cython_numpy_NeklyudovS.ipynb

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    "%load_ext cython"
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    "Essential syntax:"
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   "source": [
    "%%cython\n",
    "\n",
    "# python-level import\n",
    "import numpy as np\n",
    "\n",
    "# bring C level symbols to the namespace\n",
    "cimport numpy as cnp"
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    "# Example: apply a C function elementwise"
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    "%load_ext cython"
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   "source": [
    "%%cython -a\n",
    "\n",
    "cimport cython\n",
    "import numpy as np\n",
    "\n",
    "\n",
    "# Use the C standard library \n",
    "cdef extern from \"math.h\":\n",
    "    double sin(double x) nogil\n",
    "\n",
    "\n",
    "@cython.boundscheck(False)\n",
    "@cython.wraparound(False)\n",
    "cdef void _compute_sine(double[:] a,           # typed memoryview syntax\n",
    "                        double[::1] out) nogil:\n",
    "    cdef Py_ssize_t j\n",
    "    for j in range(a.shape[0]):\n",
    "        out[j] = sin(a[j])\n",
    "\n",
    "\n",
    "def sine(a):\n",
    "    \"\"\"Compute sin(a), elementwise.\"\"\"\n",
    "    a = np.asarray(a)\n",
    "    \n",
    "    # Store the shape and flatten the array\n",
    "    a_shape = a.shape\n",
    "    a = a.ravel()\n",
    "    \n",
    "    # allocate the output arrays in python (refcounting!)\n",
    "    cdef double[::1] out = np.empty_like(a)\n",
    "    \n",
    "    # do the work\n",
    "    _compute_sine(a, out)\n",
    "    \n",
    "    # wrap the result in a numpy array\n",
    "    return np.asarray(out).reshape(a_shape)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = np.random.random(size=20).reshape(2, 5, 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "res = sine(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(2, 5, 2)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "res.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.allclose(res, np.sin(a))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Typed memoryview of a C array\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "%%cython\n",
    "\n",
    "cimport cython\n",
    "import numpy as np\n",
    "\n",
    "cdef extern from \"math.h\":\n",
    "    double sin(double x) nogil\n",
    "    double M_PI\n",
    "\n",
    "\n",
    "# Same _compute_sine as above\n",
    "\n",
    "@cython.boundscheck(False)\n",
    "@cython.wraparound(False)\n",
    "cdef void _compute_sine(double[:] a,           # typed memoryview syntax\n",
    "                        double[::1] out) nogil:\n",
    "    cdef Py_ssize_t j\n",
    "    for j in range(a.shape[0]):\n",
    "        out[j] = sin(a[j])\n",
    "        \n",
    "\n",
    "def c_sine():\n",
    "    \"\"\"Create a C array, make a typed memoryview on it, and compute sines.\"\"\"\n",
    "    cdef double arr[8]\n",
    "    \n",
    "    for j in range(8):\n",
    "        arr[j] = (j + 0.5) * M_PI\n",
    "        \n",
    "    cdef double[::1] view = arr\n",
    "    cdef double[::1] out = np.empty(8, dtype=float)\n",
    "    _compute_sine(view, out)\n",
    "    \n",
    "    ###### BIG FAT WARNING: `copy=True`, to avoid a dangling pointer. \n",
    "    return np.array(out, copy=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 1., -1.,  1., -1.,  1., -1.,  1., -1.])"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = c_sine()\n",
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "  C_CONTIGUOUS : True\n",
       "  F_CONTIGUOUS : True\n",
       "  OWNDATA : True\n",
       "  WRITEABLE : True\n",
       "  ALIGNED : True\n",
       "  WRITEBACKIFCOPY : False\n",
       "  UPDATEIFCOPY : False"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.flags"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Best allocate memory on the Python side.** Otherwise, tie the lifetime of pointers to some python objects. (E.g., declare a `cdef class Holder` to store the pointer, or --- better --- allocate and deallocate memory in constructor and destructor)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Solve a Laplace equation, $\\nabla^2 u = 0$\n",
    "\n",
    "For simplicity, use some hard-coded boundary conditions.\n",
    "\n",
    "Travis Oliphant, http://technicaldiscovery.blogspot.ru/2011/06/speeding-up-python-numpy-cython-and.html"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "dx = 0.1\n",
    "dy = 0.1\n",
    "dx2 = dx*dx\n",
    "dy2 = dy*dy\n",
    "\n",
    "def py_update(u, dx2, xy2):\n",
    "    nx, ny = u.shape\n",
    "    for i in range(1,nx-1):\n",
    "        for j in range(1, ny-1):\n",
    "            u[i,j] = ((u[i+1, j] + u[i-1, j]) * dy2 +\n",
    "                      (u[i, j+1] + u[i, j-1]) * dx2) / (2*(dx2+dy2))\n",
    "\n",
    "            \n",
    "def calc(N, n_iter=100, func=py_update):\n",
    "    # set up the boundary condition\n",
    "    u = np.zeros((N, N), dtype=float)\n",
    "    u[0] = 1\n",
    "    \n",
    "    # loop\n",
    "    for i in range(n_iter):\n",
    "        func(u, dx2, dy2)\n",
    "        \n",
    "    return np.asarray(u)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "u = calc(10, n_iter=1000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.colorbar.Colorbar at 0x7f5bf029fa90>"
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     "execution_count": 15,
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     "output_type": "execute_result"
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     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 2 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "\n",
    "import matplotlib.pyplot as plt\n",
    "plt.style.use('ggplot')\n",
    "\n",
    "plt.imshow(u, cmap=plt.cm.Oranges)\n",
    "plt.colorbar()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "5.9 s ± 1.52 s per loop (mean ± std. dev. of 7 runs, 1 loop each)\n"
     ]
    }
   ],
   "source": [
    "%timeit calc(50, n_iter=1000)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Move the update loop to Cython"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<!DOCTYPE html>\n",
       "<!-- Generated by Cython 0.29.13 -->\n",
       "<html>\n",
       "<head>\n",
       "    <meta http-equiv=\"Content-Type\" content=\"text/html; charset=utf-8\" />\n",
       "    <title>Cython: _cython_magic_de136818a3fbe06063fa947a62dbbece.pyx</title>\n",
       "    <style type=\"text/css\">\n",
       "    \n",
       "body.cython { font-family: courier; font-size: 12; }\n",
       "\n",
       ".cython.tag  {  }\n",
       ".cython.line { margin: 0em }\n",
       ".cython.code { font-size: 9; color: #444444; display: none; margin: 0px 0px 0px 8px; border-left: 8px none; }\n",
       "\n",
       ".cython.line .run { background-color: #B0FFB0; }\n",
       ".cython.line .mis { background-color: #FFB0B0; }\n",
       ".cython.code.run  { border-left: 8px solid #B0FFB0; }\n",
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       "\n",
       ".cython.code .py_c_api  { color: red; }\n",
       ".cython.code .py_macro_api  { color: #FF7000; }\n",
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       ".cython.code .error_goto  { color: #FFA000; }\n",
       "\n",
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       "<pre class=\"cython line score-0\">&#xA0;<span class=\"\">05</span>: <span class=\"nd\">@cython</span><span class=\"o\">.</span><span class=\"n\">wraparound</span><span class=\"p\">(</span><span class=\"bp\">False</span><span class=\"p\">)</span></pre>\n",
       "<pre class=\"cython line score-0\">&#xA0;<span class=\"\">06</span>: <span class=\"nd\">@cython</span><span class=\"o\">.</span><span class=\"n\">cdivision</span><span class=\"p\">(</span><span class=\"bp\">True</span><span class=\"p\">)</span></pre>\n",
       "<pre class=\"cython line score-66\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">07</span>: <span class=\"k\">def</span> <span class=\"nf\">cy_update</span><span class=\"p\">(</span><span class=\"n\">double</span><span class=\"p\">[:,</span> <span class=\"p\">::</span><span class=\"mf\">1</span><span class=\"p\">]</span> <span class=\"n\">u</span><span class=\"p\">,</span> <span class=\"n\">double</span> <span class=\"n\">dx2</span><span class=\"p\">,</span> <span class=\"n\">double</span> <span class=\"n\">dy2</span><span class=\"p\">):</span></pre>\n",
