ch19-code-listing.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Chapter 19: Code optimization"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Robert Johansson\n",
    "\n",
    "Source code listings for [Numerical Python - A Practical Techniques Approach for Industry](http://www.apress.com/9781484205549) (ISBN 978-1-484205-54-9).\n",
    "\n",
    "The source code listings can be downloaded from http://www.apress.com/9781484205549"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "import numba"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import pyximport"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import cython"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Numba"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "np.random.seed(0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "data = np.random.randn(50000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def py_sum(data):\n",
    "    s = 0\n",
    "    for d in data:\n",
    "        s += d\n",
    "    return s"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def py_cumsum(data):\n",
    "    out = np.zeros(len(data), dtype=np.float64)\n",
    "    s = 0 \n",
    "    for n in range(len(data)):\n",
    "        s += data[n]\n",
    "        out[n] = s\n",
    "\n",
    "    return out"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%timeit py_sum(data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "assert abs(py_sum(data) - np.sum(data)) < 1e-10"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%timeit np.sum(data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%timeit py_cumsum(data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "assert np.allclose(np.cumsum(data), py_cumsum(data))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%timeit np.cumsum(data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "@numba.jit\n",
    "def jit_sum(data):\n",
    "    s = 0 \n",
    "    for d in data:\n",
    "        s += d\n",
    "\n",
    "    return s"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "assert abs(jit_sum(data) - np.sum(data)) < 1e-10"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%timeit jit_sum(data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "jit_cumsum = numba.jit()(py_cumsum)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "assert np.allclose(np.cumsum(data), jit_cumsum(data))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%timeit jit_cumsum(data)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Julia fractal"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def py_julia_fractal(z_re, z_im, j):\n",
    "    for m in range(len(z_re)):\n",
    "        for n in range(len(z_im)):\n",
    "            z = z_re[m] + 1j * z_im[n]\n",
    "            for t in range(256):\n",
    "                z = z ** 2 - 0.05 + 0.68j\n",
    "                if np.abs(z) > 2.0:\n",
    "                #if (z.real * z.real + z.imag * z.imag) > 4.0:  # a bit faster\n",
    "                    j[m, n] = t\n",
    "                    break"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "jit_julia_fractal = numba.jit(nopython=True)(py_julia_fractal)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "N = 1024\n",
    "j = np.zeros((N, N), np.int64)\n",
    "z_real = np.linspace(-1.5, 1.5, N)\n",
    "z_imag = np.linspace(-1.5, 1.5, N)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "jit_julia_fractal(z_real, z_imag, j)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false,
    "scrolled": false
   },
   "outputs": [],
   "source": [
    "fig, ax = plt.subplots(figsize=(14, 14))\n",
    "ax.imshow(j, cmap=plt.cm.RdBu_r,\n",
    "          extent=[-1.5, 1.5, -1.5, 1.5])\n",
    "ax.set_xlabel(\"$\\mathrm{Re}(z)$\", fontsize=18)\n",
    "ax.set_ylabel(\"$\\mathrm{Im}(z)$\", fontsize=18)\n",
    "fig.tight_layout()\n",
    "fig.savefig(\"ch19-numba-julia-fractal.pdf\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%timeit py_julia_fractal(z_real, z_imag, j)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%timeit jit_julia_fractal(z_real, z_imag, j)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Vectorize"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def py_Heaviside(x):\n",
    "    if x == 0.0:\n",
    "        return 0.5\n",
    "    \n",
    "    if x < 0.0:\n",
    "        return 0.0\n",
    "    else:\n",
    "        return 1.0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "x = np.linspace(-2, 2, 50001)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%timeit [py_Heaviside(xx) for xx in x]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "np_vec_Heaviside = np.vectorize(py_Heaviside)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "np_vec_Heaviside(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%timeit np_vec_Heaviside(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def np_Heaviside(x):\n",
