Herbie.ipynb

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    "#### Subtractive cancellation\n",
    "\n",
    "__MATH 420__<br>\n",
    "_Spring 2021_ <br>\n",
    "\n",
    "Herbie is an online tool (https://herbie.uwplse.org/demo/) that attempts to modify a formula into a formula that is algebraically equivalent to the input, but evaluates more accurately using floating point numbers. Herbie gives its recommendations in a variety of languages, including Julia.\n",
    "\n",
    "To test Herbie, let's try the expression $(1-\\exp(x))/x$. We will specify that $x$ is a binary64 number that is between the least and greatest binary64.\n",
    "\n",
    "When $x \\approx 0$ we have $1 - \\exp(x) \\approx 1 - 1$.  So we expect subtractive cancellation when $x$ is near zero. Actually, we have\n",
    "$$\n",
    "   \\lim_{x \\to 0} \\frac{1 - \\exp(x)}{x} = -1.\n",
    "$$\n",
    "So when we test this code for $x \\approx 0$, the result should be close to $-1$. \n",
    "\n",
    "Here is what Herbie recommends:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "herbie (generic function with 1 method)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
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   ],
   "source": [
    "function herbie(x::Float64)\n",
    "\treturn Float64(expm1(x) / Float64(-x))\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Computing $\\exp(x) - 1$ is such a common need (financial calculations, for example), that microprocessors have a built-in way to evaluate it accurately, even when $x \\approx 0$. In Julia, the function name is `expm1` (for exp minus one). Using `expm1`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "nonherbie (generic function with 1 method)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "function nonherbie(x::Float64)\n",
    "    (1-exp(x))/x\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Gadfly"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "(a,b) = (-1.e-7, 1.0e-7);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "p1 = layer(herbie,color=[colorant\"red\"],a,b);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "p2 = layer(nonherbie,color=[colorant\"white\"],a,b);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For $x$ near zero, here is a graphical comparison:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "tags": []
   },
   "outputs": [
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       "    <g transform=\"translate(95.51,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">1.40×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(95.96,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">1.50×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(96.41,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">1.60×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(96.87,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">1.70×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(97.32,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">1.80×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(97.77,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">1.90×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(98.22,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">2.00×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(98.68,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">2.10×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(99.13,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">2.20×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(99.58,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">2.30×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(100.03,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">2.40×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(100.49,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">2.50×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(100.94,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">2.60×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(101.39,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">2.70×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(101.84,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">2.80×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(102.3,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">2.90×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(102.75,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">3.00×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(103.2,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">3.10×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(103.65,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">3.20×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(104.11,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">3.30×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(104.56,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">3.40×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(105.01,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">3.50×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(105.46,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">3.60×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(105.92,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">3.70×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(106.37,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">3.80×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(106.82,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">3.90×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(107.27,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">4.00×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(107.72,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">4.10×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(108.18,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">4.20×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(108.63,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">4.30×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(109.08,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">4.40×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(109.53,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">4.50×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(109.99,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">4.60×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(110.44,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">4.70×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(110.89,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">4.80×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(111.34,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">4.90×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(111.8,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">5.00×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(112.25,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">5.10×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(112.7,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">5.20×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(113.15,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">5.30×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(113.61,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">5.40×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(114.06,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">5.50×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(114.51,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">5.60×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(114.96,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">5.70×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(115.42,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">5.80×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(115.87,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">5.90×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(116.32,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">6.00×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(116.77,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">6.10×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(117.23,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">6.20×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(117.68,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">6.30×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(118.13,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">6.40×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(118.58,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">6.50×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(119.04,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">6.60×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(119.49,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">6.70×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(119.94,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">6.80×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(120.39,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">6.90×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(120.85,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">7.00×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(121.3,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">7.10×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(121.75,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">7.20×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(122.2,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">7.30×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(122.66,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">7.40×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(123.11,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">7.50×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(123.56,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">7.60×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(124.01,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">7.70×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(124.47,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">7.80×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(124.92,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">7.90×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(125.37,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">8.00×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(125.82,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">8.10×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(126.28,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">8.20×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(126.73,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">8.30×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(127.18,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">8.40×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(127.63,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">8.50×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(128.09,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">8.60×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(128.54,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">8.70×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(128.99,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">8.80×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(129.44,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">8.90×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(129.9,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">9.00×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(130.35,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">9.10×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(130.8,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">9.20×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(131.25,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">9.30×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(131.71,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">9.40×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(132.16,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">9.50×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(132.61,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">9.60×10⁻⁸</text>\n",
       "      </g>\n",
       "    </g>\n",
       "    <g transform=\"translate(133.06,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">9.70×10⁻⁸</text>\n",
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       "    <g transform=\"translate(133.52,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">9.80×10⁻⁸</text>\n",
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       "    <g transform=\"translate(133.97,84.39)\" gadfly:scale=\"50.0\" visibility=\"hidden\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">9.90×10⁻⁸</text>\n",
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       "      <g class=\"primitive\">\n",
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       "        <text text-anchor=\"middle\" dy=\"-0em\">-1.0×10⁻⁷</text>\n",
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       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"-0em\">1.0×10⁻⁷</text>\n",
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       "f:e,unbind:b(this.node||document,d,e,f||this)}));return this};a[\"un\"+d]=A[\"un\"+d]=function(a){for(var b=this.events||[],e=b.length;e--;)if(b[e].name==d&&(b[e].f==a||!a)){b[e].unbind();b.splice(e,1);!b.length&&delete this.events;break}return this}})(v[y]);A.hover=function(a,b,d,e){return this.mouseover(a,d).mouseout(b,e||d)};A.unhover=function(a,b){return this.unmouseover(a).unmouseout(b)};var r=[];A.drag=function(b,d,f,h,n,p){function u(r,v,w){(r.originalEvent||r).preventDefault();this._drag.x=v;\n",
       "this._drag.y=w;this._drag.id=r.identifier;!q.length&&a.mousemove(e).mouseup(l);q.push({el:this,move_scope:h,start_scope:n,end_scope:p});d&&k.on(\"snap.drag.start.\"+this.id,d);b&&k.on(\"snap.drag.move.\"+this.id,b);f&&k.on(\"snap.drag.end.\"+this.id,f);k(\"snap.drag.start.\"+this.id,n||h||this,v,w,r)}if(!arguments.length){var v;return this.drag(function(a,b){this.attr({transform:v+(v?\"T\":\"t\")+[a,b]})},function(){v=this.transform().local})}this._drag={};r.push({el:this,start:u});this.mousedown(u);return this};\n",
       "A.undrag=function(){for(var b=r.length;b--;)r[b].el==this&&(this.unmousedown(r[b].start),r.splice(b,1),k.unbind(\"snap.drag.*.\"+this.id));!r.length&&a.unmousemove(e).unmouseup(l);return this}});C.plugin(function(a,v,y,C){y=y.prototype;var A=/^\\s*url\\((.+)\\)/,w=String,z=a._.$;a.filter={};y.filter=function(d){var f=this;\"svg\"!=f.type&&(f=f.paper);d=a.parse(w(d));var k=a._.id(),u=z(\"filter\");z(u,{id:k,filterUnits:\"userSpaceOnUse\"});u.appendChild(d.node);f.defs.appendChild(u);return new v(u)};k.on(\"snap.util.getattr.filter\",\n",
       "function(){k.stop();var d=z(this.node,\"filter\");if(d)return(d=w(d).match(A))&&a.select(d[1])});k.on(\"snap.util.attr.filter\",function(d){if(d instanceof v&&\"filter\"==d.type){k.stop();var f=d.node.id;f||(z(d.node,{id:d.id}),f=d.id);z(this.node,{filter:a.url(f)})}d&&\"none\"!=d||(k.stop(),this.node.removeAttribute(\"filter\"))});a.filter.blur=function(d,f){null==d&&(d=2);return a.format('<feGaussianBlur stdDeviation=\"{def}\"/>',{def:null==f?d:[d,f]})};a.filter.blur.toString=function(){return this()};a.filter.shadow=\n",
