{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Gaussian Error Propagation with SymPy"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Klaus Reygers, 2020"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "from IPython.display import display, Latex"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "def gaussian_error_propagation(f, vars):\n",
    "    \"\"\"\n",
    "    f: formula (sympy expression)\n",
    "    vars: list of independent variables and corresponding uncertainties \n",
    "    [(x1, sigma_x1), (x2, sigma_x2), ...]\n",
    "    \"\"\"\n",
    "    sum = S(0) # empty sympy expression\n",
    "    for (x, sigma) in vars:\n",
    "        sum += diff(f, x)**2 * sigma**2 \n",
    "    return sqrt(simplify(sum))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Show usage for a simple example: Volume of a cylinder with radius $r$ and height $h$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "r, h, sigma_r, sigma_h = symbols('r, h, sigma_r, sigma_h', positive=True)\n",
    "V = pi * r**2 * h # volume of a cylinder"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$V = \\pi h r^{2}, \\, \\sigma_V = \\pi r \\sqrt{4 h^{2} \\sigma_{r}^{2} + r^{2} \\sigma_{h}^{2}}$"
      ],
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sigma_V = gaussian_error_propagation(V, [(r, sigma_r), (h, sigma_h)])\n",
    "display(Latex(f\"$V = {latex(V)}, \\, \\sigma_V = {latex(sigma_V)}$\"))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plug in some numbers and print the calculated volume with its uncertaity:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "r_meas = 3 # cm\n",
    "sigma_r_meas = 0.1 # cm\n",
    "h_meas = 5 # cm\n",
    "sigma_h_meas = 0.1 # cm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$V = (141.4 \\pm 9.8) \\, \\mathrm{cm}^3$$"
      ],
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "central_value = V.subs([(r,r_meas), (h, h_meas)]).evalf()\n",
    "sigma = sigma_V.subs([(r, r_meas), (sigma_r, sigma_r_meas), (h, h_meas), (sigma_h, sigma_h_meas)]).evalf()\n",
    "display(Latex(f\"$$V = ({central_value:0.1f} \\pm {sigma:.1f}) \\, \\mathrm{{cm}}^3$$\"))"
   ]
  }
 ],
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   "pygments_lexer": "ipython3",
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