07_numbers/07-numbers.ipynb

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    "# 07 Numbers\n",
    "\n",
    "* *Computational Physics*: Ch 2.4, 2.5, 3\n",
    "* Python Tutorial [Floating Point Arithmetic: Issues and Limitations](https://docs.python.org/3/tutorial/floatingpoint.html)"
   ]
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    "## Binary representation\n",
    "\n",
    "Any number can be represented in any base as a polynomial (possibly with infinitely many terms): the digits are $0 \\leq x_k < b$ and determine the contribution of the base $b$ raise to the $k$th power.\n",
    "\n",
    "$$\n",
    "q_b = \\sum_{k=-\\infty}^{+\\infty} x_k b^k\n",
    "$$"
   ]
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    "Computers store information with two-state logic. This can be represented in a binary system with numbers 0 and 1 (i.e. base 2)\n",
    "\n",
    "$$\n",
    "b = 2\n",
    "$$"
   ]
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    "### Examples for binary numbers\n",
    "\n",
    "| base 10 | base 2 | $\\sum_{k=0}^{N} x_k 2^k$ | *N* |\n",
    "|--------:|-------:|-------------------------:|----:|\n",
    "| 0       | 0      | 0                        | 0   |\n",
    "| 1       | 1      | 1                        | 0   |\n",
    "| 2       | 10     | 2+0                      | 1   |\n",
    "| 3       | 11     | 2+1                      | 1   |\n",
    "| 4       | 100    | 4+0+0                    | 2   |\n",
    "| 5       | 101    | 4+0+1                    | 2   |\n",
    "| 10      | 1010   | 8+0+2+0                  | 3   |\n",
    "| 15      | 1111   | 8+4+2+1                  | 3   |\n",
    "| 16      | 10000  | 16+0+0+0+0               | 4   |"
   ]
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   "source": [
    "## Integers \n",
    "How can we find the base $b$-representation of a number?"
   ]
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   "source": [
    "Base 10 to base 10 (!)\n",
    "\n",
    "156\n",
    "```\n",
    "156 / 10 = 15, 6\n",
    "15  / 10 =  1, 5\n",
    "1   / 10 =  0, 1\n",
    "```"
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    "2478\n",
    "```\n",
    "2478 / 10 = 247, 8\n",
    "247  / 10 =  24, 7\n",
    "24   / 10 =   2, 4\n",
    "2    / 10 =   0, 2 \n",
    "```"
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   "source": [
    "11\n",
    "```\n",
    "11 / 10 = 1, 1\n",
    "1  / 10 = 0, 1\n",
    "```"
   ]
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   "source": [
    "What about using 2 instead of 10?\n",
    "```\n",
    "11 / 2 = 5, 1\n",
    "5  / 2 = 2, 1\n",
    "2  / 2 = 1, 0\n",
    "1  / 2 = 0, 1\n",
    "```"
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     "data": {
      "text/plain": [
       "11"
      ]
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   "source": [
    "1 * 2**3  +  0 * 2**2  +  1 * 2**1  +  1 * 2**0"
   ]
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    "Convert 11 (base 10, i.e. $1 \\times 10^1 + 1\\times 10^0$) into binary (using Python):\n",
    "\n",
    "(Note: `divmod(x, 2)` is `x // 2, x % 2`, i.e. integer division and remainder):"
   ]
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     "data": {
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       "(5, 1)"
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   "source": [
    "divmod(11, 2)"
   ]
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       "(2, 1)"
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    "divmod(_[0], 2)"
   ]
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    "(`_` is ipython's last output, i.e., `(5, 0)` so `_[0] == 5` ... this is just me being lazy and not wanting to type too much)"
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    "divmod(_[0], 2)"
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    "The binary representation of $11_{10}$ is $1011_2$ (keep dividing until there's only 1 left, then collect the 1 and all remainders in reverse order, essentially long division):\n",
    "\n",
    "(**1, 0**), (2, **1**), (5, **1**) $\\rightarrow$ **1011**$_2$"
   ]
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    "Double check by multiplying out $1011_2$:"
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    "1 * 2**3  +  0 * 2**2  +  1 * 2**1  +  1 * 2**0"
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    "or in Python"
   ]
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       "11"
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   "source": [
    "int('1011', 2)"
   ]
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   "source": [
    "0b1011"
   ]
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   "source": [
    "### Summary: Integers in binary representation\n",
    "\n",
    "**All integers are exactly representable in base 2 with a finite number of digits**.\n",
    "\n",
    "* The sign (+ or –) is represented by a single bit (0 = +, 1 = –). \n",
    "* The number of available \"bits\" (digits) determines the largest representable integer. "
   ]
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   "source": [
    "For example, with 8 bits available (a \"*byte*\"), what is the largest and smallest integer?"