       "<pre class='cython code score-66 '>/* Python wrapper */\n",
       "static PyObject *__pyx_pw_46_cython_magic_de136818a3fbe06063fa947a62dbbece_1cy_update(PyObject *__pyx_self, PyObject *__pyx_args, PyObject *__pyx_kwds); /*proto*/\n",
       "static PyMethodDef __pyx_mdef_46_cython_magic_de136818a3fbe06063fa947a62dbbece_1cy_update = {\"cy_update\", (PyCFunction)(void*)(PyCFunctionWithKeywords)__pyx_pw_46_cython_magic_de136818a3fbe06063fa947a62dbbece_1cy_update, METH_VARARGS|METH_KEYWORDS, 0};\n",
       "static PyObject *__pyx_pw_46_cython_magic_de136818a3fbe06063fa947a62dbbece_1cy_update(PyObject *__pyx_self, PyObject *__pyx_args, PyObject *__pyx_kwds) {\n",
       "  __Pyx_memviewslice __pyx_v_u = { 0, 0, { 0 }, { 0 }, { 0 } };\n",
       "  double __pyx_v_dx2;\n",
       "  double __pyx_v_dy2;\n",
       "  PyObject *__pyx_r = 0;\n",
       "  <span class='refnanny'>__Pyx_RefNannyDeclarations</span>\n",
       "  <span class='refnanny'>__Pyx_RefNannySetupContext</span>(\"cy_update (wrapper)\", 0);\n",
       "  {\n",
       "    static PyObject **__pyx_pyargnames[] = {&amp;__pyx_n_s_u,&amp;__pyx_n_s_dx2,&amp;__pyx_n_s_dy2,0};\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 = <span class='py_macro_api'>PyTuple_GET_SIZE</span>(__pyx_args);\n",
       "      switch (pos_args) {\n",
       "        case  3: values[2] = <span class='py_macro_api'>PyTuple_GET_ITEM</span>(__pyx_args, 2);\n",
       "        CYTHON_FALLTHROUGH;\n",
       "        case  2: values[1] = <span class='py_macro_api'>PyTuple_GET_ITEM</span>(__pyx_args, 1);\n",
       "        CYTHON_FALLTHROUGH;\n",
       "        case  1: values[0] = <span class='py_macro_api'>PyTuple_GET_ITEM</span>(__pyx_args, 0);\n",
       "        CYTHON_FALLTHROUGH;\n",
       "        case  0: break;\n",
       "        default: goto __pyx_L5_argtuple_error;\n",
       "      }\n",
       "      kw_args = <span class='py_c_api'>PyDict_Size</span>(__pyx_kwds);\n",
       "      switch (pos_args) {\n",
       "        case  0:\n",
       "        if (likely((values[0] = <span class='pyx_c_api'>__Pyx_PyDict_GetItemStr</span>(__pyx_kwds, __pyx_n_s_u)) != 0)) kw_args--;\n",
       "        else goto __pyx_L5_argtuple_error;\n",
       "        CYTHON_FALLTHROUGH;\n",
       "        case  1:\n",
       "        if (likely((values[1] = <span class='pyx_c_api'>__Pyx_PyDict_GetItemStr</span>(__pyx_kwds, __pyx_n_s_dx2)) != 0)) kw_args--;\n",
       "        else {\n",
       "          <span class='pyx_c_api'>__Pyx_RaiseArgtupleInvalid</span>(\"cy_update\", 1, 3, 3, 1); <span class='error_goto'>__PYX_ERR(0, 7, __pyx_L3_error)</span>\n",
       "        }\n",
       "        CYTHON_FALLTHROUGH;\n",
       "        case  2:\n",
       "        if (likely((values[2] = <span class='pyx_c_api'>__Pyx_PyDict_GetItemStr</span>(__pyx_kwds, __pyx_n_s_dy2)) != 0)) kw_args--;\n",
       "        else {\n",
       "          <span class='pyx_c_api'>__Pyx_RaiseArgtupleInvalid</span>(\"cy_update\", 1, 3, 3, 2); <span class='error_goto'>__PYX_ERR(0, 7, __pyx_L3_error)</span>\n",
       "        }\n",
       "      }\n",
       "      if (unlikely(kw_args &gt; 0)) {\n",
       "        if (unlikely(<span class='pyx_c_api'>__Pyx_ParseOptionalKeywords</span>(__pyx_kwds, __pyx_pyargnames, 0, values, pos_args, \"cy_update\") &lt; 0)) <span class='error_goto'>__PYX_ERR(0, 7, __pyx_L3_error)</span>\n",
       "      }\n",
       "    } else if (<span class='py_macro_api'>PyTuple_GET_SIZE</span>(__pyx_args) != 3) {\n",
       "      goto __pyx_L5_argtuple_error;\n",
       "    } else {\n",
       "      values[0] = <span class='py_macro_api'>PyTuple_GET_ITEM</span>(__pyx_args, 0);\n",
       "      values[1] = <span class='py_macro_api'>PyTuple_GET_ITEM</span>(__pyx_args, 1);\n",
       "      values[2] = <span class='py_macro_api'>PyTuple_GET_ITEM</span>(__pyx_args, 2);\n",
       "    }\n",
       "    __pyx_v_u = <span class='pyx_c_api'>__Pyx_PyObject_to_MemoryviewSlice_d_dc_double</span>(values[0], PyBUF_WRITABLE);<span class='error_goto'> if (unlikely(!__pyx_v_u.memview)) __PYX_ERR(0, 7, __pyx_L3_error)</span>\n",
       "    __pyx_v_dx2 = __pyx_<span class='py_c_api'>PyFloat_AsDouble</span>(values[1]); if (unlikely((__pyx_v_dx2 == (double)-1) &amp;&amp; <span class='py_c_api'>PyErr_Occurred</span>())) <span class='error_goto'>__PYX_ERR(0, 7, __pyx_L3_error)</span>\n",
       "    __pyx_v_dy2 = __pyx_<span class='py_c_api'>PyFloat_AsDouble</span>(values[2]); if (unlikely((__pyx_v_dy2 == (double)-1) &amp;&amp; <span class='py_c_api'>PyErr_Occurred</span>())) <span class='error_goto'>__PYX_ERR(0, 7, __pyx_L3_error)</span>\n",
       "  }\n",
       "  goto __pyx_L4_argument_unpacking_done;\n",
       "  __pyx_L5_argtuple_error:;\n",
       "  <span class='pyx_c_api'>__Pyx_RaiseArgtupleInvalid</span>(\"cy_update\", 1, 3, 3, <span class='py_macro_api'>PyTuple_GET_SIZE</span>(__pyx_args)); <span class='error_goto'>__PYX_ERR(0, 7, __pyx_L3_error)</span>\n",
       "  __pyx_L3_error:;\n",
       "  <span class='pyx_c_api'>__Pyx_AddTraceback</span>(\"_cython_magic_de136818a3fbe06063fa947a62dbbece.cy_update\", __pyx_clineno, __pyx_lineno, __pyx_filename);\n",
       "  <span class='refnanny'>__Pyx_RefNannyFinishContext</span>();\n",
       "  return NULL;\n",
       "  __pyx_L4_argument_unpacking_done:;\n",
       "  __pyx_r = __pyx_pf_46_cython_magic_de136818a3fbe06063fa947a62dbbece_cy_update(__pyx_self, __pyx_v_u, __pyx_v_dx2, __pyx_v_dy2);\n",
       "\n",
       "  /* function exit code */\n",
       "  <span class='refnanny'>__Pyx_RefNannyFinishContext</span>();\n",
       "  return __pyx_r;\n",
       "}\n",
       "\n",
       "static PyObject *__pyx_pf_46_cython_magic_de136818a3fbe06063fa947a62dbbece_cy_update(CYTHON_UNUSED PyObject *__pyx_self, __Pyx_memviewslice __pyx_v_u, double __pyx_v_dx2, double __pyx_v_dy2) {\n",
       "  Py_ssize_t __pyx_v_i;\n",
       "  Py_ssize_t __pyx_v_j;\n",
       "  PyObject *__pyx_r = NULL;\n",
       "  <span class='refnanny'>__Pyx_RefNannyDeclarations</span>\n",
       "  <span class='refnanny'>__Pyx_RefNannySetupContext</span>(\"cy_update\", 0);\n",
       "/* … */\n",
       "  /* function exit code */\n",
       "  __pyx_r = Py_None; <span class='pyx_macro_api'>__Pyx_INCREF</span>(Py_None);\n",
       "  __PYX_XDEC_MEMVIEW(&amp;__pyx_v_u, 1);\n",
       "  <span class='refnanny'>__Pyx_XGIVEREF</span>(__pyx_r);\n",
       "  <span class='refnanny'>__Pyx_RefNannyFinishContext</span>();\n",
       "  return __pyx_r;\n",
       "}\n",
       "/* … */\n",
       "  __pyx_tuple__19 = <span class='py_c_api'>PyTuple_Pack</span>(5, __pyx_n_s_u, __pyx_n_s_dx2, __pyx_n_s_dy2, __pyx_n_s_i, __pyx_n_s_j);<span class='error_goto'> if (unlikely(!__pyx_tuple__19)) __PYX_ERR(0, 7, __pyx_L1_error)</span>\n",
       "  <span class='refnanny'>__Pyx_GOTREF</span>(__pyx_tuple__19);\n",
       "  <span class='refnanny'>__Pyx_GIVEREF</span>(__pyx_tuple__19);\n",
       "/* … */\n",
       "  __pyx_t_1 = PyCFunction_NewEx(&amp;__pyx_mdef_46_cython_magic_de136818a3fbe06063fa947a62dbbece_1cy_update, NULL, __pyx_n_s_cython_magic_de136818a3fbe06063);<span class='error_goto'> if (unlikely(!__pyx_t_1)) __PYX_ERR(0, 7, __pyx_L1_error)</span>\n",
       "  <span class='refnanny'>__Pyx_GOTREF</span>(__pyx_t_1);\n",
       "  if (<span class='py_c_api'>PyDict_SetItem</span>(__pyx_d, __pyx_n_s_cy_update, __pyx_t_1) &lt; 0) <span class='error_goto'>__PYX_ERR(0, 7, __pyx_L1_error)</span>\n",
       "  <span class='pyx_macro_api'>__Pyx_DECREF</span>(__pyx_t_1); __pyx_t_1 = 0;\n",
       "  __pyx_codeobj__20 = (PyObject*)<span class='pyx_c_api'>__Pyx_PyCode_New</span>(3, 0, 5, 0, CO_OPTIMIZED|CO_NEWLOCALS, __pyx_empty_bytes, __pyx_empty_tuple, __pyx_empty_tuple, __pyx_tuple__19, __pyx_empty_tuple, __pyx_empty_tuple, __pyx_kp_s_home_semyon_cache_ipython_cytho, __pyx_n_s_cy_update, 7, __pyx_empty_bytes);<span class='error_goto'> if (unlikely(!__pyx_codeobj__20)) __PYX_ERR(0, 7, __pyx_L1_error)</span>\n",