       "function(d,f,k,u,p){\"string\"==typeof k&&(p=u=k,k=4);\"string\"!=typeof u&&(p=u,u=\"#000\");null==k&&(k=4);null==p&&(p=1);null==d&&(d=0,f=2);null==f&&(f=d);u=a.color(u||\"#000\");return a.format('<feGaussianBlur in=\"SourceAlpha\" stdDeviation=\"{blur}\"/><feOffset dx=\"{dx}\" dy=\"{dy}\" result=\"offsetblur\"/><feFlood flood-color=\"{color}\"/><feComposite in2=\"offsetblur\" operator=\"in\"/><feComponentTransfer><feFuncA type=\"linear\" slope=\"{opacity}\"/></feComponentTransfer><feMerge><feMergeNode/><feMergeNode in=\"SourceGraphic\"/></feMerge>',\n",
       "{color:u,dx:d,dy:f,blur:k,opacity:p})};a.filter.shadow.toString=function(){return this()};a.filter.grayscale=function(d){null==d&&(d=1);return a.format('<feColorMatrix type=\"matrix\" values=\"{a} {b} {c} 0 0 {d} {e} {f} 0 0 {g} {b} {h} 0 0 0 0 0 1 0\"/>',{a:0.2126+0.7874*(1-d),b:0.7152-0.7152*(1-d),c:0.0722-0.0722*(1-d),d:0.2126-0.2126*(1-d),e:0.7152+0.2848*(1-d),f:0.0722-0.0722*(1-d),g:0.2126-0.2126*(1-d),h:0.0722+0.9278*(1-d)})};a.filter.grayscale.toString=function(){return this()};a.filter.sepia=\n",
       "function(d){null==d&&(d=1);return a.format('<feColorMatrix type=\"matrix\" values=\"{a} {b} {c} 0 0 {d} {e} {f} 0 0 {g} {h} {i} 0 0 0 0 0 1 0\"/>',{a:0.393+0.607*(1-d),b:0.769-0.769*(1-d),c:0.189-0.189*(1-d),d:0.349-0.349*(1-d),e:0.686+0.314*(1-d),f:0.168-0.168*(1-d),g:0.272-0.272*(1-d),h:0.534-0.534*(1-d),i:0.131+0.869*(1-d)})};a.filter.sepia.toString=function(){return this()};a.filter.saturate=function(d){null==d&&(d=1);return a.format('<feColorMatrix type=\"saturate\" values=\"{amount}\"/>',{amount:1-\n",
       "d})};a.filter.saturate.toString=function(){return this()};a.filter.hueRotate=function(d){return a.format('<feColorMatrix type=\"hueRotate\" values=\"{angle}\"/>',{angle:d||0})};a.filter.hueRotate.toString=function(){return this()};a.filter.invert=function(d){null==d&&(d=1);return a.format('<feComponentTransfer><feFuncR type=\"table\" tableValues=\"{amount} {amount2}\"/><feFuncG type=\"table\" tableValues=\"{amount} {amount2}\"/><feFuncB type=\"table\" tableValues=\"{amount} {amount2}\"/></feComponentTransfer>',{amount:d,\n",
       "amount2:1-d})};a.filter.invert.toString=function(){return this()};a.filter.brightness=function(d){null==d&&(d=1);return a.format('<feComponentTransfer><feFuncR type=\"linear\" slope=\"{amount}\"/><feFuncG type=\"linear\" slope=\"{amount}\"/><feFuncB type=\"linear\" slope=\"{amount}\"/></feComponentTransfer>',{amount:d})};a.filter.brightness.toString=function(){return this()};a.filter.contrast=function(d){null==d&&(d=1);return a.format('<feComponentTransfer><feFuncR type=\"linear\" slope=\"{amount}\" intercept=\"{amount2}\"/><feFuncG type=\"linear\" slope=\"{amount}\" intercept=\"{amount2}\"/><feFuncB type=\"linear\" slope=\"{amount}\" intercept=\"{amount2}\"/></feComponentTransfer>',\n",
       "{amount:d,amount2:0.5-d/2})};a.filter.contrast.toString=function(){return this()}});return C});\n",
       "\n",
       "]]> </script>\n",
       "<script> <![CDATA[\n",
       "\n",
       "(function (glob, factory) {\n",
       "    // AMD support\n",
       "    if (typeof define === \"function\" && define.amd) {\n",
       "        // Define as an anonymous module\n",
       "        define(\"Gadfly\", [\"Snap.svg\"], function (Snap) {\n",
       "            return factory(Snap);\n",
       "        });\n",
       "    } else {\n",
       "        // Browser globals (glob is window)\n",
       "        // Snap adds itself to window\n",
       "        glob.Gadfly = factory(glob.Snap);\n",
       "    }\n",
       "}(this, function (Snap) {\n",
       "\n",
       "var Gadfly = {};\n",
       "\n",
       "// Get an x/y coordinate value in pixels\n",
       "var xPX = function(fig, x) {\n",
       "    var client_box = fig.node.getBoundingClientRect();\n",
       "    return x * fig.node.viewBox.baseVal.width / client_box.width;\n",
       "};\n",
       "\n",
       "var yPX = function(fig, y) {\n",
       "    var client_box = fig.node.getBoundingClientRect();\n",
       "    return y * fig.node.viewBox.baseVal.height / client_box.height;\n",
       "};\n",
       "\n",
       "\n",
       "Snap.plugin(function (Snap, Element, Paper, global) {\n",
       "    // Traverse upwards from a snap element to find and return the first\n",
       "    // note with the \"plotroot\" class.\n",
       "    Element.prototype.plotroot = function () {\n",
       "        var element = this;\n",
       "        while (!element.hasClass(\"plotroot\") && element.parent() != null) {\n",
       "            element = element.parent();\n",
       "        }\n",
       "        return element;\n",
       "    };\n",
       "\n",
       "    Element.prototype.svgroot = function () {\n",
       "        var element = this;\n",
       "        while (element.node.nodeName != \"svg\" && element.parent() != null) {\n",
       "            element = element.parent();\n",
       "        }\n",
       "        return element;\n",
       "    };\n",
       "\n",
       "    Element.prototype.plotbounds = function () {\n",
       "        var root = this.plotroot()\n",
       "        var bbox = root.select(\".guide.background\").node.getBBox();\n",
       "        return {\n",
       "            x0: bbox.x,\n",
       "            x1: bbox.x + bbox.width,\n",
       "            y0: bbox.y,\n",
       "            y1: bbox.y + bbox.height\n",
       "        };\n",
       "    };\n",
       "\n",
       "    Element.prototype.viewportplotbounds = function () {\n",
       "        var root = this.svgroot();\n",
       "        var bbox = root.node.getBoundingClientRect();\n",
       "        return {\n",
       "            x0: bbox.x,\n",
       "            x1: bbox.x + bbox.width,\n",
       "            y0: bbox.y,\n",
       "            y1: bbox.y + bbox.height\n",
       "        };\n",
       "    };\n",
       "\n",
       "    Element.prototype.plotcenter = function () {\n",
       "        var root = this.plotroot()\n",
       "        var bbox = root.select(\".guide.background\").node.getBBox();\n",
       "        return {\n",
       "            x: bbox.x + bbox.width / 2,\n",
       "            y: bbox.y + bbox.height / 2\n",
       "        };\n",
       "    };\n",
       "\n",
       "    // Emulate IE style mouseenter/mouseleave events, since Microsoft always\n",
       "    // does everything right.\n",
       "    // See: http://www.dynamic-tools.net/toolbox/isMouseLeaveOrEnter/\n",
       "    var events = [\"mouseenter\", \"mouseleave\"];\n",
       "\n",
       "    for (i in events) {\n",
       "        (function (event_name) {\n",
       "            var event_name = events[i];\n",
       "            Element.prototype[event_name] = function (fn, scope) {\n",
       "                if (Snap.is(fn, \"function\")) {\n",
       "                    var fn2 = function (event) {\n",
       "                        if (event.type != \"mouseover\" && event.type != \"mouseout\") {\n",
       "                            return;\n",
       "                        }\n",
       "\n",
       "                        var reltg = event.relatedTarget ? event.relatedTarget :\n",
       "                            event.type == \"mouseout\" ? event.toElement : event.fromElement;\n",
       "                        while (reltg && reltg != this.node) reltg = reltg.parentNode;\n",
       "\n",
       "                        if (reltg != this.node) {\n",
       "                            return fn.apply(this, event);\n",
       "                        }\n",
       "                    };\n",
       "\n",
       "                    if (event_name == \"mouseenter\") {\n",
       "                        this.mouseover(fn2, scope);\n",
       "                    } else {\n",
       "                        this.mouseout(fn2, scope);\n",
       "                    }\n",
       "                }\n",
       "                return this;\n",
       "            };\n",
       "        })(events[i]);\n",
       "    }\n",
       "\n",
       "\n",
       "    Element.prototype.mousewheel = function (fn, scope) {\n",
       "        if (Snap.is(fn, \"function\")) {\n",
       "            var el = this;\n",
       "            var fn2 = function (event) {\n",
       "                fn.apply(el, [event]);\n",
       "            };\n",
       "        }\n",
       "\n",
       "        this.node.addEventListener(\"wheel\", fn2);\n",
       "\n",
       "        return this;\n",
       "    };\n",
       "\n",
       "\n",
       "    // Snap's attr function can be too slow for things like panning/zooming.\n",
       "    // This is a function to directly update element attributes without going\n",
       "    // through eve.\n",
       "    Element.prototype.attribute = function(key, val) {\n",
       "        if (val === undefined) {\n",
       "            return this.node.getAttribute(key);\n",
       "        } else {\n",
       "            this.node.setAttribute(key, val);\n",
       "            return this;\n",
       "        }\n",
       "    };\n",
       "\n",
       "    Element.prototype.init_gadfly = function() {\n",
       "        this.mouseenter(Gadfly.plot_mouseover)\n",
       "            .mousemove(Gadfly.plot_mousemove)\n",
       "            .mouseleave(Gadfly.plot_mouseout)\n",
       "            .dblclick(Gadfly.plot_dblclick)\n",
       "            .mousewheel(Gadfly.guide_background_scroll)\n",
       "            .drag(Gadfly.guide_background_drag_onmove,\n",
       "                  Gadfly.guide_background_drag_onstart,\n",
       "                  Gadfly.guide_background_drag_onend);\n",
       "        this.mouseenter(function (event) {\n",
       "            init_pan_zoom(this.plotroot());\n",
       "        });\n",
       "        return this;\n",
       "    };\n",
       "});\n",
       "\n",
       "\n",
       "Gadfly.plot_mousemove = function(event, _x_px, _y_px) {\n",
       "    var root = this.plotroot();\n",
       "    var viewbounds = root.viewportplotbounds();\n",
       "\n",
       "    // (_x_px, _y_px) are offsets relative to page (event.layerX, event.layerY) rather than viewport\n",
       "    var x_px = event.clientX - viewbounds.x0;\n",
       "    var y_px = event.clientY - viewbounds.y0;\n",
       "    if (root.data(\"crosshair\")) {\n",
       "        px_per_mm = root.data(\"px_per_mm\");\n",
       "        bB = root.select('boundingbox').node.getAttribute('value').split(' ');\n",
       "        uB = root.select('unitbox').node.getAttribute('value').split(' ');\n",
       "        xscale = root.data(\"xscale\");\n",
       "        yscale = root.data(\"yscale\");\n",
       "        xtranslate = root.data(\"tx\");\n",
       "        ytranslate = root.data(\"ty\");\n",
       "\n",
       "        xisdate = uB[0].includes(\"T\");\n",
       "        yisdate = uB[1].includes(\"T\");\n",
       "        tzoffset = (new Date()).getTimezoneOffset() * 60000;\n",
       "\n",
       "        xoff_mm = bB[0].substr(0,bB[0].length-2)/1;\n",
       "        yoff_mm = bB[1].substr(0,bB[1].length-2)/1;\n",
       "        xoff_unit = xisdate ? Date.parse(uB[0]) : uB[0]/1;\n",
       "        yoff_unit = yisdate ? Date.parse(uB[1]) : uB[1]/1;\n",
       "        mm_per_xunit = bB[2].substr(0,bB[2].length-2) / uB[2];\n",
       "        mm_per_yunit = bB[3].substr(0,bB[3].length-2) / uB[3+xisdate];\n",
       "         \n",
       "        x_unit = ((x_px / px_per_mm - xtranslate) / xscale - xoff_mm) / mm_per_xunit + xoff_unit;\n",
       "        y_unit = ((y_px / px_per_mm - ytranslate) / yscale - yoff_mm) / mm_per_yunit + yoff_unit;\n",
       "\n",
       "        root.select('.crosshair').select('.primitive').select('text').node.innerHTML =\n",
       "                (xisdate ? new Date(Math.round(x_unit)-tzoffset).toISOString().slice(0,-1) : x_unit.toPrecision(3))\n",
       "                +\",\"+\n",
       "                (yisdate ? new Date(Math.round(y_unit)-tzoffset).toISOString().slice(0,-1) : y_unit.toPrecision(3));\n",
       "    };\n",
       "};\n",
       "\n",
       "Gadfly.helpscreen_visible = function(event) {\n",
       "    helpscreen_visible(this.plotroot());\n",
       "};\n",
       "var helpscreen_visible = function(root) {\n",
       "    root.select(\".helpscreen\").attr(\"visibility\", \"visible\");\n",
       "};\n",
       "\n",
       "Gadfly.helpscreen_hidden = function(event) {\n",
       "    helpscreen_hidden(this.plotroot());\n",
       "};\n",
       "var helpscreen_hidden = function(root) {\n",
       "    root.select(\".helpscreen\").attr(\"visibility\", \"hidden\");\n",
       "};\n",
       "\n",
       "Gadfly.datatip_visible = function(event) {\n",
       "    var root = this.plotroot();\n",
       "    var ilabel = this.data(\"ilabel\");\n",
       "    var element = root.select('[ilabel=\"'+ilabel+'\"]');\n",
       "    element.animate({\"fill-opacity\": 1.0}, 250);\n",
       "};\n",
       "\n",
       "Gadfly.datatip_hidden = function(event) {\n",
       "    var root = this.plotroot();\n",
       "    var ilabel = this.data(\"ilabel\");\n",
       "    var element = root.select('[ilabel=\"'+ilabel+'\"]');\n",
       "    element.animate({\"fill-opacity\": 0.0}, 250);\n",
       "};\n",
       "\n",
       "// When the plot is moused over, emphasize the grid lines.\n",
       "Gadfly.plot_mouseover = function(event) {\n",
       "    var root = this.plotroot();\n",
       "\n",
       "    var keyboard_help = function(event) {\n",
       "        if (event.which == 191) { // ?\n",
       "            helpscreen_visible(root);\n",
       "        }\n",
       "    };\n",
       "    root.data(\"keyboard_help\", keyboard_help);\n",
       "    window.addEventListener(\"keydown\", keyboard_help);\n",
       "\n",
       "    var keyboard_pan_zoom = function(event) {\n",
       "        var bounds = root.plotbounds(),\n",
       "            width = bounds.x1 - bounds.x0;\n",
       "            height = bounds.y1 - bounds.y0;\n",
       "        if (event.which == 187 || event.which == 73) { // plus or i\n",
       "            increase_zoom_by_position(root, 0.1, true);\n",
       "        } else if (event.which == 189 || event.which == 79) { // minus or o\n",
       "            increase_zoom_by_position(root, -0.1, true);\n",
       "        } else if (event.which == 39 || event.which == 76) { // right-arrow or l\n",
       "            set_plot_pan_zoom(root, root.data(\"tx\")-width/10, root.data(\"ty\"),\n",
       "                    root.data(\"xscale\"), root.data(\"yscale\"));\n",
       "        } else if (event.which == 40 || event.which == 74) { // down-arrow or j\n",
       "            set_plot_pan_zoom(root, root.data(\"tx\"), root.data(\"ty\")-height/10,\n",
       "                    root.data(\"xscale\"), root.data(\"yscale\"));\n",
       "        } else if (event.which == 37 || event.which == 72) { // left-arrow or h\n",
       "            set_plot_pan_zoom(root, root.data(\"tx\")+width/10, root.data(\"ty\"),\n",
       "                    root.data(\"xscale\"), root.data(\"yscale\"));\n",
       "        } else if (event.which == 38 || event.which == 75) { // up-arrow or k\n",
       "            set_plot_pan_zoom(root, root.data(\"tx\"), root.data(\"ty\")+height/10,\n",
