IEEE 754 Floating Point

1
IEEE 754 Floating Point

• Luddy Harrison
• CS231
• Spring 2006

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 1.325 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 Beeee
• 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
• 8

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 00000000 and 11111111 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
0
0
0 (note that there is no implicit leading 1
here)
1
100
10100000
-1 1.101 24-127 -13/8 2-123
0
11111110
00000000
1.0 2254-127 1 2127
9
Denormalized Numbers
0
00000000
00000001
0.00000001 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?)
0
11111110
00000001
1.00000001 2254-127 1.00000001 2127
Here, the mantissa has an implicit leading 1. If
we wanted 0.00000001 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
0
11111111
0
8
1
11111111
0
-8
12
Not A Number
x
11111111
?0
x
11111111
1xxxxxxx
Quiet NaN
x
11111111
0xxxxxxx ? 0
Signalling NaN