円周率の計算 チュドノフスキーの公式 ベイリー＝ボールウェイン＝プラウフの公式.ipynb

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   "source": [
    "# チュドノフスキーの公式\n",
    "\n",
    "$$\n",
    "\\frac{1}{\\pi} = 12\\sum_{n=0}^\\infty \\frac{(-1)^n(6n)!(545140134n + 13591409)}{(3n)!(n!)^3(640320)^{3n+\\frac{3}{2}}}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 117,
   "id": "bfe1bde5",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "この値に近似する 1/π ↓\n",
      "0.318309886183790671537767526745\n",
      "\n",
      "\n",
      "計算結果ほぼ同じ！\n",
      "0.318309886183790691216444201928\n"
     ]
    }
   ],
   "source": [
    "from sympy import *\n",
    "\n",
    "init_printing()\n",
    "\n",
    "def Chudnovsky(cnt):\n",
    "    sum = 0\n",
    "    for n in range(cnt):\n",
    "      numerator = ((-1)**n) * factorial(6*n) * ((545140134*n) + 13591409)\n",
    "      denominator = factorial(3*n) * (factorial(n)**3) * (640320 ** (3*n + 3/2))\n",
    "      sum += (numerator / denominator)\n",
    "    return 12 * sum\n",
    "\n",
    "\n",
    "print('この値に近似する 1/π ↓')\n",
    "print((1 / pi).evalf(30))\n",
    "print('\\n')\n",
    "print('計算結果ほぼ同じ！')\n",
    "print(Chudnovsky(10).evalf(30))\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6858469b",
   "metadata": {},
   "source": [
    "# ベイリー＝ボールウェイン＝プラウフの公式\n",
    "\n",
    "$$\n",
    "\\pi = \\sum_{n=0}^\\infty (\\frac{4}{8n+1}-\\frac{2}{8n+4}-\\frac{1}{8n+5}-\\frac{1}{8n+6})(\\frac{1}{16})^n\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 116,
   "id": "e59b45dd",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "この値に近似する π ↓\n",
      "3.14159265358979323846264338328\n",
      "\n",
      "\n",
      "計算結果ほぼ同じ！\n",
      "3.14159265358979133964112406829\n"
     ]
    }
   ],
   "source": [
    "from sympy import *\n",
    "\n",
    "init_printing()\n",
    "\n",
    "def BBP(cnt):\n",
    "    return sympify(sum([((4/(8*n+1))-(2/(8*n+4))-(1/(8*n+5))-(1/(8*n+6)))*((1/16)**n) for n in range(cnt)]))\n",
    "\n",
    "print('この値に近似する π ↓')\n",
    "print((pi).evalf(30))\n",
    "print('\\n')\n",
    "print('計算結果ほぼ同じ！')\n",
    "print(BBP(10).evalf(30))\n"
   ]
  }
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