IEEE 754 Notation
3. Creating the Bitstring
You're almost done when the number is in this form:
(-1)sign bit *
2exponent - bias
This was a clever move. Instead of one messy value, you now
have three important pieces of information that can identify the number.
- If the sign bit is a 0, then the number
is positive; (-1)0 = 1. If the sign bit is a 1, the number is
- We always factor so that the number in parentheses equals 1 +
some fraction. Since we know that the 1 is there,
the only important thing is the fraction, which you will write as a
binary string. Just multiply each digit times its place value, as in these
0.1bin = 2-1 = (1/2)
0.01bin = 2-2 = (1/4)
0.101bin = 2-1 + 2-3 = (5/8)
- The power of 2 that you got in the last step is simply an integer. Since
you don't want to deal with any messy negative numbers, you can add a
constant value (the bias) to that power.
For single-precision floating-point, the bias=127.
For double-precision, the bias=1023.
The sum of the bias and the power of 2 is the
exponent that actually goes into the IEEE 754
string. Remember, the exponent = power + bias.
(Alternatively, the power = exponent-bias).
When you have calculated these binary values, you can put them into
a 32- or 64-bit field. The digits are arranged like this:
Single precision (32-bit) form: (Bias = 127)
Double precision (64-bit) form: (Bias = 1023)
The numbers in parentheses show how many bits are required in each field.
Example: Float to IEEE 754 ->