       "                    root.data(\"xscale\"), root.data(\"yscale\"));\n",
       "        } else if (event.which == 82) { // r\n",
       "            set_plot_pan_zoom(root, 0.0, 0.0, 1.0, 1.0);\n",
       "        } else if (event.which == 191) { // ?\n",
       "            helpscreen_hidden(root);\n",
       "        } else if (event.which == 67) { // c\n",
       "            root.data(\"crosshair\",!root.data(\"crosshair\"));\n",
       "            root.select(\".crosshair\")\n",
       "                .animate({\"fill-opacity\": root.data(\"crosshair\") ? 1.0 : 0.0}, 250);\n",
       "        }\n",
       "    };\n",
       "    root.data(\"keyboard_pan_zoom\", keyboard_pan_zoom);\n",
       "    window.addEventListener(\"keyup\", keyboard_pan_zoom);\n",
       "\n",
       "    var xgridlines = root.select(\".xgridlines\"),\n",
       "        ygridlines = root.select(\".ygridlines\");\n",
       "\n",
       "    if (xgridlines) {\n",
       "        xgridlines.data(\"unfocused_strokedash\",\n",
       "                        xgridlines.attribute(\"stroke-dasharray\").replace(/(\\d)(,|$)/g, \"$1mm$2\"));\n",
       "        var destcolor = root.data(\"focused_xgrid_color\");\n",
       "        xgridlines.attribute(\"stroke-dasharray\", \"none\")\n",
       "                  .selectAll(\"path\")\n",
       "                  .animate({stroke: destcolor}, 250);\n",
       "    }\n",
       "\n",
       "    if (ygridlines) {\n",
       "        ygridlines.data(\"unfocused_strokedash\",\n",
       "                        ygridlines.attribute(\"stroke-dasharray\").replace(/(\\d)(,|$)/g, \"$1mm$2\"));\n",
       "        var destcolor = root.data(\"focused_ygrid_color\");\n",
       "        ygridlines.attribute(\"stroke-dasharray\", \"none\")\n",
       "                  .selectAll(\"path\")\n",
       "                  .animate({stroke: destcolor}, 250);\n",
       "    }\n",
       "\n",
       "    root.select(\".crosshair\")\n",
       "        .animate({\"fill-opacity\": root.data(\"crosshair\") ? 1.0 : 0.0}, 250);\n",
       "    root.select(\".questionmark\").animate({\"fill-opacity\": 1.0}, 250);\n",
       "};\n",
       "\n",
       "// Reset pan and zoom on double click\n",
       "Gadfly.plot_dblclick = function(event) {\n",
       "  set_plot_pan_zoom(this.plotroot(), 0.0, 0.0, 1.0, 1.0);\n",
       "};\n",
       "\n",
       "// Unemphasize grid lines on mouse out.\n",
       "Gadfly.plot_mouseout = function(event) {\n",
       "    var root = this.plotroot();\n",
       "\n",
       "    window.removeEventListener(\"keyup\", root.data(\"keyboard_pan_zoom\"));\n",
       "    root.data(\"keyboard_pan_zoom\", undefined);\n",
       "    window.removeEventListener(\"keydown\", root.data(\"keyboard_help\"));\n",
       "    root.data(\"keyboard_help\", undefined);\n",
       "\n",
       "    var xgridlines = root.select(\".xgridlines\"),\n",
       "        ygridlines = root.select(\".ygridlines\");\n",
       "\n",
       "    if (xgridlines) {\n",
       "        var destcolor = root.data(\"unfocused_xgrid_color\");\n",
       "        xgridlines.attribute(\"stroke-dasharray\", xgridlines.data(\"unfocused_strokedash\"))\n",
       "                  .selectAll(\"path\")\n",
       "                  .animate({stroke: destcolor}, 250);\n",
       "    }\n",
       "\n",
       "    if (ygridlines) {\n",
       "        var destcolor = root.data(\"unfocused_ygrid_color\");\n",
       "        ygridlines.attribute(\"stroke-dasharray\", ygridlines.data(\"unfocused_strokedash\"))\n",
       "                  .selectAll(\"path\")\n",
       "                  .animate({stroke: destcolor}, 250);\n",
       "    }\n",
       "\n",
       "    root.select(\".crosshair\").animate({\"fill-opacity\": 0.0}, 250);\n",
       "    root.select(\".questionmark\").animate({\"fill-opacity\": 0.0}, 250);\n",
       "    helpscreen_hidden(root);\n",
       "};\n",
       "\n",
       "\n",
       "var set_geometry_transform = function(root, tx, ty, xscale, yscale) {\n",
       "    var xscalable = root.hasClass(\"xscalable\"),\n",
       "        yscalable = root.hasClass(\"yscalable\");\n",
       "\n",
       "    var old_xscale = root.data(\"xscale\"),\n",
       "        old_yscale = root.data(\"yscale\");\n",
       "\n",
       "    var xscale = xscalable ? xscale : 1.0,\n",
       "        yscale = yscalable ? yscale : 1.0;\n",
       "\n",
       "    tx = xscalable ? tx : 0.0;\n",
       "    ty = yscalable ? ty : 0.0;\n",
       "\n",
       "    var t = new Snap.Matrix().translate(tx, ty).scale(xscale, yscale);\n",
       "    root.selectAll(\".geometry, image\").forEach(function (element, i) {\n",
       "            element.transform(t);\n",
       "        });\n",
       "\n",
       "    var t = new Snap.Matrix().scale(1.0/xscale, 1.0/yscale);\n",
       "    root.selectAll('.marker, .datatip').forEach(function (element, i) {\n",
       "        element.selectAll('.primitive').forEach(function (element, i) {\n",
       "            element.transform(t);\n",
       "        }) });\n",
       "\n",
       "    bounds = root.plotbounds();\n",
       "    px_per_mm = root.data(\"px_per_mm\");\n",
       "\n",
       "    if (yscalable) {\n",
       "        var xfixed_t = new Snap.Matrix().translate(0, ty).scale(1.0, yscale);\n",
       "        root.selectAll(\".xfixed\")\n",
       "            .forEach(function (element, i) {\n",
       "                element.transform(xfixed_t);\n",
       "            });\n",
       "\n",
       "        ylabels = root.select(\".ylabels\");\n",
       "        if (ylabels) {\n",
       "            ylabels.transform(xfixed_t)\n",
       "                   .selectAll(\"g\")\n",
       "                   .forEach(function (element, i) {\n",
       "                       if (element.attribute(\"gadfly:inscale\") == \"true\") {\n",
       "                           unscale_t = new Snap.Matrix();\n",
       "                           unscale_t.scale(1, 1/yscale);\n",
       "                           element.select(\"text\").transform(unscale_t);\n",
       " \n",
       "                           var y = element.attr(\"transform\").globalMatrix.f / px_per_mm;\n",
       "                           element.attr(\"visibility\",\n",
       "                               bounds.y0 <= y && y <= bounds.y1 ? \"visible\" : \"hidden\");\n",
       "                       }\n",
       "                   });\n",
       "        }\n",
       "    }\n",
       "\n",
       "    if (xscalable) {\n",
       "        var yfixed_t = new Snap.Matrix().translate(tx, 0).scale(xscale, 1.0);\n",
       "        var xtrans = new Snap.Matrix().translate(tx, 0);\n",
       "        root.selectAll(\".yfixed\")\n",
       "            .forEach(function (element, i) {\n",
       "                element.transform(yfixed_t);\n",
       "            });\n",
       "\n",
       "        xlabels = root.select(\".xlabels\");\n",
       "        if (xlabels) {\n",
       "            xlabels.transform(yfixed_t)\n",
       "                   .selectAll(\"g\")\n",
       "                   .forEach(function (element, i) {\n",
       "                       if (element.attribute(\"gadfly:inscale\") == \"true\") {\n",
       "                           unscale_t = new Snap.Matrix();\n",
       "                           unscale_t.scale(1/xscale, 1);\n",
       "                           element.select(\"text\").transform(unscale_t);\n",
       "\n",
       "                           var x = element.attr(\"transform\").globalMatrix.e / px_per_mm;\n",
       "                           element.attr(\"visibility\",\n",
       "                               bounds.x0 <= x && x <= bounds.x1 ? \"visible\" : \"hidden\");\n",
       "                           }\n",
       "                   });\n",
       "        }\n",
       "    }\n",
       "};\n",
       "\n",
       "\n",
       "// Find the most appropriate tick scale and update label visibility.\n",
       "var update_tickscale = function(root, scale, axis) {\n",
       "    if (!root.hasClass(axis + \"scalable\")) return;\n",
       "\n",
       "    var tickscales = root.data(axis + \"tickscales\");\n",
       "    var best_tickscale = 1.0;\n",
       "    var best_tickscale_dist = Infinity;\n",
       "    for (tickscale in tickscales) {\n",
       "        var dist = Math.abs(Math.log(tickscale) - Math.log(scale));\n",
       "        if (dist < best_tickscale_dist) {\n",
       "            best_tickscale_dist = dist;\n",
       "            best_tickscale = tickscale;\n",
       "        }\n",
       "    }\n",
       "\n",
       "    if (best_tickscale != root.data(axis + \"tickscale\")) {\n",
       "        root.data(axis + \"tickscale\", best_tickscale);\n",
       "        var mark_inscale_gridlines = function (element, i) {\n",
       "            if (element.attribute(\"gadfly:inscale\") == null) { return; }\n",
       "            var inscale = element.attr(\"gadfly:scale\") == best_tickscale;\n",
       "            element.attribute(\"gadfly:inscale\", inscale);\n",
       "            element.attr(\"visibility\", inscale ? \"visible\" : \"hidden\");\n",
       "        };\n",
       "\n",
       "        var mark_inscale_labels = function (element, i) {\n",
       "            if (element.attribute(\"gadfly:inscale\") == null) { return; }\n",
       "            var inscale = element.attr(\"gadfly:scale\") == best_tickscale;\n",
       "            element.attribute(\"gadfly:inscale\", inscale);\n",
       "            element.attr(\"visibility\", inscale ? \"visible\" : \"hidden\");\n",
       "        };\n",
       "\n",
       "        root.select(\".\" + axis + \"gridlines\").selectAll(\"g\").forEach(mark_inscale_gridlines);\n",
       "        root.select(\".\" + axis + \"labels\").selectAll(\"g\").forEach(mark_inscale_labels);\n",
       "    }\n",
       "};\n",
       "\n",
       "\n",
       "var set_plot_pan_zoom = function(root, tx, ty, xscale, yscale) {\n",
       "    var old_xscale = root.data(\"xscale\"),\n",
       "        old_yscale = root.data(\"yscale\");\n",
       "    var bounds = root.plotbounds();\n",
       "\n",
       "    var width = bounds.x1 - bounds.x0,\n",
       "        height = bounds.y1 - bounds.y0;\n",
       "\n",
       "    // compute the viewport derived from tx, ty, xscale, and yscale\n",
       "    var x_min = -width * xscale - (xscale * width - width),\n",
       "        x_max = width * xscale,\n",
       "        y_min = -height * yscale - (yscale * height - height),\n",
       "        y_max = height * yscale;\n",
       "\n",
       "    var x0 = bounds.x0 - xscale * bounds.x0,\n",
       "        y0 = bounds.y0 - yscale * bounds.y0;\n",
       "\n",
       "    var tx = Math.max(Math.min(tx - x0, x_max), x_min),\n",
       "        ty = Math.max(Math.min(ty - y0, y_max), y_min);\n",
       "\n",
       "    tx += x0;\n",
       "    ty += y0;\n",
       "\n",
       "    // when the scale changes, we may need to alter which set of\n",
       "    // ticks are being displayed\n",
       "    if (xscale != old_xscale) {\n",
       "        update_tickscale(root, xscale, \"x\");\n",
       "    }\n",
       "    if (yscale != old_yscale) {\n",
       "        update_tickscale(root, yscale, \"y\");\n",
       "    }\n",
       "\n",
       "    set_geometry_transform(root, tx, ty, xscale, yscale);\n",
       "\n",
       "    root.data(\"xscale\", xscale);\n",
       "    root.data(\"yscale\", yscale);\n",
       "    root.data(\"tx\", tx);\n",
       "    root.data(\"ty\", ty);\n",
       "};\n",
       "\n",
       "\n",
       "var scale_centered_translation = function(root, xscale, yscale) {\n",
       "    var bounds = root.plotbounds();\n",
       "\n",
       "    var width = bounds.x1 - bounds.x0,\n",
       "        height = bounds.y1 - bounds.y0;\n",
       "\n",
       "    var tx0 = root.data(\"tx\"),\n",
       "        ty0 = root.data(\"ty\");\n",
       "\n",
       "    var xscale0 = root.data(\"xscale\"),\n",
       "        yscale0 = root.data(\"yscale\");\n",
       "\n",
       "    // how off from center the current view is\n",
       "    var xoff = tx0 - (bounds.x0 * (1 - xscale0) + (width * (1 - xscale0)) / 2),\n",
       "        yoff = ty0 - (bounds.y0 * (1 - yscale0) + (height * (1 - yscale0)) / 2);\n",
       "\n",
       "    // rescale offsets\n",
       "    xoff = xoff * xscale / xscale0;\n",
       "    yoff = yoff * yscale / yscale0;\n",
       "\n",
       "    // adjust for the panel position being scaled\n",
       "    var x_edge_adjust = bounds.x0 * (1 - xscale),\n",
       "        y_edge_adjust = bounds.y0 * (1 - yscale);\n",
       "\n",
       "    return {\n",
       "        x: xoff + x_edge_adjust + (width - width * xscale) / 2,\n",
       "        y: yoff + y_edge_adjust + (height - height * yscale) / 2\n",
       "    };\n",
       "};\n",
       "\n",
       "\n",
       "// Initialize data for panning zooming if it isn't already.\n",
       "var init_pan_zoom = function(root) {\n",
       "    if (root.data(\"zoompan-ready\")) {\n",
       "        return;\n",
       "    }\n",
       "\n",
       "    root.data(\"crosshair\",false);\n",
       "\n",
       "    // The non-scaling-stroke trick. Rather than try to correct for the\n",
       "    // stroke-width when zooming, we force it to a fixed value.\n",
       "    var px_per_mm = root.node.getCTM().a;\n",
       "\n",
       "    // Drag events report deltas in pixels, which we'd like to convert to\n",
       "    // millimeters.\n",
       "    root.data(\"px_per_mm\", px_per_mm);\n",
       "\n",
       "    root.selectAll(\"path\")\n",
       "        .forEach(function (element, i) {\n",
       "        sw = element.asPX(\"stroke-width\") * px_per_mm;\n",
       "        if (sw > 0) {\n",
       "            element.attribute(\"stroke-width\", sw);\n",
       "            element.attribute(\"vector-effect\", \"non-scaling-stroke\");\n",
       "        }\n",
       "    });\n",
       "\n",
       "    // Store ticks labels original transformation\n",
       "    root.selectAll(\".xlabels > g, .ylabels > g\")\n",
       "        .forEach(function (element, i) {\n",
       "            var lm = element.transform().localMatrix;\n",
       "            element.data(\"static_transform\",\n",
       "                new Snap.Matrix(lm.a, lm.b, lm.c, lm.d, lm.e, lm.f));\n",
       "        });\n",
       "\n",
       "    var xgridlines = root.select(\".xgridlines\");\n",
       "    var ygridlines = root.select(\".ygridlines\");\n",
       "    var xlabels = root.select(\".xlabels\");\n",
       "    var ylabels = root.select(\".ylabels\");\n",
       "\n",
       "    if (root.data(\"tx\") === undefined) root.data(\"tx\", 0);\n",
       "    if (root.data(\"ty\") === undefined) root.data(\"ty\", 0);\n",
       "    if (root.data(\"xscale\") === undefined) root.data(\"xscale\", 1.0);\n",
       "    if (root.data(\"yscale\") === undefined) root.data(\"yscale\", 1.0);\n",
       "    if (root.data(\"xtickscales\") === undefined) {\n",
       "\n",
       "        // index all the tick scales that are listed\n",
       "        var xtickscales = {};\n",
       "        var ytickscales = {};\n",
       "        var add_x_tick_scales = function (element, i) {\n",
       "            if (element.attribute(\"gadfly:scale\")==null) { return; }\n",