       "</pre><pre class=\"cython line score-0\">&#xA0;<span class=\"\">08</span>:     <span class=\"k\">cdef</span> <span class=\"kt\">Py_ssize_t</span> <span class=\"nf\">i</span><span class=\"p\">,</span> <span class=\"nf\">j</span></pre>\n",
       "<pre class=\"cython line score-0\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">09</span>:     <span class=\"k\">for</span> <span class=\"n\">i</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"mf\">1</span><span class=\"p\">,</span> <span class=\"n\">u</span><span class=\"o\">.</span><span class=\"n\">shape</span><span class=\"p\">[</span><span class=\"mf\">0</span><span class=\"p\">]</span><span class=\"o\">-</span><span class=\"mf\">1</span><span class=\"p\">):</span></pre>\n",
       "<pre class='cython code score-0 '>  __pyx_t_1 = ((__pyx_v_u.shape[0]) - 1);\n",
       "  __pyx_t_2 = __pyx_t_1;\n",
       "  for (__pyx_t_3 = 1; __pyx_t_3 &lt; __pyx_t_2; __pyx_t_3+=1) {\n",
       "    __pyx_v_i = __pyx_t_3;\n",
       "</pre><pre class=\"cython line score-0\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">10</span>:         <span class=\"k\">for</span> <span class=\"n\">j</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"mf\">1</span><span class=\"p\">,</span> <span class=\"n\">u</span><span class=\"o\">.</span><span class=\"n\">shape</span><span class=\"p\">[</span><span class=\"mf\">1</span><span class=\"p\">]</span><span class=\"o\">-</span><span class=\"mf\">1</span><span class=\"p\">):</span></pre>\n",
       "<pre class='cython code score-0 '>    __pyx_t_4 = ((__pyx_v_u.shape[1]) - 1);\n",
       "    __pyx_t_5 = __pyx_t_4;\n",
       "    for (__pyx_t_6 = 1; __pyx_t_6 &lt; __pyx_t_5; __pyx_t_6+=1) {\n",
       "      __pyx_v_j = __pyx_t_6;\n",
       "</pre><pre class=\"cython line score-0\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">11</span>:             <span class=\"n\">u</span><span class=\"p\">[</span><span class=\"n\">i</span><span class=\"p\">,</span> <span class=\"n\">j</span><span class=\"p\">]</span> <span class=\"o\">=</span> <span class=\"p\">((</span><span class=\"n\">u</span><span class=\"p\">[</span><span class=\"n\">i</span><span class=\"o\">+</span><span class=\"mf\">1</span><span class=\"p\">,</span> <span class=\"n\">j</span><span class=\"p\">]</span> <span class=\"o\">+</span> <span class=\"n\">u</span><span class=\"p\">[</span><span class=\"n\">i</span><span class=\"o\">-</span><span class=\"mf\">1</span><span class=\"p\">,</span> <span class=\"n\">j</span><span class=\"p\">])</span> <span class=\"o\">*</span> <span class=\"n\">dy2</span> <span class=\"o\">+</span></pre>\n",
       "<pre class='cython code score-0 '>      __pyx_t_7 = (__pyx_v_i + 1);\n",
       "      __pyx_t_8 = __pyx_v_j;\n",
       "      __pyx_t_9 = (__pyx_v_i - 1);\n",
       "      __pyx_t_10 = __pyx_v_j;\n",
       "/* … */\n",
       "      __pyx_t_15 = __pyx_v_i;\n",
       "      __pyx_t_16 = __pyx_v_j;\n",
       "      *((double *) ( /* dim=1 */ ((char *) (((double *) ( /* dim=0 */ (__pyx_v_u.data + __pyx_t_15 * __pyx_v_u.strides[0]) )) + __pyx_t_16)) )) = (((((*((double *) ( /* dim=1 */ ((char *) (((double *) ( /* dim=0 */ (__pyx_v_u.data + __pyx_t_7 * __pyx_v_u.strides[0]) )) + __pyx_t_8)) ))) + (*((double *) ( /* dim=1 */ ((char *) (((double *) ( /* dim=0 */ (__pyx_v_u.data + __pyx_t_9 * __pyx_v_u.strides[0]) )) + __pyx_t_10)) )))) * __pyx_v_dy2) + (((*((double *) ( /* dim=1 */ ((char *) (((double *) ( /* dim=0 */ (__pyx_v_u.data + __pyx_t_11 * __pyx_v_u.strides[0]) )) + __pyx_t_12)) ))) + (*((double *) ( /* dim=1 */ ((char *) (((double *) ( /* dim=0 */ (__pyx_v_u.data + __pyx_t_13 * __pyx_v_u.strides[0]) )) + __pyx_t_14)) )))) * __pyx_v_dx2)) / (2.0 * (__pyx_v_dx2 + __pyx_v_dy2)));\n",
       "    }\n",
       "  }\n",
       "</pre><pre class=\"cython line score-0\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">12</span>:                        <span class=\"p\">(</span><span class=\"n\">u</span><span class=\"p\">[</span><span class=\"n\">i</span><span class=\"p\">,</span> <span class=\"n\">j</span><span class=\"o\">+</span><span class=\"mf\">1</span><span class=\"p\">]</span> <span class=\"o\">+</span> <span class=\"n\">u</span><span class=\"p\">[</span><span class=\"n\">i</span><span class=\"p\">,</span> <span class=\"n\">j</span><span class=\"o\">-</span><span class=\"mf\">1</span><span class=\"p\">])</span> <span class=\"o\">*</span> <span class=\"n\">dx2</span><span class=\"p\">)</span> <span class=\"o\">/</span> <span class=\"p\">(</span><span class=\"mf\">2</span><span class=\"o\">*</span><span class=\"p\">(</span><span class=\"n\">dx2</span><span class=\"o\">+</span><span class=\"n\">dy2</span><span class=\"p\">))</span></pre>\n",
       "<pre class='cython code score-0 '>      __pyx_t_11 = __pyx_v_i;\n",
       "      __pyx_t_12 = (__pyx_v_j + 1);\n",
       "      __pyx_t_13 = __pyx_v_i;\n",
       "      __pyx_t_14 = (__pyx_v_j - 1);\n",
       "</pre></div></body></html>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%%cython -a --compile-args=-O3\n",
    "\n",
    "cimport cython\n",
    "\n",
    "@cython.boundscheck(False)\n",
    "@cython.wraparound(False)\n",
    "@cython.cdivision(True)\n",
    "def cy_update(double[:, ::1] u, double dx2, double dy2):\n",
    "    cdef Py_ssize_t i, j\n",
    "    for i in range(1, u.shape[0]-1):\n",
    "        for j in range(1, u.shape[1]-1):\n",
    "            u[i, j] = ((u[i+1, j] + u[i-1, j]) * dy2 +\n",
    "                       (u[i, j+1] + u[i, j-1]) * dx2) / (2*(dx2+dy2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "708 ms ± 21 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)\n"
     ]
    }
   ],
   "source": [
    "%timeit u = calc(100, n_iter=8000, func=cy_update)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.colorbar.Colorbar at 0x7f5bf017fdd0>"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.imshow(u, cmap=plt.cm.Oranges)\n",
    "plt.colorbar()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "18.6 ms ± 412 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)\n"
     ]
    }
   ],
   "source": [
    "%timeit calc(50, n_iter=1000, func=cy_update)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Practice exercises\n",
    "\n",
    "1. Separate the boundary conditions from the `calc` function above. It should accept a preallocated array `u`.\n",
    "\n",
    "2. Modify the ODE solver for `Planet` to output the trajectory as a numpy array.\n",
    "\n",
    "3. Write a Cython function for a nearest-neighbor average of a 2D numpy array.\n",
    "\n",
    "4. Find all solutions of the equation $x = a \\, \\sin{x}$. *Hint:* cythonize the bisection root-finding algorithm, and use it on each $2\\pi$ interval."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [],
   "source": [
    "def calc_separ_u(u, n_iter=100, func=py_update):\n",
    "    # set up the boundary condition\n",
    "    #u = np.zeros((N, N), dtype=float)\n",
    "    #u[0] = 1\n",
    "    \n",
    "    # loop\n",
    "    for i in range(n_iter):\n",
    "        func(u, dx2, dy2)\n",
    "        \n",
    "    return np.asarray(u)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [],
   "source": [
    "N = 100\n",
    "start_boundaries_numpy = np.zeros((N, N), dtype=float)\n",
    "start_boundaries_numpy[0] = 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "669 ms ± 6.09 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)\n"
     ]
    }
   ],
   "source": [
    "%timeit u = calc_separ_u(start_boundaries_numpy, n_iter=8000, func=cy_update)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
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       "static int __pyx_pw_46_cython_magic_11dafdc010a400a06215a70fae45fc51_6Planet_1m_3__set__(PyObject *__pyx_v_self, PyObject *__pyx_v_value) {\n",
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       "<pre class='cython code score-11 '>/* Python wrapper */\n",
       "static int __pyx_pw_46_cython_magic_11dafdc010a400a06215a70fae45fc51_6Planet_1__init__(PyObject *__pyx_v_self, PyObject *__pyx_args, PyObject *__pyx_kwds); /*proto*/\n",
       "static int __pyx_pw_46_cython_magic_11dafdc010a400a06215a70fae45fc51_6Planet_1__init__(PyObject *__pyx_v_self, PyObject *__pyx_args, PyObject *__pyx_kwds) {\n",
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       "\n",
