{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "5fa38a35",
   "metadata": {},
   "source": [
    "# 수의 정확도, 오차\n",
    "**강좌**: *수치해석*"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d39fb010",
   "metadata": {},
   "source": [
    "## 유효 숫자\n",
    "\n",
    "과학 계산에서 수치는 실수 체계로 정의할 수 있다.\n",
    "\n",
    ":::{figure-md} Real Number\n",
    "<img src=\"https://upload.wikimedia.org/wikipedia/commons/thumb/e/e2/8723953209435270932573290759328759238723908579032857290357290q375329087459324759320958723409875iuwyeryqiytuitr.png/800px-8723953209435270932573290759328759238723908579032857290357290q375329087459324759320958723409875iuwyeryqiytuitr.png\" alt=\"number-fig\">\n",
    "\n",
    "실수 체계 (From wikimedia.org)\n",
    ":::"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d66dd18e",
   "metadata": {},
   "source": [
    "매우 긴 숫자를 다루는 것은 현실적으로 어려우며, `유효 숫자` 로 계산한다. 예를 들어 $\\pi$ 는 다음과 같은 4자리 유효 숫자로 표현할 수 있다."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "62e1f208",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.142"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import numpy as np\n",
    "\n",
    "# 4 자리 유효숫자\n",
    "np.round(np.pi, decimals=3)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "521353c2",
   "metadata": {},
   "source": [
    "실수를 표현하는 방식으로는 `Decimal Notataion` 과 `Scientific Notation` 이 있다."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "e20ab70d",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Decimal notation of $\\pi$: 3.142\n",
      "Scientific notation of $\\pi$: 3.141593e+00\n"
     ]
    }
   ],
   "source": [
    "# Decimal notation\n",
    "print(r\"Decimal notation of $\\pi$: {:.3f}\".format(np.pi))\n",
    "\n",
    "# Scientific notataion\n",
    "print(r\"Scientific notation of $\\pi$: {:e}\".format(np.pi))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dfddefb5",
   "metadata": {},
   "source": [
    ":::{note}\n",
    "컴퓨터도 모든 유효숫자를 저장할 수 없다.\n",
    ":::    "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3af773c2",
   "metadata": {},
   "source": [
    "## 오차\n",
    "컴퓨터 계산 과정에서 2가지 오차가 존재한다.\n",
    "\n",
    "- Round-off Erorr: 숫자 표기 한계로 반올림에 의한 오차\n",
    "- Truncation Error: 계산 과정에서 절단애 의한 오차\n",
    "\n",
    "즉 수치해석 결과는 엄밀해는 다음 관계를 갖는다.\n",
    "\n",
    "$$\n",
    "    Exact = Approximation + \\epsilon\n",
    "$$\n",
    "\n",
    "### 오차의 표현\n",
    "다음 수치 실험 값을 비교해보자.\n",
    "\n",
    "|       | $C_l$  | $C_d$ |\n",
    "|-------|--------|--------|\n",
    "| 실험  | 0.3501 | 0.0223 |\n",
    "| 계산  | 0.3470 | 0.0201 |\n",
    "\n",
    "$C_l$ 과 $C_d$에 대한 오차를 계산해보면 다음과 같다.\n",
    "\n",
    "|       | $C_l$  | $C_d$ |\n",
    "|-------|--------|--------|\n",
    "| $\\epsilon$  | 0.0031 | 0.0022 |\n",
    "\n",
    "(절대) 오차 $\\epsilon$ 만 비교해보면 $C_d$ 의 정확도가 더 높은 것 같다.\n",
    "\n",
    "그러나 수치 크기에 대한 오차가 없으므로 다음과 같은 상대 오차를 정의한다.\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "\\epsilon_t &= \\frac{Error}{Exact}\\\\\n",
    "\\epsilon_a &= \\frac{Error (approx)}{Approx}\\\\\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "참값을 아는 경우 참 상대오차 $\\epsilon_t$를 사용할 수 있다. \n",
    "\n",
    "위 예제에서 상대오차는 다음과 같다.\n",
    "\n",
    "|       | $C_l$  | $C_d$ |\n",
    "|-------|--------|--------|\n",
    "| $\\epsilon_t$  | 0.89% | 9.9% |\n",
    "\n",
    "즉, $C_l$의 상대오차가 더 작다.\n",
    "\n",
    "다만 실제의 경우 참값을 모르는 경우가 많으므로 근사값을 기준으로 한  $\\epsilon_a$를 사용한다."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ee1935c6",
   "metadata": {},
