CSCI206 - Computer Organization &
Programming
Floating Point Limits
zyBook: 10.9, 10.10
Review IEEE754
S
Exponent
Mantissa
Special values, else normalized numbers
Exponent
Mantissa (fraction)
Value
0
0
+/- zero
0
nonzero
denormalized number
all 1’s
0
+/- infinity
all 1’s
nonzero
NaN (not a number)
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 1111 1110 = 254
Maximum fraction is 23 1’s
Largest Single Precision Float
1111 1110 = 254
254-127 = 127
Largest Single Precision Float
Move the decimal point 23 digits to the right
subtract 23 from exponent
Largest Single Precision Float
Convert mantissa
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
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:
Denormalized
This is equivalent to:
An extra 22 bits of precision!
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:
exp = 0000 0001
The smallest nonzero denormalized:
exp = 0000 0000
m = 0000….1
Smallest Nonzero Normalized Single
biased exponent = 1
fraction = 0
0 0000 0001 00000000000000000000000
Smallest Nonzero Denormalized Single
biased exponent = 0
fraction = 0.00000000000000000000001
0 0000 0000 00000000000000000000001
*Note, even though the exponent is encoded as -127, it is
computed using the smallest “valid” exponent, which is -126.