       "static int __pyx_pf_46_cython_magic_11dafdc010a400a06215a70fae45fc51_6Planet___init__(struct __pyx_obj_46_cython_magic_11dafdc010a400a06215a70fae45fc51_Planet *__pyx_v_self) {\n",
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       "</pre><pre class=\"cython line score-0\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">11</span>:         <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">x</span> <span class=\"o\">=</span> <span class=\"mf\">1.0</span></pre>\n",
       "<pre class='cython code score-0 '>  __pyx_v_self-&gt;x = 1.0;\n",
       "</pre><pre class=\"cython line score-0\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">12</span>:         <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">y</span> <span class=\"o\">=</span> <span class=\"mf\">0</span></pre>\n",
       "<pre class='cython code score-0 '>  __pyx_v_self-&gt;y = 0.0;\n",
       "</pre><pre class=\"cython line score-0\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">13</span>:         <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">z</span> <span class=\"o\">=</span> <span class=\"mf\">0</span></pre>\n",
       "<pre class='cython code score-0 '>  __pyx_v_self-&gt;z = 0.0;\n",
       "</pre><pre class=\"cython line score-0\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">14</span>:         <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">vx</span> <span class=\"o\">=</span> <span class=\"mf\">0</span></pre>\n",
       "<pre class='cython code score-0 '>  __pyx_v_self-&gt;vx = 0.0;\n",
       "</pre><pre class=\"cython line score-0\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">15</span>:         <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">vy</span> <span class=\"o\">=</span> <span class=\"mf\">0</span></pre>\n",
       "<pre class='cython code score-0 '>  __pyx_v_self-&gt;vy = 0.0;\n",
       "</pre><pre class=\"cython line score-0\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">16</span>:         <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">vz</span> <span class=\"o\">=</span> <span class=\"mf\">1.0</span></pre>\n",
       "<pre class='cython code score-0 '>  __pyx_v_self-&gt;vz = 1.0;\n",
       "</pre><pre class=\"cython line score-0\">&#xA0;<span class=\"\">17</span>: </pre>\n",
       "<pre class=\"cython line score-0\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">18</span>:         <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">m</span> <span class=\"o\">=</span> <span class=\"mf\">1.0</span></pre>\n",
       "<pre class='cython code score-0 '>  __pyx_v_self-&gt;m = 1.0;\n",
       "</pre><pre class=\"cython line score-0\">&#xA0;<span class=\"\">19</span>: </pre>\n",
       "<pre class=\"cython line score-0\">&#xA0;<span class=\"\">20</span>: </pre>\n",
       "<pre class=\"cython line score-0\">&#xA0;<span class=\"\">21</span>: <span class=\"nd\">@cython</span><span class=\"o\">.</span><span class=\"n\">cdivision</span><span class=\"p\">(</span><span class=\"bp\">True</span><span class=\"p\">)</span></pre>\n",
       "<pre class=\"cython line score-0\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">22</span>: <span class=\"k\">cdef</span> <span class=\"kt\">void</span> <span class=\"nf\">single_step</span><span class=\"p\">(</span><span class=\"n\">Planet</span> <span class=\"n\">planet</span><span class=\"p\">,</span></pre>\n",
       "<pre class='cython code score-0 '>static void __pyx_f_46_cython_magic_11dafdc010a400a06215a70fae45fc51_single_step(struct __pyx_obj_46_cython_magic_11dafdc010a400a06215a70fae45fc51_Planet *__pyx_v_planet, double __pyx_v_dt) {\n",
       "  double __pyx_v_r;\n",
       "  double __pyx_v_r3;\n",
       "  double __pyx_v_Fx;\n",
       "  double __pyx_v_Fy;\n",
       "  double __pyx_v_Fz;\n",
       "  double __pyx_v_m;\n",
       "/* … */\n",
       "  /* function exit code */\n",
       "}\n",
       "</pre><pre class=\"cython line score-0\">&#xA0;<span class=\"\">23</span>:                       <span class=\"n\">double</span> <span class=\"n\">dt</span><span class=\"p\">)</span> <span class=\"k\">nogil</span><span class=\"p\">:</span></pre>\n",
       "<pre class=\"cython line score-0\">&#xA0;<span class=\"\">24</span>:     <span class=\"sd\">&quot;&quot;&quot;Make a single step in time, t -&gt; t+dt.&quot;&quot;&quot;</span></pre>\n",
       "<pre class=\"cython line score-0\">&#xA0;<span class=\"\">25</span>:     <span class=\"c\"># Gravitational force pulls towards origin</span></pre>\n",
       "<pre class=\"cython line score-0\">&#xA0;<span class=\"\">26</span>:     <span class=\"k\">cdef</span> <span class=\"kt\">double</span> <span class=\"nf\">r</span><span class=\"p\">,</span> <span class=\"nf\">r3</span></pre>\n",
       "<pre class=\"cython line score-0\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">27</span>:     <span class=\"n\">r</span> <span class=\"o\">=</span> <span class=\"n\">sqrt</span><span class=\"p\">(</span><span class=\"n\">planet</span><span class=\"o\">.</span><span class=\"n\">x</span><span class=\"o\">**</span><span class=\"mf\">2</span> <span class=\"o\">+</span> <span class=\"n\">planet</span><span class=\"o\">.</span><span class=\"n\">y</span><span class=\"o\">**</span><span class=\"mf\">2</span> <span class=\"o\">+</span> <span class=\"n\">planet</span><span class=\"o\">.</span><span class=\"n\">z</span><span class=\"o\">**</span><span class=\"mf\">2</span><span class=\"p\">)</span></pre>\n",
       "<pre class='cython code score-0 '>  __pyx_v_r = sqrt(((pow(__pyx_v_planet-&gt;x, 2.0) + pow(__pyx_v_planet-&gt;y, 2.0)) + pow(__pyx_v_planet-&gt;z, 2.0)));\n",
       "</pre><pre class=\"cython line score-0\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">28</span>:     <span class=\"n\">r3</span> <span class=\"o\">=</span> <span class=\"n\">r</span><span class=\"o\">*</span><span class=\"n\">r</span><span class=\"o\">*</span><span class=\"n\">r</span>                             <span class=\"c\"># XXX: check generated C code</span></pre>\n",
       "<pre class='cython code score-0 '>  __pyx_v_r3 = ((__pyx_v_r * __pyx_v_r) * __pyx_v_r);\n",
       "</pre><pre class=\"cython line score-0\">&#xA0;<span class=\"\">29</span>: </pre>\n",
       "<pre class=\"cython line score-0\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">30</span>:     <span class=\"n\">Fx</span> <span class=\"o\">=</span> <span class=\"o\">-</span><span class=\"n\">planet</span><span class=\"o\">.</span><span class=\"n\">x</span> <span class=\"o\">/</span> <span class=\"n\">r3</span></pre>\n",
       "<pre class='cython code score-0 '>  __pyx_v_Fx = ((-__pyx_v_planet-&gt;x) / __pyx_v_r3);\n",
       "</pre><pre class=\"cython line score-0\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">31</span>:     <span class=\"n\">Fy</span> <span class=\"o\">=</span> <span class=\"o\">-</span><span class=\"n\">planet</span><span class=\"o\">.</span><span class=\"n\">y</span> <span class=\"o\">/</span> <span class=\"n\">r3</span></pre>\n",
       "<pre class='cython code score-0 '>  __pyx_v_Fy = ((-__pyx_v_planet-&gt;y) / __pyx_v_r3);\n",
       "</pre><pre class=\"cython line score-0\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">32</span>:     <span class=\"n\">Fz</span> <span class=\"o\">=</span> <span class=\"o\">-</span><span class=\"n\">planet</span><span class=\"o\">.</span><span class=\"n\">z</span> <span class=\"o\">/</span> <span class=\"n\">r3</span></pre>\n",
       "<pre class='cython code score-0 '>  __pyx_v_Fz = ((-__pyx_v_planet-&gt;z) / __pyx_v_r3);\n",
       "</pre><pre class=\"cython line score-0\">&#xA0;<span class=\"\">33</span>: </pre>\n",
       "<pre class=\"cython line score-0\">&#xA0;<span class=\"\">34</span>:     <span class=\"c\"># update position</span></pre>\n",
       "<pre class=\"cython line score-0\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">35</span>:     <span class=\"n\">planet</span><span class=\"o\">.</span><span class=\"n\">x</span> <span class=\"o\">+=</span> <span class=\"n\">planet</span><span class=\"o\">.</span><span class=\"n\">vx</span> <span class=\"o\">*</span> <span class=\"n\">dt</span></pre>\n",
       "<pre class='cython code score-0 '>  __pyx_v_planet-&gt;x = (__pyx_v_planet-&gt;x + (__pyx_v_planet-&gt;vx * __pyx_v_dt));\n",
