COMP2300

IEEE 754 example

Example: Converting to IEEE 754 single precision

Convert 0.085 into IEEE 754 single-precision format:

1. Sign

   The number 0.085 is positive, so the sign bit = 0.

   (-1)⁰ = 1.

2. Write the number 0.085 in base-2 scientific notation

   We have to factor 0.085 into a number in the range [1 <= n < 2] and a power of 2.

   0.085 = (-1)⁰  *  mantissa   * 2 ^power,    or:
   0.085 / 2^power = mantissa.

   We can divide 0.085 by a power of 2 to get the mantissa:

   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. The exponent

   The power of 2 is -4, and the bias for the single-precision format is 127. This means that the exponent = 127 + (-4) = 123, or 01111011_bin

4. Write the mantissa in binary form

   Because IEEE 754 assumes numbers to be normalised, we know that they are in the interval [1 <= n < 2] and therefore we don't have to store the leading 1.

   The fraction is therefore 0.36. Unfortunately, this is not a "nice" number. All we can do is approximate the value. The IEEE 754 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+...

   0.36_ten ~ 0.01011100001010001111011 _bin.

   The binary representation of the mantissa is therefore: 01011100001010001111011.

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

5. Put the things together

   1 bit for the sign, followed by 8 bits for the exponent, and 23 bits for the fraction.

   The IEEE representation of 0.085 is therefore:

   	Sign	Exponent	Mantissa
   Decimal	0	123	0.36
   Binary	0	01111011	01011100001010001111011