1. CSCI206 - Computer Organization & Programming
Floating Point Limits
zyBook: 10.9, 10.10

2. Review IEEE754 S Exponent Mantissa
Special values, else normalized numbers
Exponent
Mantissa (fraction)
Value
+/- zero
nonzero
denormalized number
all 1’s
+/- infinity
NaN (not a number)

3. Largest Single Precision Float
8 bit exponent (bias = 127), 23 bit fraction
All 1’s in the exponent is reserved for NaN and infinity
Maximum biased exponent is = 254
Maximum fraction is 23 1’s

4. Largest Single Precision Float
= 254
= 127

5. Largest Single Precision Float
Move the decimal point 23 digits to the right
subtract 23 from exponent

6. Largest Single Precision Float
Convert mantissa

7. Smallest Nonzero Single
What we want is:
But that has exponent & fraction = 0
That value is reserved for zero!
Therefore, the closest we can get is:
either or

8. Smallest Nonzero Single
Normalized
In this case, using a normalized number is not ideal, if we could use a denormalized number we could get a much smaller value:
This is equivalent to:
An extra 22 bits of precision!
Denormalized

9. Smallest Nonzero Single
The IEEE realized this and when the exponent is zero and the fraction is > 0, the value is treated as a denormalized number.
The smallest nonzero normalized:
The smallest nonzero denormalized:
exp =
exp =
m = 0000….1

10. Smallest Nonzero Normalized Single
biased exponent = 1
fraction = 0

11. Smallest Nonzero Denormalized Single
biased exponent = 0
fraction =
*Note, even though the exponent is encoded as -127, it is computed using the smallest “valid” exponent, which is -126.