PFS:-Exponenciación-binaria.ipynb

{
 "metadata": {
  "name": "Exponenciaci\u00f3n binaria"
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Hoy voy a iniciar una serie nueva de entradas que se llamar\u00e1 'python, fuente de sabidur\u00eda'. \u00bfQu\u00e9 pretende esta serie? La idea b\u00e1sica es mostrar la influencia y realimentaci\u00f3n entre las matem\u00e1ticas y la programaci\u00f3n. La idea e inspiraci\u00f3n para estas entradas sali\u00f3 despu\u00e9s de leer esta [entrada](http://ch3m4.org/blog/2013/08/14/estudio-funcion-factorial-numpy/).\n",
      "\n",
      "Inauguraremos esta entrada con una forma de calcular la potencia entera de un n\u00famero entero. Si tuviera que crear una funci\u00f3n que calculase esto la primera idea que se me ocurre es usar la fuerza bruta (soy as\u00ed de primario)..."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def pow_fuerzabruta(x,n):\n",
      "    y = x\n",
      "    for i in range(1,n):\n",
      "        y = y * x\n",
      "    return y"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 45
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Con la anterior funci\u00f3n, la cual no he pensado mucho, lo que estamos haciendo son `n-1` multiplicaciones. A medida que aumente el exponente aumentar\u00e1 el tiempo de c\u00e1lculo. Ve\u00e1moslo en una gr\u00e1fica:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from matplotlib import pyplot as plt\n",
      "n = [5, 25, 100, 1000, 10000, 100000]       # exponente\n",
      "t = [1.13, 3.93, 17.8, 330, 15000, 1260000] # t en microsegundos\n",
      "plt.xscale('log')\n",
      "plt.yscale('log')\n",
      "plt.plot(n,t)\n",
      "plt.xlabel('valor del exponente n')\n",
      "plt.ylabel('t en microsegundos')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 61,
       "text": [
        "<matplotlib.text.Text at 0x3831170>"
       ]
      },
      {
       "output_type": "display_data",
       "png": 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wfJGmpMTbyspg7lx4/HEYNgyefNJa3BbxJ5qSEvGwbdusw3fGwMqV0Lmz3RGJ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\nIuISJQwREXGJEoaIiLhECUNERFzy/wFQUdnRp69ZewAAAABJRU5ErkJggg==\n"
      }
     ],
     "prompt_number": 61
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "[Nota: la escala de los ejes en la anterior figura es logar\u00edtmica]\n",
      "\n",
      "Esto, cuando usamos n\u00fameros muy grandes, no es muy deseable. Por ello, vamos a mostrar el algortimo de hoy, la exponenciaci\u00f3n binaria, tambi\u00e9n llamado algoritmo de izquierda a derecha binario, que nos permitir\u00e1 reducir de forma dr\u00e1stica el n\u00famero de operaciones a realizar. Su uso en Python lo pod\u00e9is ver, por ejemplo, en la siguiente implementaci\u00f3n de [`long_pow`](http://svn.python.org/view/python/tags/r271/Objects/longobject.c?view=markup) (ver a partir de l\u00ednea 3376).\n",
