Handling Real Numbers Fractions

1
Handling Real Numbers / Fractions

• Two types of real numbers are in use
• 1) Fixed Point Numbers
• e.g. 12.36 (Decimal) , 34.AD (HEX)
• -1110.11 (Binary) , 273.0 (Octal)
• 2) Floating Point Numbers
• e.g. 12.367 e 05(Decimal) gt -12.367 X 10 05
• 0.000011 e 1101 (Binary) gt 0.000011 X 2
  (-1101)
• A23.B5 e AD (Hex) gt A23.B5 X 16 AD
• 23467.0008e124 (Octal) gt 23467.0008 X 8 124

2
Fixed Point Real Numbers (Salient Features)

• General format
• ltSigngt ltInteger Partgt / 0 . ltFractional Partgt
  / 0
• ltRadix Point /
  Periodgt
• Sign may be absent (Unsigned ) assumed positive.
• Fixed position of Radix Point / Period separates
  Integer Part from the Fractional part.
• Integer Part and/or Fractional Part can be 0.
• Signed Magnitude scheme of representation is used
  because of the possible infinite precision of the
  fractional part.

3
Fixed Point Real Numbers (Salient Features) - 2

• Example
• 12.354 (Decimal) is interpreted in the
  following way
• Integer Part 12 (Unsigned) 1X10(2-1) 2 X 10
  (1-1)
• Fractional Part 354 3X 10 (-1) 5 X 10 (-2)
  4 X 10 (-3)
• -111.1101 (Binary) is interpreted in the
  following way
• Integer part -111 (-ve) 1 X 2 (2) 1 X 2 (1)
  1 X 2 (0)
• Fractional Part 1101 1 X 2 (- 1) 1 X 2(- 2)
  1 X 2 (- 3)
• A2.B3 (Hex) is interpreted in the following way
• Integer Part A2 (ve ) A 10 X 16 (1) 2 X
  16 (0)
• Fractional Part B3 B 11 X 16 (-1) 3 X 16
  (-2)

4
Floating Point Real Numbers (Salient Features) -
1

• General format
• ltSigngt lt Fixed Point Real Significand /
  Mantissa gt
• e ltexp. Signgt lt
  Integer Exponent gt.
• Sign / exp. Sign both may be absent (Unsigned )
  assumed positive in both cases.
• Convenient way to represent too small or too
  large a number
• e.g. 0.000000012 (Decimal) gt 0.12 e 07
• 1111 1111 1111 1111 (Binary) gt 1.11 e
  1111

5
Floating Point Real Numbers (Salient Features) -
2

• Examples
• 1.234 e 55 (Decimal) interpreted as
• Significand / Mantissa 1.234
• Exponent 55 gt 10 55 1.234 X 10 55
• 1234000..000 (51 Zeroes) in Total
• 1101.111 e - 101 (Binary) interpreted as
• Significand / Mantissa 1101 .111 (ve)
• Exponent -101 gt 2 -101(Binary) -5(Decimal)
  
• 1101.111 X 2 (- 5) gt 0.01101111(Binary)

6
Floating Point Real Numbers (Salient Features) -
3

• Example (Contd.)
• - 0.637 e - 06 (Octal) interpreted as
• Significand / Mantissa -0.637
• Exponent - 26 gt 8 -06 (Octal) -6 ( Decimal
  )
• - 0.637 X 8 (- 6) gt - 0.000000637(Octal)
• Position of the radix point / period floats
  within the Significand depending on the value
  (weight) of the exponent.
• e.g. 110.1111 e 101 (Binary) gt 0.1101111 e 1000
• gt 1101111.0 e 001 gt 1101.111 e 100
• -123.45 e -20 (Decimal) gt -0.12345 e -17 gt
  -1234.5 e -21

7
Signed Real Numbers (Representation Issues) - 1

• Representation of Fixed Point Numbers .
• a) Representation Format.
• b) Isolation of Integer Fractional Parts.
• Representation of Floating Point Numbers.
• i) Representing Fixed Point Significand /
  Mantissa.
• ii) Representing signed Integer Exponent.
• Choice of
• a) Universal representation format.
• b) Compact , machine compatible
  representation format.
• Measure of accuracy / limitations of the chosen
• representation format.

8
Signed Real Numbers (Representation Issues) - 2

• Salient Observations
• 1) Fractional Part size is NOT Fixed, hence
  preferred representation scheme for fixed point
  values has to be Signed Magnitude one with
  Leading / Leftmost 0 -gt ve Sign Leading (r-1)
  digit in base/radix r -gt -ve sign.
• 2) Fractional Part may needed to be either
  Truncated or Rounded Off in order to represent
  within Finite Word Size of a machine.
• 3) Floating Point provides a choice of
  universatility as well as compactness.
• e.g. ( Sign Magnitude with sign represented as
  digit )
• 123.45 (Decimal) gt 0 1.2345 e 02 gt 0
  0.12345 e 03
• - 111.0 (Binary) gt 1 1.11 e 010 gt 1
  0.111 e 011
• - 345.47 e - 08(Octal) gt 7 3.4547 e -06
  gt 7 0.34547 e - 05
• - 2.3A8 e 01 (Hex) gt F 2.3A8 e 01 gt F
  0.23A8 e 02

9
Signed Real Numbers (Representation Issues) - 3

• Compact Representation of Fixed Point
  Significand / Mantissa Normalized Significand
  /Mantissa One of the two following ways may be
  adopted
• 1) Single Digit Integer Part The format of
  representation being s d.dddd.. where s is
  the sign and d is any digit in the relevant
  base . Here the integer part consists of a single
  Non Zero digit. In some cases the Integer Part
  may be implied.
• 2) Purely Fraction The format of
  representation being
• s 0.dddd.. Here again s is the sign and d
  is any digit in the relevant base with the
  leftmost digit being NON Zero. In this case one
  can always assume a 0 integer part or alternately
  an implied integer part.
• N.B In both cases the isolation between Integer
  Part Fractional Parts are achieved as a by-
  product i.e. no special isolation techniques are
  needed.

10
Signed Real Numbers (Representation Issues) - 4

• Representation of Signed Integer Exponent
• Two choices
• 1) rs Complement Scheme One need to have
  two types of processing circuits Signed Magnitude
  for handling Significand /Mantissa rs
  Complement for processing Exponents may not be
  economical.
• 2) Biased Scheme Here one adds a fixed bias
  value B to the actual exponent Ae to convert it
  to a biased exponent Be. The choice of the bias
  value B is done in the following manner
• Be AeB 0 when Actual Bias Ae is the
  maxm. ve
• e.g. For decimal base maxm. ve Exponent (Ae)
  -99 hence the Bias value chosen B 99

11
Signed Real Numbers (Representation Issues) - 5

• Examples
• 123.45 (Decimal) gt 0 1.2345 e 02 gt 0
  1.2345 e 101
• OR gt 0 0.12345 e 102 After
  adding Bias 99 (Decimal)
• - 111.0 (Binary) gt 1 1.11 e 010 gt 1
  1.11 e 110
• OR gt 1 0.111 e 111 After
  adding Bias 100 (Binary)
• - 345.47 e - 08(Octal) gt 7 3.4547 e -06
  gt 7 3.4547 e 71
• OR gt 7 0.34547 e 72 After
  adding Bias 77 (Octal)
• - 2.3A8 e 01 (Hex) gt F 2.3A8 e 01 gt F
  2.3A8 e 100
• OR gt F 0.23A8 e 101 After adding
  Bias FF (Hex)

12
Signed Real Numbers (Representation Issues) - 6

• Conversion Steps
• ( Normalized Mantissa , Biased Exponent )
• 1) Represent the Real Number in Floating Point
  Notation (Signed Magnitude Form).
• 2) Normalize the Mantissa / Significand (
  normally into 0.dddd..) form by adjusting the
  Exponent.
• 3) Add specified bias to this adjusted
  Exponent.
• 4) Check for Overflow / Underflow ( to be
  elaborated later).

13
Signed Real Numbers (Representation Issues) - 7

• Range Precision
• Range dependent on Exponent
• The value range of any n-digit exponent
  happens to be
• -(rn 1). (rn 1) in base r.
• After biasing the range changes to 0 2(rn
  1) . Its size may also increase by at least one
  more digit.
• Increasing the number of digits in the exponent
  size
• increases the range.
• 2) Precision dictated by the Size of the
  Mantissa/ Significand
• More number of digits in the Mantissa/Significan
  d helps to increase the precision.

14
Signed Real Numbers (Representation Issues) - 8

• Illustration of Range Precision
• Mantissa/ Significand 0. dddddddd max 8
  digits.
• Unbiased Exponent 2 digits / 4 bit binary
  (with sign) .
• 1) Decimal (base 10) Bias 99 , biased
  range 100..10198
• Precision 8 digit long.
• 2) Binary (base 2) Bias 1000 , biased
  range 2 0..2 1111 binary
• (biased version can be obtained by
  inverting the MSBit of the Exponent) . Precision
  8 bit long.
• 3) Octal (base 8) Bias 77 , biased range
  8 0..8 176 Octal
• Precision 8 digit long.
• 4) Hex (base 16) Bias FF , biased range
  160..16100 Hex
• Precision 8 digit long.