       "            xtickscales[element.attribute(\"gadfly:scale\")] = true;\n",
       "        };\n",
       "        var add_y_tick_scales = function (element, i) {\n",
       "            if (element.attribute(\"gadfly:scale\")==null) { return; }\n",
       "            ytickscales[element.attribute(\"gadfly:scale\")] = true;\n",
       "        };\n",
       "\n",
       "        if (xgridlines) xgridlines.selectAll(\"g\").forEach(add_x_tick_scales);\n",
       "        if (ygridlines) ygridlines.selectAll(\"g\").forEach(add_y_tick_scales);\n",
       "        if (xlabels) xlabels.selectAll(\"g\").forEach(add_x_tick_scales);\n",
       "        if (ylabels) ylabels.selectAll(\"g\").forEach(add_y_tick_scales);\n",
       "\n",
       "        root.data(\"xtickscales\", xtickscales);\n",
       "        root.data(\"ytickscales\", ytickscales);\n",
       "        root.data(\"xtickscale\", 1.0);\n",
       "        root.data(\"ytickscale\", 1.0);  // ???\n",
       "    }\n",
       "\n",
       "    var min_scale = 1.0, max_scale = 1.0;\n",
       "    for (scale in xtickscales) {\n",
       "        min_scale = Math.min(min_scale, scale);\n",
       "        max_scale = Math.max(max_scale, scale);\n",
       "    }\n",
       "    for (scale in ytickscales) {\n",
       "        min_scale = Math.min(min_scale, scale);\n",
       "        max_scale = Math.max(max_scale, scale);\n",
       "    }\n",
       "    root.data(\"min_scale\", min_scale);\n",
       "    root.data(\"max_scale\", max_scale);\n",
       "\n",
       "    // store the original positions of labels\n",
       "    if (xlabels) {\n",
       "        xlabels.selectAll(\"g\")\n",
       "               .forEach(function (element, i) {\n",
       "                   element.data(\"x\", element.asPX(\"x\"));\n",
       "               });\n",
       "    }\n",
       "\n",
       "    if (ylabels) {\n",
       "        ylabels.selectAll(\"g\")\n",
       "               .forEach(function (element, i) {\n",
       "                   element.data(\"y\", element.asPX(\"y\"));\n",
       "               });\n",
       "    }\n",
       "\n",
       "    // mark grid lines and ticks as in or out of scale.\n",
       "    var mark_inscale = function (element, i) {\n",
       "        if (element.attribute(\"gadfly:scale\") == null) { return; }\n",
       "        element.attribute(\"gadfly:inscale\", element.attribute(\"gadfly:scale\") == 1.0);\n",
       "    };\n",
       "\n",
       "    if (xgridlines) xgridlines.selectAll(\"g\").forEach(mark_inscale);\n",
       "    if (ygridlines) ygridlines.selectAll(\"g\").forEach(mark_inscale);\n",
       "    if (xlabels) xlabels.selectAll(\"g\").forEach(mark_inscale);\n",
       "    if (ylabels) ylabels.selectAll(\"g\").forEach(mark_inscale);\n",
       "\n",
       "    // figure out the upper ond lower bounds on panning using the maximum\n",
       "    // and minimum grid lines\n",
       "    var bounds = root.plotbounds();\n",
       "    var pan_bounds = {\n",
       "        x0: 0.0,\n",
       "        y0: 0.0,\n",
       "        x1: 0.0,\n",
       "        y1: 0.0\n",
       "    };\n",
       "\n",
       "    if (xgridlines) {\n",
       "        xgridlines\n",
       "            .selectAll(\"g\")\n",
       "            .forEach(function (element, i) {\n",
       "                if (element.attribute(\"gadfly:inscale\") == \"true\") {\n",
       "                    var bbox = element.node.getBBox();\n",
       "                    if (bounds.x1 - bbox.x < pan_bounds.x0) {\n",
       "                        pan_bounds.x0 = bounds.x1 - bbox.x;\n",
       "                    }\n",
       "                    if (bounds.x0 - bbox.x > pan_bounds.x1) {\n",
       "                        pan_bounds.x1 = bounds.x0 - bbox.x;\n",
       "                    }\n",
       "                    element.attr(\"visibility\", \"visible\");\n",
       "                }\n",
       "            });\n",
       "    }\n",
       "\n",
       "    if (ygridlines) {\n",
       "        ygridlines\n",
       "            .selectAll(\"g\")\n",
       "            .forEach(function (element, i) {\n",
       "                if (element.attribute(\"gadfly:inscale\") == \"true\") {\n",
       "                    var bbox = element.node.getBBox();\n",
       "                    if (bounds.y1 - bbox.y < pan_bounds.y0) {\n",
       "                        pan_bounds.y0 = bounds.y1 - bbox.y;\n",
       "                    }\n",
       "                    if (bounds.y0 - bbox.y > pan_bounds.y1) {\n",
       "                        pan_bounds.y1 = bounds.y0 - bbox.y;\n",
       "                    }\n",
       "                    element.attr(\"visibility\", \"visible\");\n",
       "                }\n",
       "            });\n",
       "    }\n",
       "\n",
       "    // nudge these values a little\n",
       "    pan_bounds.x0 -= 5;\n",
       "    pan_bounds.x1 += 5;\n",
       "    pan_bounds.y0 -= 5;\n",
       "    pan_bounds.y1 += 5;\n",
       "    root.data(\"pan_bounds\", pan_bounds);\n",
       "\n",
       "    root.data(\"zoompan-ready\", true)\n",
       "};\n",
       "\n",
       "\n",
       "// drag actions, i.e. zooming and panning\n",
       "var pan_action = {\n",
       "    start: function(root, x, y, event) {\n",
       "        root.data(\"dx\", 0);\n",
       "        root.data(\"dy\", 0);\n",
       "        root.data(\"tx0\", root.data(\"tx\"));\n",
       "        root.data(\"ty0\", root.data(\"ty\"));\n",
       "    },\n",
       "    update: function(root, dx, dy, x, y, event) {\n",
       "        var px_per_mm = root.data(\"px_per_mm\");\n",
       "        dx /= px_per_mm;\n",
       "        dy /= px_per_mm;\n",
       "\n",
       "        var tx0 = root.data(\"tx\"),\n",
       "            ty0 = root.data(\"ty\");\n",
       "\n",
       "        var dx0 = root.data(\"dx\"),\n",
       "            dy0 = root.data(\"dy\");\n",
       "\n",
       "        root.data(\"dx\", dx);\n",
       "        root.data(\"dy\", dy);\n",
       "\n",
       "        dx = dx - dx0;\n",
       "        dy = dy - dy0;\n",
       "\n",
       "        var tx = tx0 + dx,\n",
       "            ty = ty0 + dy;\n",
       "\n",
       "        set_plot_pan_zoom(root, tx, ty, root.data(\"xscale\"), root.data(\"yscale\"));\n",
       "    },\n",
       "    end: function(root, event) {\n",
       "\n",
       "    },\n",
       "    cancel: function(root) {\n",
       "        set_plot_pan_zoom(root, root.data(\"tx0\"), root.data(\"ty0\"),\n",
       "                root.data(\"xscale\"), root.data(\"yscale\"));\n",
       "    }\n",
       "};\n",
       "\n",
       "var zoom_box;\n",
       "var zoom_action = {\n",
       "    start: function(root, _x, _y, event) {\n",
       "        var bounds = root.plotbounds();\n",
       "        // _x and _y are co-ordinates relative to page, which caused problems\n",
       "        // unless the SVG is precisely at the top-left of the page\n",
       "        var viewbounds = root.viewportplotbounds();\n",
       "        var x = event.clientX - viewbounds.x0;\n",
       "        var y = event.clientY - viewbounds.y0;\n",
       "\n",
       "        var width = bounds.x1 - bounds.x0,\n",
       "            height = bounds.y1 - bounds.y0;\n",
       "        var xscalable = root.hasClass(\"xscalable\"),\n",
       "            yscalable = root.hasClass(\"yscalable\");\n",
       "        var px_per_mm = root.data(\"px_per_mm\");\n",
       "        x = xscalable ? x / px_per_mm : bounds.x0;\n",
       "        y = yscalable ? y / px_per_mm : bounds.y0;\n",
       "        var w = xscalable ? 0 : width;\n",
       "        var h = yscalable ? 0 : height;\n",
       "        zoom_box = root.rect(x, y, w, h).attr({\n",
       "            \"fill\": \"#000\",\n",
       "            \"fill-opacity\": 0.25\n",
       "        });\n",
       "    },\n",
       "    update: function(root, dx, dy, _x, _y, event) {\n",
       "        var xscalable = root.hasClass(\"xscalable\"),\n",
       "            yscalable = root.hasClass(\"yscalable\");\n",
       "        var px_per_mm = root.data(\"px_per_mm\");\n",
       "        var bounds = root.plotbounds();\n",
       "        var viewbounds = root.viewportplotbounds();\n",
       "        var x = event.clientX - viewbounds.x0;\n",
       "        var y = event.clientY - viewbounds.y0;\n",
       "        if (yscalable) {\n",
       "            y /= px_per_mm;\n",
       "            y = Math.max(bounds.y0, y);\n",
       "            y = Math.min(bounds.y1, y);\n",
       "        } else {\n",
       "            y = bounds.y1;\n",
       "        }\n",
       "        if (xscalable) {\n",
       "            x /= px_per_mm;\n",
       "            x = Math.max(bounds.x0, x);\n",
       "            x = Math.min(bounds.x1, x);\n",
       "        } else {\n",
       "            x = bounds.x1;\n",
       "        }\n",
       "\n",
       "        dx = x - zoom_box.attr(\"x\");\n",
       "        dy = y - zoom_box.attr(\"y\");\n",
       "        var xoffset = 0,\n",
       "            yoffset = 0;\n",
       "        if (dx < 0) {\n",
       "            xoffset = dx;\n",
       "            dx = -1 * dx;\n",
       "        }\n",
       "        if (dy < 0) {\n",
       "            yoffset = dy;\n",
       "            dy = -1 * dy;\n",
       "        }\n",
       "        if (isNaN(dy)) {\n",
       "            dy = 0.0;\n",
       "        }\n",
       "        if (isNaN(dx)) {\n",
       "            dx = 0.0;\n",
       "        }\n",
       "        zoom_box.transform(\"T\" + xoffset + \",\" + yoffset);\n",
       "        zoom_box.attr(\"width\", dx);\n",
       "        zoom_box.attr(\"height\", dy);\n",
       "    },\n",
       "    end: function(root, event) {\n",
       "        var xscalable = root.hasClass(\"xscalable\"),\n",
       "            yscalable = root.hasClass(\"yscalable\");\n",
       "        var zoom_bounds = zoom_box.getBBox();\n",
       "        if (zoom_bounds.width * zoom_bounds.height <= 0) {\n",
       "            return;\n",
       "        }\n",
       "        var plot_bounds = root.plotbounds();\n",
       "        var xzoom_factor = 1.0,\n",
       "            yzoom_factor = 1.0;\n",
       "        if (xscalable) {\n",
       "            xzoom_factor = (plot_bounds.x1 - plot_bounds.x0) / zoom_bounds.width;\n",
       "        }\n",
       "        if (yscalable) {\n",
       "            yzoom_factor = (plot_bounds.y1 - plot_bounds.y0) / zoom_bounds.height;\n",
       "        }\n",
       "        var tx = (root.data(\"tx\") - zoom_bounds.x) * xzoom_factor + plot_bounds.x0,\n",
       "            ty = (root.data(\"ty\") - zoom_bounds.y) * yzoom_factor + plot_bounds.y0;\n",
       "        set_plot_pan_zoom(root, tx, ty,\n",
       "                root.data(\"xscale\") * xzoom_factor, root.data(\"yscale\") * yzoom_factor);\n",
       "        zoom_box.remove();\n",
       "    },\n",
       "    cancel: function(root) {\n",
       "        zoom_box.remove();\n",
       "    }\n",
       "};\n",
       "\n",
       "\n",
       "Gadfly.guide_background_drag_onstart = function(x, y, event) {\n",
       "    var root = this.plotroot();\n",
       "    var scalable = root.hasClass(\"xscalable\") || root.hasClass(\"yscalable\");\n",
       "    var zoomable = !event.altKey && !event.ctrlKey && event.shiftKey && scalable;\n",
       "    var panable = !event.altKey && !event.ctrlKey && !event.shiftKey && scalable;\n",
       "    var drag_action = zoomable ? zoom_action :\n",
       "                      panable  ? pan_action :\n",
       "                                 undefined;\n",
       "    root.data(\"drag_action\", drag_action);\n",
       "    if (drag_action) {\n",
       "        var cancel_drag_action = function(event) {\n",
       "            if (event.which == 27) { // esc key\n",
       "                drag_action.cancel(root);\n",
       "                root.data(\"drag_action\", undefined);\n",
       "            }\n",
       "        };\n",
       "        window.addEventListener(\"keyup\", cancel_drag_action);\n",
       "        root.data(\"cancel_drag_action\", cancel_drag_action);\n",
       "        drag_action.start(root, x, y, event);\n",
       "    }\n",
       "};\n",
       "\n",
       "\n",
       "Gadfly.guide_background_drag_onmove = function(dx, dy, x, y, event) {\n",
       "    var root = this.plotroot();\n",
       "    var drag_action = root.data(\"drag_action\");\n",
       "    if (drag_action) {\n",
       "        drag_action.update(root, dx, dy, x, y, event);\n",
       "    }\n",
       "};\n",
       "\n",
       "\n",
       "Gadfly.guide_background_drag_onend = function(event) {\n",
       "    var root = this.plotroot();\n",
       "    window.removeEventListener(\"keyup\", root.data(\"cancel_drag_action\"));\n",
       "    root.data(\"cancel_drag_action\", undefined);\n",
       "    var drag_action = root.data(\"drag_action\");\n",
       "    if (drag_action) {\n",
       "        drag_action.end(root, event);\n",
       "    }\n",
       "    root.data(\"drag_action\", undefined);\n",
       "};\n",
       "\n",
       "\n",
       "Gadfly.guide_background_scroll = function(event) {\n",
       "    if (event.shiftKey) {\n",
       "        // event.deltaY is either the number of pixels, lines, or pages scrolled past.\n",
       "        var actual_delta;\n",
       "        switch (event.deltaMode) {\n",
       "            case 0: // Chromium-based\n",
       "                actual_delta = -event.deltaY / 1000.0;\n",
       "                break;\n",
       "            case 1: // Firefox\n",
       "                actual_delta = -event.deltaY / 50.0;\n",
       "                break;\n",
       "            default:\n",
       "                actual_delta = -event.deltaY;\n",
       "        }\n",
       "        // Assumes 20 pixels/line to get reasonably consistent cross-browser behaviour.\n",
       "        increase_zoom_by_position(this.plotroot(), actual_delta);\n",
       "        event.preventDefault();\n",
       "    }\n",
       "};\n",
       "\n",
       "// Map slider position x to scale y using the function y = a*exp(b*x)+c.\n",
       "// The constants a, b, and c are solved using the constraint that the function\n",
       "// should go through the points (0; min_scale), (0.5; 1), and (1; max_scale).\n",
       "var scale_from_slider_position = function(position, min_scale, max_scale) {\n",
       "    if (min_scale==max_scale) { return 1; }\n",
       "    var a = (1 - 2 * min_scale + min_scale * min_scale) / (min_scale + max_scale - 2),\n",
       "        b = 2 * Math.log((max_scale - 1) / (1 - min_scale)),\n",
       "        c = (min_scale * max_scale - 1) / (min_scale + max_scale - 2);\n",
       "    return a * Math.exp(b * position) + c;\n",
       "}\n",
       "\n",
       "// inverse of scale_from_slider_position\n",
       "var slider_position_from_scale = function(scale, min_scale, max_scale) {\n",
       "    if (min_scale==max_scale) { return min_scale; }\n",
       "    var a = (1 - 2 * min_scale + min_scale * min_scale) / (min_scale + max_scale - 2),\n",
       "        b = 2 * Math.log((max_scale - 1) / (1 - min_scale)),\n",
       "        c = (min_scale * max_scale - 1) / (min_scale + max_scale - 2);\n",
       "    return 1 / b * Math.log((scale - c) / a);\n",
       "}\n",
