CA Lecture 3: Data Representation II

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CA Lecture 3 Data Representation (II)

• Floating Point Numbers and Character Code
• (Chapter 2 of text book)

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Disadv. of fixed point number representation

• The disadvantage of fixed point numbers need a
  great many digits to represent a practical range
  of numbers as in practice, very large and very
  small numbers can appear in any computation. E.g.
  
• To represent a number as large as a trillion2,
  the computer needs at least 40 bits to the left
  of the radix point since 240 1012.

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Disadv. of fixed point number representation
cont.

• If the same computer needs to represent one
  trillionth, then 40 bits must also be maintained
  on the right of the radix point. Thus 80 bits are
  needed per number!
• A great deal of hardware (e.g. wires) is required
  to store and manipulate numbers with 80 or more
  bits of precision, and the speed of computation
  would also be slower.

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Floating Point Numbers

• Fine precision is generally not needed when large
  numbers are used, and large numbers do not
  generally need to be represented when
  calculations are made with small numbers.
• An efficient computer can be realized when only
  as much precision is retained as is needed.
• Floating point numbers allow very large and
  very small numbers to be represented using only a
  few digits, at the expense of precision.

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Base 10 Floating Point Numbers (FPN)

• The precision is determined by the number of
  digits in the fraction 6.023 (or significand or
  mantissa which has integer and fractional parts).
• The range (represented by a power of 10, 1023) is
  determined by the number of digits in the
  exponent.

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Range and Precision in FPN

• A floating point number can be characterized by
  a triple of numbers sign, exponent, and
  significand.
• The ordering (sign bit, exponent, then
  significand) is helpful in comparing two floating
  point numbers.
• The range/precision trade-off is a major
  advantage of using a floating point
  representation.

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Normalization

• The same number can be represented in different
  floating point forms
• (3584.1)10 100 (3.5841)10 103
  (0.35841)10 104
• 
  (35.841)10 102
• Floating point numbers are usually normalized,
  in which the radix point is located in only one
  possible position for a given number.
• Usually, the normalized representation places
  the radix point immediately to the left or right
  of the leftmost, nonzero digit in the fraction,
  as in (0.35841)10 104 or (3.5841)10 103

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Normalization and The Hidden Bit in base 2

• Number zero cannot be represented in the
  normalized form. Thus zero is represented as all
  0s in the mantissa/significand.
• If the mantissa is represented as a binary,
  then there is always a leading 1 in the
  normalized mantissa, so there is no need to store
  that 1. E.g.
• (110.10)2 x 22 (0.11010)2 25 (1.1010)2
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• This missing bit is referred to as the hidden
  bit, also known as a hidden 1.

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Hidden Bit

• For the above example
• If normalized representation places the binary
  point immediately to the left of the leftmost,
  nonzero digit in the fraction, then normalized
  mantissa 0.11010 will be stored in bit pattern
  1010.
• If normalized representation places the binary
  point immediately to the right of the leftmost,
  nonzero digit in the fraction, then normalized
  mantissa 01.1010 will be stored in bit pattern
  1010.

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Representing FPN in a Computer

• Assume a computer represents floating point
  numbers in this ordering of packing the sign
  bit, 3-bit excess 4 exponent, 3 hexadecimal (base
  16) digits of significand.
• The normalized form places the radix point
  immediately to the left of the leftmost , nonzero
  digit in the fraction.
• The hexadecimal point will NOT be stored.
• Consider representing (358.0)10 in this format.
• 1st step convert the fixed point number from its
  original base into a fixed point number in the
  target base (base 16). Thus (358.0)10 (166.0)16

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Representing FPN in The Computer Cont.

• 2nd step normalize the number. Thus
• (166.)16 X 160 (.166)16 X 163
• 3rd step convert mantissa into a binary
  pattern. (166)16 (0001 0110 0110)2
• 4th step work out the exponent in excess 4 gt
  (3 4 7)10 (111)2
• 5th step fill in the bit fields, with a
  positive sign (sign bit 0), and the rest
• 0 111 . 0001 0110 0110

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Another floating point example
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Another floating point example cont.
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Reason for the use of excess 4 exponent in FPR

• The use of an excess 4 exponent instead of a 2s
  complement or a signed magnitude exponent
  simplifies comparison, addition and subtraction
  of floating point numbers.
• FPN arithmetic will be covered in the next
  lecture.

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The IEEE 754 Floating Point Standard

• IEEE stands for Institute of Electrical and
  Electronics Engineers.
• The IEEE 754 is an international standard for
  floating point numbers.
• The IEEE 754 standard must be supported by a
  computer system.
• The IEEE 754 standard can be supported by the
  hardware entirely, or can be supported by a
  mixture of hardware and software.

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The IEEE 754 Cont.

