1. Floating Point Numbers
Chapter 5

2. Exponential Notation
The following are equivalent representations of 1,234
123, x 10-2
12, x 10-1
1, x 100
x 101
x 102
x 103
x 104
The representations differ in that the decimal place – the “point” -- “floats” to the left or right (with the appropriate adjustment in the exponent).
p. 122

3. Parts of a Floating Point Number
Sign of mantissa
Location of decimal point
Mantissa
Exponent
Sign of exponent
Base
x 10-3
p. 123

4. IEEE 754 Standard
Most common standard for representing floating point numbers
Single precision: 32 bits, consisting of...
Sign bit (1 bit)
Exponent (8 bits)
Mantissa (23 bits)
Double precision: 64 bits, consisting of…
Exponent (11 bits)
Mantissa (52 bits)

5. Single Precision Format
32 bits
Mantissa (23 bits)
Exponent (8 bits)
Sign of mantissa (1 bit)

6. Double Precision Format
64 bits
Mantissa (52 bits)
Exponent (11 bits)
Sign of mantissa (1 bit) kc

7. Normalization The mantissa is normalized
Has an implied decimal place on left
Has an implied “1” on left of the decimal place
E.g.,
Mantissa:
Representation: =

8. Excess Notation
To include both positive and negative exponents, “excess-n” notation is used
Single precision: excess 127
Double precision: excess 1023
The value of the exponent stored is n larger than the actual exponent
E.g., – excess 127,
Exponent:
Representation:
135 – 127 = 8 (value)
Ed kc

9. Excess Notation - Sample -
Represent exponent of in excess 127 form: = =
Representation =
Ed kc kc

10. Excess Notation - Sample -
Represent exponent of in excess 127 form: = =
Representation =
Ed kc kc

11. Example Single precision +1.75  23 = 14.0
1.112 =
130 – 127 = 3
0 = positive mantissa
+1.75  23 = 14.0

12. Exercise – Floating Point Conversion (1)
What decimal value is represented by the following 32-bit floating point number?
Answer:
Skip answer
Answer

13. Exercise – Floating Point Conversion (1)
Answer
What decimal value is represented by the following 32-bit floating point number?
Answer:

14. Step by Step Solution To decimal form 23 1.9609375 = 15.6875
To decimal form
= 3 23 = *
( negative )

15. Step by Step Solution : Alternative Method
To decimal form
= 3
Shift “Point”
( negative ) kc

16. Exercise – Floating Point Conversion (2)
Express 3.14 as a 32-bit floating point number
Answer:
(Note: only use 10 significant bits for the mantissa)
Skip answer
Answer

17. Exercise – Floating Point Conversion (2)
Answer
Express 3.14 as a 32-bit floating point number
Answer:
(Note: only use 10 significant bits for the mantissa)

18. Detail Solution : 3.14 to IEEE double precision
3.14 To Binary (approx):
Delete implied left-most “1”
and normalize
Poof !
Exponent = position
point moved when normalized
Value is positive: Sign bit = 0

19. Packed Decimal Format - 1 of 4
* Somewhat limited use:
eg: Where precision particularly important,
as in accounting functions.
* Similar to BCD:
eg: Four bit representation, as in BCD.
-> Stores two digits per byte. kc

20. Packed Decimal Format - 2 of 4
* Stores up to 31 digits
* Last four bits for sign:
> positive
> negative
> unsigned
* Decimal Point not stored: must be
maintained by separate (application) software. kc

21. Packed Decimal Format - 3 of 4
Example:
Decimal Value: 10357, unsigned
Packed Decimal:
Byte Byte Byte 3 kc

22. Packed Decimal Format - 4 of 4
Example:
Decimal Value:
Packed Decimal:
Byte Byte Byte 3 kc

23. That’s all this time !!
Thank you