1. Handling Denormalized Numbers
IEEE Floating Point
Handling Denormalized Numbers
1/8/ L24 IEEE Floating Point Basics
Copyright Joanne DeGroat, ECE, OSU

2. Copyright 2006 - Joanne DeGroat, ECE, OSU
Lecture overview
Denormalized setup for operation with normalized
1/8/ L24 IEEE Floating Point Basics
Copyright Joanne DeGroat, ECE, OSU

3. The floating point standard
Single Precision
Value of bits stored in representation is:
If e=255 and f /= 0, then v is NaN regardless of s
If e=255 and f = 0, then v = (-1)s ¥
If 0 < e < 255, then v = (-1)s 2e-127 (1.f) – normalized number
If e = 0 and f /= 0, the v = (-1)s (0.f)
Denormalized numbers – allow for graceful underflow
If e = 0 and f = 0 the v = (-1)s 0 (zero)
1/8/ L24 IEEE Floating Point Basics
Copyright Joanne DeGroat, ECE, OSU

4. Copyright 2006 - Joanne DeGroat, ECE, OSU
Consider the example
A = 100 $42C8 0000  
S = 0
E = = 133 – 127 = 6
F =
ManA =
 B = 25 $41C8 0000  
E = = 131 – 127 = 4
ManB =
1/8/ L24 IEEE Floating Point Basics
Copyright Joanne DeGroat, ECE, OSU

5. Copyright 2006 - Joanne DeGroat, ECE, OSU
Example Continued
For A + B need to align binary pt by 2 places
ManA =
ShfManB = aligned to exp of 6
Sum is with a bin exp of 6
$ F A
1/8/ L24 IEEE Floating Point Basics
Copyright Joanne DeGroat, ECE, OSU

6. Copyright 2006 - Joanne DeGroat, ECE, OSU
Some basics
Consider or 25
This could be represented as x 24
OR x 26 to have the same value
Consider
as the fractional part of a denormalized number
This has value x 2-126
Or with a shift by 1 position x 2-127
(Note this is a shift left on a fixed binary pt)
1/8/ L24 IEEE Floating Point Basics
Copyright Joanne DeGroat, ECE, OSU

7. Copyright 2006 - Joanne DeGroat, ECE, OSU
Why do this
With the shift it has the same value but the format becomes:
v = (-1)s (x.xxxx) = (-1)s 2e-127 (x.xxxx)
where x.xxxx is the shifted fractional part
and the e here is 0
This is the same format as normalized number when e=0 and the operation now does not care that one of the inputs was denormalized.
1/8/ L24 IEEE Floating Point Basics
Copyright Joanne DeGroat, ECE, OSU

8. Copyright 2006 - Joanne DeGroat, ECE, OSU
General Rule
Fixed binary point and shift digits to
Shift left – subtract 1 from exponent to
Shift right – add 1 to exponent
Fixed binary digits and move binary point to
Move right – subtract 1 from exponent to
Move left – add 1 to exponent
1/8/ L24 IEEE Floating Point Basics
Copyright Joanne DeGroat, ECE, OSU