LAZHVlRP2g7R/qqadnvG7MtgAAAABJRU5ErkJggg==\n", | |
| "prompt_number": 6, | |
| "text": [ | |
| " 2\n", | |
| " 1 (\u03bc - x) \n", | |
| "- \u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n", | |
| " \u03c3 3 \n", | |
| " \u03c3 " | |
| ] | |
| } | |
| ], | |
| "prompt_number": 6 | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "Second derivatives:" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "ll.diff(mu, mu)\n" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "latex": [ | |
| "$$- \\frac{1}{\\sigma^{2}}$$" | |
| ], | |
| "metadata": {}, | |
| "output_type": "pyout", | |
| "png": "iVBORw0KGgoAAAANSUhEUgAAACgAAAAqBAMAAAAzGARVAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMiK7mat272ZE\nVIkJo9jjAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAArElEQVQoFWNgQAFMAihcMIc1H1OQMb0fU5CB\nYf5wE6xYb7EBM0QGr4iQMQioMDD8RwAKXPs8HTNC+QI4FDCMZA5g+YohyHWA9Q+GIAMD70csgswT\nsAg+xyLGXYAiyFj1//93hnMMR5BFnWsupwaw6N5NRBJkXMvA/YGB6///D0iC7AoMPBiu4d/AwIms\nCKw+3oGBqQFJJ5gpL8DgJYAuyHaB1QZdjIGxch5MIQCENjK5UVh8EgAAAABJRU5ErkJggg==\n", | |
| "prompt_number": 7, | |
| "text": [ | |
| "-1 \n", | |
| "\u2500\u2500\u2500\n", | |
| " 2\n", | |
| " \u03c3 " | |
| ] | |
| } | |
| ], | |
| "prompt_number": 7 | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "ll.diff(mu, sigma)" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "latex": [ | |
| "$$\\frac{1}{\\sigma^{3}} \\left(2 \\mu - 2 x\\right)$$" | |
| ], | |
| "metadata": {}, | |
| "output_type": "pyout", | |
| "png": "iVBORw0KGgoAAAANSUhEUgAAAHAAAAAqBAMAAACZ28LpAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nCJv/A/D1jnHTPoltAAAAAElFTkSuQmCC\n", | |
| "prompt_number": 8, | |
| "text": [ | |
| "2\u22c5(\u03bc - x)\n", | |
| "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n", | |
| " 3 \n", | |
| " \u03c3 " | |
| ] | |
| } | |
| ], | |
| "prompt_number": 8 | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "ll.diff(sigma, sigma)" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "latex": [ | |
| "$$\\frac{1}{\\sigma^{2}} \\left(1 - \\frac{3}{\\sigma^{2}} \\left(\\mu - x\\right)^{2}\\right)$$" | |
| ], | |
| "metadata": {}, | |
| "output_type": "pyout", | |
| "png": "iVBORw0KGgoAAAANSUhEUgAAAMQAAAAyBAMAAADxQ2lcAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nrkJggg==\n", | |
| "prompt_number": 9, | |
| "text": [ | |
| " 2\n", | |
| " 3\u22c5(\u03bc - x) \n", | |
| "1 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n", | |
| " 2 \n", | |
| " \u03c3 \n", | |
| "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n", | |
| " 2 \n", | |
| " \u03c3 " | |
| ] | |
| } | |
| ], | |
| "prompt_number": 9 | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "simplify(_)" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "latex": [ | |
| "$$\\frac{1}{\\sigma^{4}} \\left(\\sigma^{2} - 3 \\left(\\mu - x\\right)^{2}\\right)$$" | |
| ], | |
| "metadata": {}, | |
| "output_type": "pyout", | |
| "png": "iVBORw0KGgoAAAANSUhEUgAAALoAAAAqBAMAAAD2RlyBAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAzRAiu5mrdu/dZjJE\nVIl1eHuDAAAACXBIWXMAAA7EAAAOxAGVKw4bAAADw0lEQVRIDb1VTWgTQRR+2W2bn022RVBBLWwP\ntfZiA/biD7ZailAsjWL9AQ/x7MHowd7aoOBBKy20WgSVFBXEi/GmB2k8CIpog3jpoTQq1epBSkoV\nf9r43uzO7EzSBpJYH+y8733vvW8mM5NdADKPhcNamd5jrZU0gNZ5Zg3VAQb+o/qDf7FNWlSoKGv3\n1gm+EvBMNCvqLyzBVwK8Ud4tq2tXOFu+n+20ADbxflm9JsLZsr0ZrsHdbcMZmMnqE5wE88hhJ1+i\n84eNBYCaqNMmq68XUh/hlMAlAV9K/wNgZAvVdZzVsRGYCHNcog+R8kanSVp79bAQGoS2lAhKA/4Y\n1t+3e44OXU/wbh/x3GYsjlRvJNRYigJJCmZpWGFtbSxLSbQx24lx77duhlsFUwhOIxWIEO/L0KjY\nQykyWZFL6BGYSVC4zuUKkAeZN/AKR7qWeXZCiu9JmKB3CWqHyWdwWM28GTDqp/dj2jtfUDPqMsFM\nMOlGiIKD0ELr8YUVWg20OPhyORIOudePlyxyAPBo+p3lRjZiO/Myn1Xieh4ZSxwJ/4OhxqFcLoKP\noDlg/zW6EUYc3PeUnW06v6c/5twXYrRfNu+ONmNcez26wyUFMr5ECHfhYw6DnwUiqWUCY/46gHHB\n2CsVIYDO5tuWBvnuSPm5JAa38PFEoCUtJXA+y8yGwgBTgl2GnGvE2nt1CGDS4kUNN8g226HvAvpB\nfFCa/dfcrAaediqaoIGZdIY2YbC14zDgVMhOD0MVnRSpozSuQbXaGMVF1NnO6L8BNqiNLKrNQhXN\nTjuD/4vbel7NRBIsAPdGLbv5dgY1OgktCwbtQL752sFPd5hOdRT0JVMpCFkD4LXkU2X7wGrYOxMR\nm68bdqYYqw6BBMzQj/+EC/gJ1QtPlfRkugd2IdPLWZ120bbtzrv+EoWNfR8cWnXPew8Sgb89tPg5\nRpdbsq19TV8TGG/hnIGfEMfOOer7OFHE+8MrvaGcBj3OO/l24E1866jP8VwRjy9Y5TOglJriF5lx\nnjDxJJmxS8zZ1XwHKJ8BpaxBRFVRDu9w9ZYEp4r4Vuna5Zd1CKI24UAtxtXdCUVVIdB5X2EqEBYc\nffO147ncYqi3d9HuCMRFtlLwHgWenJzqwEl0Z98195JWqn4Zl34VAvOoc/d72lY7W6ko7w/FAarr\nIOgs26Z9SZ6u0PsTAHiw6tc1UFehKm8fR4AX1xPlBPMHlKjsQL+IrS0WPLYUieZ/szXNdI5Vaf2m\nIg76iBqXGXVRn3as38rr350XlxVqKWz7CxhC3OWXjyL0AAAAAElFTkSuQmCC\n", | |