       "\n",
       "var increase_zoom_by_position = function(root, delta_position, animate) {\n",
       "    var old_xscale = root.data(\"xscale\"),\n",
       "        old_yscale = root.data(\"yscale\"),\n",
       "        min_scale = root.data(\"min_scale\"),\n",
       "        max_scale = root.data(\"max_scale\");\n",
       "    var xposition = slider_position_from_scale(old_xscale, min_scale, max_scale),\n",
       "        yposition = slider_position_from_scale(old_yscale, min_scale, max_scale);\n",
       "    xposition += (root.hasClass(\"xscalable\") ? delta_position : 0.0);\n",
       "    yposition += (root.hasClass(\"yscalable\") ? delta_position : 0.0);\n",
       "    old_xscale = scale_from_slider_position(xposition, min_scale, max_scale);\n",
       "    old_yscale = scale_from_slider_position(yposition, min_scale, max_scale);\n",
       "    var new_xscale = Math.max(min_scale, Math.min(old_xscale, max_scale)),\n",
       "        new_yscale = Math.max(min_scale, Math.min(old_yscale, max_scale));\n",
       "    if (animate) {\n",
       "        Snap.animate(\n",
       "            [old_xscale, old_yscale],\n",
       "            [new_xscale, new_yscale],\n",
       "            function (new_scale) {\n",
       "                update_plot_scale(root, new_scale[0], new_scale[1]);\n",
       "            },\n",
       "            200);\n",
       "    } else {\n",
       "        update_plot_scale(root, new_xscale, new_yscale);\n",
       "    }\n",
       "}\n",
       "\n",
       "\n",
       "var update_plot_scale = function(root, new_xscale, new_yscale) {\n",
       "    var trans = scale_centered_translation(root, new_xscale, new_yscale);\n",
       "    set_plot_pan_zoom(root, trans.x, trans.y, new_xscale, new_yscale);\n",
       "};\n",
       "\n",
       "\n",
       "var toggle_color_class = function(root, color_class, ison) {\n",
       "    var escaped_color_class = color_class.replace(/([^0-9a-zA-z])/g,\"\\\\$1\");\n",
       "    var guides = root.selectAll(\".guide.\" + escaped_color_class + \",.guide .\" + escaped_color_class);\n",
       "    var geoms = root.selectAll(\".geometry.\" + escaped_color_class + \",.geometry .\" + escaped_color_class);\n",
       "    if (ison) {\n",
       "        guides.animate({opacity: 0.5}, 250);\n",
       "        geoms.animate({opacity: 0.0}, 250);\n",
       "    } else {\n",
       "        guides.animate({opacity: 1.0}, 250);\n",
       "        geoms.animate({opacity: 1.0}, 250);\n",
       "    }\n",
       "};\n",
       "\n",
       "\n",
       "Gadfly.colorkey_swatch_click = function(event) {\n",
       "    var root = this.plotroot();\n",
       "    var color_class = this.data(\"color_class\");\n",
       "\n",
       "    if (event.shiftKey) {\n",
       "        root.selectAll(\".colorkey g\")\n",
       "            .forEach(function (element) {\n",
       "                var other_color_class = element.data(\"color_class\");\n",
       "                if (typeof other_color_class !== 'undefined' && other_color_class != color_class) {\n",
       "                    toggle_color_class(root, other_color_class,\n",
       "                                       element.attr(\"opacity\") == 1.0);\n",
       "                }\n",
       "            });\n",
       "    } else {\n",
       "        toggle_color_class(root, color_class, this.attr(\"opacity\") == 1.0);\n",
       "    }\n",
       "};\n",
       "\n",
       "\n",
       "return Gadfly;\n",
       "\n",
       "}));\n",
       "\n",
       "\n",
       "//@ sourceURL=gadfly.js\n",
       "\n",
       "(function (glob, factory) {\n",
       "    // AMD support\n",
       "      if (typeof require === \"function\" && typeof define === \"function\" && define.amd) {\n",
       "        require([\"Snap.svg\", \"Gadfly\"], function (Snap, Gadfly) {\n",
       "            factory(Snap, Gadfly);\n",
       "        });\n",
       "      } else {\n",
       "          factory(glob.Snap, glob.Gadfly);\n",
       "      }\n",
       "})(window, function (Snap, Gadfly) {\n",
       "    var fig = Snap(\"#img-8f5c07e9\");\n",
       "fig.select(\"#img-8f5c07e9-5\")\n",
       "   .init_gadfly();\n",
       "fig.select(\"#img-8f5c07e9-8\")\n",
       "   .plotroot().data(\"unfocused_ygrid_color\", \"#D0D0E0\")\n",
       ";\n",
       "fig.select(\"#img-8f5c07e9-8\")\n",
       "   .plotroot().data(\"focused_ygrid_color\", \"#A0A0A0\")\n",
       ";\n",
       "fig.select(\"#img-8f5c07e9-242\")\n",
       "   .plotroot().data(\"unfocused_xgrid_color\", \"#D0D0E0\")\n",
       ";\n",
       "fig.select(\"#img-8f5c07e9-242\")\n",
       "   .plotroot().data(\"focused_xgrid_color\", \"#A0A0A0\")\n",
       ";\n",
       "fig.select(\"#img-8f5c07e9-494\")\n",
       "   .mouseenter(Gadfly.helpscreen_visible)\n",
       ".mouseleave(Gadfly.helpscreen_hidden)\n",
       ";\n",
       "    });\n",
       "]]> </script>\n",
       "</svg>\n",
       "\n",
       "</html>\n"
      ],
      "text/plain": [
       "Plot(...)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(p1,p2, Guide.ylabel(\"Herbie & nonHerbie\"), Guide.xlabel(\"x\"))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The non Herbie code has _spurious_ oscillations for $x$ near zero, but the Herbie code looks like a line. The wild oscillations are rubbish, making Herbie's suggestion _far_ more accurate near zero.\n",
    "\n",
    "Away from zero, the two functions agree to a high degree."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "<?xml version=\"1.0\" encoding=\"UTF-8\"?>\n",
       "<svg xmlns=\"http://www.w3.org/2000/svg\"\n",
       "     xmlns:xlink=\"http://www.w3.org/1999/xlink\"\n",
       "     xmlns:gadfly=\"http://www.gadflyjl.org/ns\"\n",
       "     version=\"1.2\"\n",
       "     width=\"141.42mm\" height=\"100mm\" viewBox=\"0 0 141.42 100\"\n",
       "     stroke=\"none\"\n",
       "     fill=\"#000000\"\n",
       "     stroke-width=\"0.3\"\n",
       "     font-size=\"3.88\"\n",
       ">\n",
       "<defs>\n",
       "  <marker id=\"arrow\" markerWidth=\"15\" markerHeight=\"7\" refX=\"5\" refY=\"3.5\" orient=\"auto\" markerUnits=\"strokeWidth\">\n",
       "    <path d=\"M0,0 L15,3.5 L0,7 z\" stroke=\"context-stroke\" fill=\"context-stroke\"/>\n",
       "  </marker>\n",
       "</defs>\n",
       "<g class=\"plotroot xscalable yscalable\" id=\"img-59ff4f69-1\">\n",
       "  <g font-size=\"3.88\" font-family=\"'PT Sans','Helvetica Neue','Helvetica',sans-serif\" fill=\"#564A55\" stroke=\"#000000\" stroke-opacity=\"0.000\" id=\"img-59ff4f69-2\">\n",
       "    <g transform=\"translate(85.34,88.39)\">\n",
       "      <g class=\"primitive\">\n",
       "        <text text-anchor=\"middle\" dy=\"0.6em\">x</text>\n",
       "      </g>\n",
       "    </g>\n",
       "  </g>\n",
       "  <g class=\"guide xlabels\" font-size=\"2.82\" font-family=\"'PT Sans Caption','Helvetica Neue','Helvetica',sans-serif\" fill=\"#6C606B\" id=\"img-59ff4f69-3\">\n",
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       "function(d,f,k,u,p){\"string\"==typeof k&&(p=u=k,k=4);\"string\"!=typeof u&&(p=u,u=\"#000\");null==k&&(k=4);null==p&&(p=1);null==d&&(d=0,f=2);null==f&&(f=d);u=a.color(u||\"#000\");return a.format('<feGaussianBlur in=\"SourceAlpha\" stdDeviation=\"{blur}\"/><feOffset dx=\"{dx}\" dy=\"{dy}\" result=\"offsetblur\"/><feFlood flood-color=\"{color}\"/><feComposite in2=\"offsetblur\" operator=\"in\"/><feComponentTransfer><feFuncA type=\"linear\" slope=\"{opacity}\"/></feComponentTransfer><feMerge><feMergeNode/><feMergeNode in=\"SourceGraphic\"/></feMerge>',\n",
       "{color:u,dx:d,dy:f,blur:k,opacity:p})};a.filter.shadow.toString=function(){return this()};a.filter.grayscale=function(d){null==d&&(d=1);return a.format('<feColorMatrix type=\"matrix\" values=\"{a} {b} {c} 0 0 {d} {e} {f} 0 0 {g} {b} {h} 0 0 0 0 0 1 0\"/>',{a:0.2126+0.7874*(1-d),b:0.7152-0.7152*(1-d),c:0.0722-0.0722*(1-d),d:0.2126-0.2126*(1-d),e:0.7152+0.2848*(1-d),f:0.0722-0.0722*(1-d),g:0.2126-0.2126*(1-d),h:0.0722+0.9278*(1-d)})};a.filter.grayscale.toString=function(){return this()};a.filter.sepia=\n",
       "function(d){null==d&&(d=1);return a.format('<feColorMatrix type=\"matrix\" values=\"{a} {b} {c} 0 0 {d} {e} {f} 0 0 {g} {h} {i} 0 0 0 0 0 1 0\"/>',{a:0.393+0.607*(1-d),b:0.769-0.769*(1-d),c:0.189-0.189*(1-d),d:0.349-0.349*(1-d),e:0.686+0.314*(1-d),f:0.168-0.168*(1-d),g:0.272-0.272*(1-d),h:0.534-0.534*(1-d),i:0.131+0.869*(1-d)})};a.filter.sepia.toString=function(){return this()};a.filter.saturate=function(d){null==d&&(d=1);return a.format('<feColorMatrix type=\"saturate\" values=\"{amount}\"/>',{amount:1-\n",
       "d})};a.filter.saturate.toString=function(){return this()};a.filter.hueRotate=function(d){return a.format('<feColorMatrix type=\"hueRotate\" values=\"{angle}\"/>',{angle:d||0})};a.filter.hueRotate.toString=function(){return this()};a.filter.invert=function(d){null==d&&(d=1);return a.format('<feComponentTransfer><feFuncR type=\"table\" tableValues=\"{amount} {amount2}\"/><feFuncG type=\"table\" tableValues=\"{amount} {amount2}\"/><feFuncB type=\"table\" tableValues=\"{amount} {amount2}\"/></feComponentTransfer>',{amount:d,\n",
       "amount2:1-d})};a.filter.invert.toString=function(){return this()};a.filter.brightness=function(d){null==d&&(d=1);return a.format('<feComponentTransfer><feFuncR type=\"linear\" slope=\"{amount}\"/><feFuncG type=\"linear\" slope=\"{amount}\"/><feFuncB type=\"linear\" slope=\"{amount}\"/></feComponentTransfer>',{amount:d})};a.filter.brightness.toString=function(){return this()};a.filter.contrast=function(d){null==d&&(d=1);return a.format('<feComponentTransfer><feFuncR type=\"linear\" slope=\"{amount}\" intercept=\"{amount2}\"/><feFuncG type=\"linear\" slope=\"{amount}\" intercept=\"{amount2}\"/><feFuncB type=\"linear\" slope=\"{amount}\" intercept=\"{amount2}\"/></feComponentTransfer>',\n",
       "{amount:d,amount2:0.5-d/2})};a.filter.contrast.toString=function(){return this()}});return C});\n",
       "\n",
       "]]> </script>\n",
       "<script> <![CDATA[\n",
       "\n",
       "(function (glob, factory) {\n",
       "    // AMD support\n",
       "    if (typeof define === \"function\" && define.amd) {\n",
       "        // Define as an anonymous module\n",
       "        define(\"Gadfly\", [\"Snap.svg\"], function (Snap) {\n",
       "            return factory(Snap);\n",
       "        });\n",
       "    } else {\n",
       "        // Browser globals (glob is window)\n",
       "        // Snap adds itself to window\n",
       "        glob.Gadfly = factory(glob.Snap);\n",
       "    }\n",
       "}(this, function (Snap) {\n",
       "\n",
       "var Gadfly = {};\n",
       "\n",
       "// Get an x/y coordinate value in pixels\n",
       "var xPX = function(fig, x) {\n",
       "    var client_box = fig.node.getBoundingClientRect();\n",
       "    return x * fig.node.viewBox.baseVal.width / client_box.width;\n",
       "};\n",
       "\n",
       "var yPX = function(fig, y) {\n",
       "    var client_box = fig.node.getBoundingClientRect();\n",
       "    return y * fig.node.viewBox.baseVal.height / client_box.height;\n",
       "};\n",
       "\n",
       "\n",
       "Snap.plugin(function (Snap, Element, Paper, global) {\n",
       "    // Traverse upwards from a snap element to find and return the first\n",
       "    // note with the \"plotroot\" class.\n",
       "    Element.prototype.plotroot = function () {\n",
       "        var element = this;\n",
       "        while (!element.hasClass(\"plotroot\") && element.parent() != null) {\n",
       "            element = element.parent();\n",
       "        }\n",
       "        return element;\n",
       "    };\n",
       "\n",
       "    Element.prototype.svgroot = function () {\n",
       "        var element = this;\n",
       "        while (element.node.nodeName != \"svg\" && element.parent() != null) {\n",
       "            element = element.parent();\n",
       "        }\n",
       "        return element;\n",
       "    };\n",
       "\n",
       "    Element.prototype.plotbounds = function () {\n",
       "        var root = this.plotroot()\n",
       "        var bbox = root.select(\".guide.background\").node.getBBox();\n",
       "        return {\n",
       "            x0: bbox.x,\n",
       "            x1: bbox.x + bbox.width,\n",
       "            y0: bbox.y,\n",
       "            y1: bbox.y + bbox.height\n",
       "        };\n",
       "    };\n",
       "\n",
       "    Element.prototype.viewportplotbounds = function () {\n",
       "        var root = this.svgroot();\n",
       "        var bbox = root.node.getBoundingClientRect();\n",
       "        return {\n",
       "            x0: bbox.x,\n",
       "            x1: bbox.x + bbox.width,\n",
       "            y0: bbox.y,\n",
       "            y1: bbox.y + bbox.height\n",
       "        };\n",
       "    };\n",
       "\n",
       "    Element.prototype.plotcenter = function () {\n",
       "        var root = this.plotroot()\n",
       "        var bbox = root.select(\".guide.background\").node.getBBox();\n",
       "        return {\n",
       "            x: bbox.x + bbox.width / 2,\n",
       "            y: bbox.y + bbox.height / 2\n",
       "        };\n",
       "    };\n",
       "\n",
       "    // Emulate IE style mouseenter/mouseleave events, since Microsoft always\n",
       "    // does everything right.\n",
       "    // See: http://www.dynamic-tools.net/toolbox/isMouseLeaveOrEnter/\n",
       "    var events = [\"mouseenter\", \"mouseleave\"];\n",
       "\n",
       "    for (i in events) {\n",
       "        (function (event_name) {\n",
       "            var event_name = events[i];\n",
       "            Element.prototype[event_name] = function (fn, scope) {\n",
       "                if (Snap.is(fn, \"function\")) {\n",
       "                    var fn2 = function (event) {\n",
       "                        if (event.type != \"mouseover\" && event.type != \"mouseout\") {\n",
       "                            return;\n",
       "                        }\n",
       "\n",
       "                        var reltg = event.relatedTarget ? event.relatedTarget :\n",
       "                            event.type == \"mouseout\" ? event.toElement : event.fromElement;\n",
       "                        while (reltg && reltg != this.node) reltg = reltg.parentNode;\n",
       "\n",
       "                        if (reltg != this.node) {\n",
       "                            return fn.apply(this, event);\n",
       "                        }\n",
       "                    };\n",
       "\n",
       "                    if (event_name == \"mouseenter\") {\n",
       "                        this.mouseover(fn2, scope);\n",
       "                    } else {\n",
       "                        this.mouseout(fn2, scope);\n",
       "                    }\n",
       "                }\n",
       "                return this;\n",
       "            };\n",
       "        })(events[i]);\n",
       "    }\n",
       "\n",
       "\n",
       "    Element.prototype.mousewheel = function (fn, scope) {\n",
       "        if (Snap.is(fn, \"function\")) {\n",
       "            var el = this;\n",
       "            var fn2 = function (event) {\n",
       "                fn.apply(el, [event]);\n",
       "            };\n",
       "        }\n",
       "\n",