   ]
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    {
     "data": {
      "text/plain": [
       "127"
      ]
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   "source": [
    "0b1111111  # 7 bits for number, 1 for sign (not included)"
   ]
  },
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     "data": {
      "text/plain": [
       "-127"
      ]
     },
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   "source": [
    "-0b1111111"
   ]
  },
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   "source": [
    "### Sidenote: using numpy to quickly convert integers\n",
    "\n",
    "If you want to properly sum all terms, use numpy arrays and the element-wise operations:"
   ]
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   "source": [
    "import numpy as np"
   ]
  },
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    "nbits = 7\n",
    "exponents = np.arange(nbits)\n",
    "bases = 2*np.ones(nbits)  # base 2\n",
    "digits = np.ones(nbits)   # all 1, for 1111111 (127 in binary)"
   ]
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    {
     "data": {
      "text/plain": [
       "(array([0, 1, 2, 3, 4, 5, 6]),\n",
       " array([2., 2., 2., 2., 2., 2., 2.]),\n",
       " array([1., 1., 1., 1., 1., 1., 1.]))"
      ]
     },
     "execution_count": 12,
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   "source": [
    "exponents, bases, digits"
   ]
  },
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     "data": {
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       "127.0"
      ]
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   "source": [
    "np.sum(digits * bases**exponents)"
   ]
  },
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    "### Examples: limits of integers\n",
    "\n",
    "What is the smallest and largest integer that you can represent\n",
    "\n",
    "1. if you have 4 bits available and only consider non-negative (\"unsigned\") integers?\n",
    "2. if you have 32 bits and consider positive and negative integers?\n",
    "3. if you have 64 bits and consider positive and negative integers?"
   ]
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   "source": [
    "Smallest and largest 4 bit unsigned integer:"
   ]
  },
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       "0"
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   ],
   "source": [
    "0b0000"
   ]
  },
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   "execution_count": 15,
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "15"
      ]
     },
     "execution_count": 15,
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   ],
   "source": [
    "0b1111"
   ]
  },
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   "source": [
    "Smallest and largest 32-bit signed integer (int32):"
   ]
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   "source": [
    "1 bit is sign, 31 bits are available, so the highest number has 31 ones (111...11111). The *next highest* number is 1000...000, a one with 32 bits and 31 zeroes, i.e., $2^{31}$."
   ]
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   "source": [
    "Thus, the highest number is $2^{31} - 1$:"
   ]
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    {
     "data": {
      "text/plain": [
       "2147483647"
      ]
     },
     "execution_count": 16,
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   "source": [
    "2**31 - 1"
   ]
  },
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    "(and the smallest number is just $-(2^{31} - 1)$)"
   ]
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   "source": [
    "And int64 (signed):"
   ]
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   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-9223372036854775807 9223372036854775807\n"
     ]
    }
   ],
   "source": [
    "max64 = 2**(64-1) - 1\n",
    "print(-max64, max64)"
   ]
  },
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   "source": [
    "### Python's arbitrary precision integers "
   ]
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    "tags": []
   },
   "source": [
    "In Python, integers *have arbitrary precision*: integer arithmetic (`+`, `-`, `*`, `//`) is exact and will not overflow. Thus the following code will run forever (until memory is exhausted); if you run it, you can stop the evaluation with the ''Kernel / Interrupt'' menu command in the notebook and then investigate `n` and `nbits`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "b439cebb-9584-46b8-a655-a224cabec700"
    },
    "scrolled": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [
    {
     "ename": "KeyboardInterrupt",
     "evalue": "",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mKeyboardInterrupt\u001b[0m                         Traceback (most recent call last)",
      "Cell \u001b[0;32mIn[18], line 3\u001b[0m\n\u001b[1;32m      1\u001b[0m n \u001b[38;5;241m=\u001b[39m \u001b[38;5;241m1\u001b[39m \n\u001b[1;32m      2\u001b[0m nbits \u001b[38;5;241m=\u001b[39m \u001b[38;5;241m1\u001b[39m\n\u001b[0;32m----> 3\u001b[0m \u001b[38;5;28;01mwhile\u001b[39;00m \u001b[38;5;28;01mTrue\u001b[39;00m:\n\u001b[1;32m      4\u001b[0m     n \u001b[38;5;241m*\u001b[39m\u001b[38;5;241m=\u001b[39m \u001b[38;5;241m2\u001b[39m\n\u001b[1;32m      5\u001b[0m     nbits \u001b[38;5;241m+\u001b[39m\u001b[38;5;241m=\u001b[39m \u001b[38;5;241m1\u001b[39m\n",
      "\u001b[0;31mKeyboardInterrupt\u001b[0m: "
     ]
    }
   ],
   "source": [
    "n = 1 \n",
    "nbits = 1\n",
    "while True:\n",
    "    n *= 2\n",
    "    nbits += 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "f521aada-df67-47b1-960a-9b5eb2291861"
    },
    "slideshow": {
     "slide_type": "subslide"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "int"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "b8005015-6e5e-4ddd-a781-9c31e44a9171"