    "    return (x > 0.0) + (x == 0.0)/2.0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%timeit np_Heaviside(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "@numba.vectorize([numba.float32(numba.float32),\n",
    "                  numba.float64(numba.float64)])\n",
    "def jit_Heaviside(x):\n",
    "    if x == 0.0:\n",
    "        return 0.5\n",
    "    \n",
    "    if x < 0:\n",
    "        return 0.0\n",
    "    else:\n",
    "        return 1.0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%timeit jit_Heaviside(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "jit_Heaviside([-1, -0.5, 0.0, 0.5, 1.0])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Cython"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "!rm cy_sum.*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%%writefile cy_sum.pyx\n",
    "\n",
    "def cy_sum(data):\n",
    "    s = 0.0\n",
    "    for d in data:\n",
    "        s += d\n",
    "    return s"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "!cython cy_sum.pyx"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false,
    "scrolled": false
   },
   "outputs": [],
   "source": [
    "# 5 lines of python code -> 1470 lines of C code ...\n",
    "!wc cy_sum.c"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%%writefile setup.py\n",
    "\n",
    "from distutils.core import setup\n",
    "from Cython.Build import cythonize\n",
    "\n",
    "import numpy as np\n",
    "setup(ext_modules=cythonize('cy_sum.pyx'),\n",
    "      include_dirs=[np.get_include()],\n",
    "      requires=['Cython', 'numpy'] )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false,
    "scrolled": false
   },
   "outputs": [],
   "source": [
    "!python setup.py build_ext --inplace > /dev/null"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from cy_sum import cy_sum"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "cy_sum(data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%timeit cy_sum(data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%timeit py_sum(data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%%writefile cy_cumsum.pyx\n",
    "\n",
    "cimport numpy\n",
    "import numpy\n",
    "\n",
    "def cy_cumsum(data):\n",
    "    out = numpy.zeros_like(data)\n",
    "    s = 0 \n",
    "    for n in range(len(data)):\n",
    "        s += data[n]\n",
    "        out[n] = s\n",
    "\n",
    "    return out"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "pyximport.install(setup_args={'include_dirs': np.get_include()});"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "pyximport.install(setup_args=dict(include_dirs=np.get_include()));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from cy_cumsum import cy_cumsum"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%timeit cy_cumsum(data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%timeit py_cumsum(data)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Using IPython cython command"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%load_ext cython"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%%cython -a\n",
    "def cy_sum(data):\n",
    "    s = 0.0\n",
    "    for d in data:\n",
    "        s += d\n",
    "    return s"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%timeit cy_sum(data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%timeit py_sum(data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "assert np.allclose(np.sum(data), cy_sum(data))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%%cython -a\n",
    "cimport numpy\n",
    "cimport cython\n",
    "\n",
    "@cython.boundscheck(False)\n",
    "@cython.wraparound(False)\n",
    "def cy_sum(numpy.ndarray[numpy.float64_t, ndim=1] data):\n",
    "    cdef numpy.float64_t s = 0.0\n",
    "    #cdef int n, N = data.shape[0]\n",
    "    cdef int n, N = len(data)\n",
    "    for n in range(N):\n",
    "        s += data[n]\n",
    "    return s"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%timeit cy_sum(data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%timeit jit_sum(data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%timeit np.sum(data)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Cummulative sum"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%%cython -a\n",
    "cimport numpy\n",
    "import numpy\n",
    "cimport cython\n",
    "\n",
    "ctypedef numpy.float64_t FTYPE_t\n",
    "\n",
    "@cython.boundscheck(False)\n",
    "@cython.wraparound(False)\n",
    "def cy_cumsum(numpy.ndarray[FTYPE_t, ndim=1] data):\n",
    "    cdef int n, N = data.size\n",
    "    cdef numpy.ndarray[FTYPE_t, ndim=1] out = numpy.zeros(N, dtype=data.dtype)\n",
    "    cdef numpy.float64_t s = 0.0\n",
    "    for n in range(N):\n",
    "        s += data[n]\n",
    "        out[n] = s\n",
    "    return out"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%timeit py_cumsum(data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%timeit cy_cumsum(data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%timeit jit_cumsum(data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%timeit np.cumsum(data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "assert np.allclose(cy_cumsum(data), np.cumsum(data))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Fused types"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "py_sum([1.0, 2.0, 3.0, 4.0, 5.0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "py_sum([1, 2, 3, 4, 5])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "cy_sum(np.array([1.0, 2.0, 3.0, 4.0, 5.0]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "cy_sum(np.array([1, 2, 3, 4, 5]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%%cython -a\n",
    "cimport numpy\n",
    "cimport cython\n",
    "\n",
    "ctypedef fused I_OR_F_t:\n",
    "    numpy.int64_t \n",
    "    numpy.float64_t \n",
    "\n",
    "@cython.boundscheck(False)\n",
    "@cython.wraparound(False)\n",
    "def cy_fused_sum(numpy.ndarray[I_OR_F_t, ndim=1] data):\n",
    "    cdef I_OR_F_t s = 0\n",
    "    cdef int n, N = data.size\n",
    "    for n in range(N):\n",
    "        s += data[n]\n",
    "    return s"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "cy_fused_sum(np.array([1.0, 2.0, 3.0, 4.0, 5.0]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "cy_fused_sum(np.array([1, 2, 3, 4, 5]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Julia fractal"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%%cython -a\n",
    "cimport numpy\n",
    "cimport cython\n",
    "\n",
    "ctypedef numpy.int64_t ITYPE_t\n",
    "ctypedef numpy.float64_t FTYPE_t\n",
    "\n",
    "cpdef inline double abs2(double complex z):\n",
    "    return z.real * z.real + z.imag * z.imag\n",
    "\n",
    "@cython.boundscheck(False)\n",
    "@cython.wraparound(False)\n",
    "def cy_julia_fractal(numpy.ndarray[FTYPE_t, ndim=1] z_re, \n",
    "                     numpy.ndarray[FTYPE_t, ndim=1] z_im, \n",
    "                     numpy.ndarray[ITYPE_t, ndim=2] j):\n",
    "    cdef int m, n, t, M = z_re.size, N = z_im.size\n",
    "    cdef double complex z\n",
    "    for m in range(M):\n",
    "        for n in range(N):\n",
    "            z = z_re[m] + 1.0j * z_im[n]\n",
    "            for t in range(256):\n",
    "                z = z ** 2 - 0.05 + 0.68j\n",
    "                if abs2(z) > 4.0:\n",
    "                    j[m, n] = t\n",
    "                    break"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "N = 1024"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "j = np.zeros((N, N), dtype=np.int64)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "z_real = np.linspace(-1.5, 1.5, N)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "z_imag = np.linspace(-1.5, 1.5, N)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%timeit cy_julia_fractal(z_real, z_imag, j)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%timeit jit_julia_fractal(z_real, z_imag, j)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "j1 = np.zeros((N, N), dtype=np.int64)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "cy_julia_fractal(z_real, z_imag, j1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "j2 = np.zeros((N, N), dtype=np.int64)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "jit_julia_fractal(z_real, z_imag, j2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "assert np.allclose(j1, j2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Calling C function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%%cython\n",
    "\n",
    "cdef extern from \"math.h\":\n",
    "     double acos(double)\n",
    "\n",
    "def cy_acos1(double x):\n",
    "    return acos(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%timeit cy_acos1(0.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%%cython\n",
    "\n",
    "from libc.math cimport acos\n",
    "\n",
    "def cy_acos2(double x):\n",
    "    return acos(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%timeit cy_acos2(0.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from numpy import arccos"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%timeit arccos(0.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from math import acos"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%timeit acos(0.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "assert cy_acos1(0.5) == acos(0.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "assert cy_acos2(0.5) == acos(0.5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Versions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%reload_ext version_information"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%version_information numpy, cython, numba, matplotlib"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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   "codemirror_mode": {
    "name": "ipython",
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