Computing Pi via Leibniz, Nilakantha, Machin's formulas

Leibniz_Nilakantha_Arctan_Computing_Pi.ipynb

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    "## 3 Approaches to computing Pi\n",
    "\n",
    "**Leibniz** - 4_000_000 terms and does not converge for 12 digits in ~75ms\n",
    "\n",
    "**Nilakantha** - 10_000 terms to converge for 12 digits in ~870 µs\n",
    " \n",
    "**Machin's Arctan** - 10 terms to fully converge for 12 digits in ~55 µs\n"
   ]
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   "source": [
    "# Target: first 12 digits of π\n",
    "true_pi = 3.141592653589793"
   ]
  },
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    "def leibniz_pi(terms=5_000_000):\n",
    "    \"\"\"Leibniz formula: π/4 = 1 - 1/3 + 1/5 - 1/7 + ...\n",
    "        This converges EXTREMELY slow, and will still have an error ratio at above 4m\n",
    "    \"\"\"\n",
    "    pi_approx = 0\n",
    "    sign = 1\n",
    "    for i in range(terms):\n",
    "        denominator = 2 * i + 1\n",
    "        pi_approx += sign * (1 / denominator)\n",
    "        sign = -sign\n",
    "        if i % 200_000 == 0 and i > 0:\n",
    "            current = 4 * pi_approx\n",
    "            print(f\"Leibniz ({i} terms): {current:.12f}\")\n",
    "    return 4 * pi_approx"
   ]
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     "text": [
      "Leibniz Method:\n",
      "Leibniz (200000 terms): 3.141597653565\n",
      "Leibniz (400000 terms): 3.141595153583\n",
      "Leibniz (600000 terms): 3.141594320254\n",
      "Leibniz (800000 terms): 3.141593903588\n",
      "Leibniz (1000000 terms): 3.141593653589\n",
      "Leibniz (1200000 terms): 3.141593486922\n",
      "Leibniz (1400000 terms): 3.141593367875\n",
      "Leibniz (1600000 terms): 3.141593278589\n",
      "Leibniz (1800000 terms): 3.141593209145\n",
      "Leibniz (2000000 terms): 3.141593153589\n",
      "Leibniz (2200000 terms): 3.141593108135\n",
      "Leibniz (2400000 terms): 3.141593070256\n",
      "Leibniz (2600000 terms): 3.141593038205\n",
      "Leibniz (2800000 terms): 3.141593010732\n",
      "Leibniz (3000000 terms): 3.141592986923\n",
      "Leibniz (3200000 terms): 3.141592966090\n",
      "Leibniz (3400000 terms): 3.141592947707\n",
      "Leibniz (3600000 terms): 3.141592931367\n",
      "Leibniz (3800000 terms): 3.141592916747\n",
      "Leibniz (4000000 terms): 3.141592903590\n",
      "Leibniz (4200000 terms): 3.141592891685\n",
      "Leibniz (4400000 terms): 3.141592880862\n",
      "Leibniz (4600000 terms): 3.141592870981\n",
      "Leibniz (4800000 terms): 3.141592861923\n",
      "Final Leibniz   : 3.141592453590\n",
      "\n",
      "True π (12digit): 3.141592653590\n",
      "\n",
      "CPU times: user 351 ms, sys: 0 ns, total: 351 ms\n",
      "Wall time: 351 ms\n"
     ]
    }
   ],
   "source": [
    "%%time\n",
    "print(\"Leibniz Method:\")\n",
    "leibniz_result = leibniz_pi()\n",
    "\n",
    "print(f\"Final Leibniz   : {leibniz_result:.12f}\\n\")\n",
    "print(f\"True π (12digit): {true_pi:.12f}\\n\")"
   ]
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    "def nilakantha_pi(terms=10000):\n",
    "    \"\"\"Nilakantha series: π = 3 + 4/(2*3*4) - 4/(4*5*6) + ...\"\"\"\n",
    "    pi = 3.0\n",
    "    sign = 1\n",