    },
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "526784"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "int.bit_length(n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "cc73d924-f587-4fc2-8320-4608cb08b20f"
    },
    "scrolled": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "526784"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nbits"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "4bdfee4f-8d8f-470a-aeb5-6a757f581f03"
    },
    "slideshow": {
     "slide_type": "subslide"
    },
    "tags": []
   },
   "source": [
    "### NumPy has fixed precision integers\n",
    "NumPy data types (dtypes) are fixed precision. Overflows \"wrap around\":"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "068a07a2-636f-4985-a370-11d6536f4173"
    },
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "70eb1399-ee1e-4f2f-a563-331082675475"
    },
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([32767], dtype=int16)"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.array([2**15-1], dtype=np.int16)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "4645022f-d3aa-4199-b0a0-31cbab79f48e"
    },
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/var/folders/sm/37rm_wm16tq9qsf4n5md98ph0000gp/T/ipykernel_10570/4237367081.py:1: DeprecationWarning: NumPy will stop allowing conversion of out-of-bound Python integers to integer arrays.  The conversion of 32768 to int16 will fail in the future.\n",
      "For the old behavior, usually:\n",
      "    np.array(value).astype(dtype)\n",
      "will give the desired result (the cast overflows).\n",
      "  np.array([2**15], dtype=np.int16)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "array([-32768], dtype=int16)"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.array([2**15], dtype=np.int16)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "a7a8b65d-460b-48a1-b5a8-7370f549879a"
    },
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/var/folders/sm/37rm_wm16tq9qsf4n5md98ph0000gp/T/ipykernel_10570/3574435361.py:1: DeprecationWarning: NumPy will stop allowing conversion of out-of-bound Python integers to integer arrays.  The conversion of 32769 to int16 will fail in the future.\n",
      "For the old behavior, usually:\n",
      "    np.array(value).astype(dtype)\n",
      "will give the desired result (the cast overflows).\n",
      "  np.array([2**15 + 1], dtype=np.int16)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "array([-32767], dtype=int16)"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.array([2**15 + 1], dtype=np.int16)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "77ff4a67-fe07-4675-838a-91b996f6f16c"
    },
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "## Binary fractions\n",
    "Decimal fractions can be represented as binary fractions:\n",
    "\n",
    "Convert $0.125_{10}$ to base 2:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "9bb9458c-97ca-4c43-b234-f2f4b4264940"
    },
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.25"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "0.125 * 2  # 0.0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "8a8a2267-a1eb-4a50-bd8c-5c0de40c6e5e"
    },
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.5"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "_ * 2      # 0.00"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "5cc5b87f-c925-47e4-8c1a-49a6f0ee4896"
    },
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "_ * 2      # 0.001"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "a6a8ff4b-73da-4bde-9d50-722f85bd07de"
    },
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "source": [
    "Thus the binary representation of $0.125_{10}$ is $0.001_2$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "b197b603-9088-4206-83ab-b384fd6496a6"
    },
    "slideshow": {
     "slide_type": "subslide"
    },
    "tags": []
   },
   "source": [
    "General recipe:\n",
    "1. multiply by 2\n",
    "2. if you get a number < 1, add a digit 0 to the right\n",
    "3. if you get a number ≥ 1, add a digit 1 to the right and then repeat the algorithm 1–3 with the remainder"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "d89b6907-11ab-4c5c-8d8d-5b4eecbe52f7"
    },
    "slideshow": {
     "slide_type": "subslide"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.625"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "0.3125 * 2    # 0.0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.25"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "_ * 2         # 0.01"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.25"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "_ - 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.5"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "_ * 2   # 0.010"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "_ * 2         # 0.0101"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "692848dc-d215-472a-8192-0864590a227a"
    },
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "source": [
    "Thus, 0.3125 is $0.0101_2$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "a611d7cb-864a-4d73-9c92-58e865c01826"
    },
    "slideshow": {
     "slide_type": "subslide"
    },
    "tags": []
   },
   "source": [
    "What is the binary representation of decimal $0.1 = \\frac{1}{10}$?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "cac5a3ae-35cd-4a2b-ae61-b029ebf55498"
    },
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.2"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "0.1 * 2  # 0.0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "c11b3fc7-4100-4c6f-adcf-de0b4458ada6"
    },
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.4"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "_ * 2   # 0.00"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "bcf6df92-62c5-4860-b6b4-16e013b57c97"