       "        this.node.addEventListener(\"wheel\", fn2);\n",
       "\n",
       "        return this;\n",
       "    };\n",
       "\n",
       "\n",
       "    // Snap's attr function can be too slow for things like panning/zooming.\n",
       "    // This is a function to directly update element attributes without going\n",
       "    // through eve.\n",
       "    Element.prototype.attribute = function(key, val) {\n",
       "        if (val === undefined) {\n",
       "            return this.node.getAttribute(key);\n",
       "        } else {\n",
       "            this.node.setAttribute(key, val);\n",
       "            return this;\n",
       "        }\n",
       "    };\n",
       "\n",
       "    Element.prototype.init_gadfly = function() {\n",
       "        this.mouseenter(Gadfly.plot_mouseover)\n",
       "            .mousemove(Gadfly.plot_mousemove)\n",
       "            .mouseleave(Gadfly.plot_mouseout)\n",
       "            .dblclick(Gadfly.plot_dblclick)\n",
       "            .mousewheel(Gadfly.guide_background_scroll)\n",
       "            .drag(Gadfly.guide_background_drag_onmove,\n",
       "                  Gadfly.guide_background_drag_onstart,\n",
       "                  Gadfly.guide_background_drag_onend);\n",
       "        this.mouseenter(function (event) {\n",
       "            init_pan_zoom(this.plotroot());\n",
       "        });\n",
       "        return this;\n",
       "    };\n",
       "});\n",
       "\n",
       "\n",
       "Gadfly.plot_mousemove = function(event, _x_px, _y_px) {\n",
       "    var root = this.plotroot();\n",
       "    var viewbounds = root.viewportplotbounds();\n",
       "\n",
       "    // (_x_px, _y_px) are offsets relative to page (event.layerX, event.layerY) rather than viewport\n",
       "    var x_px = event.clientX - viewbounds.x0;\n",
       "    var y_px = event.clientY - viewbounds.y0;\n",
       "    if (root.data(\"crosshair\")) {\n",
       "        px_per_mm = root.data(\"px_per_mm\");\n",
       "        bB = root.select('boundingbox').node.getAttribute('value').split(' ');\n",
       "        uB = root.select('unitbox').node.getAttribute('value').split(' ');\n",
       "        xscale = root.data(\"xscale\");\n",
       "        yscale = root.data(\"yscale\");\n",
       "        xtranslate = root.data(\"tx\");\n",
       "        ytranslate = root.data(\"ty\");\n",
       "\n",
       "        xisdate = uB[0].includes(\"T\");\n",
       "        yisdate = uB[1].includes(\"T\");\n",
       "        tzoffset = (new Date()).getTimezoneOffset() * 60000;\n",
       "\n",
       "        xoff_mm = bB[0].substr(0,bB[0].length-2)/1;\n",
       "        yoff_mm = bB[1].substr(0,bB[1].length-2)/1;\n",
       "        xoff_unit = xisdate ? Date.parse(uB[0]) : uB[0]/1;\n",
       "        yoff_unit = yisdate ? Date.parse(uB[1]) : uB[1]/1;\n",
       "        mm_per_xunit = bB[2].substr(0,bB[2].length-2) / uB[2];\n",
       "        mm_per_yunit = bB[3].substr(0,bB[3].length-2) / uB[3+xisdate];\n",
       "         \n",
       "        x_unit = ((x_px / px_per_mm - xtranslate) / xscale - xoff_mm) / mm_per_xunit + xoff_unit;\n",
       "        y_unit = ((y_px / px_per_mm - ytranslate) / yscale - yoff_mm) / mm_per_yunit + yoff_unit;\n",
       "\n",
       "        root.select('.crosshair').select('.primitive').select('text').node.innerHTML =\n",
       "                (xisdate ? new Date(Math.round(x_unit)-tzoffset).toISOString().slice(0,-1) : x_unit.toPrecision(3))\n",
       "                +\",\"+\n",
       "                (yisdate ? new Date(Math.round(y_unit)-tzoffset).toISOString().slice(0,-1) : y_unit.toPrecision(3));\n",
       "    };\n",
       "};\n",
       "\n",
       "Gadfly.helpscreen_visible = function(event) {\n",
       "    helpscreen_visible(this.plotroot());\n",
       "};\n",
       "var helpscreen_visible = function(root) {\n",
       "    root.select(\".helpscreen\").attr(\"visibility\", \"visible\");\n",
       "};\n",
       "\n",
       "Gadfly.helpscreen_hidden = function(event) {\n",
       "    helpscreen_hidden(this.plotroot());\n",
       "};\n",
       "var helpscreen_hidden = function(root) {\n",
       "    root.select(\".helpscreen\").attr(\"visibility\", \"hidden\");\n",
       "};\n",
       "\n",
       "Gadfly.datatip_visible = function(event) {\n",
       "    var root = this.plotroot();\n",
       "    var ilabel = this.data(\"ilabel\");\n",
       "    var element = root.select('[ilabel=\"'+ilabel+'\"]');\n",
       "    element.animate({\"fill-opacity\": 1.0}, 250);\n",
       "};\n",
       "\n",
       "Gadfly.datatip_hidden = function(event) {\n",
       "    var root = this.plotroot();\n",
       "    var ilabel = this.data(\"ilabel\");\n",
       "    var element = root.select('[ilabel=\"'+ilabel+'\"]');\n",
       "    element.animate({\"fill-opacity\": 0.0}, 250);\n",
       "};\n",
       "\n",
       "// When the plot is moused over, emphasize the grid lines.\n",
       "Gadfly.plot_mouseover = function(event) {\n",
       "    var root = this.plotroot();\n",
       "\n",
       "    var keyboard_help = function(event) {\n",
       "        if (event.which == 191) { // ?\n",
       "            helpscreen_visible(root);\n",
       "        }\n",
       "    };\n",
       "    root.data(\"keyboard_help\", keyboard_help);\n",
       "    window.addEventListener(\"keydown\", keyboard_help);\n",
       "\n",
       "    var keyboard_pan_zoom = function(event) {\n",
       "        var bounds = root.plotbounds(),\n",
       "            width = bounds.x1 - bounds.x0;\n",
       "            height = bounds.y1 - bounds.y0;\n",
       "        if (event.which == 187 || event.which == 73) { // plus or i\n",
       "            increase_zoom_by_position(root, 0.1, true);\n",
       "        } else if (event.which == 189 || event.which == 79) { // minus or o\n",
       "            increase_zoom_by_position(root, -0.1, true);\n",
       "        } else if (event.which == 39 || event.which == 76) { // right-arrow or l\n",
       "            set_plot_pan_zoom(root, root.data(\"tx\")-width/10, root.data(\"ty\"),\n",
       "                    root.data(\"xscale\"), root.data(\"yscale\"));\n",
       "        } else if (event.which == 40 || event.which == 74) { // down-arrow or j\n",
       "            set_plot_pan_zoom(root, root.data(\"tx\"), root.data(\"ty\")-height/10,\n",
       "                    root.data(\"xscale\"), root.data(\"yscale\"));\n",
       "        } else if (event.which == 37 || event.which == 72) { // left-arrow or h\n",
       "            set_plot_pan_zoom(root, root.data(\"tx\")+width/10, root.data(\"ty\"),\n",
       "                    root.data(\"xscale\"), root.data(\"yscale\"));\n",
       "        } else if (event.which == 38 || event.which == 75) { // up-arrow or k\n",
       "            set_plot_pan_zoom(root, root.data(\"tx\"), root.data(\"ty\")+height/10,\n",
       "                    root.data(\"xscale\"), root.data(\"yscale\"));\n",
       "        } else if (event.which == 82) { // r\n",
       "            set_plot_pan_zoom(root, 0.0, 0.0, 1.0, 1.0);\n",
       "        } else if (event.which == 191) { // ?\n",
       "            helpscreen_hidden(root);\n",
       "        } else if (event.which == 67) { // c\n",
       "            root.data(\"crosshair\",!root.data(\"crosshair\"));\n",
       "            root.select(\".crosshair\")\n",
       "                .animate({\"fill-opacity\": root.data(\"crosshair\") ? 1.0 : 0.0}, 250);\n",
       "        }\n",
       "    };\n",
       "    root.data(\"keyboard_pan_zoom\", keyboard_pan_zoom);\n",
       "    window.addEventListener(\"keyup\", keyboard_pan_zoom);\n",
       "\n",
       "    var xgridlines = root.select(\".xgridlines\"),\n",
       "        ygridlines = root.select(\".ygridlines\");\n",
       "\n",
       "    if (xgridlines) {\n",
       "        xgridlines.data(\"unfocused_strokedash\",\n",
       "                        xgridlines.attribute(\"stroke-dasharray\").replace(/(\\d)(,|$)/g, \"$1mm$2\"));\n",
       "        var destcolor = root.data(\"focused_xgrid_color\");\n",
       "        xgridlines.attribute(\"stroke-dasharray\", \"none\")\n",
       "                  .selectAll(\"path\")\n",
       "                  .animate({stroke: destcolor}, 250);\n",
       "    }\n",
       "\n",
       "    if (ygridlines) {\n",
       "        ygridlines.data(\"unfocused_strokedash\",\n",
       "                        ygridlines.attribute(\"stroke-dasharray\").replace(/(\\d)(,|$)/g, \"$1mm$2\"));\n",
       "        var destcolor = root.data(\"focused_ygrid_color\");\n",
       "        ygridlines.attribute(\"stroke-dasharray\", \"none\")\n",
       "                  .selectAll(\"path\")\n",
       "                  .animate({stroke: destcolor}, 250);\n",
       "    }\n",
       "\n",
       "    root.select(\".crosshair\")\n",
       "        .animate({\"fill-opacity\": root.data(\"crosshair\") ? 1.0 : 0.0}, 250);\n",
       "    root.select(\".questionmark\").animate({\"fill-opacity\": 1.0}, 250);\n",
       "};\n",
       "\n",
       "// Reset pan and zoom on double click\n",
       "Gadfly.plot_dblclick = function(event) {\n",
       "  set_plot_pan_zoom(this.plotroot(), 0.0, 0.0, 1.0, 1.0);\n",
       "};\n",
       "\n",
       "// Unemphasize grid lines on mouse out.\n",
       "Gadfly.plot_mouseout = function(event) {\n",
       "    var root = this.plotroot();\n",
       "\n",
       "    window.removeEventListener(\"keyup\", root.data(\"keyboard_pan_zoom\"));\n",
       "    root.data(\"keyboard_pan_zoom\", undefined);\n",
       "    window.removeEventListener(\"keydown\", root.data(\"keyboard_help\"));\n",
       "    root.data(\"keyboard_help\", undefined);\n",
       "\n",
       "    var xgridlines = root.select(\".xgridlines\"),\n",
       "        ygridlines = root.select(\".ygridlines\");\n",
       "\n",
       "    if (xgridlines) {\n",
       "        var destcolor = root.data(\"unfocused_xgrid_color\");\n",
       "        xgridlines.attribute(\"stroke-dasharray\", xgridlines.data(\"unfocused_strokedash\"))\n",
       "                  .selectAll(\"path\")\n",
       "                  .animate({stroke: destcolor}, 250);\n",
       "    }\n",
       "\n",
       "    if (ygridlines) {\n",
       "        var destcolor = root.data(\"unfocused_ygrid_color\");\n",
       "        ygridlines.attribute(\"stroke-dasharray\", ygridlines.data(\"unfocused_strokedash\"))\n",
       "                  .selectAll(\"path\")\n",
       "                  .animate({stroke: destcolor}, 250);\n",
       "    }\n",
       "\n",
       "    root.select(\".crosshair\").animate({\"fill-opacity\": 0.0}, 250);\n",
       "    root.select(\".questionmark\").animate({\"fill-opacity\": 0.0}, 250);\n",
       "    helpscreen_hidden(root);\n",
       "};\n",
       "\n",
       "\n",
       "var set_geometry_transform = function(root, tx, ty, xscale, yscale) {\n",
       "    var xscalable = root.hasClass(\"xscalable\"),\n",
       "        yscalable = root.hasClass(\"yscalable\");\n",
       "\n",
       "    var old_xscale = root.data(\"xscale\"),\n",
       "        old_yscale = root.data(\"yscale\");\n",
       "\n",
       "    var xscale = xscalable ? xscale : 1.0,\n",
       "        yscale = yscalable ? yscale : 1.0;\n",
       "\n",
       "    tx = xscalable ? tx : 0.0;\n",
       "    ty = yscalable ? ty : 0.0;\n",
       "\n",
       "    var t = new Snap.Matrix().translate(tx, ty).scale(xscale, yscale);\n",
       "    root.selectAll(\".geometry, image\").forEach(function (element, i) {\n",
       "            element.transform(t);\n",
       "        });\n",
       "\n",
       "    var t = new Snap.Matrix().scale(1.0/xscale, 1.0/yscale);\n",
       "    root.selectAll('.marker, .datatip').forEach(function (element, i) {\n",
       "        element.selectAll('.primitive').forEach(function (element, i) {\n",
       "            element.transform(t);\n",
       "        }) });\n",
       "\n",
       "    bounds = root.plotbounds();\n",
       "    px_per_mm = root.data(\"px_per_mm\");\n",
       "\n",
       "    if (yscalable) {\n",
       "        var xfixed_t = new Snap.Matrix().translate(0, ty).scale(1.0, yscale);\n",
       "        root.selectAll(\".xfixed\")\n",
       "            .forEach(function (element, i) {\n",
       "                element.transform(xfixed_t);\n",
       "            });\n",
       "\n",
       "        ylabels = root.select(\".ylabels\");\n",
       "        if (ylabels) {\n",
       "            ylabels.transform(xfixed_t)\n",
       "                   .selectAll(\"g\")\n",
       "                   .forEach(function (element, i) {\n",
       "                       if (element.attribute(\"gadfly:inscale\") == \"true\") {\n",
       "                           unscale_t = new Snap.Matrix();\n",
       "                           unscale_t.scale(1, 1/yscale);\n",
       "                           element.select(\"text\").transform(unscale_t);\n",
       " \n",
       "                           var y = element.attr(\"transform\").globalMatrix.f / px_per_mm;\n",
       "                           element.attr(\"visibility\",\n",
       "                               bounds.y0 <= y && y <= bounds.y1 ? \"visible\" : \"hidden\");\n",
       "                       }\n",
       "                   });\n",
       "        }\n",
       "    }\n",
       "\n",
       "    if (xscalable) {\n",
       "        var yfixed_t = new Snap.Matrix().translate(tx, 0).scale(xscale, 1.0);\n",
       "        var xtrans = new Snap.Matrix().translate(tx, 0);\n",
       "        root.selectAll(\".yfixed\")\n",
       "            .forEach(function (element, i) {\n",
       "                element.transform(yfixed_t);\n",
       "            });\n",
       "\n",
       "        xlabels = root.select(\".xlabels\");\n",
       "        if (xlabels) {\n",
       "            xlabels.transform(yfixed_t)\n",
       "                   .selectAll(\"g\")\n",
       "                   .forEach(function (element, i) {\n",
       "                       if (element.attribute(\"gadfly:inscale\") == \"true\") {\n",
       "                           unscale_t = new Snap.Matrix();\n",
       "                           unscale_t.scale(1/xscale, 1);\n",
       "                           element.select(\"text\").transform(unscale_t);\n",
       "\n",
       "                           var x = element.attr(\"transform\").globalMatrix.e / px_per_mm;\n",
       "                           element.attr(\"visibility\",\n",
       "                               bounds.x0 <= x && x <= bounds.x1 ? \"visible\" : \"hidden\");\n",
       "                           }\n",
       "                   });\n",
       "        }\n",
       "    }\n",
       "};\n",
       "\n",
       "\n",
       "// Find the most appropriate tick scale and update label visibility.\n",
       "var update_tickscale = function(root, scale, axis) {\n",
       "    if (!root.hasClass(axis + \"scalable\")) return;\n",
       "\n",
       "    var tickscales = root.data(axis + \"tickscales\");\n",
       "    var best_tickscale = 1.0;\n",
       "    var best_tickscale_dist = Infinity;\n",
       "    for (tickscale in tickscales) {\n",
       "        var dist = Math.abs(Math.log(tickscale) - Math.log(scale));\n",
       "        if (dist < best_tickscale_dist) {\n",