       "</pre><pre class=\"cython line score-0\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">36</span>:     <span class=\"n\">planet</span><span class=\"o\">.</span><span class=\"n\">y</span> <span class=\"o\">+=</span> <span class=\"n\">planet</span><span class=\"o\">.</span><span class=\"n\">vy</span> <span class=\"o\">*</span> <span class=\"n\">dt</span></pre>\n",
       "<pre class='cython code score-0 '>  __pyx_v_planet-&gt;y = (__pyx_v_planet-&gt;y + (__pyx_v_planet-&gt;vy * __pyx_v_dt));\n",
       "</pre><pre class=\"cython line score-0\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">37</span>:     <span class=\"n\">planet</span><span class=\"o\">.</span><span class=\"n\">z</span> <span class=\"o\">+=</span> <span class=\"n\">planet</span><span class=\"o\">.</span><span class=\"n\">vz</span> <span class=\"o\">*</span> <span class=\"n\">dt</span></pre>\n",
       "<pre class='cython code score-0 '>  __pyx_v_planet-&gt;z = (__pyx_v_planet-&gt;z + (__pyx_v_planet-&gt;vz * __pyx_v_dt));\n",
       "</pre><pre class=\"cython line score-0\">&#xA0;<span class=\"\">38</span>: </pre>\n",
       "<pre class=\"cython line score-0\">&#xA0;<span class=\"\">39</span>:     <span class=\"c\"># update velocity</span></pre>\n",
       "<pre class=\"cython line score-0\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">40</span>:     <span class=\"n\">m</span> <span class=\"o\">=</span> <span class=\"n\">planet</span><span class=\"o\">.</span><span class=\"n\">m</span></pre>\n",
       "<pre class='cython code score-0 '>  __pyx_t_1 = __pyx_v_planet-&gt;m;\n",
       "  __pyx_v_m = __pyx_t_1;\n",
       "</pre><pre class=\"cython line score-0\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">41</span>:     <span class=\"n\">planet</span><span class=\"o\">.</span><span class=\"n\">vx</span> <span class=\"o\">+=</span> <span class=\"n\">Fx</span> <span class=\"o\">*</span> <span class=\"n\">dt</span> <span class=\"o\">/</span> <span class=\"n\">m</span></pre>\n",
       "<pre class='cython code score-0 '>  __pyx_v_planet-&gt;vx = (__pyx_v_planet-&gt;vx + ((__pyx_v_Fx * __pyx_v_dt) / __pyx_v_m));\n",
       "</pre><pre class=\"cython line score-0\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">42</span>:     <span class=\"n\">planet</span><span class=\"o\">.</span><span class=\"n\">vy</span> <span class=\"o\">+=</span> <span class=\"n\">Fy</span> <span class=\"o\">*</span> <span class=\"n\">dt</span> <span class=\"o\">/</span> <span class=\"n\">m</span></pre>\n",
       "<pre class='cython code score-0 '>  __pyx_v_planet-&gt;vy = (__pyx_v_planet-&gt;vy + ((__pyx_v_Fy * __pyx_v_dt) / __pyx_v_m));\n",
       "</pre><pre class=\"cython line score-0\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">43</span>:     <span class=\"n\">planet</span><span class=\"o\">.</span><span class=\"n\">vz</span> <span class=\"o\">+=</span> <span class=\"n\">Fz</span> <span class=\"o\">*</span> <span class=\"n\">dt</span> <span class=\"o\">/</span> <span class=\"n\">m</span></pre>\n",
       "<pre class='cython code score-0 '>  __pyx_v_planet-&gt;vz = (__pyx_v_planet-&gt;vz + ((__pyx_v_Fz * __pyx_v_dt) / __pyx_v_m));\n",
       "</pre><pre class=\"cython line score-0\">&#xA0;<span class=\"\">44</span>: </pre>\n",
       "<pre class=\"cython line score-0\">&#xA0;<span class=\"\">45</span>: </pre>\n",
       "<pre class=\"cython line score-0\">&#xA0;<span class=\"\">46</span>: <span class=\"nd\">@cython</span><span class=\"o\">.</span><span class=\"n\">cdivision</span><span class=\"p\">(</span><span class=\"bp\">True</span><span class=\"p\">)</span></pre>\n",
       "<pre class=\"cython line score-68\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">47</span>: <span class=\"k\">def</span> <span class=\"nf\">propagate</span><span class=\"p\">(</span><span class=\"n\">planet</span><span class=\"p\">,</span></pre>\n",
       "<pre class='cython code score-68 '>/* Python wrapper */\n",
       "static PyObject *__pyx_pw_46_cython_magic_11dafdc010a400a06215a70fae45fc51_1propagate(PyObject *__pyx_self, PyObject *__pyx_args, PyObject *__pyx_kwds); /*proto*/\n",
       "static char __pyx_doc_46_cython_magic_11dafdc010a400a06215a70fae45fc51_propagate[] = \"Make a number of time steps.\";\n",
       "static PyMethodDef __pyx_mdef_46_cython_magic_11dafdc010a400a06215a70fae45fc51_1propagate = {\"propagate\", (PyCFunction)(void*)(PyCFunctionWithKeywords)__pyx_pw_46_cython_magic_11dafdc010a400a06215a70fae45fc51_1propagate, METH_VARARGS|METH_KEYWORDS, __pyx_doc_46_cython_magic_11dafdc010a400a06215a70fae45fc51_propagate};\n",
       "static PyObject *__pyx_pw_46_cython_magic_11dafdc010a400a06215a70fae45fc51_1propagate(PyObject *__pyx_self, PyObject *__pyx_args, PyObject *__pyx_kwds) {\n",
       "  PyObject *__pyx_v_planet = 0;\n",
       "  double __pyx_v_time_span;\n",
       "  int __pyx_v_num_steps;\n",
       "  PyObject *__pyx_r = 0;\n",
       "  <span class='refnanny'>__Pyx_RefNannyDeclarations</span>\n",
       "  <span class='refnanny'>__Pyx_RefNannySetupContext</span>(\"propagate (wrapper)\", 0);\n",
       "  {\n",
       "    static PyObject **__pyx_pyargnames[] = {&amp;__pyx_n_s_planet,&amp;__pyx_n_s_time_span,&amp;__pyx_n_s_num_steps,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 = <span class='py_macro_api'>PyTuple_GET_SIZE</span>(__pyx_args);\n",
       "      switch (pos_args) {\n",
       "        case  3: values[2] = <span class='py_macro_api'>PyTuple_GET_ITEM</span>(__pyx_args, 2);\n",
       "        CYTHON_FALLTHROUGH;\n",
       "        case  2: values[1] = <span class='py_macro_api'>PyTuple_GET_ITEM</span>(__pyx_args, 1);\n",
       "        CYTHON_FALLTHROUGH;\n",
       "        case  1: values[0] = <span class='py_macro_api'>PyTuple_GET_ITEM</span>(__pyx_args, 0);\n",
       "        CYTHON_FALLTHROUGH;\n",
       "        case  0: break;\n",
       "        default: goto __pyx_L5_argtuple_error;\n",
       "      }\n",
       "      kw_args = <span class='py_c_api'>PyDict_Size</span>(__pyx_kwds);\n",
       "      switch (pos_args) {\n",
       "        case  0:\n",
       "        if (likely((values[0] = <span class='pyx_c_api'>__Pyx_PyDict_GetItemStr</span>(__pyx_kwds, __pyx_n_s_planet)) != 0)) kw_args--;\n",
       "        else goto __pyx_L5_argtuple_error;\n",
       "        CYTHON_FALLTHROUGH;\n",
       "        case  1:\n",
       "        if (likely((values[1] = <span class='pyx_c_api'>__Pyx_PyDict_GetItemStr</span>(__pyx_kwds, __pyx_n_s_time_span)) != 0)) kw_args--;\n",
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       "          <span class='pyx_c_api'>__Pyx_RaiseArgtupleInvalid</span>(\"propagate\", 1, 3, 3, 1); <span class='error_goto'>__PYX_ERR(0, 47, __pyx_L3_error)</span>\n",
       "        }\n",
       "        CYTHON_FALLTHROUGH;\n",
       "        case  2:\n",
       "        if (likely((values[2] = <span class='pyx_c_api'>__Pyx_PyDict_GetItemStr</span>(__pyx_kwds, __pyx_n_s_num_steps)) != 0)) kw_args--;\n",
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       "          <span class='pyx_c_api'>__Pyx_RaiseArgtupleInvalid</span>(\"propagate\", 1, 3, 3, 2); <span class='error_goto'>__PYX_ERR(0, 47, __pyx_L3_error)</span>\n",
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       "      }\n",
       "      if (unlikely(kw_args &gt; 0)) {\n",
       "        if (unlikely(<span class='pyx_c_api'>__Pyx_ParseOptionalKeywords</span>(__pyx_kwds, __pyx_pyargnames, 0, values, pos_args, \"propagate\") &lt; 0)) <span class='error_goto'>__PYX_ERR(0, 47, __pyx_L3_error)</span>\n",
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       "    } else if (<span class='py_macro_api'>PyTuple_GET_SIZE</span>(__pyx_args) != 3) {\n",
       "      goto __pyx_L5_argtuple_error;\n",
       "    } else {\n",
       "      values[0] = <span class='py_macro_api'>PyTuple_GET_ITEM</span>(__pyx_args, 0);\n",
       "      values[1] = <span class='py_macro_api'>PyTuple_GET_ITEM</span>(__pyx_args, 1);\n",
       "      values[2] = <span class='py_macro_api'>PyTuple_GET_ITEM</span>(__pyx_args, 2);\n",
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       "    __pyx_v_planet = values[0];\n",