    },
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.8"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "_ *  2  # 0.000"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "41f5f8cf-a807-4332-a695-48f1759bc4f7"
    },
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.6"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "_ * 2  # 0.0001"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": "subslide"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.6000000000000001"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "_ - 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "256c0ec6-e480-456e-82ec-2b384a8205e5"
    },
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.2000000000000002"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "_ * 2  # 0.00011"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.20000000000000018"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "_ - 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "87506381-29d5-4437-ae55-4e617565f7ad"
    },
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.40000000000000036"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "_ * 2  # 0.000110"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "d1420898-d687-4547-bd57-20c76b718ef4"
    },
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.8000000000000007"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "_ * 2  # 0.0001100 "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "cba6d1e4-881c-49de-a927-15d502e7c10b"
    },
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.6000000000000014"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "_ * 2  # 0.00011001 "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "71e2a420-fa68-4f76-85c4-eaf16f56610b"
    },
    "slideshow": {
     "slide_type": "subslide"
    },
    "tags": []
   },
   "source": [
    "... etc: this is an infinitely repeating fraction and the binary representation of $0.1_{10}$ is $0.000 1100 1100 1100 ..._2$ = $0.000\\overline{1100}_2$.\n",
    "\n",
    "**Thus, with a finite number of bits, 0.1 is not exactly representable in the computer.**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "ccff977d-1bad-4a54-920a-67d32c4da08c"
    },
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "source": [
    "The number 0.1 is not stored exactly in the computer. `print` only shows you a convenient approximation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "42b221fb-993f-4255-a5c7-92d4670e5809"
    },
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.1\n"
     ]
    }
   ],
   "source": [
    "print(0.1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "bc4af227-7435-4493-984f-9994630a7bfc"
    },
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.1000000000000000055511151231257827021181583404541015625\n"
     ]
    }
   ],
   "source": [
    "print(\"{0:.55f}\".format(0.1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "bec99fe1-c645-4771-b4f5-7d3393995f23"
    },
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "## Problems with floating point arithmetic\n",
    "\n",
    "Only a subset of all real numbers can be represented with **floating point numbers of finite bit size**. Almost all floating point numbers are not exact:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "60706432-dcfd-4788-a8df-87b980cd9776"
    },
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "0.1 + 0.1 + 0.1 == 0.3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "2c599aa3-cf3f-4e65-ae3c-ed64b8593eec"
    },
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "source": [
    "... which should have yielded `True`! But because the machine representation of 0.1 is not exact, the equality cannot be fulfilled."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbpresent": {
     "id": "f4542400-ceb8-4998-ac88-0aafbd7002bd"
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Representation of floats: IEEE 754\n",
    "\n",
    "Floating point numbers are stored in \"scientific notation\": e.g. $c = 2.88792458 \\times 10^8$ m/s\n",
    "  * **mantissa**: $2.88792458$\n",
    "  * **exponent**: $+8$\n",
    "  * **sign**: +\n",
    "\n",
    "Format: \n",
    "$$\n",
    "x = (-1)^s \\times 1.f \\times 2_{10}^{e - \\mathrm{bias}}\n",
    "$$\n",
    "\n",
    "($f$ is $M$ bits long. The leading 1 in the mantissa is assumed and not stored: \"ghost\" or \"phantom\" bit. The $2_{10}$ is in decimal but for the remainder, I'll leave out the subscript 10 as it should be clear from context.)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbpresent": {
     "id": "af94b571-be80-4e33-a0fc-caf38cb96d2e"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Format: \n",
    "\n",
    "$$\n",
    "x = (-1)^s \\times 1.f \\times 2^{e - \\mathrm{bias}}\n",
    "$$\n",
    "\n",
    "Note: \n",
    "* In IEEE 754, the highest value of $e$ in the exponent is reserved and not used, e.g. for a 32-bit *float* (see below) the exponent has $(30 - 23) + 1 = 8$ bit and hence the highest number for $e$ is $(2^8 - 1) - 1 = 255 - 1 = 254$. Taking the *bias* into account (for *float*, *bias* = 127), the largest value for the exponent is $2^{254 - 127} = 2^{127}$.\n",
    "* The case of $e=0$ is also special. In this case, the format is $$x = (-1)^s \\times 0.f \\times 2^{-\\mathrm{bias}}$$ i.e. the \"ghost 1\" becomes a zero, gaining a additional order of magnitude."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbpresent": {
     "id": "db5901b9-55cf-4f94-a886-c59e3f090ad7"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### IEEE float (32 bit)\n",
    "\n",
    "IEEE *float* uses **32 bits**\n",
    "  * $\\mathrm{bias} = 127_{10}$\n",
    "  * bits\n",
    "    <table>\n",
    "    <tr><td></td><td>s</td><td>e</td><td>f</td></tr>\n",
    "    <tr><td>bit position</td><td>31</td><td>30–23</td><td>22–0</td></tr>\n",
    "    </table>\n",
    "  * range: $1.4 \\times 10^{-45} \\leq |x_{(32)}| \\leq 3.4 \\times 10^{38}$     \n",
    "  * **six or seven decimal places of significance** (1 in $2^{23}$)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "90a705cc-2e10-44d4-949d-9d509e449e4c"
    },
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.1920928955078125e-07"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1/2**23"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "3c3732d0-ac76-4b16-abb8-168a9b107f27"
    },
    "slideshow": {
     "slide_type": "subslide"
    },
    "tags": []
   },
   "source": [
    "### IEEE double (64 bit)\n",
    "Python floating point numbers are 64-bit doubles. NumPy has dtypes `float32` and `float64`.\n",
    "\n",
    "\n",
    "IEEE *double* uses **64 bits**\n",
    "  * $\\mathrm{bias} = 1023_{10}$\n",
    "  * bits\n",
    "    <table>\n",
    "    <tr><td></td><td>s</td><td>e</td><td>f</td></tr>\n",
    "    <tr><td>bit position</td><td>63</td><td>62–52</td><td>51–0</td></tr>\n",
    "    </table>\n",
    "  * range: $4.9 \\times 10^{-324} \\leq |x_{(64)}| \\leq 1.8 \\times 10^{308}$     \n",
    "  * **about 16 decimal places of significance** (1 in $2^{52}$)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "e8ae2f22-b95e-4238-ae0c-bac8927275a4"
    },
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.220446049250313e-16"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1/2**52"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "dd679719-d7cd-4661-85e0-856f6b8ef972"
    },
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "source": [
    "For numerical calculations, *doubles* are typically required."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbpresent": {
     "id": "3ffc0bfe-f352-4852-8415-6b562e34aa63"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Special numbers\n",
    "IEEE 754 also introduces special \"numbers\" that can result from floating point arithmetic\n",
    "* `NaN` (not a number)\n",
    "* `+INF` and `-INF` (infinity)\n",
    "* `-0` (signed zero)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbpresent": {
     "id": "bd921889-a302-48c6-bfae-e3bd4f5f0897"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Python itself does not use the IEEE special numbers"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "69e4fcaa-80ce-491a-a7ad-343cb460424e"
    },
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [
    {
     "ename": "ZeroDivisionError",
     "evalue": "division by zero",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mZeroDivisionError\u001b[0m                         Traceback (most recent call last)",
      "Cell \u001b[0;32mIn[49], line 1\u001b[0m\n\u001b[0;32m----> 1\u001b[0m \u001b[38;5;241;43m1\u001b[39;49m\u001b[38;5;241;43m/\u001b[39;49m\u001b[38;5;241;43m0\u001b[39;49m\n",
      "\u001b[0;31mZeroDivisionError\u001b[0m: division by zero"
     ]
    }
   ],
   "source": [
    "1/0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "aa0d3a24-9355-482e-a944-a439536f7461"
    },
    "slideshow": {
     "slide_type": "subslide"
    },
    "tags": []
   },
   "source": [
    "But numpy does:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "1b9d6aaa-d8ee-4db5-8d8e-7594083c924c"
    },
    "scrolled": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/var/folders/sm/37rm_wm16tq9qsf4n5md98ph0000gp/T/ipykernel_10570/2672139334.py:1: RuntimeWarning: divide by zero encountered in divide\n",
      "  np.array([1, -1])/np.zeros(2)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "array([ inf, -inf])"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.array([1, -1])/np.zeros(2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "33f6354b-147b-416b-987f-a02912424342"
    },
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "source": [
    "But beware, you cannot use `INF` to \"take limits\". It is purely a sign that something bad happened somewhere..."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "And **not a number**, `nan`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "62636dd1-0dd1-485b-a979-620ffe456de0"
    },
    "scrolled": true,
    "slideshow": {
     "slide_type": "-"
    },
    "tags": []
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/var/folders/sm/37rm_wm16tq9qsf4n5md98ph0000gp/T/ipykernel_10570/1297332888.py:1: RuntimeWarning: invalid value encountered in divide\n",
      "  np.zeros(2)/np.zeros(2)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "array([nan, nan])"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.zeros(2)/np.zeros(2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "3b5b1758-bd76-4847-b32c-878996fcd407"
    },
    "slideshow": {
     "slide_type": "subslide"
    },
    "tags": []
   },
   "source": [
    "### Overflow and underflow\n",
    "\n",
    "* underflow: typically just set to zero (and that works well most of the time)\n",
    "* overflow: raises exception or just set to `inf`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "4873ac11-247b-4f69-b668-2332840691c5"
    },
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.79e+308"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "big = 1.79e308\n",
    "big"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "66270b9d-bb6e-4b94-8cf7-eca82b30d794"
    },
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "inf"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "2 * big"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "9058fabf-8ff7-4c20-817c-fcce56af2c37"
    },
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/var/folders/sm/37rm_wm16tq9qsf4n5md98ph0000gp/T/ipykernel_10570/3296897378.py:1: RuntimeWarning: overflow encountered in multiply\n",
      "  2 * np.array([big], dtype=np.float64)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "array([inf])"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "2 * np.array([big], dtype=np.float64)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "93e580a2-3d10-4919-b91f-84872af916c0"