   "source": [
    "#### $e^x$ 근사 계산 예제\n",
    "Taylor expansion (Maclaurin expansion)을 사용하면 $e^x$ 는 다음과 같이 근사할 수 있다.\n",
    "\n",
    "$$\n",
    "e^x = 1 + x + \\frac{x^2}{2} + \\frac{x^3}{3!} +... + \\frac{x^n}{n!} + ...\n",
    "$$\n",
    "\n",
    "$|x| < 1$ 인 경우, 이전 항이 그 다음항보다 크다. n번째 항 까지만 계산할 경우 오차의 크기는 다음과 같다.\n",
    "\n",
    "$$\n",
    "\\epsilon =  \\frac{x^{n+1}}{n+1!} + \\frac{x^{n+2}}{n+2!} + ... \\approx C x^{n+1}\n",
    "$$\n",
    "\n",
    "두번째항 까지만 근사하면 아래와 같다.\n",
    "\n",
    "$$\n",
    "e^x \\approx 1 + x\n",
    "$$\n",
    "\n",
    "$x=0.5$ 에 대해 절대오차는 다음과 같다.\n",
    "\n",
    "$$\n",
    "\\epsilon = e^{0.5} - (1+0.5) = 1.4872\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "80c1fce6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.1487212707001282"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.exp(0.5) - (1.5)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6b8e7970",
   "metadata": {},
   "source": [
    "상대 오차는 다음과 같다.\n",
    "\n",
    "$$\n",
    "\\epsilon_t = \\frac{e^{0.5} - 1.5}{e^{0.5}}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "994d7650",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "9.02040104310499"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(np.exp(0.5) - (1.5)) / np.exp(0.5)*100"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fed1ed65",
   "metadata": {},
   "source": [
    "참값 $e^{0.5}$를 모르는 경우 이전 근사값 (첫번째 항만 근사)와 현재 근사값간의 차이를 근사 오차 Error (appox) 로 해서 상대오차를 계산한다.\n",
    "\n",
    "$$\n",
    "\\epsilon_a = \\frac{1.5 - 1}{1.5}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "587609a2",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "33.33333333333333"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(1.5 - 1) / (1.5)*100"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e1bc78f6",
   "metadata": {},
   "source": [
    "근사식의 항을 늘리면서 두 상대오차를 계산하면 다음과 같다."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "cc1b8aee",
   "metadata": {},
   "outputs": [],
   "source": [
    "def factorial(n):\n",
    "    \"\"\" Factorial 계산\n",
    "    Parameters\n",
    "    ----------\n",
    "    n : integer\n",
    "        n\n",
    "    \"\"\"\n",
    "    fac = 1\n",
    "    for i in range(1, n+1):\n",
    "        fac *= i\n",
    "        \n",
    "    return fac\n",
    "\n",
    "\n",
    "def approx_exp(n, x):\n",
    "    \"\"\" Exponential 함수 근사\n",
    "    Parameters\n",
    "    n : integer\n",
    "        항의 계수\n",
    "    x : float\n",
    "        값\n",
    "    \"\"\"    \n",
    "    exp = 0\n",
    "    for i in range(n):\n",
    "        exp += 1/factorial(i)*x**i\n",
    "    \n",
    "    ## Pythoniac\n",
    "    # exp = sum([factorial(i)*x**i for i in range(n)])\n",
    "    return exp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "3240eb1f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "9.02040104310499"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def epsilon_t(n, x):\n",
    "    \"\"\"참 상대오차 계산\n",
    "    Parameters\n",
    "    ----------\n",
    "    n : integer\n",
    "        항의 계수\n",
    "    x : float\n",
    "        값\n",
    "    \"\"\"\n",
    "    # Exact\n",
    "    exact = np.exp(x)\n",
    "    \n",
    "    # Approx\n",
    "    approx = approx_exp(n, x)\n",
    "    \n",
    "    return (exact - approx)/exact*100\n",
    "\n",
    "\n",
    "epsilon_t(2, 0.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "29dcb2c7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "33.33333333333333"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def epsilon_a(n, x):\n",