• There are two primary formats in the IEEE 754
  standard single precision and double precision.
• Both single and double formats are in base 2.
• Uses hidden bit.
• The single-precision format occupies 32 bits.
• The double-precision format occupies 64 bits.
• Normalization binary point is on the right
  side of the leftmost non-zero digit.

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The IEEE 754 Cont.
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Single Precision Format

• The sign bit is on the leftmost position and
  indicates a positive or negative number for a 0
  or a 1, respectively.
• The 8-bit excess 127 exponent follows. Bit
  patterns 00000000 and 11111111 are reserved for
  special cases.
• The 23-bit base 2 fraction follows.
• A hidden bit is to the left of the binary
  point. Totally there are 12324-bit significand.

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Double Precision Format

• The sign bit is the same as in SPF.
• The 11-bit excess 1023 exponent follows. Again,
  all 0s and all 1s are reserved for special
  cases.
• The 52-bit base 2 fraction follows.
• A hidden bit is to the left of the binary
  point. Totally there are 15253-bit significand.
• The number is normalised unless denormalised
  numbers are supported.

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The IEEE 754 - Cont.

• A so-called clean zero is represented by the
  reserved bit pattern 00000000 in the exponent and
  all 0s in the fraction. The sign bit can be 0 or
  1. Thus there are 2 representations for zero.
• Infinity has a representation in which the
  exponent contains the reserved bit pattern
  11111111, the fraction contains all 0s, and the
  sign bit can be 0 or 1.
• A zero number divided by a zero will produce
  NaN (Not a number).
• In a NaN, the exponent contains the reserved
  bit pattern 11111111, the fraction is nonzero,
  and the sign bit can be 0 or 1.

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IEEE-754 Examples
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IEEE-754 Conversion Example

• Represent -12.62510 in single precision
  IEEE-754 format.
• Step 1 Convert to target base. -12.62510
  -1100.1012
• Step 2 Normalize. -1100.1012 -1.1001012 X
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• Step 3 Fill in bit fields. Sign is negative,
  so sign bit is 1. Exponent is in excess 127 (not
  excess 128!), so exponent is represented as the
  unsigned integer 3 127 130. Leading 1 of
  significand is hidden, so final bit pattern is
• 1 1000 0010 . 1001 0100 0000 0000 0000 000

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The ASCII Character Set

• The American Standard Code for Information
  Interchange (ASCII) is a 7-bit code, commonly
  stored in 8-bit bytes.
• The characters in positions 00-1F and position
  7F are special control characters.

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The ASCII Character Set
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The Unicode Character Set

• Unicode is a 16-bit coding.
• Unicode supports a great breadth of the worlds
  character sets.
• Unicode is an evolving (or changing) standard.

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The Unicode Character Set
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Tutorial Two Questions

• 1. For the following single-precision IEEE 754
  bit patterns show the numerical value as a base 2
  significand with an exponent
• (a) 0 10000011 01100000000000000000000
• (b) 1 10000000 00000000000000000000000
• (c) 1 00000000 00000000000000000000000
• (d) 1 11111111 00000000000000000000000
• (e) 0 11111111 11010000000000000000000
• (f) 0 00000001 10010000000000000000000

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Tutorial Two Questions

• 2. Convert the following decimal numbers to IEEE
  single-precision floating point
• 128
• -32.75
• 18.125
• 0.0625

3. A hidden 1 representation will not work for
base 16. Why not?
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Tutorial Two Questions

• 4. Consider a system with the following
  floating point format 1 sign bit, followed by a
  3-bit exponent, followed by a 4-bit mantissa. The
  exponent is in twos complement. The magnitude of
  the mantissa is a fixed point number, with a
  hidden bit. Answer the following questions for
  this number system, giving answers in
  conventional decimal form
• 4a)     What is the largest possible exponent?
• 4b)     What is the largest possible mantissa if
  the hidden 1 is to the left of the binary point?

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Tutorial Two Questions

• 4c)      What is the smallest positive normalized
  mantissa if the hidden 1 is to the left of the
  binary point?
• 4d)     What is the smallest positive normalized
  mantissa if the hidden 1 is to the right of the
  binary point?
• 4e)      What is the smallest positive number in
  this system if the hidden 1 is to the left of the
  binary point?
• 4f)       How many different numbers can be
  represented by the system?

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Tutorial Two Questions

• The IBM short floating point representation uses
  base 8, one sign bit, a seven-bit excess 64
  exponent, and a normalized 24-bit fraction.
  (Normalisation the point is on the left side of
  the leftmost non-zero digit.)
• 5a. Represent (14.3)6 in this notation.
• 5b. What number is represented by the following
  bit pattern?
• 1 0111111 01110000 00000000 00000000
• Show your answer in decimal. Note The spaces
  are included in the number for readability only.