| "prompt_number": 10, | |
| "text": [ | |
| " 2 2\n", | |
| "\u03c3 - 3\u22c5(\u03bc - x) \n", | |
| "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n", | |
| " 4 \n", | |
| " \u03c3 " | |
| ] | |
| } | |
| ], | |
| "prompt_number": 10 | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "Let's find the point where the gradient is zero for $\\mu$ and $\\sigma$" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "solve(f.diff(mu), mu)" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "latex": [ | |
| "$$\\begin{bmatrix}x\\end{bmatrix}$$" | |
| ], | |
| "metadata": {}, | |
| "output_type": "pyout", | |
| "png": "iVBORw0KGgoAAAANSUhEUgAAABYAAAAZBAMAAADHzMPdAAAAKlBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADmU0mKAAAADXRSTlMAu90iEHarRIlmzVQyXQFQhAAA\nAAlwSFlzAAAOxAAADsQBlSsOGwAAAF5JREFUGBljEFIyYAABdiUNBgUwC0SYUZPNEppc2go104Nh\nUoAWlN3C0OuwGcp2YFiIZO9VBJvnDgMLVM0W1gsMIRA2+23GC8wTIGyW1JCImdR2M8gBQGDGoAQP\nHzUAIM4WtLB8TSEAAAAASUVORK5CYII=\n", | |
| "prompt_number": 11, | |
| "text": [ | |
| "[x]" | |
| ] | |
| } | |
| ], | |
| "prompt_number": 11 | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "solve(f.diff(sigma), sigma)" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "latex": [ | |
| "$$\\begin{bmatrix}- \\mu + x, & \\mu - x\\end{bmatrix}$$" | |
| ], | |
| "metadata": {}, | |
| "output_type": "pyout", | |
| "png": "iVBORw0KGgoAAAANSUhEUgAAAJgAAAAZBAMAAAA/APpvAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n+QtCEFzfw9ZCtQAAAABJRU5ErkJggg==\n", | |
| "prompt_number": 12, | |
| "text": [ | |
| "[-\u03bc + x, \u03bc - x]" | |
| ] | |
| } | |
| ], | |
| "prompt_number": 12 | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "What if we want to use these calculations in numerical optimization code?" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "d2f_sigma2 = lambdify((mu, sigma, x), f.diff(sigma, sigma))" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [], | |
| "prompt_number": 13 | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "d2f_sigma2(1.2, 2, 3)" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "latex": [ | |
| "$$-0.0463620287292$$" | |
| ], | |
| "metadata": {}, | |
| "output_type": "pyout", | |
| "png": "iVBORw0KGgoAAAANSUhEUgAAAKUAAAAPBAMAAAB3ghJhAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nY/IXed4JxwCefm8AAAAASUVORK5CYII=\n", | |
| "prompt_number": 14, | |
| "text": [ | |
| "-0.0463620287292" | |
| ] | |
| } | |
| ], | |
| "prompt_number": 14 | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "timeit d2f_sigma2(1.2, 2, 3)" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "stream": "stdout", | |
| "text": [ | |
| "1000000 loops, best of 3: 1.61 \u00b5s per loop\n" | |
| ] | |
| } | |
| ], | |
| "prompt_number": 15 | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "from sympy.utilities import codegen" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [], | |
| "prompt_number": 17 | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "codegen.ccode(f)" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "metadata": {}, | |
| "output_type": "pyout", | |
| "prompt_number": 18, | |
| "text": [ | |
| "'(1.0L/2.0L)*sqrt(2)*exp(-1.0L/2.0L*pow(-mu + x, 2)/pow(sigma, 2))/(sqrt(M_PI)*sigma)'" | |
| ] | |
| } | |
| ], | |
| "prompt_number": 18 | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| " print codegen.codegen(('d2f_dsigma2', f), 'C', 'd2f_dsigma2')[0][1]" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "stream": "stdout", | |
| "text": [ | |
| "/******************************************************************************\n", | |
| " * Code generated with sympy 0.7.4.1 *\n", | |
| " * *\n", | |
| " * See http://www.sympy.org/ for more information. *\n", | |
