Floating_Point_Representation.ipynb

{"cells":[{"cell_type":"markdown","metadata":{"id":"1WWUD7qirAJL"},"source":["Make sure you remove `raise NotImplementedError()` and fill in any place that says `# YOUR CODE HERE`, as well as your `NAME`, `ID`, and `LAB_SECTION` below:"]},{"cell_type":"code","execution_count":1,"metadata":{"executionInfo":{"elapsed":510,"status":"ok","timestamp":1635250365787,"user":{"displayName":"MD. ASHIFUL HAQUE","photoUrl":"https://lh3.googleusercontent.com/a-/AOh14Gj1WZSCA1WBvN3MoZuFTfdjPfNbA236Bk3KxGvO=s64","userId":"18054771173752848147"},"user_tz":-360},"id":"5xN2aMZ0rAJN"},"outputs":[],"source":["NAME = \"ARNOB MAJUMDER\"\n","ID = \"20301089\"\n","SECTION = \"07\""]},{"cell_type":"markdown","metadata":{"id":"iiq6Xum6rAJO"},"source":["---"]},{"cell_type":"markdown","metadata":{"deletable":false,"editable":false,"id":"Io3ii0VArAJO","nbgrader":{"cell_type":"markdown","checksum":"7c4e9d0fccc2e8dd26580075f9ceb52d","grade":false,"grade_id":"intro_8bit","locked":true,"schema_version":3,"solution":false,"task":false}},"source":["# Floating Point Representation\n","\n","## 8 bit Floating Point Representation\n","\n","We will implement an 8 bit Floating point number system in this part, where First bit is a sign bit, next three bits are exponents,\n","and the rest 4 bits are mantissas (significands).\n","\n","Numbers are divided into two categories: normals and subnormals.\n","\n","For normals the encoding is:\n","\\begin{equation}\n","(-1)^{sign} \\times 2^{(e-3)} \\times 1.fraction\n","\\end{equation}\n","For subnormals the encoding is:\n","\\begin{equation}\n","(-1)^{sign} \\times 2^{(1-3)} \\times 0.fraction\n","\\end{equation}\n","\n","Example:\n","\\begin{aligned}\n","    0 001 0000 &= 2^{(1-3)} \\times 1.0 = 0.25\\\\\n","    0 010 0000 &= 2^{(2-3)} \\times 1.0 = 0.5\\\\\n","    0 011 0000 &= 2^{(3-3)} \\times 1.0 = 1.0\\\\\n","    0 011 0100 &= 2^{(3-3)} \\times 1.25 = 1.25\n","\\end{aligned}\n","\n","If the exponent is 0 then the number is considered subnormal.\n","\n","Example:\n","\\begin{aligned}\n","    0 000 0000 &= 2^{-2} \\times 0   = 0.0\\\\\n","    0 000 1000 &= 2^{-2} \\times 0.5 = 0.125\\\\\n","    0 000 1100 &= 2^{-2} \\times 0.75 = 0.1875\n","\\end{aligned}\n","\n","If the exponent is 7 then and mantissa is zero then the number is \n","considered infinity (inf), otherwise it is not-a-number (nan).\n","\n","Example:\n","\\begin{aligned}\n","    0 111 0000 &= +\\infty\\\\\n","    1 111 0000 &= -\\infty\\\\\n","    X 111 XXXX &= NaN\n","\\end{aligned}\n","\n","### Instructions\n","In this task, you need to implement the `Float8()` class. Some parts of the class are already written for you. You only need to modify `__init__(self, bitstring)` and `calculate_value(self)` fuctions, where it says `# YOUR CODE HERE`. You may ignore `raise NotImplementedError()`."]},{"cell_type":"code","execution_count":2,"metadata":{"deletable":false,"editable":false,"executionInfo":{"elapsed":11,"status":"ok","timestamp":1635250370732,"user":{"displayName":"MD. ASHIFUL HAQUE","photoUrl":"https://lh3.googleusercontent.com/a-/AOh14Gj1WZSCA1WBvN3MoZuFTfdjPfNbA236Bk3KxGvO=s64","userId":"18054771173752848147"},"user_tz":-360},"id":"qIGtiRs-rAJP","nbgrader":{"cell_type":"code","checksum":"ac5f73bdfe958d67ad35fcd7def0142b","grade":false,"grade_id":"import","locked":true,"schema_version":3,"solution":false,"task":false}},"outputs":[],"source":["import math\n","import itertools\n","import matplotlib.pyplot as plt\n","import numpy as np"]},{"cell_type":"code","execution_count":19,"metadata":{"deletable":false,"executionInfo":{"elapsed":512,"status":"ok","timestamp":1635253098255,"user":{"displayName":"MD. ASHIFUL HAQUE","photoUrl":"https://lh3.googleusercontent.com/a-/AOh14Gj1WZSCA1WBvN3MoZuFTfdjPfNbA236Bk3KxGvO=s64","userId":"18054771173752848147"},"user_tz":-360},"id":"ubVFqV6mrAJP","nbgrader":{"cell_type":"code","checksum":"98f9407d030b5a7e7abf8599b885afad","grade":false,"grade_id":"8bit_ans","locked":false,"schema_version":3,"solution":true,"task":false}},"outputs":[],"source":["class Float8():\n","    \n","    def __init__(self, bitstring):\n","        '''Constructor\n","        takes a 8-bit string of 0's and 1's as input and stores the sub-strings\n","        accordingly.