Floating Point Numbers

1
Lecture 5
ITEC 1000 Introduction to Information Technology

• Floating Point Numbers

www.governmentauctions.org
Prof. Peter Khaiter
2
Lecture Template

• Floating Point Numbers
• Exponential Notation
• Excess-50 Notation
• Overflow and Underflow
• Floating Point Calculations
• Normalization in Floating Point
• IEEE 754 Standard
• Packed Decimal Format
• Programming Considerations

3
Floating Point Numbers

• Real numbers
• Used in computer when the number
• is outside the integer range of the computer (too
  large or too small)
• contains a decimal fraction
• the range in PCs
• r
• or more

4
Exponential Notation

• The following are equivalent representations of
  1,234

123,400.0 x 10-2 12,340.0 x 10-1 1,234.0
x 100 123.4 x 101 12.34 x 102
1.234 x 103 0.1234 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).
5
Exponential Notation

• Also called scientific notation
• 4 specifications required for a number
• Sign ( in example)
• Magnitude or mantissa (12345)
• Sign of the exponent ( in 105)
• Magnitude of the exponent (5)
• Plus
• Base of the exponent (10)
• Location of decimal point (or other base) radix
  point

6
Parts of a Floating Point Number
-0.9876 x 10-3
7
Floating Point Format Specification

• Integer format (8-bit word)
• 7 decimal digits and a sign
• Range -9,999,999 lt I lt 9,999,999
• Floating point format (8-bit word)

8
Format

• Mantissa stored in sign-magnitude format
• Assume decimal point located at the beginning of
  mantissa
• Exponent stored in Excess-N notation
  Complementary notation
• Pick middle value as offset where N is the middle
  value 0..99 e.g., excess-50

9
Excess-50 notation

• Excess-N representation R N EE
• Example1 N 50, EE 38, R 88
• Example2 N 50, EE -38, R 12
• Excess-50 Magnitude range

10
Overflow and Underflow

• Possible for the number to be too large or too
  small for representation

0.00001 x 10-50 10-55
11
Floating Point Format Excess-50

• First digit represents the sign of mantissa
• 0 is used as a sign
• 5 is used as a -sign (arbitrarily)
• Two next digits represent exponent in excess-50
• Five last digits represent mantissa
• fixed decimal point located at the beginning

12
Examples
13
Normalization

• Shift numbers left by increasing the exponent
  until leading zeros eliminated
• Converting decimal number into standard format
• Provide number with exponent (0 if not yet
  specified)
• Increase/decrease exponent to shift decimal point
  to proper position
• Decrease exponent to eliminate leading zeros on
  mantissa
• Correct precision by adding 0s or
  discarding/rounding least significant digits

14
Example 1 246.8035
Sign
Excess-50 exponent
Mantissa
15
Example 2 1255 x 10-3
16
Example 3 - 0.00000075
17
Floating Point Calculations

• Addition and subtraction
• Exponent and mantissa treated separately
• Exponents of numbers must agree
• Align decimal points
• Least significant digits may be lost
• Mantissa overflow requires exponent again shifted
  right

18
Example
Precision lost
19
Multiplication and Division

• Mantissas multiplied or divided
• Exponents added or subtracted
• Normalization necessary to
• Restore location of decimal point
• Maintain precision of the result
• Adjust excess value since added twice
• Example 2 numbers with exponent 53 represented
  in excess-50 notation
• 53 53 106
• Since 50 added twice, subtract 106 50 56

• Maintaining precision
• Normalizing and rounding multiplication

20
Example
21
Floating Point in the Computer

• Replace digits with 0 and 1 bits
• Typical floating point format
• 32 bits provide range 10-38 to 1038
• 8-bit exponent 256 levels
• Excess-128 notation
• 23 bits of mantissa approximately 7 decimal
  digits of precision

22
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
• Sign bit (1 bit)
• Exponent (11 bits)
• Mantissa (52 bits)