    "    \"\"\"근사 상대오차 계산\n",
    "    Parameters\n",
    "    ----------\n",
    "    n : integer\n",
    "        항의 계수\n",
    "    x : float\n",
    "        값\n",
    "    \"\"\"\n",
    "    # n번째와 n-1번째 항까지 근사값\n",
    "    approx_n = approx_exp(n, x)\n",
    "    approx_n1 = approx_exp(n-1, x)\n",
    "    \n",
    "    return (approx_n - approx_n1) / approx_n*100\n",
    "\n",
    "epsilon_a(2, 0.5)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bae1f088",
   "metadata": {},
   "source": [
    ":::{note}\n",
    "Docstring (문서화)는 매우 중요하다. 규칙을 사용할 필요가 있다. 이 강의에서는 [Numpydoc](https://numpydoc.readthedocs.io/en/latest/format.html#) 를 따른다.\n",
    ":::   "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "4ead11c0",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "terms 2, $\\epsilon_t$ : 9.020e+00, $\\epsilon_a$ : 3.333e+01\n",
      "terms 3, $\\epsilon_t$ : 1.439e+00, $\\epsilon_a$ : 7.692e+00\n",
      "terms 4, $\\epsilon_t$ : 1.752e-01, $\\epsilon_a$ : 1.266e+00\n",
      "terms 5, $\\epsilon_t$ : 1.721e-02, $\\epsilon_a$ : 1.580e-01\n",
      "terms 6, $\\epsilon_t$ : 1.416e-03, $\\epsilon_a$ : 1.580e-02\n",
      "terms 7, $\\epsilon_t$ : 1.002e-04, $\\epsilon_a$ : 1.316e-03\n",
      "terms 8, $\\epsilon_t$ : 6.220e-06, $\\epsilon_a$ : 9.402e-05\n",
      "terms 9, $\\epsilon_t$ : 3.435e-07, $\\epsilon_a$ : 5.876e-06\n"
     ]
    }
   ],
   "source": [
    "x = 0.5\n",
    "err_t = []\n",
    "err_a = []\n",
    "\n",
    "for n in range(2, 10):\n",
    "    err_tn = epsilon_t(n, x)\n",
    "    err_an = epsilon_a(n, x)\n",
    "    print(r\"terms {}, $\\epsilon_t$ : {:.3e}, $\\epsilon_a$ : {:.3e}\".format(n, err_tn, err_an))\n",
    "    err_t.append(err_tn)\n",
    "    err_a.append(err_an)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "5dbde361",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7f9f26415050>"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 960x720 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "from matplotlib import pyplot as plt\n",
    "\n",
    "plt.style.use('ggplot')\n",
    "plt.rcParams['figure.dpi'] = 150    \n",
    "    \n",
    "plt.semilogy(range(2, 10), err_t)\n",
    "plt.semilogy(range(2, 10), err_a)\n",
    "\n",
    "plt.xlabel(\"Number of terms (n)\")\n",
    "plt.ylabel(\"Error\")\n",
    "plt.legend([\"$\\epsilon_t$\", \"$\\epsilon_a$\"])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dd0d1b8a",
   "metadata": {},
   "source": [
    "#### DIY #1\n",
    "$\\cos(x)$ 과 $\\sin(x)$ 함수는 Taylor expansion에 의해 다음과 같이 표현할 수 있다.\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "\\cos x &= 1 - \\frac{x^2}{2!} + \\frac{x^4}{4!} - \\frac{x^6}{6!} + ... \\\\\n",
    "\\sin x &= x - \\frac{x^3}{3!} + \\frac{x^5}{5!} - \\frac{x^7}{7!} + ...\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "여기서 $x$의 단위는 radian이다.\n",
    "\n",
    "$x$가 5도 각도일 때 근사항의 갯수에 따른 절대오차, 참 상대오차, 근사 상대 오차를 구하고 이를 그래프로 표현하시오.\n",
    "\n",
    "Tip) `np.deg2rad`, `np.rad2deg` 함수 참고할 것"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "32c6df91",
   "metadata": {},
   "source": [
    "## Round-off Error\n",
    "### 컴퓨터에수 수의 표현\n",
    "컴퓨터는 여러개의 스위치(?, bit)를 이용해서 2진법으로 숫자를 표현한다. \n",
    "\n",
    ":::{figure-md} Binary number\n",
    "<img src=\"https://upload.wikimedia.org/wikipedia/commons/b/bc/Value_of_digits_in_the_Binary_numeral_system.svg\" alt=\"number-fig\">\n",
    "\n",