\n","        Usage: Float8('00011110')\n","        '''\n","        \n","        # Make sure the input consists of exactly 8-bits.\n","        assert(len(bitstring)==8)\n","        \n","        self.sign = bitstring[0]\n","        self.exponent = bitstring[1:4]\n","        self.mantissa = bitstring[4:8]\n","        \n","        self.val = self.calculate_value()\n","        \n","    def __str__(self):\n","        return f'Sign bit value: {self.sign}\\n' + \\\n","            f'Exponent value: {self.exponent}\\n' + \\\n","            f'Mantissa value: {self.mantissa}\\n' + \\\n","            f'Floating value: {self.val}\\n'\n","    \n","    def tobitstring(self):\n","        return self.sign + self.exponent + self.mantissa\n","    \n","    def toformattedstring(self):\n","        return ' '.join([self.sign, self.exponent, self.mantissa])\n","    \n","    def calculate_value(self):\n","        '''Calculate the value of the number from bits'''\n","\n","        val = 0.0\n","        \n","        if self.exponent == '111':\n","          if self.mantissa == '0000':\n","            val = math.inf\n","          else:\n","            val = math.nan\n","        elif self.exponent == '000':\n","          for exp, bits in enumerate(self.mantissa):\n","            val += (int(bits)*(2**(-(exp+1))))\n","          val *= (2**(-2))\n","        else:\n","          val = 1.0\n","          for exp, bits in enumerate(self.mantissa):\n","            val += (int(bits)*(2**(-(exp+1))))\n","          expo=0\n","          for exp, bits in enumerate(reversed(self.exponent)):\n","            expo += (int(bits)*(2**exp))\n","          val *= (2**(expo-3))\n","\n","        if self.sign == '1':\n","          val *= -1\n","        \n","        return val"]},{"cell_type":"code","execution_count":20,"metadata":{"colab":{"base_uri":"https://localhost:8080/"},"deletable":false,"editable":false,"executionInfo":{"elapsed":505,"status":"ok","timestamp":1635253102071,"user":{"displayName":"MD. ASHIFUL HAQUE","photoUrl":"https://lh3.googleusercontent.com/a-/AOh14Gj1WZSCA1WBvN3MoZuFTfdjPfNbA236Bk3KxGvO=s64","userId":"18054771173752848147"},"user_tz":-360},"id":"dnTMKYNKrAJQ","nbgrader":{"cell_type":"code","checksum":"6ca28ff8f98269b69d905fdcd45d338c","grade":true,"grade_id":"8bit_test","locked":true,"points":10,"schema_version":3,"solution":false,"task":false},"outputId":"4b020a95-8d26-4851-a4e5-ea5311ee1901"},"outputs":[{"name":"stdout","output_type":"stream","text":["28 out of 28 outputs matched for 8-bit floating points\n"]}],"source":["'''28 test cases for 8-bit floating points'''\n","import numpy as np\n","\n","data = [ '00000000', '00000001', '00001001', '00010000',\n","         '00010001', '00011000', '00011011', '00100000',\n","         '00101101', '00110000', '00110101', '01000011',\n","         '01000000', '01010000', '01011100', '01100000',\n","         '01110111', '01110000', '10000000', '10000001',\n","         '11110001', '11110000', '10110001', '10111101',\n","         '11100000', '11101011', '11010000', '11000000']\n","results = ['(0, 1)', '(1, 64)', '(9, 64)', '(1, 4)', '(17, 64)', '(3, 8)', '(27, 64)',\n","          '(1, 2)', '(29, 32)', '(1, 1)', '(21, 16)', '(19, 8)', '(2, 1)', '(4, 1)',\n","          '(7, 1)', '(8, 1)', 'nan', 'inf', '(0, 1)', '(-1, 64)', 'nan', '-inf',\n","          '(-17, 16)', '(-29, 16)', '(-8, 1)', '(-27, 2)', '(-4, 1)', '(-2, 1)']\n","\n","test = [Float8(x).val for x in data]\n","\n","for index