| " * *\n", | |
| " * This file is part of 'project' *\n", | |
| " ******************************************************************************/\n", | |
| "#include \"d2f_dsigma2.h\"\n", | |
| "#include <math.h>\n", | |
| "\n", | |
| "double d2f_dsigma2(double mu, double sigma, double x) {\n", | |
| "\n", | |
| " return (1.0L/2.0L)*sqrt(2)*exp(-1.0L/2.0L*pow(-mu + x, 2)/pow(sigma, 2))/(sqrt(M_PI)*sigma);\n", | |
| "\n", | |
| "}\n", | |
| "\n" | |
| ] | |
| } | |
| ], | |
| "prompt_number": 19 | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "Too lazy to compile a C extension. Do it for me!" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "from sympy.utilities import autowrap" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [], | |
| "prompt_number": 20 | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "fw = autowrap.autowrap(f.diff(sigma, sigma))" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [], | |
| "prompt_number": 21 | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "timeit fw(1.2, 2, 3)" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "stream": "stdout", | |
| "text": [ | |
| "1000000 loops, best of 3: 214 ns per loop\n" | |
| ] | |
| } | |
| ], | |
| "prompt_number": 22 | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "logpdf = autowrap.autowrap(ll)" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [], | |
| "prompt_number": 24 | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "logpdf(2.3, 6., 1)" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "latex": [ | |
| "$$-2.73417022465$$" | |
| ], | |
| "metadata": {}, | |
| "output_type": "pyout", | |
| "png": "iVBORw0KGgoAAAANSUhEUgAAAJEAAAAPBAMAAAAWmjAjAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nv8VfhqXApTA0Rd8AAAAASUVORK5CYII=\n", | |
| "prompt_number": 27, | |
| "text": [ | |
| "-2.73417022465" | |
| ] | |
| } | |
| ], | |
| "prompt_number": 27 | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "from scipy.stats.distributions import norm\n", | |
| "s_logpdf = norm.logpdf" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [], | |
| "prompt_number": 40 | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "s_logpdf(1, 2.3, 6.)" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "latex": [ | |
| "$$-2.73417022465$$" | |
| ], | |
| "metadata": {}, | |
| "output_type": "pyout", | |
| "png": "iVBORw0KGgoAAAANSUhEUgAAAJEAAAAPBAMAAAAWmjAjAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nv8VfhqXApTA0Rd8AAAAASUVORK5CYII=\n", | |
| "prompt_number": 41, | |
| "text": [ | |
| "-2.73417022465" | |
| ] | |
| } | |
| ], | |
| "prompt_number": 41 | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "%timeit logpdf(2.3, 6., 1)" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "stream": "stdout", | |
| "text": [ | |
| "1000000 loops, best of 3: 197 ns per loop\n" | |
| ] | |
| } | |
| ], | |
| "prompt_number": 42 | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "%timeit norm.logpdf(2.3, 6., 1)" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "stream": "stdout", | |
| "text": [ | |
| "10000 loops, best of 3: 86 \u00b5s per loop\n" | |
| ] | |
| } | |
| ], | |
| "prompt_number": 43 | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "x = np.arange(100)\n", | |
| "vlogpdf = np.vectorize(logpdf)" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [], | |
| "prompt_number": 45 | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "timeit vlogpdf(2.3, 6, x)" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "stream": "stdout", | |
| "text": [ | |
| "10000 loops, best of 3: 51.8 \u00b5s per loop\n" | |
| ] | |
| } | |
| ], | |
| "prompt_number": 47 | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "timeit s_logpdf(2.3, 6., x)" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "stream": "stdout", | |
| "text": [ | |
| "10000 loops, best of 3: 108 \u00b5s per loop\n" | |
| ] | |
| } | |
| ], | |
| "prompt_number": 48 | |
| } | |
| ], | |
| "metadata": {} | |
| } | |
| ] | |
| } |
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