      "\n",
      "Podemos pensar que estas cosas son muy modernas y sofisticadas pero, en este caso, no es as\u00ed. Los primeros or\u00edgenes de esta implementaci\u00f3n para calcular potencias parece que ser\u00eda anterior a 2200 a\u00f1os atr\u00e1s. Seg\u00fan el libro de [Donald Knuth titulado *Seminumerical Algorithms, volume 2 of The Art of Computer Programming*, p\u00e1gina 441](http://www.goodreads.com/book/show/112246.Art_of_Computer_Programming_Volume_2), podemos leer:\n",
      "\n",
      "> El m\u00e9todo es bastante antiguo; apareci\u00f3 antes del a\u00f1o 200 A.C. en *Pingala's Hindu classic Chandah-sutra* [ver B. Datta y A.N. Singh, *History of Hindu Mathematics 1*, 1935]; sin embargo, parece que existen otras referencias a este m\u00e9todo fuera de la India a lo largo de los 1000 a\u00f1os posteriores. Una discusi\u00f3n muy clara sobre como calcular `2n` de forma eficiente para valores de `n` arbitrarios ser\u00eda debida a al-Uqlidisi de Damscus en 952 D.C.; pod\u00e9is ver *The Arithmetic of al-Uglidisi* por A.S. Saidan (1975), p. 341-342, donde las ideas generales est\u00e1n ilustradas para el caso de n=51. Ver tambi\u00e9n *al-Biruni's Chronology of Ancient Nations (1879)*, p. 132-136; durante el siglo XI, los avances del mundo \u00e1rabe tuvieron una gran influencia.\n",
      "\n",
      "M\u00e1s informaci\u00f3n hist\u00f3rica sobre los or\u00edgenes la pod\u00e9is encontrar en este [art\u00edculo](http://arxiv.org/abs/math/0703658).\n",
      "\n",
      "Vamos a ver como funciona, m\u00e1s o menos, este algoritmo. La idea b\u00e1sica detr\u00e1s del mismo es la siguiente (ver [wikipedia](http://es.wikipedia.org/wiki/Exponenciaci%C3%B3n_binaria) para una explicaci\u00f3n m\u00e1s completa):\n",
      "\n",
      "Las potencias m\u00e1s r\u00e1pidas de calcular son las que tienen la siguiente forma ${x^2}^n$. Estos es debido a que se pueden encontrar las potencias calculando el cuadrado del n\u00famero de forma repetida.\n",
      "\n",
      "Vamos a explicar el mismo ejemplo que al-Uqlidisi de Damscus (unos diez siglos m\u00e1s tarde...) considerando un n\u00famero elevado a 51. Por tanto, puesdo calcular $5^2$ con solo una multiplicaci\u00f3n, $5^4 = {5^2}^2$ con solo dos multiplicaciones, $5^8 = {{5^2}^2}^2$ en tres multiplicaciones,...\n",
      "\n",
      "Creo que vais viendo por donde van los tiros. El algoritmo explicado de forma muy b\u00e1sica ser\u00eda para el ejemplo $5^{51}$:\n",
      "\n",
      "* Obtener el exponente como suma de potencias de 2\n",
      "\n",
      "En este caso ser\u00eda $51 = 32 + 16 + 2 + 1$\n",
      "\n",
      "* Encontrar la base de potencias de dos que aparecen en el exponente\n",
      "\n",
      "$5^1 = 5$\n",
      "\n",
      "$5^2 = 5 \u00b7 5 = 25$\n",
      "\n",
      "$5^4 = 25 \u00b7 25 = 625$\n",
      "\n",
