{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# The trapezoidal rule: vectorization and perormance\n",
    "\n",
    "Here we discuss in detail the performance profile of various solutions to the exercise from our Numpy introduction, the trapezoidal rule for the numerical approximation of definite integrals.\n",
    "\n",
    "If we denote by $x_{i}$ ($i=0,\\ldots,n,$ with $x_{0}=a$ and $x_{n}=b$) the abscissas\n",
    "where the function is sampled, then\n",
    "\n",
    "$$\\int_{a}^{b}f(x)dx\\approx\\frac{1}{2}\\sum_{i=1}^{n}\\left(x_{i}-x_{i-1}\\right)\\left(f(x_{i})+f(x_{i-1})\\right).$$\n",
    "\n",
    "The common case of using equally spaced abscissas with spacing $h=(b-a)/n$ reads:\n",
    "\n",
    "$$\\int_{a}^{b}f(x)dx\\approx\\frac{h}{2}\\sum_{i=1}^{n}\\left(f(x_{i})+f(x_{i-1})\\right).$$\n",
    "\n",
    "One frequently receives the function values already precomputed, $y_{i}=f(x_{i}),$ so the formula becomes\n",
    "\n",
    "$$\\int_{a}^{b}f(x)dx\\approx\\frac{1}{2}\\sum_{i=1}^{n}\\left(x_{i}-x_{i-1}\\right)\\left(y_{i}+y_{i-1}\\right).$$\n",
    "\n",
    "In this exercise, you'll need to write two functions, `trapz` and `trapzf`. `trapz` applies the trapezoid formula to pre-computed values, implementing equation trapz, while `trapzf` takes a function $f$ as input, as well as the total number of samples to evaluate, and computes the equation above.\n",
    "\n",
    "Test it and show that it produces correct values for some simple integrals you can compute analytically or compare your answers against `scipy.integrate.trapz`, using as test function $f(x)$:\n",
    "\n",
    "$$\n",
    "f(x) = (x-3)(x-5)(x-7)+85\n",
    "$$\n",
    "\n",
    "integrated between $a=1$ and $b=9$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "import scipy.integrate as sint"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Reference value (trapezoid): 680.0\n"
     ]
    }
   ],
   "source": [
    "def f(x):\n",
    "    return (x-3)*(x-5)*(x-7)+85\n",
    "\n",
    "a, b, n = (1, 9, 200)\n",
    "x = np.linspace(a, b, n)\n",
    "y = f(x)\n",
    "\n",
    "print(\"Reference value (trapezoid):\", sint.trapz(y, x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Our functions\n",
      "680.0\n",
      "680.0\n"
     ]
    }
   ],
   "source": [
    "def trapz(y, x):\n",
    "    return 0.5*((x[1:]-x[:-1])*(y[1:]+y[:-1])).sum()\n",
    "\n",
    "def trapzf(f, a, b, n):\n",
    "    x = np.linspace(a, b, n)\n",
    "    # Sample the input function at all values of x\n",
    "    y = f(x)\n",
    "    # Note that in the equations above, the list had n+1 points. Since here we're using\n",
    "    # n as the total number of points, the actual size of the interval h should \n",
    "    # use n-1 in the formula:\n",
    "    h = (b-a)/(n-1)\n",
    "    # Compute the trapezoid rule sum for the final result\n",
    "    return (h/2)*(y[1:]+y[:-1]).sum()\n",
    "\n",
    "print(\"Our functions\")\n",
    "print(trapz(y, x))\n",
    "print(trapzf(f, a, b, n))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Some performance notes\n",
    "\n",
    "Let's compare the time it takes to compute the fully vectorized form in our `trapz` with a more manual implementation that does the computation with a pure Python loop."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "680.00000000000011"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def trapzl(y, x):\n",
    "    \"Pure python version of trapezoid rule.\"\n",
    "    s = 0\n",
    "    for i in range(1, len(x)):\n",
    "        s += (x[i]-x[i-1])*(y[i]+y[i-1])\n",
    "    return s/2\n",
    "\n",
    "trapzl(y, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's compute this over a much finer grid to really see the performance difference"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "35.2 µs ± 2.61 µs per loop (mean ± std. dev. of 7 runs, 10000 loops each)\n",
      "10.8 ms ± 444 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)\n"
     ]
    }
   ],
   "source": [
    "a, b, n = (1, 9, 20_000)\n",
    "x = np.linspace(a, b, n)\n",
    "y = f(x)\n",
    "\n",
    "tv = %timeit -o trapz(y, x)\n",
    "tl = %timeit -o trapzl(y, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The output of `%timeit` is a special `TimeitResult` object that contains all the information about the timing run in a set of attributes. Try typing `tv.<TAB>` to see some of its attributes."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<TimeitResult : 35.2 µs ± 2.61 µs per loop (mean ± std. dev. of 7 runs, 10000 loops each)>"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tv"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In particular, we can look at the average run time, which as we can see, is in seconds:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.5238153685661796e-05"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tv.average"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "With this, we can now see how much faster the Numpy version was at 20,000 sample points:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "306.7110725043031"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tl.average/tv.average"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's see what the performance difference is across a range of sizes (note this will take a while to run). We're actually going to need the same computation of $x$ and $f(x)$ over and over, so let's do it once and retrieve them later on:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "a, b = 1, 9\n",
    "\n",
    "xy = []\n",
    "sizes = np.logspace(1, 7, 7)\n",
    "for n in sizes:\n",
    "    x = np.linspace(a, b, int(n))\n",
    "    y = f(x)\n",
    "    xy.append((x, y))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3.43 µs ± 147 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)\n",
      "5.46 µs ± 136 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)\n",
      "3.72 µs ± 219 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)\n",
      "52.4 µs ± 1.63 µs per loop (mean ± std. dev. of 7 runs, 10000 loops each)\n",
      "5.49 µs ± 89 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)\n",
      "514 µs ± 11.6 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)\n",
      "17.3 µs ± 338 ns per loop (mean ± std. dev. of 7 runs, 10000 loops each)\n",
      "5.29 ms ± 143 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)\n",
      "144 µs ± 3.32 µs per loop (mean ± std. dev. of 7 runs, 10000 loops each)\n",
      "51.7 ms ± 1.03 ms per loop (mean ± std. dev. of 7 runs, 10 loops each)\n",
      "3.56 ms ± 225 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)\n",
      "548 ms ± 10.5 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)\n",
      "71.7 ms ± 672 µs per loop (mean ± std. dev. of 7 runs, 10 loops each)\n",
      "5.18 s ± 50.6 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)\n"
     ]
    }
   ],
   "source": [
    "tv, tl = [], []\n",
    "tve, tle = [], []\n",
    "for x, y in xy:\n",
    "    t1 = %timeit -o trapz(y, x)\n",
    "    tv.append(t1.average)\n",
    "    tve.append(t1.stdev)\n",
    "    t2 = %timeit -o trapzl(y, x)\n",
    "    tl.append(t2.average)\n",
    "    tle.append(t2.stdev)\n",
    "\n",
    "tva = np.array(tv); tvea = np.array(tve)\n",
    "tla = np.array(tl); tlea = np.array(tle)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let's plot these, along with proper error bars (remember your error propagation formulas, assuming here that the measurements are normal IID for simplicity):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def rat_err(p, dp, q, dq):\n",
    "    ratio = p/q\n",
    "    err = ratio*np.sqrt((dp/p)**2 + (dq/q)**2)\n",
    "    return ratio, err"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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N8bVx92HS1+/l5kEdqFfHGsZFGuvEL3TKTBAl4z64P2NUNb6S40EUAg+o6unA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\nGGOMX/8POywGG5WOrgwAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x110765ba8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ratio, err = rat_err(tla, tlea, tva, tvea)\n",
    "\n",
    "plt.errorbar(sizes, ratio, yerr=err, fmt = 'o-')\n",
    "plt.xscale('log')\n",
    "\n",
    "plt.grid(True)\n",
    "plt.xlabel(\"Input size\")\n",
    "plt.ylabel(\"Timing ratio Python/Numpy\")\n",
    "plt.title(\"Python loop relative slowness\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A few observations:\n",
    "\n",
    "- At very small sizes the difference isn't very dramatic. The overhead of the Python function call and basic arithmetic dominates.\n",
    "\n",
    "- For input sizes ranging from the thousands to the million elements, the vectorized version can be ~ 100x to ~300x fastr than the pure Python loop.  This is not uncommon in real-world applications.\n",
    "\n",
    "- At very large sizes (at a million) the performance profile changes.\n",
    "\n",
    "- There's significant uncertainty for one of the values but little for the others, and even accounting for that uncertainty, the conclusion doesn't fundamentally change.\n",
    "\n",
    "In general it's extremely hard to predict exact performance ratios, which often vary in hard to understand ways with problem size, as hardware, cache and algorithm specifics intersect.  But the results of this example aren't suprising: whether exactly 100x or 350x slower, the fact remains that the vectorized solution vastly outperforms a pure Python loop.  This is a key aspect of Numpy as a key piece of high-performance Python.\n",
    "\n",
    "For an in-depth look at the \"vectorized mindset\" for Numpy, Nicolas Rougier has written the excellent (and free!) book [\"From Python to Numpy\"](http://www.labri.fr/perso/nrougier/from-python-to-numpy)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Numba\n",