       "    __pyx_v_time_span = __pyx_<span class='py_c_api'>PyFloat_AsDouble</span>(values[1]); if (unlikely((__pyx_v_time_span == (double)-1) &amp;&amp; <span class='py_c_api'>PyErr_Occurred</span>())) <span class='error_goto'>__PYX_ERR(0, 48, __pyx_L3_error)</span>\n",
       "    __pyx_v_num_steps = <span class='pyx_c_api'>__Pyx_PyInt_As_int</span>(values[2]); if (unlikely((__pyx_v_num_steps == (int)-1) &amp;&amp; <span class='py_c_api'>PyErr_Occurred</span>())) <span class='error_goto'>__PYX_ERR(0, 49, __pyx_L3_error)</span>\n",
       "  }\n",
       "  goto __pyx_L4_argument_unpacking_done;\n",
       "  __pyx_L5_argtuple_error:;\n",
       "  <span class='pyx_c_api'>__Pyx_RaiseArgtupleInvalid</span>(\"propagate\", 1, 3, 3, <span class='py_macro_api'>PyTuple_GET_SIZE</span>(__pyx_args)); <span class='error_goto'>__PYX_ERR(0, 47, __pyx_L3_error)</span>\n",
       "  __pyx_L3_error:;\n",
       "  <span class='pyx_c_api'>__Pyx_AddTraceback</span>(\"_cython_magic_11dafdc010a400a06215a70fae45fc51.propagate\", __pyx_clineno, __pyx_lineno, __pyx_filename);\n",
       "  <span class='refnanny'>__Pyx_RefNannyFinishContext</span>();\n",
       "  return NULL;\n",
       "  __pyx_L4_argument_unpacking_done:;\n",
       "  __pyx_r = __pyx_pf_46_cython_magic_11dafdc010a400a06215a70fae45fc51_propagate(__pyx_self, __pyx_v_planet, __pyx_v_time_span, __pyx_v_num_steps);\n",
       "\n",
       "  /* function exit code */\n",
       "  <span class='refnanny'>__Pyx_RefNannyFinishContext</span>();\n",
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       "\n",
       "static PyObject *__pyx_pf_46_cython_magic_11dafdc010a400a06215a70fae45fc51_propagate(CYTHON_UNUSED PyObject *__pyx_self, PyObject *__pyx_v_planet, double __pyx_v_time_span, int __pyx_v_num_steps) {\n",
       "  double __pyx_v_dt;\n",
       "  int __pyx_v_j;\n",
       "  PyObject *__pyx_v_trajectory = NULL;\n",
       "  PyObject *__pyx_r = NULL;\n",
       "  <span class='refnanny'>__Pyx_RefNannyDeclarations</span>\n",
       "  <span class='refnanny'>__Pyx_RefNannySetupContext</span>(\"propagate\", 0);\n",
       "/* … */\n",
       "  /* function exit code */\n",
       "  __pyx_r = Py_None; <span class='pyx_macro_api'>__Pyx_INCREF</span>(Py_None);\n",
       "  goto __pyx_L0;\n",
       "  __pyx_L1_error:;\n",
       "  <span class='pyx_macro_api'>__Pyx_XDECREF</span>(__pyx_t_1);\n",
       "  <span class='pyx_macro_api'>__Pyx_XDECREF</span>(__pyx_t_2);\n",
       "  <span class='pyx_macro_api'>__Pyx_XDECREF</span>(__pyx_t_3);\n",
       "  <span class='pyx_macro_api'>__Pyx_XDECREF</span>(__pyx_t_4);\n",
       "  <span class='pyx_c_api'>__Pyx_AddTraceback</span>(\"_cython_magic_11dafdc010a400a06215a70fae45fc51.propagate\", __pyx_clineno, __pyx_lineno, __pyx_filename);\n",
       "  __pyx_r = NULL;\n",
       "  __pyx_L0:;\n",
       "  <span class='pyx_macro_api'>__Pyx_XDECREF</span>(__pyx_v_trajectory);\n",
       "  <span class='refnanny'>__Pyx_XGIVEREF</span>(__pyx_r);\n",
       "  <span class='refnanny'>__Pyx_RefNannyFinishContext</span>();\n",
       "  return __pyx_r;\n",
       "}\n",
       "/* … */\n",
       "  __pyx_tuple_ = <span class='py_c_api'>PyTuple_Pack</span>(6, __pyx_n_s_planet, __pyx_n_s_time_span, __pyx_n_s_num_steps, __pyx_n_s_dt, __pyx_n_s_j, __pyx_n_s_trajectory);<span class='error_goto'> if (unlikely(!__pyx_tuple_)) __PYX_ERR(0, 47, __pyx_L1_error)</span>\n",
       "  <span class='refnanny'>__Pyx_GOTREF</span>(__pyx_tuple_);\n",
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    "%%cython -a\n",
    "\n",
    "from libc.math cimport sqrt\n",
    "\n",
    "cimport cython\n",
    "\n",
    "import numpy as np\n",
    "\n",
    "cdef class Planet(object):\n",
    "    cdef public double x, y, z, vx, vy, vz, m\n",
    "    def __init__(self):\n",
    "        self.x = 1.0\n",
    "        self.y = 0\n",
    "        self.z = 0\n",
    "        self.vx = 0\n",
    "        self.vy = 0\n",
    "        self.vz = 1.0\n",
    "        \n",
    "        self.m = 1.0\n",
    "\n",
    "\n",
    "@cython.cdivision(True)\n",
    "cdef void single_step(Planet planet,\n",
    "                      double dt) nogil:\n",
    "    \"\"\"Make a single step in time, t -> t+dt.\"\"\"\n",
    "    # Gravitational force pulls towards origin\n",
    "    cdef double r, r3\n",
    "    r = sqrt(planet.x**2 + planet.y**2 + planet.z**2)\n",
    "    r3 = r*r*r                             # XXX: check generated C code\n",
    "    \n",
    "    Fx = -planet.x / r3\n",
    "    Fy = -planet.y / r3\n",
    "    Fz = -planet.z / r3\n",
    "    \n",
    "    # update position\n",
    "    planet.x += planet.vx * dt\n",
    "    planet.y += planet.vy * dt\n",
    "    planet.z += planet.vz * dt\n",
    "    \n",
    "    # update velocity\n",
    "    m = planet.m\n",
    "    planet.vx += Fx * dt / m\n",
    "    planet.vy += Fy * dt / m\n",
    "    planet.vz += Fz * dt / m\n",
    "\n",
    "\n",
    "@cython.cdivision(True)\n",
    "def propagate(planet,\n",
    "              double time_span,\n",
    "              int num_steps):\n",
    "    \"\"\"Make a number of time steps.\"\"\"\n",
    "\n",
    "    cdef double dt = time_span / num_steps\n",
    "    cdef int j\n",
    "    trajectory = np.zeros((3, num_steps))\n",
    "    for j in range(num_steps):\n",
    "        trajectory[0][j] = planet.x\n",
    "        trajectory[1][j] = planet.y\n",
    "        trajectory[2][j] = planet.z\n",
    "        single_step(planet, dt)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "56.7 µs ± 1.78 µs per loop (mean ± std. dev. of 7 runs, 10000 loops each)\n"
     ]
    }
   ],
   "source": [
    "planet = Planet()\n",
    "%timeit trajectory = propagate(planet, 1, 100)"
   ]
  },
  {
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   "metadata": {},
   "source": [
    "3"
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       "<pre class=\"cython line score-8\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">03</span>: <span class=\"k\">import</span> <span class=\"nn\">numpy</span> <span class=\"k\">as</span> <span class=\"nn\">np</span></pre>\n",
       "<pre class='cython code score-8 '>  __pyx_t_1 = <span class='pyx_c_api'>__Pyx_Import</span>(__pyx_n_s_numpy, 0, 0);<span class='error_goto'> if (unlikely(!__pyx_t_1)) __PYX_ERR(0, 3, __pyx_L1_error)</span>\n",
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       "<pre class='cython code score-8 '>  __pyx_t_1 = <span class='pyx_c_api'>__Pyx_Import</span>(__pyx_n_s_array, 0, 0);<span class='error_goto'> if (unlikely(!__pyx_t_1)) __PYX_ERR(0, 4, __pyx_L1_error)</span>\n",
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       "<pre class=\"cython line score-0\">&#xA0;<span class=\"\">06</span>:     <span class=\"n\">double</span> <span class=\"n\">sin</span><span class=\"p\">(</span><span class=\"n\">double</span> <span class=\"n\">x</span><span class=\"p\">)</span> <span class=\"k\">nogil</span></pre>\n",
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       "<pre class='cython code score-0 '>static double __pyx_f_46_cython_magic_a297fda3e7a45a1389aefc007915c04d__compute_sinea(double __pyx_v_x, struct __pyx_opt_args_46_cython_magic_a297fda3e7a45a1389aefc007915c04d__compute_sinea *__pyx_optional_args) {\n",
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       "<pre class='cython code score-0 '>  __pyx_v_out = (__pyx_v_a * sin(__pyx_v_x));\n",
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       "static PyMethodDef __pyx_mdef_46_cython_magic_a297fda3e7a45a1389aefc007915c04d_1compute_sina = {\"compute_sina\", (PyCFunction)(void*)(PyCFunctionWithKeywords)__pyx_pw_46_cython_magic_a297fda3e7a45a1389aefc007915c04d_1compute_sina, METH_VARARGS|METH_KEYWORDS, 0};\n",
       "static PyObject *__pyx_pw_46_cython_magic_a297fda3e7a45a1389aefc007915c04d_1compute_sina(PyObject *__pyx_self, PyObject *__pyx_args, PyObject *__pyx_kwds) {\n",
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       "        default: goto __pyx_L5_argtuple_error;\n",
       "      }\n",
       "      kw_args = <span class='py_c_api'>PyDict_Size</span>(__pyx_kwds);\n",
       "      switch (pos_args) {\n",
       "        case  0:\n",
       "        if (likely((values[0] = <span class='pyx_c_api'>__Pyx_PyDict_GetItemStr</span>(__pyx_kwds, __pyx_n_s_left)) != 0)) kw_args--;\n",