15
Signed Real Numbers (Representation Issues) - 9

• Overflow Under Flow cases ( Example)
• Decimal 8 digit Normalized 0.ddd..
  Significand , 3 digit biased Expont
• Largest ? 0.99999999 X 10 198 represented as
  0/9 99999999 e 198
• Smallest ? 0.1 X 10 0 represented as 0/9
  10000000 e 000
• Now consider representing ? d.0 X 10 198 where d
  represents any non zero digit . Clearly if we
  want to normalize the mantissa , the exponent
  will go beyond the specified range. Exponent
  Overflow .
• Again one cannot represent ? 0.0d X 10 0 where d
  represents any non zero digit . Clearly if we
  want to normalize the mantissa , the exponent
  will go below the specified range. Exponent
  Underflow .

16
IEEE 754 standard for representing binary Real
Numbers - 1

• Significand / Mantissa
• Format 0/1 1.dddd..
• The integer part (1.) is always implied except
  for Zero
• -ZERO
• Leftmost bit of the fractional part need not be
  non zero.
• Size of the Significand may be one of the
  following
• Size in bits Type of Value
• -------------------------------------------------
• 23 Single Precision
• 52 Double Precision
• 63 Extended
  Precision

17
IEEE 754 standard for representing binary Real
Numbers - 2

• Biased Exponent
• Size of the Biased Exponent may be one of the
  following
• Size in bits Biased Exponent Range
  Type of Value
• --------------------------------------------------
  -------------------------------
• 8 1 - 255
  Single Precision
• 11 1 - 2047
  Double Precision
• 15 1 2047 X 16
  Extended Precision

18
IEEE 754 Single Precision Floating Point Number (
Salient Features )

• a) NaN (Not a Number ) Exceeds Significant
  Precision
• X 11111111 1. ltAny Valuegt
  
• ltSigngt lt8 bit Biased Exp.gt lt Impliedgt lt 23
  bits Mantissa gt
• NaN ltAny Operatorgt Any Value gt NaN
• b) Positive Values
• i) ? ( ve Infinity) Any further Processing
  may yield ? OR NaN
• 0 11111111 1. 0000000..0
  Binding Upper Limit
• ltSigngt lt8 bit Biased Exp.gt lt Impliedgt lt 23
  bits Mantissa gt
• ii) Ve Normalized 0 11111110 ? 00000001
  1. 11..1 ?00..00
• 
  lt 8 bit Exponent Range gt lt
  Mantissa Rangegt
• iii) Ve De-Normalized 0 00000000 0.
  11..1 ? 00..01
• 
  lt 8 bit Exponentgt lt Mantissa Rangegt

19
IEEE 754 Single Precision Floating Point Number
(Salient Features) -2

• c) Zero 0 00000000 0. 000000000
• d) Zero 1 00000000 0. 000000000
• e) Negative Values
• i) -Ve De-Normalized 1 00000000 0.
  11..1 ? 00..01
• 
  lt 8 bit Exponentgt lt Mantissa Rangegt
• ii) -Ve Normalized 1 11111110 ? 00000001
  1. 11..1 ? 00..00
• 
  lt 8 bit Exponent Range gt lt
  Mantissa Rangegt
• iii) -? ( - ve Infinity)
• 1 11111111 1. 0000000..0
  Binding Lower Limit
• ltSigngt lt8 bit Biased Exp.gt lt Impliedgt lt 23
  bits Mantissa gt

20
IEEE 754 Floating Point Number (Common Features)

• A) Rounding Modes of the Significand
• i) Round towards ? ( for ve values)
• ii) Round towards ? ( for ve values).
• iii) Truncate ( for ANY value).
• iv) Round to the nearest even (for ANY value) .

21
IEEE 754 Floating Point Number (Common Features)
- 2

• B) Usage of De Normalized Numbers
• 0 / 1 00000000..00 0.
  11..1 ? 00..01
• ltSigngt lt Exponentgt lt
  Impliedgt lt Mantissa Rangegt
• 1) Enables the machine to handle small magnitude
  numbers ( values nearer to zero) more
  predictably.
• 2) During processing some intermediate values
  may become De Normalized momentarily which
  otherwise may force one to abandon processing in
  the middle.
• C) Separate processing units along with
  dedicated processing instructions normally exist
  in all modern day CPU (as a Co-processor part)
  to handle real values.