in range(len(test)):\n","    d = test[index]\n","    try:\n","        test[index] = str(d.as_integer_ratio())\n","    except Exception:\n","        test[index] = str(d)\n","\n","np.testing.assert_equal(np.array(test), np.array(results))\n","print('28 out of 28 outputs matched for 8-bit floating points')"]},{"cell_type":"markdown","metadata":{"deletable":false,"editable":false,"id":"D2V7P4eyrAJR","nbgrader":{"cell_type":"markdown","checksum":"56c84b861684b011679fc374962c97d8","grade":false,"grade_id":"intro_16bit","locked":true,"schema_version":3,"solution":false,"task":false}},"source":["## 16 Bit Floating Point Representation\n","\n","Over here, we will implement a 16 bit Floating point number system,\n","where the first bit is a sign bit, next four bits are exponents,\n","and the rest 11 bits are mantissas (significands).\n","\n","Numbers are divided into two categories: normals and subnormals.\n","\n","For normals the encoding is:\n","\\begin{equation}\n","(-1)^{sign} \\times 2^{(e-7)} \\times 1.fraction\n","\\end{equation}\n","For subnormals the encoding is:\n","\\begin{equation}\n","(-1)^{sign} \\times 2^{(1-7)} \\times 0.fraction\n","\\end{equation}\n","\n","Example:\n","\\begin{aligned}\n","    0 0101 00000000000 &= 2^{(5-7)} \\times 1.0  = 0.25\\\\\n","    0 0110 00000000000 &= 2^{(6-7)} \\times 1.0  = 0.5\\\\\n","    0 0111 00000000000 &= 2^{(3-7)} \\times 1.0  = 1.0\\\\\n","    0 0111 01000000000 &= 2^{(3-7)} \\times 1.25 = 1.25\n","\\end{aligned}\n","\n","If the exponent is 0 then the number is considered subnormal.\n","\n","Example:\n","\\begin{aligned}\n","    0 0000 00000000000 &= 2^{-6} \\times 0    = +0.0\\\\\n","    0 0000 10000000000 &= 2^{-6} \\times 0.5  = +0.0078125\\\\\n","    0 0000 11000000000 &= 2^{-6} \\times 0.75 = +0.01171875\n","\\end{aligned}\n","\n","If the exponent is 15 (all 1's) then and mantissa is zero then the number is considered infinity (inf), otherwise it is not-a-number (nan).\n","    \n","Example:\n","\\begin{aligned}\n","    0 1111 00000000000 &= +\\infty\\\\\n","    1 1111 00000000000 &= -\\infty\\\\\n","    X 1111 XXXXXXXXXXX &= NaN\n","\\end{aligned}\n","\n","### Instructions\n","In this task, you need to implement the `Float16()` class. Some parts of the class are already written for you. You only need to modify `__init__(self, bitstring)` and `calculate_value(self)` fuctions, where it says `# YOUR CODE HERE`. You may ignore `raise NotImplementedError()`."]},{"cell_type":"code","execution_count":28,"metadata":{"deletable":false,"executionInfo":{"elapsed":740,"status":"ok","timestamp":1635254405328,"user":{"displayName":"MD. ASHIFUL HAQUE","photoUrl":"https://lh3.googleusercontent.com/a-/AOh14Gj1WZSCA1WBvN3MoZuFTfdjPfNbA236Bk3KxGvO=s64","userId":"18054771173752848147"},"user_tz":-360},"id":"FEARBhuYrAJS","nbgrader":{"cell_type":"code","checksum":"9176054a6507a9547cb18a21136d733e","grade":false,"grade_id":"16bit_ans","locked":false,"schema_version":3,"solution":true,"task":false}},"outputs":[],"source":["class Float16():\n","    \n","\n","    def __init__(self, bitstring):\n","        '''Constructor\n","        takes a 16-bit string of 0's and 1's as input and stores the sub-strings\n","        accordingly.\n","        Usage: Float16('0001111000011110')\n","        '''\n","\n","        # Make sure the input consists of exactly 16-bits.