1. Luddy Harrison CS433G Spring 2007
IEEE 754 Floating Point
Luddy Harrison
CS433G Spring 2007

2. What is represented Real numbers
5.6745
1.23 × 1019
Remember however that the representation is finite, so only a subset of the reals can be represented
No trancendentals
Limited range
Limited precision (number of digits)

3. Normalizing Numbers
In Scientific Notation, we generally choose one digit to the left of the decimal point
13.25 × 1010 becomes × 1011
Normalizing means
Shifting the decimal point until we have the right number of digits to its left (normally one)
Adding or subtracting from the exponent to reflect the shift

4. Binary Floating Point
A binary number in scientific notation is called a floating point number
Examples:
1.001 × 217
0.001 × 2-13

5. Parts of a floating point number
±1.mmmmmmm × B±eeee
A signed fixed-point fraction (±1.mmmmmmm) called the mantissa
For non-zero mantissas, the leading 1 is implicit
That is, it is not present in the representation (bit pattern), but it is assumed to be there when interpreting the bit pattern
See the previous lecture for the meaning of fixed point fractions
An implicit base B
A unsigned integer (±eeee) called the exponent
An implicit bias. The actual exponent is eeee – bias
Some bit patterns are reserved for special values
Not A Number ±∞

6. About IEEE 754
This standard defines several floating point types and the meaning of operations (+, ×, etc.) on them
Single
Double
Extended Precision
It deals at length with the thorny questions of
Erroneous and exceptional results
Rounding and conversion

7. 32-bit Single Precision S E M 1 8 23 -1S × 1.M × 2E - 127
E is an unsigned twos-complement integer. A bias of 127 is used, so that the actual exponent is E – 127.
Exponents and are reserved for special purposes
The sign bit of the mantissa is separated from magnitude bits of the mantissa. The mantissa is therefore an unsigned fixed point fraction with an implicit 1 to the left of the binary point.
All zero bits (S, E, and M) means zero (0). In this case there is no leading 1 mantissa bit implied.

8. Some examples = 0 (note that there is no implicit leading 1 here) 1
= 0 (note that there is no implicit leading 1 here) 1
100
1010…0000
= -1 × × = -13/8 × 2-123
0000…0000
= 1.0 × = 1 × 2127

9. Denormalized Numbers 00000000 0000…0001 = 0.0000…0001 × 2-126
0000…0001
= …0001 × 2-126
An exponent field of zero is special; it indicates that there is no implicit leading 1 on the mantissa. This allows very small numbers to be represented. Note that we cannot normalize this value. (Why?) Zero is effectively a denorm (and it cannot be normalized – why?)
0000…0001
= …0001 × = …0001 × 2127
Here, the mantissa has an implicit leading 1. If we wanted …0001 × 2127 we could obtain it by writing 1.0 × 2104.

10. 64-bit Double Precision S E M 1 11 52 -1S × 1.M × 2E - 1023
E is an unsigned twos-complement integer. A bias of 1023 is used, so that the actual exponent is E – 1023.
As before, an exponent of all 0 bits or all 1 bits is reserved for special values.
As before, the mantissa is an unsigned fixed point fraction with an implicit 1 to the left of the binary point. The sign of the entire number is held separately in S.
A representation of all zero bits (S, E, and M) means zero (0). In this case there is no leading 1 mantissa bit implied.

11. Infinity
= +∞ 1
= -∞

12. Not A Number x 11111111 ≠0 x 11111111 1xxx…xxxx Quiet NaN x 11111111
Signalling NaN