    "2진법 (From wikimedia.org)\n",
    ":::\n",
    "\n",
    "크게 정수 (Integer) 와 부동 소숫점 (floating point) 로 표현한다.\n",
    "\n",
    "### 정수\n",
    "\n",
    "16bit 컴퓨터에서 10진법으로 표현할 수 있는 정수의 범위를 생각해보자.\n",
    "\n",
    "가장 큰 수\n",
    "\n",
    "$$\n",
    "1\\times 2^{14} + 1\\times 2^{14} + ... + 1\\times 2^1 + 1 = 2^{15} -1 = 32,767\n",
    "$$\n",
    "\n",
    "즉 (-32,767, 32767) 까지 수를 표현할 수 있다.\n",
    "\n",
    "\n",
    "### Fixed Point vs Floating Point\n",
    "Fixed Point는 Decimcal notation 처럼 정수부와 소수부로 구분해서 표현하는 방식이다.\n",
    "\n",
    "예를 들어 16bit 컴퓨터에서 1개의 bit는 부호, 7개는 정수부, 8개는 소수부로 표현하는 방식이다. 수의 표현에 많은 한계가 있다.\n",
    "\n",
    "Floating point (부동소숫점)은 Scientific notation과 같이 부호, 지수부 (exponent), 가수부 (mantissa or fraction) 로 구분한다. \n",
    "\n",
    "$$\n",
    "m \\times b^e\n",
    "$$\n",
    "\n",
    "정규화를 위해 $1/b \\leq m < 1$ 로 제한된다.\n",
    "\n",
    "IEEE754 규격을 가장 널리 사용하는 표준이다.\n",
    "\n",
    ":::{figure-md} floating\n",
    "<img src=\"https://upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Float_example.svg/2880px-Float_example.svg.png\" alt=\"ieee754-fig\">\n",
    "\n",
    "Floating Point (From Wikipedia)\n",
    ":::"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "df8f37f5",
   "metadata": {},
   "source": [
    "#### 7bit floating point\n",
    "다음과 같이 표현한다.\n",
    "- 첫번째 bit : 부호\n",
    "- 2~4 번째 bit : 지수의 부호 (1 bit), 지수의 크기 (2bit)\n",
    "- 5~7 번재 bit : 가수 (3bit)\n",
    "\n",
    "가장 작은 양수: $0111100_{(2)} = +2^{-(1\\times2 + 1)}\\times(1\\times2^{-1} + 0\\times2^{-2} + 0\\times2^{-3}) = +0.5\\times2^{-3}=0.0625$\n",
    "\n",
    "그 다음으로 표현 가능한 수는 아래와 같다.\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "0111101_{(2)} &= +2^{-(1\\times2 + 1)}\\times(1\\times2^{-1} + 0\\times2^{-2} + 1\\times2^{-3}) = 0.078125 \\\\\n",
    "0111110_{(2)} &= +2^{-(1\\times2 + 1)}\\times(1\\times2^{-1} + 1\\times2^{-2} + 0\\times2^{-3}) = 0.093750 \\\\\n",
    "0111111_{(2)} &= +2^{-(1\\times2 + 1)}\\times(1\\times2^{-1} + 1\\times2^{-2} + 1\\times2^{-3}) = 0.109375\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "즉 $0.015625(=2^{-6})$ 씩 증가시킬 수 있다. \n",
    "\n",
    "그 다음 큰 수는 지수를 증가시켜야 한다.\n",
    "\n",
    "$$\n",
    "0110100_{(2)} = +2^{-(1\\times2 + 0)}\\times(1\\times2^{-1} + 0\\times2^{-2} + 0\\times2^{-3}) = 0.12500\n",
    "$$\n",
    "\n",
    "그 다음 큰 수는 아래와 같다.\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "0110101_{(2)} &= +2^{-(1\\times2 + 0)}\\times(1\\times2^{-1} + 0\\times2^{-2} + 1\\times2^{-3}) = 0.156250 \\\\\n",
    "0110110_{(2)} &= +2^{-(1\\times2 + 0)}\\times(1\\times2^{-1} + 1\\times2^{-2} + 0\\times2^{-3}) = 0.187500 \\\\\n",
    "0110111_{(2)} &= +2^{-(1\\times2 + 0)}\\times(1\\times2^{-1} + 1\\times2^{-2} + 1\\times2^{-3}) = 0.218750\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "즉 $0.03215(=2^{-5})$ 씩 증가시킬 수 있다.\n",
    "\n",
    "가장 큰 수는 다음과 같다.\n",
    "\n",
    "$$\n",
    "0011111_{(2)} = +2^{+(1\\times2 + 1)}\\times(1\\times2^{-1} + 1\\times2^{-2} + \\times2^{-3}) = 7\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1a09c155",
   "metadata": {},
   "source": [
    "#### Underflow, Overflow, Machine epsilon\n",
    "위 예제와 같이 floating point는 넓은 범위의 숫자를 표현할 수 있으나 모든 숫자를 표현할 수 없다.\n",
    "\n",
    ":::{figure-md} floating_dist\n",
    "<img src=\"https://upload.wikimedia.org/wikipedia/commons/b/b6/FloatingPointPrecisionAugmented.png\">\n",
    "\n",
    "Distriution of Floating Point (From Wikipedia)\n",
    ":::\n",
    "\n",
    "다음 3가지 특징이 있다.\n",