    "\n",
    "Let's have a look at how [Numba](https://numba.pydata.org) can help in this type of loop-intensive code."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "680.0000000000001"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from numba import jit\n",
    "\n",
    "trapzn = jit(trapzl)\n",
    "\n",
    "x = np.linspace(a, b, 200)\n",
    "y = f(x)\n",
    "\n",
    "trapzn(y, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "34.6 µs ± 1.39 µs per loop (mean ± std. dev. of 7 runs, 10000 loops each)\n",
      "22.5 µs ± 732 ns per loop (mean ± std. dev. of 7 runs, 10000 loops each)\n"
     ]
    }
   ],
   "source": [
    "x = np.linspace(a, b, 20_000)\n",
    "y = f(x)\n",
    "\n",
    "%timeit trapz(y, x)\n",
    "%timeit trapzn(y, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "That's pretty impressive! By simply *compiling* our slow, Python code, we get it to beat the Numpy version at a size of 200.\n",
    "\n",
    "Let's collect Numba performance numbers similarly to the code above, and make a plot comparing Numba to Numpy at the same sizes:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "413 ns ± 4.92 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops each)\n",
      "512 ns ± 11.7 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops each)\n",
      "1.56 µs ± 25.3 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops each)\n",
      "11.3 µs ± 150 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)\n",
      "110 µs ± 1.3 µs per loop (mean ± std. dev. of 7 runs, 10000 loops each)\n",
      "1.16 ms ± 14.6 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)\n",
      "13 ms ± 948 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)\n"
     ]
    }
   ],
   "source": [
    "tn, tne = [], []\n",
    "for x, y in xy:\n",
    "    t = %timeit -o trapzn(y, x)\n",
    "    tn.append(t.average)\n",
    "    tne.append(t.stdev)\n",
    "\n",
    "tna = np.array(tn)\n",
    "tnea = np.array(tne)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's look at the Numba data alongside the Python one:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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auxeYUgHUsE4MY/u2YmzfVqzbfZB3lmcye3kmn677iToxEVzcvSmXpiTSs0Vd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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1530aaaef0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "rp, ep = rat_err(tla, tlea, tva, tvea)\n",
    "rn, en = rat_err(tna, tnea, tva, tvea)\n",
    "\n",
    "plt.errorbar(sizes, rp, yerr=ep, fmt = 'o-', label=\"Python\")\n",
    "plt.errorbar(sizes, rn, yerr=en, fmt = 'o-', label=\"Numba\")\n",
    "plt.xscale('log')\n",
    "plt.yscale('log')\n",
    "\n",
    "plt.grid(True)\n",
    "plt.axhline(1, color='red')\n",
    "plt.legend()\n",
    "plt.xlabel(\"Input size\")\n",
    "plt.ylabel(\"Timing ratios\")\n",
    "plt.title(\"Performance relative to vectorized Numpy\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's quickly see the Numba/Numpy ratios (the raw numbers):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.12029788  0.13762593  0.28367041  0.65578777  0.76074653  0.32554678\n",
      "  0.18141704]\n",
      "[ 8.31269874  7.26607237  3.52521785  1.52488357  1.31449828  3.07175515\n",
      "  5.51216148]\n"
     ]
    }
   ],
   "source": [
    "print(tna/tva)\n",
    "print(tva/tna)  # Let's look at the reverse relation, which may be a bit easier to interpret"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Some quick thoughts:\n",
    "\n",
    "- The Numba performance is *always* better than Numpy.  Note that this is a pretty ideal case for Numpy, and we did exactly *nothing* to the Python code other than compile it with `numba.jit`.  We're basically getting better-than-numpy performance for free.  Nubma can be a bit like magic ([this blog post](https://jakevdp.github.io/blog/2015/02/24/optimizing-python-with-numpy-and-numba/) by Jake VanderPlas contains much more detail).\n",
    "\n",
    "- In some cases, the difference is quite remarkable. On the other hand, where Numpy performs best comapred to raw Python, Numba still beats it but not by too much.  This probably indicates Numpy is already nearly optimal in that range.\n",
    "\n",
    "\n",
    "Let's look at the actual times instead, which in this case can be informative:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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aAvP9F5YxJpQcz8ph2LvLWJucyv8N7kzvdo0K31EVVn0Ac0ZBVD247Qs468Ly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\nQNcAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1530bd3d68>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.errorbar(sizes, tva, yerr=tvea, fmt='o-', label=\"Numpy\")\n",
    "plt.errorbar(sizes, tna, yerr=tnea, fmt='o-', label=\"Numba\")\n",
    "plt.xscale('log')\n",
    "plt.yscale('log')\n",
    "\n",
    "plt.legend()\n",
    "plt.grid(True)\n",