\n","        assert(len(bitstring)==16)\n","        self.sign = bitstring[0]\n","        self.exponent = bitstring[1:5]\n","        self.mantissa = bitstring[5:]\n","   \n","        self.val = self.calculate_value()\n","        \n","    def __str__(self):\n","        return f'Sign bit value: {self.sign}\\n' + \\\n","            f'Exponent value: {self.exponent}\\n' + \\\n","            f'Mantissa value: {self.mantissa}\\n' + \\\n","            f'Floating value: {self.val}\\n'\n","    \n","    def tobitstring(self):\n","        return self.sign + self.exponent + self.mantissa\n","    \n","    def toformattedstring(self):\n","        return ' '.join([self.sign, self.exponent, self.mantissa])\n","    \n","    def calculate_value(self):\n","        '''Calculate the value of the number from bits'''\n","\n","        # Initialize with zero\n","        val = 0.0\n","        \n","        if self.exponent == '1111':\n","          if self.mantissa == '00000000000':\n","            val = math.inf\n","          else:\n","            val = math.nan\n","        elif self.exponent == '0000':\n","          for exp, bits in enumerate(self.mantissa):\n","            val += (int(bits)*(2**(-(exp+1))))\n","          val *= (2**(-6))\n","        else:\n","          val = 1.0\n","          for exp, bits in enumerate(self.mantissa):\n","            val += (int(bits)*(2**(-(exp+1))))\n","          expo=0\n","          for exp, bits in enumerate(reversed(self.exponent)):\n","            expo += (int(bits)*(2**exp))\n","          val *= (2**(expo-7))\n","\n","        if self.sign == '1':\n","          val *= -1\n","\n","        return val"]},{"cell_type":"code","execution_count":29,"metadata":{"colab":{"base_uri":"https://localhost:8080/"},"deletable":false,"editable":false,"executionInfo":{"elapsed":13,"status":"ok","timestamp":1635254408619,"user":{"displayName":"MD. ASHIFUL HAQUE","photoUrl":"https://lh3.googleusercontent.com/a-/AOh14Gj1WZSCA1WBvN3MoZuFTfdjPfNbA236Bk3KxGvO=s64","userId":"18054771173752848147"},"user_tz":-360},"id":"SpIZzEIyrAJT","nbgrader":{"cell_type":"code","checksum":"0c6dcd10e961fa37e25e8cba3164cea1","grade":true,"grade_id":"16bit_test","locked":true,"points":10,"schema_version":3,"solution":false,"task":false},"outputId":"96e5ef33-1233-4d34-ac12-212782dae0b8"},"outputs":[{"name":"stdout","output_type":"stream","text":["28 out of 28 outputs matched for 16-bit floating points\n"]}],"source":["'''28 test cases for 16-bit floating points'''\n","data = [ '0011100000000010', '0100000000000000', '1100000000000000', '0100010000000000',\n","         '1100010000000000', '0100100000000000', '1100100000000000', '0100101000000000',\n","         '1100101000000000', '0100110000000000', '1100110000000000', '0101101110000000',\n","         '0010010000000000', '0000000000000001', '0000011111111111', '0000100000000000',\n","         '0111011111111111', '0000000000000000', '1000000000000000', '0111100000000000',\n","         '1111100000000000', '0111100000000001', '0111110000000001', '0111111111111111',\n","         '0010101010101011', '0100010010010001', '0011100000000000', '0011100000000001']\n","results = ['(1025, 1024)', '(2, 1)', '(-2, 1)', '(3, 1)', '(-3, 1)', '(4, 1)', '(-4, 1)',\n","           '(5, 1)', '(-5, 1)', '(6, 1)', '(-6, 1)', '(23, 1)', '(3, 16)', '(1, 131072)',\n","           '(2047, 131072)', '(1, 64)', '(4095, 16)', '(0, 1)', '(0, 1)', 'inf', '-inf',\n","           'nan', 'nan', 'nan', '(2731, 8192)', '(3217, 1024)', '(1, 1)', '(2049, 2048)']\n","\n","test = [Float16(x).val for x in data]\n","\n","for index in range(len(test)):\n","    d = test[index]\n","    try:\n","        test[index] = str(d.as_integer_ratio())\n","    except Exception:\n","        test[index] = str(d)\n","\n","np.testing.assert_equal(np.array(test), np.array(results))\n","print('28 out of 28 outputs matched for 16-bit floating points')"]}],"metadata":{"colab":{"collapsed_sections":[],"name":"Floating_Point_Representation.ipynb","provenance":[{"file_id":"1Nbm-GTNRbg9Ia6-SXKvy9sb8WwcJn5qb","timestamp":1635236246163}]},"kernelspec":{"display_name":"Python 3","language":"python","name":"python3"},"language_info":{"codemirror_mode":{"name":"ipython","version":3},"file_extension":".py","mimetype":"text/x-python","name":"python","nbconvert_exporter":"python","pygments_lexer":"ipython3","version":"3.7.4"}},"nbformat":4,"nbformat_minor":0}