{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Quadruple precision computations (new in 1.3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Qudruple precision floats\n",
    "The class real128 in pyaudi ia a multiple precision float, basically wrapping the GNU C++ *__float128* type. It can be used outside of the whole differential algebra machinery (the gduals) to perform quadruple precision computations"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from pyaudi import real128\n",
    "from pyaudi import sin, acos"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3.33333333333333333333333333333333317e-01\n"
     ]
    }
   ],
   "source": [
    "# Some simple math\n",
    "x = real128(\"3\")\n",
    "print(1/x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.41120008059867222100744802808110299e-01\n"
     ]
    }
   ],
   "source": [
    "# Trigonometry\n",
    "print(sin(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Double precision:\t 3.141592653589793\n",
      "Quad precision:\t\t 3.14159265358979323846264338327950280e+00\n"
     ]
    }
   ],
   "source": [
    "# The value of pi in double precision\n",
    "print(\"Double precision:\\t\", 2*acos(0))\n",
    "# The value of pi in quadruple precision\n",
    "print(\"Quad precision:\\t\\t\", 2*acos(real128(\"0\")))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Qudruple precision gdual\n",
    "The gdual can also perform computations in quadruple precision.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "from pyaudi import gdual_real128 as gdual\n",
    "from pyaudi import sin, acos, sqrt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "A quadruple precision gdual computation: \n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "\\[ -4.35605126899062478098382207778725168e-08{dy}-3.07827623008670817856190093496965737e-08{dx}+1.13010764054002057997221799187876804e-07{dy}^{2}+9.23482869026012453568570280490897241e-09+1.45201708966354159366127402592908378e-07{dx}{dy}+5.64350642182563166069681838077770461e-08{dx}^{2}+\\mathcal{O}\\left(3\\right) \\]"
      ],
      "text/plain": [
       "-4.35605126899062478098382207778725168e-08*dy-3.07827623008670817856190093496965737e-08*dx+1.13010764054002057997221799187876804e-07*dy**2+9.23482869026012453568570280490897241e-09+1.45201708966354159366127402592908378e-07*dx*dy+5.64350642182563166069681838077770461e-08*dx**2"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def easy_fun(x,y):\n",
    "    return (1/x/y)**10\n",
    "\n",
    "x = gdual(\"3\", \"x\", 2)\n",
    "y = gdual(\"2.12\", \"y\", 2)\n",
    "res = easy_fun(x,y)\n",
    "print(\"A quadruple precision gdual computation: \")\n",
    "res"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "All quantities, that is the function value and its derivatives are in quadruple precision. Lets extract $\\frac{\\partial^2 f}{\\partial x \\partial y}$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "value:  1.45201708966354159366127402592908378e-07\n",
      "type:  <class 'pyaudi.core.real128'>\n"
     ]
    }
   ],
   "source": [
    "der = res.get_derivative({\"dx\": 1, \"dy\": 1})\n",
    "print(\"value: \", der)\n",
    "print(\"type: \", type(der))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<img src=\"http://images6.fanpop.com/image/photos/38700000/That-s-All-Folks-the-looney-tunes-show-38740983-500-281.jpg\"/>"
      ],
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from IPython.display import Image\n",
    "from IPython.core.display import HTML \n",
    "Image(url= \"http://images6.fanpop.com/image/photos/38700000/That-s-All-Folks-the-looney-tunes-show-38740983-500-281.jpg\")"
   ]
  }
 ],
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