    "plt.xlabel(\"Input size\")\n",
    "plt.ylabel(\"Time (s)\")\n",
    "plt.title(\"Total runtime\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We see that the runtime profile is more smoothly linear (as it should be) for the Numba code."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise: Numba atop Numpy?\n",
    "\n",
    "Let's make `trapznv` which is a Numba-fied version of `trapz`, the Numpy version. The question is: can Numba be of any benefit on a function that is already a single Numpy expression? Let's see..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "680.0000000000001"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "trapznv = jit(trapz)\n",
    "\n",
    "x = np.linspace(a, b, 200)\n",
    "y = f(x)\n",
    "\n",
    "trapznv(y, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now that we see it's correct, let's look at performance in just one case:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "4.19 µs ± 97.7 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)\n",
      "1.04 µs ± 22.1 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops each)\n"
     ]
    }
   ],
   "source": [
    "%timeit trapz(y, x)\n",
    "%timeit trapznv(y, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "631 ns ± 3.56 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops each)\n",
      "839 ns ± 13.1 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops each)\n",
      "2.84 µs ± 30.6 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)\n",
      "23.5 µs ± 287 ns per loop (mean ± std. dev. of 7 runs, 10000 loops each)\n",
      "228 µs ± 3.59 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)\n",
      "3.49 ms ± 125 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)\n",
      "43.7 ms ± 842 µs per loop (mean ± std. dev. of 7 runs, 10 loops each)\n"
     ]
    }
   ],
   "source": [
    "tnv, tnve = [], []\n",
    "for x, y in xy:\n",
    "    t = %timeit -o trapznv(y, x)\n",
    "    tnv.append(t.average)\n",
    "    tnve.append(t.stdev)\n",
    "\n",
    "tnva = np.array(tnv)\n",
    "tnvea = np.array(tnve)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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FY659bynTv9/s953XmiBUQFq/+xDvLt7OxNPa0ytBmyJV/WjZJJJZ1w7hgt6t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\nAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1531028438>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "r, e = rat_err(tnva, tnvea, tva, tvea)\n",
    "\n",
    "plt.errorbar(sizes, r, yerr=e, fmt='o-', label=\"Python\")\n",
    "plt.xscale('log')\n",
    "plt.yscale('log')\n",
    "\n",
    "plt.grid(True)\n",
    "plt.axhline(1, color='red')\n",
    "plt.legend()\n",
    "plt.xlabel(\"Input size\")\n",
    "plt.ylabel(\"Timing ratio\")\n",
    "plt.title(\"Numbifyed Numpy vs regular Numpy\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Numerical accuracy\n",
    "\n",
    "We should finally take a quick look at accuracy considerations. In this case we know the exact answer (680), so we can see how each method compares. As $n \\rightarrow \\infty$ we expect them all to converge to 680, but let's see if any of them accumulates error worse than the others.  Remember, since the Numba version is algorithmically nothing but the Python code, any differences between Python and Numba would be due to Numba's compilation/optimizations:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def err(val, exact=680):\n",
    "    return abs((val-exact)/exact)\n",
    "\n",
    "# This time, let's make empty arrays and fill them in\n",
    "epy = np.empty_like(sizes)\n",
    "enpy = np.empty_like(sizes)\n",
    "enba = np.empty_like(sizes)\n",
    "\n",
    "for i, (x, y) in enumerate(xy):\n",
    "    epy[i] = err(trapzl(y, x))\n",
    "    enpy[i] = err(trapz(y, x))\n",
    "    enba[i] = err(trapzn(y, x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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MHjyYX//618WvK/qanZ1NXl5eudvPycn5xffMn5w5c8av81WFJ9vS+KfP6LjreYwI33X5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\nAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1531021278>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.loglog(sizes, enpy, 'o-', label='Numpy')\n",
    "plt.loglog(sizes, epy, 'o-', label='Python')\n",
    "plt.loglog(sizes, enba, 'o', label='Numba')\n",
    "plt.legend()\n",
    "plt.grid()\n",
    "plt.xlabel(\"Input size\")\n",
    "plt.ylabel(\"Relative error\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Fortunately, we see that Numba produces identical results to Pure Python (good news: its optimizations don't change the results).  Furthermore, while we see a small loss of accuracy relative to Numpy's [high accuracy summation algorithm](https://github.com/numpy/numpy/pull/3685), the error is $\\cal{O}(10^{-14})$, which is still close to floating-point precision."
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}