      "$5^8 = 625 \u00b7 625 = 390625$\n",
      "\n",
      "$5^{16} = 390625 \u00b7 390625 = 152587890625$\n",
      "\n",
      "$5^{32} = 152587890625 \u00b7 152587890625 = 23283064365386962890625$\n",
      "\n",
      "* Y, por \u00faltimo, se multiplicar\u00edan de forma conjunta las potencias ya encontradas\n",
      "\n",
      "$5^{51} = 5^{32} \u00b7 5^{16} \u00b7 5^2 \u00b7 5 = 444089209850062616169452667236328125$\n",
      "\n",
      "De esta forma, en lugar de multiplicar 5 por si mismo 50 veces solo hemos de realizar 8 multiplicaciones.\n",
      "\n",
      "El que vamos a implementar es un algoritmo de exponenciaci\u00f3n binaria pero de derecha a izquierda. Pod\u00e9is ver los detalles en la [wikipedia](http://en.wikipedia.org/wiki/Modular_exponentiation#Right-to-left_binary_method)."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def my_pow(x,n):\n",
      "    r = 1\n",
      "    y = x\n",
      "    while n > 1:\n",
      "        if n%2:\n",
      "            r = r * y\n",
      "        n = int(n/2)\n",
      "        y = y * y\n",
      "        print(n,r,y)\n",
      "    r = r*y\n",
      "    return r"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Si calculamos los tiempos que hemos obtenido con la nueva funci\u00f3n (`my_pow`) y los comparamos con la anterior implementaci\u00f3n (`pow_fuerzabruta`) veremos la ganancia que hemos obtenido:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from matplotlib import pyplot as plt\n",
      "n = [5, 25, 100, 1000, 10000, 100000]       # exponente\n",
      "t_bruta = [1.13, 3.93, 17.8, 330, 15000, 1260000] # t  para la primera funci\u00f3n\n",
      "t_bin = [1.34, 2.52, 3.68, 19.8, 765, 32800] # t para la segunda funci\u00f3n\n",
      "plt.plot(n,t_bruta, label='bruta')\n",
      "plt.plot(n,t_bin, label='bin')\n",
      "plt.legend()\n",
      "plt.xlabel('valor del exponente n')\n",
      "plt.ylabel('t en microsegundos')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 86,
       "text": [
        "<matplotlib.text.Text at 0x4fbc970>"
       ]
      },
      {
       "output_type": "display_data",
       "png": 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HpdxBg0mrpqaGgoIC3n//fe6++26AJv08Syn16zZtgu7doaQEvv0W7rvP2REp\nd9Jgl/fnnnuO+Ph4br75ZiIjI9m/fz8Wi8URsSmlXEhZmTGo7QcfwJtvwj33ODsi5Y4uqiOGu9CO\nGErVtWEDPPQQxMbCX/4ClzBWtnIDLtERIycnh/vuu4927drRrl07Bg8eXGdKEKVU01VaCuPHG2MF\nvv02/O//asJSztVg0ho7diwJCQnk5+eTn5/PgAEDGDt2rCNiU0o50QcfQGgo/O53Rs/AuDhnR6TU\nBdweDA8P5+uvv26wrSnR24PKnRUVwdSpsGWLUVn16+fsiFRj4RK3B729vVmyZAk1NTVUV1fz7rvv\n0rZtW7sGpZRyjlWrICwMvL2N0dk1YSlX02CldfDgQR599FG++OILAG666SbeeOMNOnbs6JAAnUEr\nLeVujhyByZPhq69g4UK4+WZnR6QaI0dcO3+1y3t1dTV/+tOf+Oc//2nXIJRSziECy5cbtwNHj4ZF\ni+CKK5wdlVLn96tJy8PDg4MHD1JRUVFnehKlVONXUGDMc5WdDWvXQmSksyNSqmENvlzcuXNnbrnl\nFhISErjyyisBowR8/PHH7R6cUuryEzEGt50+3Xj3avlyo4egUo1Bg0nL398ff39/amtr+eWXXxwR\nk1LKTnJzjXeu8vIgPR169nR2REpdHB0Rox7aEUM1NSKQkgJPPQWPPQYzZkCLFs6OSjU1LtHlPTY2\nltLSUtvvxcXFxMfH2zUopdTlc+CA8WLwW2/Bxx/Ds89qwlKNV4NJ6+jRo3idMW5LmzZtKCwstGtQ\nSqnfrrYWFiyAG26AO+6ArVuNd7CUaswaTFrNmzfn4MGDtt8PHDhAs2YNrvarSktLGTJkCF27diUk\nJISsrCyKi4uJjY0lMDCQuLi4OtXdnDlzsFgsBAcHk5GRYWvfsWMHYWFhWCwWpkyZYmuvqKggMTER\ni8VCVFRUnfhTU1MJDAwkMDCQxYsX/6bjUMpV/fvfcPvt8O67sHmzcTvQo8En2Eo1AtKA9evXS4cO\nHeT++++X+++/Xzp06CDr169vaLVfNXr0aElJSRERkaqqKiktLZXp06fL3LlzRUQkKSlJZsyYISIi\ne/bskfDwcKmsrBSr1Sr+/v5SW1srIiIRERGSlZUlIiL9+/e3xbVgwQKZNGmSiIikpaVJYmKiiIgU\nFRVJly5dpKSkREpKSmyfz3YBp0Upl1RdLTJ/voi3t/Hf1dXOjki5E0dcOy9oD0eOHJG1a9fKP//5\nTzl69OhXzZGSAAAgAElEQVRv2mFpaal07tz5nPagoCA5fPiwiIgUFBRIUFCQiIi89NJLkpSUZFsu\nPj5etm7dKvn5+RIcHGxrX7ZsmTz88MO2Zb744gsRMZJi27ZtRURk6dKlMnHiRNs6Dz/8sCxbtuyc\nWDRpqcbou+9EbrxR5NZbRfbtc3Y0yh054tp53hsG3333HV27dmXHjh2YTCauu+46AA4dOsShQ4fo\n1avXJVV2VquVdu3aMXbsWL7++mt69+7Nq6++SmFhIT4+PgD4+PjYnpvl5+cTFRVlW99sNpOXl4en\npydms9nW7ufnR15eHgB5eXl06NABMF6QbtWqFUVFReTn59dZ5/S26jNz5kzb5+joaKKjoy/peJWy\nt+pqmDcPXnkFZs0yXhj+jXfwlbogmZmZZGZmOnSf501a8+fP5+233+aJJ57AZDKd8/2mTZsuaYfV\n1dXs3LmTv/3tb0RERDB16lSSkpLqLGMymerdpyOdmbSUclXffgvjxkHLlvDll9C5s7MjUu7k7D/o\nZ82aZfd9njdpvf322wCXPYuazWbMZjMREREADBkyhDlz5uDr68vhw4fx9fWloKCA9u3bA0YFlZOT\nY1s/NzcXs9mMn59fnckoT7efXufQoUNcd911VFdXc+zYMby9vfHz86tzPDk5Odx+++2X9fiUcoSq\nKpg7F157DV56CSZMACf/naeUQzR4E6G6upo1a9bw2muvMX/+fObNm8f8+fMveYe+vr506NCBffv2\nAfDhhx/SrVs3BgwYQGpqKmD08Lv33nsBSEhIIC0tjcrKSqxWK9nZ2URGRuLr60vLli3JyspCRFiy\nZAkDBw60rXN6WytXriQmJgaAuLg4MjIyKC0tpaSkhI0bN+o7Z6rR2bXLGCdwyxbYuRMefFATlnIf\nDXaCHTBgAFdccQVhYWG/uav7aW+88Qb3338/lZWV+Pv7884771BTU8OwYcNISUmhU6dOvP/++wCE\nhIQwbNgwQkJC8PDwIDk52XbrMDk5mQceeICTJ09y1113ceeddwIwfvx4Ro0ahcViwdvbm7S0NMB4\nx+zZZ5+1VXnPP/98nXfQlHJlFRXw4ovGS8KvvGKMyq7JSrmbBodx6t69O7t373ZUPC5Bh3FSrubL\nL2HsWPD3hzffhFP9opRyKS4xjFNcXBwbNmywaxBKqfqdPGm8GDxgADz9NKxerQlLubcGbw/edNNN\n3HfffdTW1uLp6QkY2fTnn3+2e3BKubMtW4yegd27w+7dcKpvklJurcHbg506dWLt2rWEhoZetmda\nrk5vDypnOnHCqKqWL4c33oDBg50dkVIXxiVuD3bs2JFu3bq5TcJSypk++cSorI4ehW++0YSl1Nku\naObifv360b9/f1qcms9AZy5W6vIqK4M//hHWrIHkZEhIcHZESrmmC0panTt3prKyksrKSkfEpJRb\n2bjReNfq9tuNES70LQylzk9nLq6HPtNSjnDsGDz5JGRkGO9enXrNUKlGyyWeaSmlLr916yA0FJo3\nN55dacJS6sLotHBKOVBxMUybZkzMmJpq3BJUSl04rbSUcpDVq43p7lu1Mt670oSl1MVrsNI6cuQI\nb7/9NgcOHKC6uhow7lsuXLjQ7sEp1RQcPQqPPQY7dkBaGvTt6+yIlGq8GkxaAwcO5NZbbyU2Ntb2\nrpaz57pSqjEQgRUrYMoU+MMfICUFrrzS2VEp1bg12HuwR48efPXVV46KxyVo70H1Wx0+DP/1X/Dd\nd/DOO9Cnj7MjUsr+XKL34D333MO//vUvuwahVFMhAu++C+HhEBxszHelCUupy6fBSuvqq6/mxIkT\ntGjRwm0GzNVKS12KvDyYOBEOHjSqq969nR2RUo7lEpXWL7/8Qm1tLeXl5ZSVlVFWVtakE5ZSF0sE\nFi6Enj3hhhtg+3ZNWErZS4MdMWpra3nvvfewWq0899xzHDp0iMOHDxMZGemI+JRyaYcOGUMw/fQT\nfPihMditUsp+Gqy0HnnkEbZu3crSpUsB43bhI488YvfAlHJltbXwP/9jVFTR0ZCVpQlLKUdosNLK\nyspi165d9OzZE4A2bdpQVVVl98CUclU//ggTJhjzXn3yCYSEODsipdxHg5VWixYtqKmpsf1+9OhR\nnVtLuaXaWnj9dYiMhLvvhs8/14SllKM1WGlNnjyZ++67jyNHjvCnP/2JlStX8uKLLzoiNqVcxr59\nMG4cmEywZQsEBjo7IqXc0wVNTfLdd9/x0UcfARATE0PXrl3tHpgzaZd3dVpNDcyfD3PnwvPPGy8M\n640GperniGunzqdVD01aCmDvXhg7Fq6+Gt5+G7p0cXZESrk2l3hPy15qamro2bMnAwYMAKC4uJjY\n2FgCAwOJi4ujtLTUtuycOXOwWCwEBweTkZFha9+xYwdhYWFYLBamTJlia6+oqCAxMRGLxUJUVBQH\nDx60fZeamkpgYCCBgYEsXrzYAUeqGpuqKnjpJbjtNuOW4IcfasJSylU4LWm99tprhISE2AbfTUpK\nIjY2ln379hETE0NSUhIAe/fuZfny5ezdu5f09HQeeeQRWyafNGkSKSkpZGdnk52dTXp6OgApKSl4\ne3uTnZ3NtGnTmDFjBmAkxhdeeIFt27axbds2Zs2aVSc5KvX118awS59+aozK/vDDxnMspZRraDBp\nnb7gN9R2MXJzc1m3bh0TJkywJaC1a9cyZswYAMaMGcPq1asBWLNmDSNGjMDT05NOnToREBBAVlYW\nBQUFlJWV2V5yHj16tG2dM7c1ePBg2/O4DRs2EBcXh5eXF15eXsTGxtoSnXJvlZUwcybExsLkybB+\nPXTs6OyolFJna7D3YEZGBnPnzq3Ttm7dunPaLsa0adN45ZVX6gwHVVhYiI+PDwA+Pj4UFhYCkJ+f\nT1RUlG05s9lMXl4enp6emM1mW7ufnx95eXkA5OXl0aFDBwA8PDxo1aoVRUVF5Ofn11nn9LbqM3Pm\nTNvn6OhooqOjL/l4lWvbscN4dnX99bBrF/j5OTsipRqHzMxMMjMzHbrP8yatN998k+TkZPbv309Y\nWJitvaysjJtvvvmSd/jBBx/Qvn17evbsed6DNZlMTp+z68ykpZqm8nKYNcsYN3D+fBg5Um8FKnUx\nzv6DftasWXbf53mT1siRI+nfvz9//OMfmTt3ru023jXXXIO3t/cl73DLli2sXbuWdevWUV5ezs8/\n/8yoUaPw8fHh8OHD+Pr6UlBQQPv27QGjgsrJybGtn5ubi9lsxs/Pj9zc3HPaT69z6NAhrrvuOqqr\nqzl27Bje3t74+fnVSZQ5OTncrnOeu6UvvjA6WYSEwO7dcKrIV0q5OnGizMxMueeee0REZPr06ZKU\nlCQiInPmzJEZM2aIiMiePXskPDxcKioq5Mcff5QuXbpIbW2tiIhERkbKF198IbW1tdK/f39Zv369\niIgsWLBAJk6cKCIiy5Ytk8TERBERKSoqks6dO0tJSYkUFxfbPp/NyadF2dHx4yKPPy7i6yvy/vvO\njkappsUR184Gn2nZ2+nbgH/84x8ZNmwYKSkpdOrUiffffx+AkJAQhg0bRkhICB4eHiQnJ9vWSU5O\n5oEHHuDkyZPcdddd3HnnnQCMHz+eUaNGYbFY8Pb2Ji0tDTDGTXz22WeJiIgA4Pnnn8fLy8vRh6yc\n5NNPYfx4iIiAb76Btm2dHZFS6mLpy8X10JeLm5ZffoGnnoJVqyA5GQYOdHZESjVNTfrlYqUc4aOP\nICwMysrg2281YSnV2Dn99qBS9vDzzzB9uvG+1VtvQf/+zo5IKXU5aKWlmpz0dAgNBRHj2ZUmLKWa\nDq20VJNRUgKPPw6Zmca7V3fc4eyIlFKXm1ZaqklYu9Z4dnXVVUZ1pQlLqaZJKy3VqBUVwWOPwbZt\nsHQp3HqrsyNSStmTVlqq0Vq50nh25etrjM6uCUuppk8rLdXoFBbCo48aXdhXrYIbb3R2REopR9FK\nSzUaIsYtwO7dISDAGJFdE5ZS7kUrLdUo5OfDpEnw44/wr3/BDTc4OyKllDNopaVcmggsWgQ9ehg/\nO3ZowlLKnWmlpVxWTg489JDxDCsjw0haSin3ppWWcjki8Pe/Q69ecMstkJWlCUspZdBKS7kUqxUm\nTDAGuM3MhG7dnB2RUsqVaKWlXEJtLfztbxAZCXfeCVu2aMJSSp1LKy3ldNnZxuSMNTXw2WcQFOTs\niJRSrkorLeU0NTUwf77xrtXgwcbMwpqwlFK/Rist5RTffQfjxsHvf290tPD3d3ZESqnGQCst5VDV\n1TBnjjFO4OjRxszCmrCUUhdKKy3lMN98A2PHQps2sH07XH+9syNSSjU2Wmkpu6ushBdegJgYeOQR\n2LBBE5ZS6tJopaXsaudOo7rq0MEY4NbPz9kRKaUaM620lF1UVMAzz0D//vDkk/DPf2rCUkr9dg5P\nWjk5OfTr149u3boRGhrK66+/DkBxcTGxsbEEBgYSFxdHaWmpbZ05c+ZgsVgIDg4mIyPD1r5jxw7C\nwsKwWCxMmTLF1l5RUUFiYiIWi4WoqCgOHjxo+y41NZXAwEACAwNZvHixA47Y/WRlGUMw7d1rTM44\nahSYTM6OSinVJIiDFRQUyK5du0REpKysTAIDA2Xv3r0yffp0mTt3roiIJCUlyYwZM0REZM+ePRIe\nHi6VlZVitVrF399famtrRUQkIiJCsrKyRESkf//+sn79ehERWbBggUyaNElERNLS0iQxMVFERIqK\niqRLly5SUlIiJSUlts9nc8JpaRJOnBB58kkRHx+RtDSRU/8zKaXchCOunQ6vtHx9felxavTTq6++\nmq5du5KXl8fatWsZM2YMAGPGjGH16tUArFmzhhEjRuDp6UmnTp0ICAggKyuLgoICysrKiIyMBGD0\n6NG2dc7c1uDBg/noo48A2LBhA3FxcXh5eeHl5UVsbCzp6ekOPf6m6vPPjUFtc3KMXoKJiVpdKaUu\nP6d2xDhw4AC7du2iT58+FBYW4uPjA4CPjw+FhYUA5OfnExUVZVvHbDaTl5eHp6cnZrPZ1u7n50de\nXh4AeXl5dOjQAQAPDw9atWpFUVER+fn5ddY5va36zJw50/Y5Ojqa6Ojoy3LMTc3x4/CnP8HKlcbY\ngffd5+yIlFKOkpmZSWZmpkP36bSk9csvvzB48GBee+01rrnmmjrfmUwmTE7+M/3MpKXqt2mTMSL7\nzTcb1VWbNs6OSCnlSGf/QT9r1iy779MpvQerqqoYPHgwo0aN4t577wWM6urw4cMAFBQU0L59e8Co\noHJycmzr5ubmYjab8fPzIzc395z20+scOnQIgOrqao4dO4a3t/c528rJyalTeakLU1YGkyYZI1q8\n/josXqwJSynlGA5PWiLC+PHjCQkJYerUqbb2hIQEUlNTAaOH3+lklpCQQFpaGpWVlVitVrKzs4mM\njMTX15eWLVuSlZWFiLBkyRIGDhx4zrZWrlxJTEwMAHFxcWRkZFBaWkpJSQkbN24kPj7ekYff6GVk\nQGgoVFXBt9/C3Xc7OyKllFuxe1ePs2zevFlMJpOEh4dLjx49pEePHrJ+/XopKiqSmJgYsVgsEhsb\nW6dX3+zZs8Xf31+CgoIkPT3d1r59+3YJDQ0Vf39/mTx5sq29vLxchg4dKgEBAdKnTx+xWq227xYu\nXCgBAQESEBAgixYtqjdGJ5wWl1dSIjJunMj114tkZDg7GqWUK3LEtdN0akfqDCaTCT0t//HBB8bt\nwIQESEqCsx5BKqUU4Jhrpw7jpM6ruBimTDFmEV6yBLQDpVLK2XQYJ1WvVauMZ1dt28Lu3ZqwlFKu\nQSstVcfRo/Doo/DVV7BihdGdXSmlXIVWWgoAEUhLg7Aw6NTJSFqasJRSrkYrLUVBgTHPVXY2rF0L\np0bGUkopl6OVlhsTMV4M7tHDeH61Y4cmLKWUa9NKy03l5sLDD0NeHqSnQ8+ezo5IKaUappWWmxGB\n//1fI0lFRcGXX2rCUko1HlppuZEDB+DBB6G0FD7+2Oh0oZRSjYlWWm6gthaSkyEiAu64A7Zu1YSl\nlGqctNJq4vbvh/HjobISNm+G4GBnR6SUUpdOK60mqqYGXn3VeG41cKAmLKVU06CVVhP0/fdGdeXh\nYdwKDAhwdkRKKXV5aKXVhFRXw8svQ9++MHKkMbOwJiylVFOilVYT8e23MG4ctGpldGPv1MnZESml\n1OWnlVYjV1UFL74It98ODz1kzCysCUsp1VRppdWIffUVjB0L115rDMHUoYOzI1JKKfvSSqsRqqiA\n556DuDiYOhX+9S9NWEop96CVViPz5ZdGdRUQAF9/bVRZSinlLjRpNRLl5TBzJixaZLx/lZgIJpOz\no1JKKcfSpNUIbNli9Azs3h1274b27Z0dkVLqUogItVJLdW01VbVVVNdWG59rzvh8qr2+tktul9+w\n7kW0O4JJRMQhe2pETCYTrnBaTpyAp5+G5cvhjTdg8GBnR6TU5XcxF/KLbXeFC/nZ7c1MzfBo5oFn\nM088mnkYn5uf8dke7Q7aV9ur2tr92umWlVZ6ejpTp06lpqaGCRMmMGPGDGeHVEdlJXz2mdGFPSoK\nvvkGvL2dE0tmZibR0dHO2bmLceS5sOeF/Ffb61mmvraCbwrw6urV6C/kV3pciUeL37bNr7/4mqhb\noi54+WYm7f/2W7hd0qqpqeHRRx/lww8/xM/Pj4iICBISEujatatd9ldVBcXF8NNP5/4UFdXfXl5u\nvGv117/CgAF2CeuCuVrSqqmtcdqF/JNFnxBZEWnfv/5d5ULewLqrNq1iZI+Rly2GxnwhX/fNOsIG\n67QJjuJ2SWvbtm0EBATQ6dQbuMOHD2fNmjUXlLSqq40EdL5kU1/78ePQpg20bWtUS23b/ufHbDam\nuj/9++nvW7a8uE4Wpy/kF3shvZBld+TvYMG2BRf0F7kj2k0m0zkXQEddyK/wuAJzS7PDbrm48oX8\n+3bfc2/wvc4OQ7kht0taeXl5dDjjpSaz2UxWVtY5y7Wbcg8iUFMjVFYZP9XVgqen4Nni1I/nGf/t\nJXi0Ezw8BQ8PwctTaOshNG8unP5PpQh5CLlyqkUEKRck59TPme3Ib7qQn+/CeLEX0YJfCtj7095L\n/ou8KV3Iqz6u4vEbH3fa/pVSbtgR4x//+Afp6em8/fbbALz77rtkZWXxxhtv2JYxaV9ypZS6JNoR\n4zLz8/MjJyfH9ntOTg5ms7nOMm6Wx5VSqtFw3ZvmdnLDDTeQnZ3NgQMHqKysZPny5SQkJDg7LKWU\nUhfA7SotDw8P/va3vxEfH09NTQ3jx4+3W89BpZRSl5fbVVoA/fv354cffuDf//43Tz31lK09PT2d\n4OBgLBYLc+fOdWKEl1dOTg79+vWjW7duhIaG8vrrrwNQXFxMbGwsgYGBxMXFUVpaaltnzpw5WCwW\ngoODycjIsLXv2LGDsLAwLBYLU6ZMsbVXVFSQmJiIxWIhKiqKgwcPOu4AL0FNTQ09e/ZkwKl3Ctz1\nXJSWljJkyBC6du1KSEgIWVlZbnsu5syZQ7du3QgLC2PkyJFUVFS4zbkYN24cPj4+hIX9p+u+o449\nNTWVwMBAAgMDWbx4ccPBihIRkerqavH39xer1SqVlZUSHh4ue/fudXZYl0VBQYHs2rVLRETKysok\nMDBQ9u7dK9OnT5e5c+eKiEhSUpLMmDFDRET27Nkj4eHhUllZKVarVfz9/aW2tlZERCIiIiQrK0tE\nRPr37y/r168XEZEFCxbIpEmTREQkLS1NEhMTHXqMF2vevHkycuRIGTBggIiI256L0aNHS0pKioiI\nVFVVSWlpqVueC6vVKp07d5by8