23
Single Precision Format
32 bits
24
Double Precision Format
64 bits
25
IEEE 754 Standard
26
IEEE 754 Standard

• 32-bit Floating Point Value Definition

27
Normalization in Floating Point

• Mantissa
• Must always start with 1
• Leading bit is not stored
• Implied that it is located to the left of the
  binary point
• Normalized Form 1.MMMMMMM
• E.g.
• Mantissa
• Actual value
• Exponent
• Formatted using Excess-127 notation
• Base 2 is implied
• Range 2-126 to 2127

10100000000000000000000 1.1012 1.62510
28
Excess Notation Example
Represent exponent of 1410 in excess-127 form
12710 011111112 1410
000011102 Representation 100011012
14110
29
Excess Notation Example
Represent exponent of -810 in excess 127 form
12710 011111112 - 810 -
000010002 Representation
011101112
11910
30
Single Precision Example
0 10000010 11000000000000000000000
31
Single Precision Exercise

• What decimal value is represented by the
  following 32-bit floating point number?
• Answer

1 10000010 11110110000000000000000
Skip answer
Answer
32
Single Precision Exercise
Answer

• What decimal value is represented by the
  following 32-bit floating point number?
• Answer -15.6875

1 10000010 11110110000000000000000
33
Step by Step Solution
1 10000010 11110110000000000000000
To decimal form
130 - 127 3
1.11110110000000000000000000
1 .5 .25 .125 .0625 0 .015625
.0078125
1.9609375
23
15.6875

- 15.6875
( negative )
34
Step by Step Solution Alternative Method
1 10000010 11110110000000000000000
To decimal form
130 - 127 3
1.11110110000000000000000000
1111.10110000000000000000000
Shift Point
- 15.6875
( negative )
35
IBM floating point formats
36
Alpha floating point formats
37
Exercise Floating Point Conversion

• Express 3.14 as a 32-bit floating point number
• Answer
• (Note only use 10 significant bits for the
  mantissa)

Skip answer
Answer
38
Exercise Floating Point Conversion
Answer

• Express 3.14 as a 32-bit floating point number
• Answer
• (Note only use 10 significant bits for the
  mantissa)

0 10000000 10010001111000000000000
39
Detail Solution 3.14 to IEEE double precision
3.14 To Binary (approx)
11.001000111101
Delete implied left-most 1 and normalize
1001000111101
Prove !
Exponent 127 1 position point moved when
normalized
10000000
Value is positive Sign bit 0
0 10000000 10010001111010000000000
40
Packed Decimal Format

• Limited use e.g where precision particularly
  important, as in accounting and business
  functions.
• Similar to BCD e.g four bit representation, as
  in BCD.
• -gt Stores two digits per byte.
• Supported by business-oriented languages like
  COBOL
• Implemented in IBM System 370/390 and Compaq
  Alpha

41
Packed Decimal Format

• Each decimal digit is stored in BCD
• Two digits in a byte
• The most significant digit stored first, in the
  high-order bits of the first byte
• Can store up to 31 digits in 16 bytes
• The sign is stored in the low-order bits of the
  last byte
• Binary 1100 represents
• Binary 1101 represents -
• Binary 1111 represents unsigned number
• Decimal point not stored must be maintained by
  application software

42
Packed Decimal Format Example 1
Decimal Value 1 0 3 5 7, unsigned Packed
Decimal 0001 0000 0011 0101 0111 1111
Byte 1 Byte 2 Byte 3
43
Packed Decimal Format Example 2
Decimal Value - 9 0 4 1 3 Packed
Decimal 1001 0000 0100 0001 0011 1101
Byte 1 Byte 2 Byte 3
44
Integer vs. Floating Point Programming
Considerations

• Integer advantages
• Easier for computer to perform
• Potential for higher precision
• Faster to execute
• Fewer storage locations to save time and space
• Most high-level languages provide 2 or more
  different integer word sizes/formats
• Short integer (16 bits)
• Long integer (64 bits)

45
Integer vs. Floating Point Programming
Considerations

• Real numbers, if
• Variable or constant has fractional part
• Numbers take on very large or very small values
  outside integer range
• Program should use least precision sufficient for
  the task
• Higher precision formats require more storage
• Packed decimal attractive alternative for
  business applications

46
Thank you!
Reading Lecture slides and notes, Chapter 5