       "            best_tickscale_dist = dist;\n",
       "            best_tickscale = tickscale;\n",
       "        }\n",
       "    }\n",
       "\n",
       "    if (best_tickscale != root.data(axis + \"tickscale\")) {\n",
       "        root.data(axis + \"tickscale\", best_tickscale);\n",
       "        var mark_inscale_gridlines = function (element, i) {\n",
       "            if (element.attribute(\"gadfly:inscale\") == null) { return; }\n",
       "            var inscale = element.attr(\"gadfly:scale\") == best_tickscale;\n",
       "            element.attribute(\"gadfly:inscale\", inscale);\n",
       "            element.attr(\"visibility\", inscale ? \"visible\" : \"hidden\");\n",
       "        };\n",
       "\n",
       "        var mark_inscale_labels = function (element, i) {\n",
       "            if (element.attribute(\"gadfly:inscale\") == null) { return; }\n",
       "            var inscale = element.attr(\"gadfly:scale\") == best_tickscale;\n",
       "            element.attribute(\"gadfly:inscale\", inscale);\n",
       "            element.attr(\"visibility\", inscale ? \"visible\" : \"hidden\");\n",
       "        };\n",
       "\n",
       "        root.select(\".\" + axis + \"gridlines\").selectAll(\"g\").forEach(mark_inscale_gridlines);\n",
       "        root.select(\".\" + axis + \"labels\").selectAll(\"g\").forEach(mark_inscale_labels);\n",
       "    }\n",
       "};\n",
       "\n",
       "\n",
       "var set_plot_pan_zoom = function(root, tx, ty, xscale, yscale) {\n",
       "    var old_xscale = root.data(\"xscale\"),\n",
       "        old_yscale = root.data(\"yscale\");\n",
       "    var bounds = root.plotbounds();\n",
       "\n",
       "    var width = bounds.x1 - bounds.x0,\n",
       "        height = bounds.y1 - bounds.y0;\n",
       "\n",
       "    // compute the viewport derived from tx, ty, xscale, and yscale\n",
       "    var x_min = -width * xscale - (xscale * width - width),\n",
       "        x_max = width * xscale,\n",
       "        y_min = -height * yscale - (yscale * height - height),\n",
       "        y_max = height * yscale;\n",
       "\n",
       "    var x0 = bounds.x0 - xscale * bounds.x0,\n",
       "        y0 = bounds.y0 - yscale * bounds.y0;\n",
       "\n",
       "    var tx = Math.max(Math.min(tx - x0, x_max), x_min),\n",
       "        ty = Math.max(Math.min(ty - y0, y_max), y_min);\n",
       "\n",
       "    tx += x0;\n",
       "    ty += y0;\n",
       "\n",
       "    // when the scale changes, we may need to alter which set of\n",
       "    // ticks are being displayed\n",
       "    if (xscale != old_xscale) {\n",
       "        update_tickscale(root, xscale, \"x\");\n",
       "    }\n",
       "    if (yscale != old_yscale) {\n",
       "        update_tickscale(root, yscale, \"y\");\n",
       "    }\n",
       "\n",
       "    set_geometry_transform(root, tx, ty, xscale, yscale);\n",
       "\n",
       "    root.data(\"xscale\", xscale);\n",
       "    root.data(\"yscale\", yscale);\n",
       "    root.data(\"tx\", tx);\n",
       "    root.data(\"ty\", ty);\n",
       "};\n",
       "\n",
       "\n",
       "var scale_centered_translation = function(root, xscale, yscale) {\n",
       "    var bounds = root.plotbounds();\n",
       "\n",
       "    var width = bounds.x1 - bounds.x0,\n",
       "        height = bounds.y1 - bounds.y0;\n",
       "\n",
       "    var tx0 = root.data(\"tx\"),\n",
       "        ty0 = root.data(\"ty\");\n",
       "\n",
       "    var xscale0 = root.data(\"xscale\"),\n",
       "        yscale0 = root.data(\"yscale\");\n",
       "\n",
       "    // how off from center the current view is\n",
       "    var xoff = tx0 - (bounds.x0 * (1 - xscale0) + (width * (1 - xscale0)) / 2),\n",
       "        yoff = ty0 - (bounds.y0 * (1 - yscale0) + (height * (1 - yscale0)) / 2);\n",
       "\n",
       "    // rescale offsets\n",
       "    xoff = xoff * xscale / xscale0;\n",
       "    yoff = yoff * yscale / yscale0;\n",
       "\n",
       "    // adjust for the panel position being scaled\n",
       "    var x_edge_adjust = bounds.x0 * (1 - xscale),\n",
       "        y_edge_adjust = bounds.y0 * (1 - yscale);\n",
       "\n",
       "    return {\n",
       "        x: xoff + x_edge_adjust + (width - width * xscale) / 2,\n",
       "        y: yoff + y_edge_adjust + (height - height * yscale) / 2\n",
       "    };\n",
       "};\n",
       "\n",
       "\n",
       "// Initialize data for panning zooming if it isn't already.\n",
       "var init_pan_zoom = function(root) {\n",
       "    if (root.data(\"zoompan-ready\")) {\n",
       "        return;\n",
       "    }\n",
       "\n",
       "    root.data(\"crosshair\",false);\n",
       "\n",
       "    // The non-scaling-stroke trick. Rather than try to correct for the\n",
       "    // stroke-width when zooming, we force it to a fixed value.\n",
       "    var px_per_mm = root.node.getCTM().a;\n",
       "\n",
       "    // Drag events report deltas in pixels, which we'd like to convert to\n",
       "    // millimeters.\n",
       "    root.data(\"px_per_mm\", px_per_mm);\n",
       "\n",
       "    root.selectAll(\"path\")\n",
       "        .forEach(function (element, i) {\n",
       "        sw = element.asPX(\"stroke-width\") * px_per_mm;\n",
       "        if (sw > 0) {\n",
       "            element.attribute(\"stroke-width\", sw);\n",
       "            element.attribute(\"vector-effect\", \"non-scaling-stroke\");\n",
       "        }\n",
       "    });\n",
       "\n",
       "    // Store ticks labels original transformation\n",
       "    root.selectAll(\".xlabels > g, .ylabels > g\")\n",
       "        .forEach(function (element, i) {\n",
       "            var lm = element.transform().localMatrix;\n",
       "            element.data(\"static_transform\",\n",
       "                new Snap.Matrix(lm.a, lm.b, lm.c, lm.d, lm.e, lm.f));\n",
       "        });\n",
       "\n",
       "    var xgridlines = root.select(\".xgridlines\");\n",
       "    var ygridlines = root.select(\".ygridlines\");\n",
       "    var xlabels = root.select(\".xlabels\");\n",
       "    var ylabels = root.select(\".ylabels\");\n",
       "\n",
       "    if (root.data(\"tx\") === undefined) root.data(\"tx\", 0);\n",
       "    if (root.data(\"ty\") === undefined) root.data(\"ty\", 0);\n",
       "    if (root.data(\"xscale\") === undefined) root.data(\"xscale\", 1.0);\n",
       "    if (root.data(\"yscale\") === undefined) root.data(\"yscale\", 1.0);\n",
       "    if (root.data(\"xtickscales\") === undefined) {\n",
       "\n",
       "        // index all the tick scales that are listed\n",
       "        var xtickscales = {};\n",
       "        var ytickscales = {};\n",
       "        var add_x_tick_scales = function (element, i) {\n",
       "            if (element.attribute(\"gadfly:scale\")==null) { return; }\n",
       "            xtickscales[element.attribute(\"gadfly:scale\")] = true;\n",
       "        };\n",
       "        var add_y_tick_scales = function (element, i) {\n",
       "            if (element.attribute(\"gadfly:scale\")==null) { return; }\n",
       "            ytickscales[element.attribute(\"gadfly:scale\")] = true;\n",
       "        };\n",
       "\n",
       "        if (xgridlines) xgridlines.selectAll(\"g\").forEach(add_x_tick_scales);\n",
       "        if (ygridlines) ygridlines.selectAll(\"g\").forEach(add_y_tick_scales);\n",
       "        if (xlabels) xlabels.selectAll(\"g\").forEach(add_x_tick_scales);\n",
       "        if (ylabels) ylabels.selectAll(\"g\").forEach(add_y_tick_scales);\n",
       "\n",
       "        root.data(\"xtickscales\", xtickscales);\n",
       "        root.data(\"ytickscales\", ytickscales);\n",
       "        root.data(\"xtickscale\", 1.0);\n",
       "        root.data(\"ytickscale\", 1.0);  // ???\n",
       "    }\n",
       "\n",
       "    var min_scale = 1.0, max_scale = 1.0;\n",
       "    for (scale in xtickscales) {\n",
       "        min_scale = Math.min(min_scale, scale);\n",
       "        max_scale = Math.max(max_scale, scale);\n",
       "    }\n",
       "    for (scale in ytickscales) {\n",
       "        min_scale = Math.min(min_scale, scale);\n",
       "        max_scale = Math.max(max_scale, scale);\n",
       "    }\n",
       "    root.data(\"min_scale\", min_scale);\n",
       "    root.data(\"max_scale\", max_scale);\n",
       "\n",
       "    // store the original positions of labels\n",
       "    if (xlabels) {\n",
       "        xlabels.selectAll(\"g\")\n",
       "               .forEach(function (element, i) {\n",
       "                   element.data(\"x\", element.asPX(\"x\"));\n",
       "               });\n",
       "    }\n",
       "\n",
       "    if (ylabels) {\n",
       "        ylabels.selectAll(\"g\")\n",
       "               .forEach(function (element, i) {\n",
       "                   element.data(\"y\", element.asPX(\"y\"));\n",
       "               });\n",
       "    }\n",
       "\n",
       "    // mark grid lines and ticks as in or out of scale.\n",
       "    var mark_inscale = function (element, i) {\n",
       "        if (element.attribute(\"gadfly:scale\") == null) { return; }\n",
       "        element.attribute(\"gadfly:inscale\", element.attribute(\"gadfly:scale\") == 1.0);\n",
       "    };\n",
       "\n",
       "    if (xgridlines) xgridlines.selectAll(\"g\").forEach(mark_inscale);\n",
       "    if (ygridlines) ygridlines.selectAll(\"g\").forEach(mark_inscale);\n",
       "    if (xlabels) xlabels.selectAll(\"g\").forEach(mark_inscale);\n",
       "    if (ylabels) ylabels.selectAll(\"g\").forEach(mark_inscale);\n",
       "\n",
       "    // figure out the upper ond lower bounds on panning using the maximum\n",
       "    // and minimum grid lines\n",
       "    var bounds = root.plotbounds();\n",
       "    var pan_bounds = {\n",
       "        x0: 0.0,\n",
       "        y0: 0.0,\n",
       "        x1: 0.0,\n",
       "        y1: 0.0\n",
       "    };\n",
       "\n",
       "    if (xgridlines) {\n",
       "        xgridlines\n",
       "            .selectAll(\"g\")\n",
       "            .forEach(function (element, i) {\n",
       "                if (element.attribute(\"gadfly:inscale\") == \"true\") {\n",
       "                    var bbox = element.node.getBBox();\n",
       "                    if (bounds.x1 - bbox.x < pan_bounds.x0) {\n",
       "                        pan_bounds.x0 = bounds.x1 - bbox.x;\n",
       "                    }\n",
       "                    if (bounds.x0 - bbox.x > pan_bounds.x1) {\n",
       "                        pan_bounds.x1 = bounds.x0 - bbox.x;\n",
       "                    }\n",
       "                    element.attr(\"visibility\", \"visible\");\n",
       "                }\n",
       "            });\n",
       "    }\n",
       "\n",
       "    if (ygridlines) {\n",
       "        ygridlines\n",
       "            .selectAll(\"g\")\n",
       "            .forEach(function (element, i) {\n",
       "                if (element.attribute(\"gadfly:inscale\") == \"true\") {\n",
       "                    var bbox = element.node.getBBox();\n",
       "                    if (bounds.y1 - bbox.y < pan_bounds.y0) {\n",
       "                        pan_bounds.y0 = bounds.y1 - bbox.y;\n",
       "                    }\n",
       "                    if (bounds.y0 - bbox.y > pan_bounds.y1) {\n",
       "                        pan_bounds.y1 = bounds.y0 - bbox.y;\n",
       "                    }\n",
       "                    element.attr(\"visibility\", \"visible\");\n",
       "                }\n",
       "            });\n",
       "    }\n",
       "\n",
       "    // nudge these values a little\n",
       "    pan_bounds.x0 -= 5;\n",
       "    pan_bounds.x1 += 5;\n",
       "    pan_bounds.y0 -= 5;\n",
       "    pan_bounds.y1 += 5;\n",
       "    root.data(\"pan_bounds\", pan_bounds);\n",
       "\n",
       "    root.data(\"zoompan-ready\", true)\n",
       "};\n",
       "\n",
       "\n",
       "// drag actions, i.e. zooming and panning\n",
       "var pan_action = {\n",
       "    start: function(root, x, y, event) {\n",
       "        root.data(\"dx\", 0);\n",
       "        root.data(\"dy\", 0);\n",
       "        root.data(\"tx0\", root.data(\"tx\"));\n",
       "        root.data(\"ty0\", root.data(\"ty\"));\n",
       "    },\n",
       "    update: function(root, dx, dy, x, y, event) {\n",
       "        var px_per_mm = root.data(\"px_per_mm\");\n",
       "        dx /= px_per_mm;\n",
       "        dy /= px_per_mm;\n",
       "\n",
       "        var tx0 = root.data(\"tx\"),\n",
       "            ty0 = root.data(\"ty\");\n",
       "\n",
       "        var dx0 = root.data(\"dx\"),\n",
       "            dy0 = root.data(\"dy\");\n",
       "\n",
       "        root.data(\"dx\", dx);\n",
       "        root.data(\"dy\", dy);\n",
       "\n",
       "        dx = dx - dx0;\n",
       "        dy = dy - dy0;\n",
       "\n",
       "        var tx = tx0 + dx,\n",
       "            ty = ty0 + dy;\n",
       "\n",
       "        set_plot_pan_zoom(root, tx, ty, root.data(\"xscale\"), root.data(\"yscale\"));\n",
       "    },\n",
       "    end: function(root, event) {\n",
       "\n",
       "    },\n",
       "    cancel: function(root) {\n",
       "        set_plot_pan_zoom(root, root.data(\"tx0\"), root.data(\"ty0\"),\n",
       "                root.data(\"xscale\"), root.data(\"yscale\"));\n",
       "    }\n",
       "};\n",
       "\n",
       "var zoom_box;\n",
       "var zoom_action = {\n",
       "    start: function(root, _x, _y, event) {\n",
       "        var bounds = root.plotbounds();\n",
       "        // _x and _y are co-ordinates relative to page, which caused problems\n",
       "        // unless the SVG is precisely at the top-left of the page\n",
       "        var viewbounds = root.viewportplotbounds();\n",
       "        var x = event.clientX - viewbounds.x0;\n",
       "        var y = event.clientY - viewbounds.y0;\n",
       "\n",
       "        var width = bounds.x1 - bounds.x0,\n",
       "            height = bounds.y1 - bounds.y0;\n",
       "        var xscalable = root.hasClass(\"xscalable\"),\n",
       "            yscalable = root.hasClass(\"yscalable\");\n",
       "        var px_per_mm = root.data(\"px_per_mm\");\n",
       "        x = xscalable ? x / px_per_mm : bounds.x0;\n",
       "        y = yscalable ? y / px_per_mm : bounds.y0;\n",
       "        var w = xscalable ? 0 : width;\n",
       "        var h = yscalable ? 0 : height;\n",
       "        zoom_box = root.rect(x, y, w, h).attr({\n",
       "            \"fill\": \"#000\",\n",
       "            \"fill-opacity\": 0.25\n",
       "        });\n",
       "    },\n",
       "    update: function(root, dx, dy, _x, _y, event) {\n",
       "        var xscalable = root.hasClass(\"xscalable\"),\n",
       "            yscalable = root.hasClass(\"yscalable\");\n",
       "        var px_per_mm = root.data(\"px_per_mm\");\n",
       "        var bounds = root.plotbounds();\n",
       "        var viewbounds = root.viewportplotbounds();\n",
       "        var x = event.clientX - viewbounds.x0;\n",
       "        var y = event.clientY - viewbounds.y0;\n",
       "        if (yscalable) {\n",
       "            y /= px_per_mm;\n",