    "\n",
    "1. 표현할 수 있는 수의 한계가 있음\n",
    "- Overflow: 허용된 범위를 벗어난 큰 수 표현하지 못함\n",
    "- Underflow: 매우 작은 숫자를 표현하지 못함\n",
    "\n",
    "2. 주어진 범위 안에넛 표현할 수 있는 수자의 개수는 유한함\n",
    "- 근사값으로 표현 (Quantizing error, round-off error)\n",
    "- Machine epsilon\n",
    "\n",
    "3. 수 사이의 간격 $\\Delta x$는 수의 크가기 증가함에 따라 커짐"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3769a635",
   "metadata": {},
   "source": [
    "#### 단정밀도, 배정밀도\n",
    "일반적으로 컴퓨터의 기본 단위 (word)는 32비트로 한다. 이를 기준으로 해서 다음과 같은 정밀도를 생각한다.\n",
    "\n",
    "\n",
    "| 정밀도 | 크기   |\n",
    "|---------|--------|\n",
    "| 단정밀도 | 32bit |\n",
    "| 배정밀도 | 64bit |\n",
    "| 반정밀도 | 16bit |\n",
    "\n",
    "FP32, FP64, FP16 등으로 표현하기도 한다. 현대 수치해석에서 기본 단위는 배정밀도이다. \n",
    "\n",
    "Numpy 에서는 `np.float32`, `np.float64`, `np.float16` 등으로 표현한다.\n",
    "\n",
    "`np.finfo` 이용하면 여러 특징을 살펴볼 수 있다."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "40420e00",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "finfo(resolution=1e-15, min=-1.7976931348623157e+308, max=1.7976931348623157e+308, dtype=float64)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.finfo(1.0)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "05c7a705",
   "metadata": {},
   "source": [
    "#### 연산의 오류\n",
    "연산에서 매우 작은 수 또는 매우 큰 수는 표현할 수 없어서 오류가 발생한다"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "b10dba73",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Machine parameters for float16\n",
      "---------------------------------------------------------------\n",
      "precision =   3   resolution = 1.00040e-03\n",
      "machep =    -10   eps =        9.76562e-04\n",
      "negep =     -11   epsneg =     4.88281e-04\n",
      "minexp =    -14   tiny =       6.10352e-05\n",
      "maxexp =     16   max =        6.55040e+04\n",
      "nexp =        5   min =        -max\n",
      "smallest_normal = 6.10352e-05   smallest_subnormal = 5.96046e-08\n",
      "---------------------------------------------------------------\n",
      "\n"
     ]
    }
   ],
   "source": [
    "# FP16 숫자 특징\n",
    "print(np.finfo(np.float16))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "e5c81832",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.001"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# FP16에서 매우 작은 수 더하기\n",
    "a = np.float16(1.0)\n",
    "\n",
    "np.float16(a + 9e-4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "5f8426c7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# FP16에서 비슷한 크기의 숫자 빼기 (밸샘의 무효화)\n",
    "a = np.float16(0.5)\n",
    "b = np.float16(0.4999)\n",
    "\n",
    "np.float16(b -a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "13d46365",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/tmp/ipykernel_296/3507621214.py:5: RuntimeWarning: overflow encountered in scalar add\n",
      "  np.float16(a + b)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "inf"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Overflow\n",
    "a = np.float16(4e4)\n",
    "b = np.float16(3e4)\n",
    "\n",
    "np.float16(a + b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "c5d4d60a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Underflow\n",
    "a = np.float16(1e-7)\n",
    "\n",
    "np.float16(a/10)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "12ba3ba1",
   "metadata": {},
   "source": [
    "#### DIY #2\n",
    "단정밀도, 배정밀도에 대해서 뺄샘의 무효화, Overflow, Underflow 상황을 만들어 보시오."
   ]
  }
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