nIRERk2bJgsWrTIbc7Fp59+Kjt37pTQ0FBbmyOOvaioSLp06SIl\nJSVSUlJi+/xrNGmdsmXLFomPj7f9PmfOHJkzZ44TI7KfgQMHysaNGyUoKEgOHz4sIkZiCwoKEhGR\nl156SZKSkmzLx8fHy9atWyU/P1+Cg4Nt7cuWLZOHH37YtswXX3whIsbFr23bto46nIuWk5MjMTEx\n8vHHH8s999wjIuKW56K0tFQ6d+58Trs7nouioiIJDAyU4uJiqaqqknvuuUcyMjLc6lxYrdY6ScsR\nx7506VKZOHGibZ2HH35Yli1b9qtxuuXtwfrU9/5WXl6eEyOyjwMHDrBr1y769OlDYWEhPj4+APj4\n+FBYWAhAfn5+nR6Vp8/F2e1+fn62c3Tm+fPw8KBVq1YUFxc76rAuyrRp03jllVdo1uw///d3x3Nh\ntVpp164dY8eOpVevXjz44IMcP37cLc9FmzZteOKJJ+jYsSPXXXcdXl5exMbGuuW5OM3ex15UVHTe\nbf0aTVqnuMO7Wb/88guDBw/mtdde45prrqnznclkcotz8MEHH9C+fXt69ux53lcb3OVcVFdXs3Pn\nTh555BF27tzJVVddRVJSUp1l3OVc7N+/n1dffZUDBw6Qn5/PL7/8wrvvvltnGXc5F/VxpWPXpHXK\nhby/1ZhVVVUxePBgRo0axb33GsPv+Pj4cPjwYQAKCgpof2rOk7PPRW5uLmazGT8/P3Jzc89pP73O\noUOHAONieOzYMdq0aeOQY7sYW7ZsYe3atXTu3JkRI0bw8ccfM2rUKLc8F2azGbPZTEREBABDhgxh\n586d+Pr6ut252L59OzfddBPe3t54eHgwaNAgtm7d6pbn4jR7/5vw9va+pOuuJq1TmvL7WyLC+PHj\nCQkJYerUqbb2hIQEUlNTAaMHz+lklpCQQFpaGpWVlVitVrKzs4mMjMTX15eWLVuSlZWFiLBkyRIG\nDhx4zrZWrlxJTEyMg4/ywrz00kvk5ORgtVpJS0vj9ttvZ8mSJW55Lnx9fenQoQP79u0D4MMPP6Rb\nt24MGDDA7c5FcHAwX3zxBSdPnkRE+PDDDwkJCXHLc3GaI/5NxMXFkZGRQWlpKSUlJWzcuJH4+Phf\nD+xSHtg1VevWrZPAwEDx9/eXl156ydnhXDabN28Wk8kk4eHh0qNHD+nRo4esX79eioqKJCYmRiwW\ni8TGxtbptTN79mzx9/eXoKAgSU9Pt7Vv375dQkNDxd/fXyZPnmxrLy8vl6FDh0pAQID06dNHrFar\nIw/xkmRmZtp6D7rrufjqq6/khhtukO7du8t9990npaWlbnsu5s6dKyEhIRIaGiqjR4+WyspKtzkX\nw4cPl2uvvVY8PT3FbDbLwoULHXbsCxculICAAAkICJBFixY1GKvbjT2olFKq8dLbg0oppRoNTVpK\nKaUaDU1aSimlGg1NWkoppRoNTVpKnXL11Vdftm0tWrSIyZMn/+ZlXN0nn3zC1q1bnR2GciOatJQ6\n5WLe+K+urnbo/lzVpk2b2LJli7PDUG5Ek5Zqkp566imSk5Ntv8+cOZN58+Zx/Phx7rjjDnr37k33\n7t1Zu3btOeuKCNOnTycsLIzu3bvz/vvvA5CZmUnfvn0ZOHAg3bp1O2e9d955h6CgIPr06VPnQn70\n6FGGDBlCZGQkkZGRDV7kjx8/zrhx4+jTpw+9evWyxTh16lT+/Oc/A7BhwwZuu+02RIQHHniAiRMn\nEhERQVBQEP/6178AKC8vZ+zYsXTv3p1evXqRmZkJGBXeoEGD6N+/P4GBgcyYMcO274yMDG666SZ6\n9+7NsGHDOH78OACdOnVi5syZtvP2ww8/cODAAd566y3++te/0rNnTz7//PMLOtZf279SDfrtr6Up\n5Xp27dolt912m+33kJAQyc3Nlerqavn5559FROTo0aMSEBBgW+bqq68WEZGVK1dKbGys1NbWSmFh\noXTs2FEKCgpk06ZNctVVV8mBAwfO2V9+fr507NhRfvrpJ6msrJSbb77Z9nLliBEj5LPPPhMRkYMH\nD0rXrl1FROSdd96RRx999JxtPfXUU/Luu++KiEhJSYkEBgbKiRMn5MSJE9KtWzf5+OOPJSgoSH78\n8UcRERkzZoz0799fRESys7PFbDZLeXm5/OUvf5Hx48eLiMj3338vHTt2lPLycnnnnXekS5cu8vPP\nP0t5eblcf/31kpubK0ePHpVbb71VTpw4ISLGdBQvvPCCiIh06tRJ/va3v4mISHJyskyYMEFERGbO\nnCnz5s2zxX6+Yz3T+fav1IVwu5mLlXvo0aMHR44coaCggCNHjtC6dWv8/PyoqqriqaeeYvPmzTRr\n1oz8/HyOHDliG1cN4LPPPmPkyJGYTCbat2/PbbfdxpdffknLli2JjIzk+uuvP2d/WVlZ9OvXD29v\nbwASExPrDI/03Xff2ZYtKyuzVTD1ycjI4J///Cd/+ctfAGMCvUOHDhEUFMTbb79N3759ee211+jc\nuTNg3GYcNmwYAAEBAXTp0oXvv/+ezz//nMceewyAoKAgrr/+evbt24fJZCImJsY2aHJISAgHDhyg\npKSEvXv3ctNNNwFQWVlp+wwwaNAgAHr16sWqVats7XLG+AT1HeuJEye48sorbW3n27+fn995z4lS\np2nSUk3W0KFDWblyJYcPH2b48OEAvPfee/z000/s3LmT5s2b07lzZ8rLy+usZzKZzhkB/vTzp6uu\nuqrefZ29jojY1hERsrKyaNGiRb3brM+qVauwWCzntO/evZt27do1OH3Dmfuuz+9+9zvb5+bNm9ue\n0cXGxrJ06dJfXefM5c92vmNtaP81NTW/urxSp+kzLdVkJSYmsmzZMlauXMnQoUMB+Pnnn2nfvj3N\nmzdn06ZN9U553rdvX5YvX05tbS1Hjx7l008/JTIy8rwJACAyMpJPPvmE4uJiqqqqWLFihe27uLg4\nXn/9ddvvX331FXD+hBIfH19n+V27dgFw8OBB5s+fz65du1i/fj3btm2zbWfFihWICPv37+fHH38k\nODiYvn378t577wGwb98+Dh06RHBwcL37NZlMREVF8fnnn7N//37AeLaWnZ193mMGuOaaaygrK2vw\nWM9U3/5/7dwqdSZNWqrJCgkJ4ZdffsFsNtsms7v//vvZvn073bt3Z8mSJXTt2tW2/Onq5L777qN7\n9/TZmd8AAAEUSURBVO6Eh4cTExPDK6+8Qvv27X91TqFrr72WmTNncuONN3LLLbfU6ajx+uuvs337\ndsLDw+nWrRt///vfbfurb3vPPvssVVVVdO/endDQUJ5//nkAJkyYwLx58/D19SUlJYUJEyZQUVGB\nyWSiY8eOREZGctddd/HWW2/RokULHnnkEWpra+nevTvDhw8nNTUVT0/P8+63bdu2LFq0iBEjRhAe\nHs5NN93EDz/8cM5yZ64/YMAA/u///s/WEeN8x3q+9c8+90o1RAfMVaqRGzt2LAMGDLA9c1KqKdNK\nSymlVKOhlZZSSqlGQystpZRSjYYmLaWUUo2GJi2llFKNhiYtpZRSjYYmLaWUUo2GJi2llFKNxv8D\ncoIbkwcWZfwAAAAASUVORK5CYII=\n"
      }
     ],
     "prompt_number": 86
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "[Nota: Ahora, las escalas de los ejes son lineales]\n",
      "\n",
      "Vemos que la ganancia, dependiendo del valor del exponente, es cada vez m\u00e1s importante a medida que aumenta el valor del mismo."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "for i, t1, t2 in zip(n, t_bruta, t_bin):\n",
      "    print('Ganancia para n = {0:>6}, {1:>10.2f}'.format(i, t1/t2))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Ganancia para n =      5,       0.84\n",
        "Ganancia para n =     25,       1.56\n",
        "Ganancia para n =    100,       4.84\n",
        "Ganancia para n =   1000,      16.67\n",
        "Ganancia para n =  10000,      19.00\n",
        "Ganancia para n = 100000,      38.00\n"
       ]
      }
     ],
     "prompt_number": 89
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Para n\u00fameros peque\u00f1os vemos que incluso puede llegar a ser peor pero para n\u00fameros grandes la ganancia es significativa llegando a valores en torno a 40 para un n\u00famero `n` de orden $10^5$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Por \u00faltimo, solo comentar que las funciones `pow` y `math.pow` son m\u00e1s eficientes que lo que he hecho ([programadas en C](http://hg.python.org/cpython/file/41de6f0e62fd/Modules/mathmodule.c)). La idea de la entrada solo es mostrar que, gracias al trabajo y saber colectivo, la inteligencia que se esconde en las tripas de Python (y el resto de lenguajes de programaci\u00f3n) es mucha y en el d\u00eda a d\u00eda ni nos damos cuenta de ello.\n",
      "\n",
      "P.D.: Las funciones implementadas no contemplan exponentes negativos ni 0. He intentado simplificar el c\u00f3digo al m\u00e1ximo para que se vea el bosque en lugar de los \u00e1rboles."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Esta entrada la puedes descargar en formato notebook desde [nuestro repositorio en github](https://github.com/Pybonacci/notebooks)."
     ]
    }
   ],
   "metadata": {}
  }
 ]
}