       "            y = Math.max(bounds.y0, y);\n",
       "            y = Math.min(bounds.y1, y);\n",
       "        } else {\n",
       "            y = bounds.y1;\n",
       "        }\n",
       "        if (xscalable) {\n",
       "            x /= px_per_mm;\n",
       "            x = Math.max(bounds.x0, x);\n",
       "            x = Math.min(bounds.x1, x);\n",
       "        } else {\n",
       "            x = bounds.x1;\n",
       "        }\n",
       "\n",
       "        dx = x - zoom_box.attr(\"x\");\n",
       "        dy = y - zoom_box.attr(\"y\");\n",
       "        var xoffset = 0,\n",
       "            yoffset = 0;\n",
       "        if (dx < 0) {\n",
       "            xoffset = dx;\n",
       "            dx = -1 * dx;\n",
       "        }\n",
       "        if (dy < 0) {\n",
       "            yoffset = dy;\n",
       "            dy = -1 * dy;\n",
       "        }\n",
       "        if (isNaN(dy)) {\n",
       "            dy = 0.0;\n",
       "        }\n",
       "        if (isNaN(dx)) {\n",
       "            dx = 0.0;\n",
       "        }\n",
       "        zoom_box.transform(\"T\" + xoffset + \",\" + yoffset);\n",
       "        zoom_box.attr(\"width\", dx);\n",
       "        zoom_box.attr(\"height\", dy);\n",
       "    },\n",
       "    end: function(root, event) {\n",
       "        var xscalable = root.hasClass(\"xscalable\"),\n",
       "            yscalable = root.hasClass(\"yscalable\");\n",
       "        var zoom_bounds = zoom_box.getBBox();\n",
       "        if (zoom_bounds.width * zoom_bounds.height <= 0) {\n",
       "            return;\n",
       "        }\n",
       "        var plot_bounds = root.plotbounds();\n",
       "        var xzoom_factor = 1.0,\n",
       "            yzoom_factor = 1.0;\n",
       "        if (xscalable) {\n",
       "            xzoom_factor = (plot_bounds.x1 - plot_bounds.x0) / zoom_bounds.width;\n",
       "        }\n",
       "        if (yscalable) {\n",
       "            yzoom_factor = (plot_bounds.y1 - plot_bounds.y0) / zoom_bounds.height;\n",
       "        }\n",
       "        var tx = (root.data(\"tx\") - zoom_bounds.x) * xzoom_factor + plot_bounds.x0,\n",
       "            ty = (root.data(\"ty\") - zoom_bounds.y) * yzoom_factor + plot_bounds.y0;\n",
       "        set_plot_pan_zoom(root, tx, ty,\n",
       "                root.data(\"xscale\") * xzoom_factor, root.data(\"yscale\") * yzoom_factor);\n",
       "        zoom_box.remove();\n",
       "    },\n",
       "    cancel: function(root) {\n",
       "        zoom_box.remove();\n",
       "    }\n",
       "};\n",
       "\n",
       "\n",
       "Gadfly.guide_background_drag_onstart = function(x, y, event) {\n",
       "    var root = this.plotroot();\n",
       "    var scalable = root.hasClass(\"xscalable\") || root.hasClass(\"yscalable\");\n",
       "    var zoomable = !event.altKey && !event.ctrlKey && event.shiftKey && scalable;\n",
       "    var panable = !event.altKey && !event.ctrlKey && !event.shiftKey && scalable;\n",
       "    var drag_action = zoomable ? zoom_action :\n",
       "                      panable  ? pan_action :\n",
       "                                 undefined;\n",
       "    root.data(\"drag_action\", drag_action);\n",
       "    if (drag_action) {\n",
       "        var cancel_drag_action = function(event) {\n",
       "            if (event.which == 27) { // esc key\n",
       "                drag_action.cancel(root);\n",
       "                root.data(\"drag_action\", undefined);\n",
       "            }\n",
       "        };\n",
       "        window.addEventListener(\"keyup\", cancel_drag_action);\n",
       "        root.data(\"cancel_drag_action\", cancel_drag_action);\n",
       "        drag_action.start(root, x, y, event);\n",
       "    }\n",
       "};\n",
       "\n",
       "\n",
       "Gadfly.guide_background_drag_onmove = function(dx, dy, x, y, event) {\n",
       "    var root = this.plotroot();\n",
       "    var drag_action = root.data(\"drag_action\");\n",
       "    if (drag_action) {\n",
       "        drag_action.update(root, dx, dy, x, y, event);\n",
       "    }\n",
       "};\n",
       "\n",
       "\n",
       "Gadfly.guide_background_drag_onend = function(event) {\n",
       "    var root = this.plotroot();\n",
       "    window.removeEventListener(\"keyup\", root.data(\"cancel_drag_action\"));\n",
       "    root.data(\"cancel_drag_action\", undefined);\n",
       "    var drag_action = root.data(\"drag_action\");\n",
       "    if (drag_action) {\n",
       "        drag_action.end(root, event);\n",
       "    }\n",
       "    root.data(\"drag_action\", undefined);\n",
       "};\n",
       "\n",
       "\n",
       "Gadfly.guide_background_scroll = function(event) {\n",
       "    if (event.shiftKey) {\n",
       "        // event.deltaY is either the number of pixels, lines, or pages scrolled past.\n",
       "        var actual_delta;\n",
       "        switch (event.deltaMode) {\n",
       "            case 0: // Chromium-based\n",
       "                actual_delta = -event.deltaY / 1000.0;\n",
       "                break;\n",
       "            case 1: // Firefox\n",
       "                actual_delta = -event.deltaY / 50.0;\n",
       "                break;\n",
       "            default:\n",
       "                actual_delta = -event.deltaY;\n",
       "        }\n",
       "        // Assumes 20 pixels/line to get reasonably consistent cross-browser behaviour.\n",
       "        increase_zoom_by_position(this.plotroot(), actual_delta);\n",
       "        event.preventDefault();\n",
       "    }\n",
       "};\n",
       "\n",
       "// Map slider position x to scale y using the function y = a*exp(b*x)+c.\n",
       "// The constants a, b, and c are solved using the constraint that the function\n",
       "// should go through the points (0; min_scale), (0.5; 1), and (1; max_scale).\n",
       "var scale_from_slider_position = function(position, min_scale, max_scale) {\n",
       "    if (min_scale==max_scale) { return 1; }\n",
       "    var a = (1 - 2 * min_scale + min_scale * min_scale) / (min_scale + max_scale - 2),\n",
       "        b = 2 * Math.log((max_scale - 1) / (1 - min_scale)),\n",
       "        c = (min_scale * max_scale - 1) / (min_scale + max_scale - 2);\n",
       "    return a * Math.exp(b * position) + c;\n",
       "}\n",
       "\n",
       "// inverse of scale_from_slider_position\n",
       "var slider_position_from_scale = function(scale, min_scale, max_scale) {\n",
       "    if (min_scale==max_scale) { return min_scale; }\n",
       "    var a = (1 - 2 * min_scale + min_scale * min_scale) / (min_scale + max_scale - 2),\n",
       "        b = 2 * Math.log((max_scale - 1) / (1 - min_scale)),\n",
       "        c = (min_scale * max_scale - 1) / (min_scale + max_scale - 2);\n",
       "    return 1 / b * Math.log((scale - c) / a);\n",
       "}\n",
       "\n",
       "var increase_zoom_by_position = function(root, delta_position, animate) {\n",
       "    var old_xscale = root.data(\"xscale\"),\n",
       "        old_yscale = root.data(\"yscale\"),\n",
       "        min_scale = root.data(\"min_scale\"),\n",
       "        max_scale = root.data(\"max_scale\");\n",
       "    var xposition = slider_position_from_scale(old_xscale, min_scale, max_scale),\n",
       "        yposition = slider_position_from_scale(old_yscale, min_scale, max_scale);\n",
       "    xposition += (root.hasClass(\"xscalable\") ? delta_position : 0.0);\n",
       "    yposition += (root.hasClass(\"yscalable\") ? delta_position : 0.0);\n",
       "    old_xscale = scale_from_slider_position(xposition, min_scale, max_scale);\n",
       "    old_yscale = scale_from_slider_position(yposition, min_scale, max_scale);\n",
       "    var new_xscale = Math.max(min_scale, Math.min(old_xscale, max_scale)),\n",
       "        new_yscale = Math.max(min_scale, Math.min(old_yscale, max_scale));\n",
       "    if (animate) {\n",
       "        Snap.animate(\n",
       "            [old_xscale, old_yscale],\n",
       "            [new_xscale, new_yscale],\n",
       "            function (new_scale) {\n",
       "                update_plot_scale(root, new_scale[0], new_scale[1]);\n",
       "            },\n",
       "            200);\n",
       "    } else {\n",
       "        update_plot_scale(root, new_xscale, new_yscale);\n",
       "    }\n",
       "}\n",
       "\n",
       "\n",
       "var update_plot_scale = function(root, new_xscale, new_yscale) {\n",
       "    var trans = scale_centered_translation(root, new_xscale, new_yscale);\n",
       "    set_plot_pan_zoom(root, trans.x, trans.y, new_xscale, new_yscale);\n",
       "};\n",
       "\n",
       "\n",
       "var toggle_color_class = function(root, color_class, ison) {\n",
       "    var escaped_color_class = color_class.replace(/([^0-9a-zA-z])/g,\"\\\\$1\");\n",
       "    var guides = root.selectAll(\".guide.\" + escaped_color_class + \",.guide .\" + escaped_color_class);\n",
       "    var geoms = root.selectAll(\".geometry.\" + escaped_color_class + \",.geometry .\" + escaped_color_class);\n",
       "    if (ison) {\n",
       "        guides.animate({opacity: 0.5}, 250);\n",
       "        geoms.animate({opacity: 0.0}, 250);\n",
       "    } else {\n",
       "        guides.animate({opacity: 1.0}, 250);\n",
       "        geoms.animate({opacity: 1.0}, 250);\n",
       "    }\n",
       "};\n",
       "\n",
       "\n",
       "Gadfly.colorkey_swatch_click = function(event) {\n",
       "    var root = this.plotroot();\n",
       "    var color_class = this.data(\"color_class\");\n",
       "\n",
       "    if (event.shiftKey) {\n",
       "        root.selectAll(\".colorkey g\")\n",
       "            .forEach(function (element) {\n",
       "                var other_color_class = element.data(\"color_class\");\n",
       "                if (typeof other_color_class !== 'undefined' && other_color_class != color_class) {\n",
       "                    toggle_color_class(root, other_color_class,\n",
       "                                       element.attr(\"opacity\") == 1.0);\n",
       "                }\n",
       "            });\n",
       "    } else {\n",
       "        toggle_color_class(root, color_class, this.attr(\"opacity\") == 1.0);\n",
       "    }\n",
       "};\n",
       "\n",
       "\n",
       "return Gadfly;\n",
       "\n",
       "}));\n",
       "\n",
       "\n",
       "//@ sourceURL=gadfly.js\n",
       "\n",
       "(function (glob, factory) {\n",
       "    // AMD support\n",
       "      if (typeof require === \"function\" && typeof define === \"function\" && define.amd) {\n",
       "        require([\"Snap.svg\", \"Gadfly\"], function (Snap, Gadfly) {\n",
       "            factory(Snap, Gadfly);\n",
       "        });\n",
       "      } else {\n",
       "          factory(glob.Snap, glob.Gadfly);\n",
       "      }\n",
       "})(window, function (Snap, Gadfly) {\n",
       "    var fig = Snap(\"#img-e289550e\");\n",
       "fig.select(\"#img-e289550e-5\")\n",
       "   .init_gadfly();\n",
       "fig.select(\"#img-e289550e-8\")\n",
       "   .plotroot().data(\"unfocused_ygrid_color\", \"#D0D0E0\")\n",
       ";\n",
       "fig.select(\"#img-e289550e-8\")\n",
       "   .plotroot().data(\"focused_ygrid_color\", \"#A0A0A0\")\n",
       ";\n",
       "fig.select(\"#img-e289550e-281\")\n",
       "   .plotroot().data(\"unfocused_xgrid_color\", \"#D0D0E0\")\n",
       ";\n",
       "fig.select(\"#img-e289550e-281\")\n",
       "   .plotroot().data(\"focused_xgrid_color\", \"#A0A0A0\")\n",
       ";\n",
       "fig.select(\"#img-e289550e-529\")\n",
       "   .mouseenter(Gadfly.helpscreen_visible)\n",
       ".mouseleave(Gadfly.helpscreen_hidden)\n",
       ";\n",
       "    });\n",
       "]]> </script>\n",
       "</svg>\n",
       "\n",
       "</html>\n"
      ],
      "text/plain": [
       "Plot(...)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot([x -> herbie(x) - nonherbie(x)], 1,10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here is an example of using `expm1` in a problem from finance. The function `payment` determines the payment for a loan amount of $P$ for an annual interest rate or $r$ and requiring $n$ years to pay off the loan. A simple implementation is"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "payment (generic function with 1 method)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "function payment(P::Number, r::Number, n::Integer)\n",
    "    P*r*(1+r)^n/((1+r)^n-1)\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We have\n",
    "$$\n",
    "(1+r)^n-1 = \\exp(n \\log(1+r)) - 1 = \\mathrm{expm1(n \\log(1+r))}\n",
    "$$\n",
    "Further, using $\\log(1+r) = \\mathrm{log1p(r)}$, we have\n",
    "$$\n",
    "(1+r)^n-1 =\\mathrm{expm1(n \\, \\mathrm{log1p}(r))}\n",
    "$$\n",
    "So an alternative formula for the payment is"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "payment2 (generic function with 1 method)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "function payment2(P::Number, r::Number, n::Integer)\n",
    "    w = expm1(n*log1p(r))    \n",
    "    P*r*(w+1)/w\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Does the fancy `expm1` function ever make a difference? Using `Float64` numbers and choosing $P = 10^0, r = 10^{-9}$ and $n = 2$, we have"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [],
   "source": [
    "P = 1.0e9;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0e-9"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "r = 1.0e-9"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "n = 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4.999999596298179e8"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "p1 = payment(P,r,n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5.0000000075e8"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "p2 = payment2(P,r,n)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "OK, the difference is \\$41.12 for a billion dollar loan. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "41.12018209695816"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "p2-p1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For typical home mortgage payments, the difference between the payments determined by these two functions isn't worth mentioning; for example, to purchase a fifteen-year loan for \\$350,000 home with an APR of 7\\%, we have"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [],
   "source": [
    "P,r,n = 350000.0, 0.07, 15;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "38428.118645352275"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "p1 = payment(P,r,n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "38428.1186453523"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "p2 = payment2(P,r,n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-2.1827872842550278e-11"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "p1-p2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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  "kernelspec": {
   "display_name": "Julia 1.10.0",
   "language": "julia",
   "name": "julia-1.10"
  },
  "language_info": {
   "file_extension": ".jl",
   "mimetype": "application/julia",
   "name": "julia",
   "version": "1.10.0"
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