IEEE 754 Notation

Example: Converting to IEEE 754 Form


Put 0.085 in single-precision format

  1. The first step is to look at the sign of the number.
    Because 0.085 is positive, the sign bit =0.

    (-1)0 = 1.



  2. Write 0.085 in base-2 scientific notation.
    This means that we must factor it into a number in the range [1 <= n < 2] and a power of 2.

    0.085 = (-1)0  *  (1+fraction)   * 2 power,    or:
    0.085 / 2power = (1+fraction).

    So we can divide 0.085 by a power of 2 to get the (1 + fraction).

    0.085 / 2-1 = 0.17
    0.085 / 2-2 = 0.34
    0.085 / 2-3 = 0.68
    0.085 / 2-4 = 1.36

    Therefore, 0.085 = 1.36 * 2-4



  3. Find the exponent.
    The power of 2 is -4, and the bias for the single-precision format is 127. This means that the exponent = 123ten, or 01111011bin



  4. Write the fraction in binary form
    The fraction = 0.36 . Unfortunately, this is not a "pretty" number, like those shown in the book. The best we can do is to approximate the value. Single-precision format allows 23 bits for the fraction.

    Binary fractions look like this:

    0.1 = (1/2) = 2-1
    0.01 = (1/4) = 2-2
    0.001 = (1/8) = 2-3

    To approximate 0.36, we can say:

    0.36 = (0/2) + (1/4) + (0/8) + (1/16) + (1/32) +...
    0.36 = 2-2 + 2-4 + 2-5+...

    0.36ten ~ 0.01011100001010001111011bin .

    The binary string we need is: 01011100001010001111011.

    It's important to notice that you will not get 0.36 exactly. This is why floating-point numbers have error when you put them in IEEE 754 format.



  5. Now put the binary strings in the correct order -
    1 bit for the sign, followed by 8 for the exponent, and 23 for the fraction. The answer is:

    Sign Exponent Fraction
    Decimal 0 123 0.36
    Binary 0 01111011 01011100001010001111011

Example: IEEE 754 to Float ->