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       "        CYTHON_FALLTHROUGH;\n",
       "        case  1:\n",
       "        if (likely((values[1] = <span class='pyx_c_api'>__Pyx_PyDict_GetItemStr</span>(__pyx_kwds, __pyx_n_s_right)) != 0)) kw_args--;\n",
       "        else {\n",
       "          <span class='pyx_c_api'>__Pyx_RaiseArgtupleInvalid</span>(\"compute_sina\", 0, 3, 4, 1); <span class='error_goto'>__PYX_ERR(0, 21, __pyx_L3_error)</span>\n",
       "        }\n",
       "        CYTHON_FALLTHROUGH;\n",
       "        case  2:\n",
       "        if (likely((values[2] = <span class='pyx_c_api'>__Pyx_PyDict_GetItemStr</span>(__pyx_kwds, __pyx_n_s_num)) != 0)) kw_args--;\n",
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       "          <span class='pyx_c_api'>__Pyx_RaiseArgtupleInvalid</span>(\"compute_sina\", 0, 3, 4, 2); <span class='error_goto'>__PYX_ERR(0, 21, __pyx_L3_error)</span>\n",
       "        }\n",
       "        CYTHON_FALLTHROUGH;\n",
       "        case  3:\n",
       "        if (kw_args &gt; 0) {\n",
       "          PyObject* value = <span class='pyx_c_api'>__Pyx_PyDict_GetItemStr</span>(__pyx_kwds, __pyx_n_s_eps);\n",
       "          if (value) { values[3] = value; kw_args--; }\n",
       "        }\n",
       "      }\n",
       "      if (unlikely(kw_args &gt; 0)) {\n",
       "        if (unlikely(<span class='pyx_c_api'>__Pyx_ParseOptionalKeywords</span>(__pyx_kwds, __pyx_pyargnames, 0, values, pos_args, \"compute_sina\") &lt; 0)) <span class='error_goto'>__PYX_ERR(0, 21, __pyx_L3_error)</span>\n",
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       "      switch (<span class='py_macro_api'>PyTuple_GET_SIZE</span>(__pyx_args)) {\n",
       "        case  4: values[3] = <span class='py_macro_api'>PyTuple_GET_ITEM</span>(__pyx_args, 3);\n",
       "        CYTHON_FALLTHROUGH;\n",
       "        case  3: values[2] = <span class='py_macro_api'>PyTuple_GET_ITEM</span>(__pyx_args, 2);\n",
       "        values[1] = <span class='py_macro_api'>PyTuple_GET_ITEM</span>(__pyx_args, 1);\n",
       "        values[0] = <span class='py_macro_api'>PyTuple_GET_ITEM</span>(__pyx_args, 0);\n",
       "        break;\n",
       "        default: goto __pyx_L5_argtuple_error;\n",
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       "    }\n",
       "    __pyx_v_left = __pyx_<span class='py_c_api'>PyFloat_AsDouble</span>(values[0]); if (unlikely((__pyx_v_left == (double)-1) &amp;&amp; <span class='py_c_api'>PyErr_Occurred</span>())) <span class='error_goto'>__PYX_ERR(0, 21, __pyx_L3_error)</span>\n",
       "    __pyx_v_right = __pyx_<span class='py_c_api'>PyFloat_AsDouble</span>(values[1]); if (unlikely((__pyx_v_right == (double)-1) &amp;&amp; <span class='py_c_api'>PyErr_Occurred</span>())) <span class='error_goto'>__PYX_ERR(0, 21, __pyx_L3_error)</span>\n",
       "    __pyx_v_num = <span class='pyx_c_api'>__Pyx_PyInt_As_int</span>(values[2]); if (unlikely((__pyx_v_num == (int)-1) &amp;&amp; <span class='py_c_api'>PyErr_Occurred</span>())) <span class='error_goto'>__PYX_ERR(0, 21, __pyx_L3_error)</span>\n",
       "    if (values[3]) {\n",
       "      __pyx_v_eps = __pyx_<span class='py_c_api'>PyFloat_AsDouble</span>(values[3]); if (unlikely((__pyx_v_eps == (double)-1) &amp;&amp; <span class='py_c_api'>PyErr_Occurred</span>())) <span class='error_goto'>__PYX_ERR(0, 21, __pyx_L3_error)</span>\n",
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       "    }\n",
       "  }\n",
       "  goto __pyx_L4_argument_unpacking_done;\n",
       "  __pyx_L5_argtuple_error:;\n",
       "  <span class='pyx_c_api'>__Pyx_RaiseArgtupleInvalid</span>(\"compute_sina\", 0, 3, 4, <span class='py_macro_api'>PyTuple_GET_SIZE</span>(__pyx_args)); <span class='error_goto'>__PYX_ERR(0, 21, __pyx_L3_error)</span>\n",
       "  __pyx_L3_error:;\n",
       "  <span class='pyx_c_api'>__Pyx_AddTraceback</span>(\"_cython_magic_a297fda3e7a45a1389aefc007915c04d.compute_sina\", __pyx_clineno, __pyx_lineno, __pyx_filename);\n",
       "  <span class='refnanny'>__Pyx_RefNannyFinishContext</span>();\n",
       "  return NULL;\n",
       "  __pyx_L4_argument_unpacking_done:;\n",
       "  __pyx_r = __pyx_pf_46_cython_magic_a297fda3e7a45a1389aefc007915c04d_compute_sina(__pyx_self, __pyx_v_left, __pyx_v_right, __pyx_v_num, __pyx_v_eps);\n",
       "\n",
       "  /* function exit code */\n",
       "  <span class='refnanny'>__Pyx_RefNannyFinishContext</span>();\n",
       "  return __pyx_r;\n",
       "}\n",
       "\n",
       "static PyObject *__pyx_pf_46_cython_magic_a297fda3e7a45a1389aefc007915c04d_compute_sina(CYTHON_UNUSED PyObject *__pyx_self, double __pyx_v_left, double __pyx_v_right, CYTHON_UNUSED int __pyx_v_num, double __pyx_v_eps) {\n",
       "  double __pyx_v_l;\n",
       "  double __pyx_v_r;\n",
       "  PyObject *__pyx_v_x_array = 0;\n",
       "  double __pyx_v_mid;\n",
       "  PyObject *__pyx_r = NULL;\n",
       "  <span class='refnanny'>__Pyx_RefNannyDeclarations</span>\n",
       "  <span class='refnanny'>__Pyx_RefNannySetupContext</span>(\"compute_sina\", 0);\n",
       "/* … */\n",
       "  /* function exit code */\n",
       "  __pyx_L1_error:;\n",
       "  <span class='pyx_macro_api'>__Pyx_XDECREF</span>(__pyx_t_1);\n",
       "  <span class='pyx_c_api'>__Pyx_AddTraceback</span>(\"_cython_magic_a297fda3e7a45a1389aefc007915c04d.compute_sina\", __pyx_clineno, __pyx_lineno, __pyx_filename);\n",
       "  __pyx_r = NULL;\n",
       "  __pyx_L0:;\n",
       "  <span class='pyx_macro_api'>__Pyx_XDECREF</span>(__pyx_v_x_array);\n",
       "  <span class='refnanny'>__Pyx_XGIVEREF</span>(__pyx_r);\n",
       "  <span class='refnanny'>__Pyx_RefNannyFinishContext</span>();\n",
       "  return __pyx_r;\n",
       "}\n",
       "/* … */\n",
       "  __pyx_tuple_ = <span class='py_c_api'>PyTuple_Pack</span>(8, __pyx_n_s_left, __pyx_n_s_right, __pyx_n_s_num, __pyx_n_s_eps, __pyx_n_s_l, __pyx_n_s_r, __pyx_n_s_x_array, __pyx_n_s_mid);<span class='error_goto'> if (unlikely(!__pyx_tuple_)) __PYX_ERR(0, 21, __pyx_L1_error)</span>\n",
       "  <span class='refnanny'>__Pyx_GOTREF</span>(__pyx_tuple_);\n",
       "  <span class='refnanny'>__Pyx_GIVEREF</span>(__pyx_tuple_);\n",
       "/* … */\n",
       "  __pyx_t_1 = PyCFunction_NewEx(&amp;__pyx_mdef_46_cython_magic_a297fda3e7a45a1389aefc007915c04d_1compute_sina, NULL, __pyx_n_s_cython_magic_a297fda3e7a45a1389);<span class='error_goto'> if (unlikely(!__pyx_t_1)) __PYX_ERR(0, 21, __pyx_L1_error)</span>\n",
       "  <span class='refnanny'>__Pyx_GOTREF</span>(__pyx_t_1);\n",
       "  if (<span class='py_c_api'>PyDict_SetItem</span>(__pyx_d, __pyx_n_s_compute_sina, __pyx_t_1) &lt; 0) <span class='error_goto'>__PYX_ERR(0, 21, __pyx_L1_error)</span>\n",
       "  <span class='pyx_macro_api'>__Pyx_DECREF</span>(__pyx_t_1); __pyx_t_1 = 0;\n",
       "</pre><pre class=\"cython line score-0\">&#xA0;<span class=\"\">22</span>:     <span class=\"c\">#num = (right - left)//M_PI</span></pre>\n",
       "<pre class=\"cython line score-0\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">23</span>:     <span class=\"k\">cdef</span> <span class=\"kt\">double</span> <span class=\"nf\">l</span> <span class=\"o\">=</span> <span class=\"n\">left</span></pre>\n",
       "<pre class='cython code score-0 '>  __pyx_v_l = __pyx_v_left;\n",
       "</pre><pre class=\"cython line score-0\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">24</span>:     <span class=\"k\">cdef</span> <span class=\"kt\">double</span> <span class=\"nf\">r</span> <span class=\"o\">=</span> <span class=\"n\">left</span> <span class=\"o\">+</span> <span class=\"n\">M_PI</span></pre>\n",
       "<pre class='cython code score-0 '>  __pyx_v_r = (__pyx_v_left + M_PI);\n",
       "</pre><pre class=\"cython line score-5\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">25</span>:     <span class=\"k\">cdef</span> <span class=\"kt\">list</span> <span class=\"nf\">x_array</span> <span class=\"o\">=</span> <span class=\"p\">[]</span></pre>\n",
       "<pre class='cython code score-5 '>  __pyx_t_1 = <span class='py_c_api'>PyList_New</span>(0);<span class='error_goto'> if (unlikely(!__pyx_t_1)) __PYX_ERR(0, 25, __pyx_L1_error)</span>\n",
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       "  __pyx_v_x_array = ((PyObject*)__pyx_t_1);\n",