    },
    "slideshow": {
     "slide_type": "subslide"
    },
    "tags": []
   },
   "source": [
    "... but you can (maybe*) just use an even bigger data type:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "664d2a53-f8b4-46d9-8cba-197a80207007"
    },
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([3.58e+308], dtype=float128)"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "2 * np.array([big], dtype=np.float128)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "source": [
    "(*Note: float128 is 80 bit on Linux/macOS and 64 bit on Windows... see [Numpy data types: Extended Precisions](https://docs.scipy.org/doc/numpy/user/basics.types.html#extended-precision))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "1e44225a-62fc-4473-b260-9122fc9418ca"
    },
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "### Insignificant digits"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "source": [
    "Even though 0.2 and 0.4 are not exactly representable, their difference appears to be equal to 0.2, as would be expected from algebra:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": "-"
    },
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.2\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = 0.4 - 0.2\n",
    "print(a)\n",
    "a == 0.2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "source": [
    "What about adding 1000 to each operand? Should not change the equation..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "1ba45ec2-d307-460f-8ec2-e7f5d1578c05"
    },
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.1999999999999318\n"
     ]
    }
   ],
   "source": [
    "A = 1000.4 - 1000.2\n",
    "print(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "57f50a5d-b0d4-4a88-991a-afcd173714c8"
    },
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A == 0.2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "8efd7e27-9314-4790-9544-2017618f1dd6"
    },
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "source": [
    "... oops!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "a64135c7-990b-469b-8c60-8d522eacd089"
    },
    "slideshow": {
     "slide_type": "subslide"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "700.0"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = 700\n",
    "y = 1e-14\n",
    "x - y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "948dd07e-4b78-43e2-8efe-1daac7a59e18"
    },
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x - y < 700"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "8011b45e-3181-4f12-b0d2-0e1e46c2111f"
    },
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "source": [
    "... ooops"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "89fbbed7-384a-4926-9901-cf0e9cc2b692"
    },
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "## Machine precision\n",
    "Only a limited number of floating point numbers can be represented. This *limited precision* affects calculations:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "09461108-573f-4ac1-b7f0-7f2c604935ae"
    },
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5.0"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = 5  + 1e-16\n",
    "x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "939a478a-83eb-4c35-9adc-ba79299c9591"
    },
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x == 5"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "0d6185d9-53ff-4265-a1bb-3603e16178f7"
    },
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "source": [
    "... oops."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "8553cb11-0754-4b01-86d6-252873a6846a"
    },
    "slideshow": {
     "slide_type": "subslide"
    },
    "tags": []
   },
   "source": [
    "**Machine precision** $\\epsilon_m$ is defined as the maximum number that can be added to 1 in the computer without changing that number 1:\n",
    "\n",
    "$$\n",
    "1_c + \\epsilon_m := 1_c\n",
    "$$\n",
    "\n",
    "Thus, the *floating point representation* $x_c$ of an arbitrary number $x$ is \"in the vicinity of $x$\"\n",
    "\n",
    "$$\n",
    "x_c = x(1\\pm\\epsilon), \\quad |\\epsilon| \\leq \\epsilon_m\n",
    "$$\n",
    "\n",
    "where we don't know the true value of $\\epsilon$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbpresent": {
     "id": "42ba93a9-cbdc-4cb5-b281-9d8ff7925c58"
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Thus except for powers of 2 (which are represented exactly) **all floating point numbers contain an unknown error in the 6th decimal place (32 bit floats) or 15th decimal (64 bit doubles)**. \n",
    "\n",
    "This error should be treated as a random error because we don't know its magnitude."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "8474338a-d69a-4f30-b8b1-73e20e0681d0"
    },
    "slideshow": {
     "slide_type": "subslide"
    },
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "machine precision reached for 52 bits\n",
      "eps = 1.1102230246251565e-16, 1 + eps = 1.0\n"
     ]
    }
   ],
   "source": [
    "N = 100\n",
    "eps = 1\n",
    "for nbits in range(N):\n",
    "    eps /= 2\n",
    "    one_plus_eps = 1.0 + eps\n",
    "    # print(\"eps = {0}, 1 + eps = {1}\".format(eps, one_plus_eps))\n",
    "    if one_plus_eps == 1.0:\n",
    "        print(\"machine precision reached for {0} bits\".format(nbits))\n",
    "        print(\"eps = {0}, 1 + eps = {1}\".format(eps, one_plus_eps))\n",
    "        break\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "4cdfc17e-8e9e-40ca-ba9f-801ca904c1ef"
    },
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "source": [
    "Compare to our estimate for the precision of float64:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "bd394be6-b41a-42ca-a441-a6871294ba54"
    },
    "slideshow": {
     "slide_type": "-"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.220446049250313e-16"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1/2**52"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "4d893a2a-6697-4106-97a3-9d3cedda8196"
    },
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "## Appendix"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Floating point binary to decimal "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbpresent": {
     "id": "4bdc548a-84a8-444f-887f-7c920168061f"
    }
   },
   "source": [
    "A quick hack to convert a floating point binary representation to a floating point number."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "92ee3a78-19fa-427a-9a45-c244a1af741f"
    },
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [],
   "source": [
    "bits = \"1010.0001100110011001100110011001100110011001100110011\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "b789ba2c-ffa1-4c1a-94ee-da49efc65761"
    },
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [],
   "source": [
    "import math\n",
    "def bits2number(bits):\n",
    "    if '.' in bits:\n",
    "        integer, fraction = bits.split('.')\n",
    "    else:\n",
    "        integer = bits\n",
    "        fraction = \"\"\n",
    "    powers = [int(bit) * 2**n for n, bit in enumerate(reversed(integer))]\n",
    "    powers.extend([int(bit) * 2**(-n) for n, bit in enumerate(fraction, start=1)])\n",
    "    return math.fsum(powers)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "957186a6-c447-4a68-be16-a222ae0c0ca4"
    },
    "slideshow": {
     "slide_type": "subslide"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10.1"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bits2number(bits)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "85d457c7-a1e7-4f8b-9c49-afbceb35ab8e"
    },
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "15.0"
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bits2number('1111')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "556735ca-7177-43e8-b751-64a269841ac2"
    },
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.09999999999999964"
      ]
     },
     "execution_count": 69,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bits2number('0.0001100110011001100110011001100110011001100110011')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "c8e89824-6653-447d-bfeb-91b07f76635f"
    },
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.09375"
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bits2number('0.0001100')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "f4629844-0cfb-4b03-bf9e-37d91209f4a3"
    },
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.3125"
      ]
     },
     "execution_count": 71,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bits2number('0.0101')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "2b6be023-56ef-4b4d-b486-1d6213852f65"
    },
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.65625"
      ]
     },
     "execution_count": 72,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bits2number(\"10.10101\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "0410b25b-3969-460b-8f27-ee6eb7045621"
    },
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.4999999999990905"
      ]
     },
     "execution_count": 73,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bits2number('0.0111111111111111111111111111111111111111')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "0b5777ce-5a2b-450a-b508-87b4a858ebdc"
    },
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.7998046875"
      ]
     },
     "execution_count": 74,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bits2number('0.110011001100')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "d8bdc59e-f378-4e97-ae51-8ded024b1745"
    },
    "slideshow": {
     "slide_type": "subslide"
    },
    "tags": []
   },
   "source": [
    "Python can convert to binary using the `struct` module:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "c890ea7a-5ef7-4198-bf22-8b240f4dee48"
    },
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "00000001101100101011001\n"
     ]
    }
   ],
   "source": [
    "x = 6.0e-8\n",
    "\n",
    "import struct\n",
    "fpack = struct.pack('f', x)    # pack float into bytes\n",
    "fint = struct.unpack('i', fpack)[0] # unpack to int\n",
    "m_bits = bin(fint)[-23:]            # mantissa bits\n",
    "print(m_bits)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "1033df42-daf1-4391-8b28-a30b197d0a04"
    },
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "source": [
    "With phantom bit:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "34c317f8-0810-46de-89f3-ee4e9adf3d94"
    },
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.00000001101100101011001\n"
     ]
    }
   ],
   "source": [
    "mantissa_bits = '1.' + m_bits\n",
    "print(mantissa_bits)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {
    "editable": true,
    "nbpresent": {
     "id": "69151f0f-c2f2-4c1d-8bbc-71b690a4f28c"
    },
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "6e-08 = 0.50331648 -23\n"
     ]
    }
   ],
   "source": [
    "import math\n",
    "mn, ex = math.frexp(x)\n",
    "print(x, '=', mn, ex)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "source": [
    "### remainder multiplication \n",
    "\n",
    "Define the \"multmod\" operation (equivalent to \"divmod\"):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [],
   "source": [
    "import math\n",
    "\n",
    "def multmod(a, b):\n",
    "    p = a*b\n",
    "    p_frac, p_int = math.modf(p)\n",
    "    return int(p_int), p_frac"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "source": [
    "This allows us to manually run the algorithm for converting a decimal fraction to a binary fraction:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1, 0.10000000000000009)"
