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Floating Point Numbers

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• General Format of Decimal floating point numbers representation:
  • Floating point numbers can be written as an exponent and a mantissa

  • Examples: 314.159 = 0.314159 x 10³ = 314159 x 10^-3

  • We do not need to record the fact that the exponent used is 10, and we can represent the number 314.159 using a pair of numbers:

    314.159 = (314.159, 0) = (0.314159, 3) = (314159, -3)

    The first number is the mantissa and the second is the exponent.

• Binary floating point numbers:
  • Binary floating point numbers can also be written as an exponent and a mantissa:

  • Examples: 1010.1011 = 0.10101011 x 2⁴ = 10101011 x 2^-4

  • SImilarly, we do not need to record the fact that the exponent used is 2, and we can represent the number 1010.1011 using a pair of binary numbers:

    1010.1011 = (1010.1011, 0) = (0.10101011, 00000100 (4) ) = (10101011, 11111100 (-4) )

    The first number is the mantissa and the second is the exponent.

• The IEEE Standards:
  • IEEE is a standard making organization (Institute of Electric and Electronics Engineers).

  • IEEE has defined a number of floating point number formats (single precision and double precision)

  • The IEEE Single Precision represention:
    • Uses 32 bits in the representation.
    • The mantissa uses the sign/magnitude representation
    • The exponent uses a modified 2's complement representation (which is similar to the 2's complement representation)

           S EEEEEEEE MMMMMMMMMMMMMMMMMMMMMMM
           0 1      8 9                    31
       

    • The first bit is the sign bit, S of the mantissa
    • The next eight bits are the exponent bits, 'E',
    • The final 23 bits are the mantissa 'M':
    • The mantissa is normalized so that 1.0 <= M < 2.0

  • The IEEE Double Precision represention:
    • Uses 64 bits, and it is similar to the single precision format

          S EEEEEEEEEEE FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF
          0 1        11 12                                                63
       

    • The first bit is the sign bit, S of the mantissa
    • The next 11 bits are the exponent bits, 'E',
    • The final 52 bits are the mantissa 'M':
    • The mantissa is normalized so that 1.0 <= M < 2.0

• Explore further....

  Here is a nice webpage where you can construct floating point representations: click here

  When you explore the above webpage, you must know that there are 2 things that are strange in the IEEE float representation

  1. Because the first bit of the mantissa is always 1 (stop, think, why is that so ? because we made it so: 1.0 <= mantissa < 2.0), it is not stored.

     So if the mantissa bits in a single precision representation are 01010101010101010101010, the actual mantissa bits are 1.01010101010101010101010 (and the mantissa is between 1.0 and 2.0)

  2. The exponent uses a modified 2's complement encoding, called "excess" encoding.

     The single precision exponent uses the "excess 127" encoding which uses 01111111 to represent 0:

  Bit pattern:
      01111100   01111101   01111110   01111111   10000000   10000001   10000010
      ----+----------+----------+----------+----------+----------+----------+-
         -3         -2         -1          0          1          2          3
  Value represented: