COMP201 Computer Systems Floating Point Numbers

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COMP201 Computer SystemsFloating Point Numbers
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Floating Point Numbers

• Representations considered so far have a limited
  range dependent on the number of bits
• 16 bits 0 to 65535 for unsigned -32768 to 32767
  for 2's complement.
• 32 bits 0 to 4294967295 for unsigned -2147483648
  to 2147483647 2's Complement
• How do we represent the following numbers
• mass of an electron 0.00000000000000000000000000
  000910956Kg
• Mass of the earth 5975000000000000000000000Kg
• Both these numbers have very few significant
  digits but are well beyond the range above.

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Floating Point Numbers

• We normally use scientific notation
• Mass of electron 9.10956 x 10-31 Kg
• Mass of earth 5.975 x 1024 Kg
• These numbers can be split into two components
• Mantissa
• Exponent
• e.g. 9.10956 x 10-31 Kg mantissa
  9.10956 exponent -31

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Floating Point Numbers

• Also need to be able to represent negative
  numbers
• Same approach can be taken with binary numbers
  i.e. a x re
• where a is the mantissa, r is the base( Radix)
  and e is the exponent
• Notice, we are trading off precision and possibly
  making computation times longer, in exchange for
  being able to represent a larger range of
  numbers.

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Defining a floating point number

• Several things must be defined, in order to
  define a floating point number
• Size of mantissa
• Sign of mantissa
• Size of exponent
• Sign of exponent
• Number base in use

• 9.10956
• Positive
• 31
• Negative
• 10

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Floating Point Numbers

• Consider the following using eight digits and
  radix 10

MSD
LSD
Mantissa (5 digits)
Exponent (2 digits)
The high order bit is the sign bit (0 for and
1 for - )
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Floating Point Numbers

• But this has serious limitations!
• Range of numbers which can be represented is 100
  1099 since only two digits are devoted to
  exponent.
• Precision is only 5 digits
• No provision for negative exponents

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Floating Point Numbers (Continued)

• One solution for the limitations of range and
  negative exponents, is to decrease the positive
  range to 49, by applying an offset of 50
  (excess-50 notation). Then, the range of numbers
  which can be expressed becomes
• 0.00001 x 10-50 to 0.99999 x 1049
• Often implementations require that the most
  significant digit not be zero so restrict the
  most negative value to 0.10000 x 10-50 and the
  all-0s represents 0.

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From the text
Excess-50 notation
Range of represented numbers
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Examples (from textbook)

• Problem Convert 246.8035 to the standard format
• Write as 0.2468035 x 103
• Truncate at 5 digits 0.24680 x 103
• Final number 05324680

• Problem Convert -0.00000075 to FP
  format.
• Write as -0.75 x 10-6
• Zero fill to 5 digits
• Final number
• 54475000

NOTE Plus is represented by 0 minus by 5 in
this format.
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Floating point in the computer

• Within the computer, we work in base 2, and use a
  format such as
• NOTE Bit order depends upon the particular
  computer

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Numbers represented

• Using 32 bits to represent a number, with 1 sign
  bit, 8 bits for exponent, and the remaining (23
  bits) for mantissa.
• In order to represent negative exponents, use
  excess-128 notation.
• Range of numbers represented is approximately
  10-38 to 1038 (in decimal terms)

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But this is not the end!

• The precision of the mantissa can be improved
  from 23 bits to 24 bits by noticing that the MSB
  of the mantissa in a normalized binary number is
  always 1
• And so, it can be implied, rather than expressed
  directly. The added complications (see below)
  are felt to be a good tradeoff for the added
  precision.
• This complication is the fact that certain
  numbers are too small to be normalized, and the
  number 0.0 cannot be represented at all!
• The solution is to utilize certain codes for
  these special cases.

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Floating point numbers the IEEE standard

• IEEE Standard 754
• Most (but not all) computer manufactures use
  IEEE-754 format
• Number represented (-1)S (1.M)2(E -
  Bias)2 main formats single and double

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And there are special cases to be considered

• Exponent Mantissa
• 0 /- 0
• 0 not 0
• 1-254 any
• 255 /- 0
• 255 not 0

• Zero, represented by
• /- 2-126 x 0.M
• 2E-127 x 1.M
• /- infinity
• Special condition

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Special cases (continued)

• The thing to remember about the special cases, is
  that they are SPECIAL, that is, not expected to
  occur.
• They cover numbers outside the range expected, by
  using some unlikely-to-be used codes
• Special code for 0.0
• Special code for numbers too small to be
  normalized
• Special code for infinity
• What you are expected to remember is that these
  special codes exist, and they limit the range of
  numbers that can be represented by the standard.

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Final ranges represented

• 2-126 to 2127
• Approx. 10-38 to 1038
• Similarly, the double-precision used sixty-four
  bits, and represents a range of approximately
  10-300 to 10300

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Floating Point Numbers

• IEEE Standard 754
• Example What Decimal number does the following
  IEEE floating point number represent? 0
  10000001 10110000000000000000000Sign 0
  (positive)Exponent 2 (0x81 or 129 127
  2)Mantissa 1 .10110000000000000000000

Final answer 110.112 or 6.7510
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More

• What is the IEEE f.p. number
• 01000001111111100000000000000000
• Converted to decimal?
• Begin by dividing up the number into the fields
  of sign, exponent and mantissa
• 0 10000011 11111100000000000000000
• Then, convert the exponent ( 10000011 131) and
  subtract the bias (127) to get the shift (4)
• Add in the implied one to the mantissa
• 1.11111100000000000000000
• And shift 4 places, to get the final number
  11111.11, or 31.75 in decimal.

Mantissa (23 bits)
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And more

• What is the decimal number 32.125 converted to
  IEEE fp format?
• First, convert the number to binary
• 100000.001
• Then, normalize for 1.xxxx format
• 1.00000001 shift 5
• Add in the bias (127) and convert the total (132)
  to binary (10000100 or 0x84)
• Assemble final number
• 0 1000010 00000001000000000000000

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Floating Point Application

• Floating point arithmetic is very important in
  certain types of application
• Primarily scientific computation
• Many applications use no floating point
  arithmetic at all
• Communications
• Operating systems
• Most graphics

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Floating point implementation

• Floating point arithmetic may be implemented in
  software or hardware.
• It is much more complex than integer arithmetic.
• Hardware instruction implementations require more
  gates than equivalent integer instructions and
  are often slower
• Software implementations are generally very slow
• CPU Floating performance can be very different
  from Integer performance