      ]
     },
     "execution_count": 79,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "multmod(0.55, 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0, 0.20000000000000018)"
      ]
     },
     "execution_count": 80,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "multmod(_[1], 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0, 0.40000000000000036)"
      ]
     },
     "execution_count": 81,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "multmod(_[1], 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0, 0.8000000000000007)"
      ]
     },
     "execution_count": 82,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "multmod(_[1], 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1, 0.6000000000000014)"
      ]
     },
     "execution_count": 83,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "multmod(_[1], 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1, 0.20000000000000284)"
      ]
     },
     "execution_count": 84,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "multmod(_[1], 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0, 0.4000000000000057)"
      ]
     },
     "execution_count": 85,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "multmod(_[1], 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0, 0.8000000000000114)"
      ]
     },
     "execution_count": 86,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "multmod(_[1], 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1, 0.6000000000000227)"
      ]
     },
     "execution_count": 87,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "multmod(_[1], 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1, 0.20000000000004547)"
      ]
     },
     "execution_count": 88,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "multmod(_[1], 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0, 0.40000000000009095)"
      ]
     },
     "execution_count": 89,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "multmod(_[1], 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0, 0.8000000000001819)"
      ]
     },
     "execution_count": 90,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "multmod(_[1], 2)"
   ]
  },
  {
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    "Collect the integer parts in forward order as the fractional binary expansion after the point:\n",
    "\n",
    "$0.55_{10}$ = $0.10001100110011\\dots_2$ = $0.10\\overline{0011}_2$"
   ]
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   "source": [
    "### conversion of decimal floats to binary representation\n",
    "\n",
    "Using our `multmod()` function we can implement the algorithm for binary fractions. Similarly, we use `divmod()` for the integer part."
   ]
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   "source": [
    "import math\n",
    "\n",
    "def multmod(a, b):\n",
    "    p = a*b\n",
    "    p_frac, p_int = math.modf(p)\n",
    "    return int(p_int), p_frac\n",
    "\n",
    "def frac10to2(x, nmax=15):\n",
    "    if not abs(x) < 1:\n",
    "        raise ValueError(f\"{x} is not the fractional part\")\n",
    "    remainder = x\n",
    "    digits = []\n",
    "    for k in range(nmax):\n",
    "        digit, remainder = multmod(remainder, 2)\n",
    "        digits.append(digit)\n",
    "    return tuple(digits)\n",
    "\n",
    "def int10to2(x):\n",
    "    if int(x) != x:\n",
    "        raise ValueError(f\"{x} must be integer\")\n",
    "    digits = []\n",
    "    p, remainder = divmod(int(x), 2)\n",
    "    while p > 1:\n",
    "        digits.append(remainder)\n",
    "        p, remainder = divmod(p, 2)\n",
    "    return tuple([p, remainder] + digits[::-1])\n",
    "\n",
    "def decimal2binary(x, nmax=15):\n",
    "    \"\"\"Return binary representation of floating point number x.\n",
    "    \n",
    "    Parameters\n",
    "    ----------\n",
    "    x : float\n",
    "        floating point number to be converted\n",
    "    nmax : int\n",
    "         maximum power in the fractional expansion, 2**(-nmax),\n",
    "         i.e., the fractional part will always contain nmax\n",
    "         digits (zero padded)\n",
    "    \n",
    "    Returns\n",
    "    -------\n",
    "    binary_digits: (int_tuple, fraction_tuple)\n",
    "         Resulting tuple contains a tuple for the integer part and for the\n",
    "         fractional part. The `fraction_tuple` always contains `nmax` numbers\n",
    "         and may not exactly represent the fractional part.\n",
    "         The `int_tuple` contains the leading 1 and as many following digits\n",
    "         as necessary to exactly represent the integer part.\n",
    "         \n",
    "    Note\n",
    "    ----\n",
    "    Almost all fractional numbers cannot be exactly represented with a finite\n",
    "    number of fractional digits.\n",
    "    \"\"\"\n",
    "    x_frac, x_int = math.modf(x)\n",
    "    return int10to2(x_int), frac10to2(x_frac, nmax=nmax)"
   ]
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    {
     "data": {
      "text/plain": [
       "(1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0)"
      ]
     },
     "execution_count": 92,
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    "frac10to2(0.55)"
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       "(1, 0, 1, 0)"
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    "int10to2(10)"
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   "execution_count": 94,
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    {
     "data": {
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       "((1, 0, 1, 0), (1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0))"
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    "decimal2binary(4.25, nmax=2)"
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