       "  __pyx_t_1 = 0;\n",
       "</pre><pre class=\"cython line score-0\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">26</span>:     <span class=\"k\">while</span> <span class=\"n\">r</span> <span class=\"o\">&lt;</span> <span class=\"n\">right</span><span class=\"p\">:</span></pre>\n",
       "<pre class='cython code score-0 '>  while (1) {\n",
       "    __pyx_t_2 = ((__pyx_v_r &lt; __pyx_v_right) != 0);\n",
       "    if (!__pyx_t_2) break;\n",
       "</pre><pre class=\"cython line score-0\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">27</span>:         <span class=\"n\">mid</span> <span class=\"o\">=</span> <span class=\"p\">(</span><span class=\"n\">l</span> <span class=\"o\">+</span> <span class=\"n\">r</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"mf\">2</span></pre>\n",
       "<pre class='cython code score-0 '>    __pyx_v_mid = ((__pyx_v_l + __pyx_v_r) / 2.0);\n",
       "</pre><pre class=\"cython line score-0\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">28</span>:         <span class=\"k\">while</span> <span class=\"nb\">abs</span><span class=\"p\">(</span><span class=\"n\">l</span> <span class=\"o\">-</span> <span class=\"n\">r</span><span class=\"p\">)</span> <span class=\"o\">&gt;</span> <span class=\"n\">eps</span><span class=\"p\">:</span></pre>\n",
       "<pre class='cython code score-0 '>    while (1) {\n",
       "      __pyx_t_2 = ((fabs((__pyx_v_l - __pyx_v_r)) &gt; __pyx_v_eps) != 0);\n",
       "      if (!__pyx_t_2) break;\n",
       "</pre><pre class=\"cython line score-0\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">29</span>:             <span class=\"k\">if</span> <span class=\"n\">_compute_sinea</span><span class=\"p\">(</span><span class=\"n\">mid</span><span class=\"p\">)</span><span class=\"o\">*</span><span class=\"n\">_compute_sinea</span><span class=\"p\">(</span><span class=\"n\">l</span><span class=\"p\">)</span> <span class=\"o\">&gt;</span> <span class=\"mf\">0</span><span class=\"p\">:</span></pre>\n",
       "<pre class='cython code score-0 '>      __pyx_t_2 = (((__pyx_f_46_cython_magic_a297fda3e7a45a1389aefc007915c04d__compute_sinea(__pyx_v_mid, NULL) * __pyx_f_46_cython_magic_a297fda3e7a45a1389aefc007915c04d__compute_sinea(__pyx_v_l, NULL)) &gt; 0.0) != 0);\n",
       "      if (__pyx_t_2) {\n",
       "/* … */\n",
       "      }\n",
       "</pre><pre class=\"cython line score-0\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">30</span>:                 <span class=\"n\">l</span> <span class=\"o\">=</span> <span class=\"n\">mid</span></pre>\n",
       "<pre class='cython code score-0 '>        __pyx_v_l = __pyx_v_mid;\n",
       "</pre><pre class=\"cython line score-0\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">31</span>:             <span class=\"k\">if</span> <span class=\"n\">_compute_sinea</span><span class=\"p\">(</span><span class=\"n\">r</span><span class=\"p\">)</span><span class=\"o\">*</span><span class=\"n\">_compute_sinea</span><span class=\"p\">(</span><span class=\"n\">mid</span><span class=\"p\">)</span> <span class=\"o\">&gt;</span> <span class=\"mf\">0</span><span class=\"p\">:</span></pre>\n",
       "<pre class='cython code score-0 '>      __pyx_t_2 = (((__pyx_f_46_cython_magic_a297fda3e7a45a1389aefc007915c04d__compute_sinea(__pyx_v_r, NULL) * __pyx_f_46_cython_magic_a297fda3e7a45a1389aefc007915c04d__compute_sinea(__pyx_v_mid, NULL)) &gt; 0.0) != 0);\n",
       "      if (__pyx_t_2) {\n",
       "/* … */\n",
       "      }\n",
       "</pre><pre class=\"cython line score-0\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">32</span>:                 <span class=\"n\">r</span> <span class=\"o\">=</span> <span class=\"n\">mid</span></pre>\n",
       "<pre class='cython code score-0 '>        __pyx_v_r = __pyx_v_mid;\n",
       "</pre><pre class=\"cython line score-0\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">33</span>:             <span class=\"n\">mid</span> <span class=\"o\">=</span> <span class=\"p\">(</span><span class=\"n\">l</span> <span class=\"o\">+</span> <span class=\"n\">r</span><span class=\"p\">)</span><span class=\"o\">/</span><span class=\"mf\">2</span></pre>\n",
       "<pre class='cython code score-0 '>      __pyx_v_mid = ((__pyx_v_l + __pyx_v_r) / 2.0);\n",
       "    }\n",
       "</pre><pre class=\"cython line score-0\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">34</span>:         <span class=\"n\">l</span> <span class=\"o\">=</span> <span class=\"n\">l</span> <span class=\"o\">+</span> <span class=\"n\">M_PI</span></pre>\n",
       "<pre class='cython code score-0 '>    __pyx_v_l = (__pyx_v_l + M_PI);\n",
       "</pre><pre class=\"cython line score-0\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">35</span>:         <span class=\"n\">r</span> <span class=\"o\">=</span> <span class=\"n\">r</span> <span class=\"o\">+</span> <span class=\"n\">M_PI</span></pre>\n",
       "<pre class='cython code score-0 '>    __pyx_v_r = (__pyx_v_r + M_PI);\n",
       "</pre><pre class=\"cython line score-8\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">36</span>:         <span class=\"n\">x_array</span><span class=\"o\">.</span><span class=\"n\">append</span><span class=\"p\">(</span><span class=\"n\">mid</span><span class=\"p\">)</span></pre>\n",
       "<pre class='cython code score-8 '>    __pyx_t_1 = <span class='py_c_api'>PyFloat_FromDouble</span>(__pyx_v_mid);<span class='error_goto'> if (unlikely(!__pyx_t_1)) __PYX_ERR(0, 36, __pyx_L1_error)</span>\n",
       "    <span class='refnanny'>__Pyx_GOTREF</span>(__pyx_t_1);\n",
       "    __pyx_t_3 = <span class='pyx_c_api'>__Pyx_PyList_Append</span>(__pyx_v_x_array, __pyx_t_1);<span class='error_goto'> if (unlikely(__pyx_t_3 == ((int)-1))) __PYX_ERR(0, 36, __pyx_L1_error)</span>\n",
       "    <span class='pyx_macro_api'>__Pyx_DECREF</span>(__pyx_t_1); __pyx_t_1 = 0;\n",
       "  }\n",
       "</pre><pre class=\"cython line score-2\" onclick=\"(function(s){s.display=s.display==='block'?'none':'block'})(this.nextElementSibling.style)\">+<span class=\"\">37</span>:     <span class=\"k\">return</span> <span class=\"n\">x_array</span></pre>\n",
       "<pre class='cython code score-2 '>  <span class='pyx_macro_api'>__Pyx_XDECREF</span>(__pyx_r);\n",
       "  <span class='pyx_macro_api'>__Pyx_INCREF</span>(__pyx_v_x_array);\n",
       "  __pyx_r = __pyx_v_x_array;\n",
       "  goto __pyx_L0;\n",
       "</pre></div></body></html>"
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      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%%cython -a\n",
    "\n",
    "cimport cython\n",
    "import numpy as np\n",
    "import array\n",
    "cdef extern from \"math.h\":\n",
    "    double sin(double x) nogil\n",
    "    double M_PI\n",
    "\n",
    "\n",
    "# Same _compute_sine as above\n",
    "\n",
    "@cython.boundscheck(False)\n",
    "@cython.wraparound(False)\n",
    "cdef double _compute_sinea(double x,           # typed memoryview syntax\n",
    "                        double a = 2.) nogil:\n",
    "    out = a*sin(x)\n",
    "    return out\n",
    "\n",
    "@cython.boundscheck(False)\n",
    "@cython.wraparound(False)  \n",
    "def compute_sina(double left, double right, int num, double eps = 0.000001):\n",
    "    #num = (right - left)//M_PI\n",
    "    cdef double l = left\n",
    "    cdef double r = left + M_PI\n",
    "    cdef list x_array = []\n",
    "    while r < right:\n",
    "        mid = (l + r)/2\n",
    "        while abs(l - r) > eps:\n",
    "            if _compute_sinea(mid)*_compute_sinea(l) > 0:\n",
    "                l = mid\n",
    "            if _compute_sinea(r)*_compute_sinea(mid) > 0:\n",
    "                r = mid\n",
    "            mid = (l + r)/2\n",
    "        l = l + M_PI\n",
    "        r = r + M_PI\n",
    "        x_array.append(mid)\n",
    "    return x_array"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.64 µs ± 9.46 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops each)\n"
     ]
    }
   ],
   "source": [
    "left = -6.29\n",
    "right = 6.29\n",
    "%timeit compute_sina(left, right, abs((right - left)//np.pi))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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