    "    for n in range(2, terms * 2, 2):\n",
    "        denominator = n * (n + 1) * (n + 2)\n",
    "        pi += sign * (4 / denominator)\n",
    "        sign = -sign\n",
    "        if n % 1500 == 0:\n",
    "            print(f\"Nilakantha ({n//2} terms): {pi:.12f}\")\n",
    "    return pi"
   ]
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      "Nilakantha Method:\n",
      "Nilakantha (750 terms): 3.141592653000\n",
      "Nilakantha (1500 terms): 3.141592653516\n",
      "Nilakantha (2250 terms): 3.141592653568\n",
      "Nilakantha (3000 terms): 3.141592653581\n",
      "Nilakantha (3750 terms): 3.141592653585\n",
      "Nilakantha (4500 terms): 3.141592653587\n",
      "Nilakantha (5250 terms): 3.141592653588\n",
      "Nilakantha (6000 terms): 3.141592653589\n",
      "Nilakantha (6750 terms): 3.141592653589\n",
      "Nilakantha (7500 terms): 3.141592653589\n",
      "Nilakantha (8250 terms): 3.141592653589\n",
      "Nilakantha (9000 terms): 3.141592653589\n",
      "Nilakantha (9750 terms): 3.141592653590\n",
      "Final Nilakantha: 3.141592653590\n",
      "\n",
      "True π (12digit): 3.141592653590\n",
      "\n",
      "CPU times: user 892 µs, sys: 0 ns, total: 892 µs\n",
      "Wall time: 850 µs\n"
     ]
    }
   ],
   "source": [
    "%%time\n",
    "print(\"Nilakantha Method:\")\n",
    "nilakantha_result = nilakantha_pi()\n",
    "print(f\"Final Nilakantha: {nilakantha_result:.12f}\\n\")\n",
    "print(f\"True π (12digit): {true_pi:.12f}\\n\")"
   ]
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    "def arctan_pi(terms=10):\n",
    "    \"\"\"Machin's formula: π = 4 * (4 * arctan(1/5) - arctan(1/239))\"\"\"\n",
    "    def arctan(x, n_terms):\n",
    "        result = 0\n",
    "        for n in range(n_terms):\n",
    "            power = 2 * n + 1\n",
    "            result += (-1)**n * (x**power) / power\n",
    "        return result\n",
    "    \n",
    "    # Calculate arctan values once\n",
    "    atan_1_5 = arctan(1/5, terms)\n",
    "    atan_1_239 = arctan(1/239, terms)\n",
    "    \n",
    "    # Print progress using partial sums\n",
    "    for t in range(2, terms + 1, 2):\n",
    "        partial_5 = arctan(1/5, t)\n",
    "        partial_239 = arctan(1/239, t)\n",
    "        partial_pi = 4 * (4 * partial_5 - partial_239)\n",
    "        print(f\"Arctan ({t} terms): {partial_pi:.12f}\")\n",
    "    \n",
    "    # Final result\n",
    "    return 4 * (4 * atan_1_5 - atan_1_239)"
   ]
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     "text": [
      "Arctan Method:\n",
      "Arctan (2 terms): 3.140597029326\n",
      "Arctan (4 terms): 3.141591772182\n",
      "Arctan (6 terms): 3.141592652615\n",
      "Arctan (8 terms): 3.141592653589\n",
      "Arctan (10 terms): 3.141592653590\n",
      "Final Arctan    : 3.141592653590\n",
      "\n",
      "True π (12digit): 3.141592653590\n",
      "\n",
      "CPU times: user 114 µs, sys: 10 µs, total: 124 µs\n",
      "Wall time: 112 µs\n"
     ]
    }
   ],
   "source": [
    "%%time\n",
    "print(\"Arctan Method:\")\n",
    "arctan_result = arctan_pi()\n",
    "print(f\"Final Arctan    : {arctan_result:.12f}\\n\")\n",
    "print(f\"True π (12digit): {true_pi:.12f}\\n\")"
   ]
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