Floating point numbers

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Floating point numbers
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Computable reals

• computable numbers may be described briefly as
  the real numbers whose expressions as a decimal
  are calculable by finite means.(A. M. Turing, On
  Computable Numbers with an Application to the
  Entschiedungsproblem, Proc. London Mathematical
  Soc., Ser. 2 , Vol 42, pages 230-265, 1936-7.)

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Look first at decimal reals

• A real number may be approximated by a decimal
  expansion with a determinate decimal point.
• As more digits are added to the decimal expansion
  the precision rises.
• Any effective calculation is always finite if
  it were not then the calculation would go on for
  ever.
• There is thus a limit to the precision that the
  reals can be represented as.

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Transcendental numbers

• In principle, transcendental numbers such as Pi
  or root 2 have no finite representation
• We are always dealing with approximations to
  them.
• We can still treat Pi as a real rather than a
  rational because there is always an algorithmic
  step by which we can add another digit to its
  expansion.

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First solution

• Store the numbers in memory just as they are
  printed as a string of characters.
• 249.75
• Would be stored as 6 bytes as shown below
• Note that decimal numbers are in the range 30H to
  39H as ascii codes

Full stop char
Char for 3
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Implications

• The number strings can be of variable length.
• This allows arbitrary precision.
• This representation is used in systems like
  Mathematica which requires very high accuracy.

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Example with Mathematica

• 5!
• Out1120
• In210!
• Out23628800
• In350!
• Out33041409320171337804361260816606476884437764
  1568960512000000000000

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Decimal byte arithmetic

• 9 8 17 decimal
• 39H38H71H hexadecimal ascii
• 5756113 decimal ascii
• Adjust by taking 30H48 away -gt 41H65
• If greater than 939H57 take away 100AH and
  carry 1
• Thus 41H-0Ah 65-105537H so the answer would
  be 31H,37H 17

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Representing variables

• Variables are represented as pointers to
  character strings in this system
• A249.75

A
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Advantages

• Arbitrarily precise
• Needs no special hardware
• Disadvantages
• Slow
• Needs complex memory management

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Binary Coded Decimal (BCD) or Calculator style
floating point

• Note that 249.75 can be represented as 2.4975 x
  102
• Store this 2 digits to a byte to fixed precision
  as follows

exponent
mantissa
24
97
50
02
Each digit uses 4 bits
32 bits overall
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Normalise

• Convert N to format with one digit in front of
  the decimal point as follows
• If Ngt10 then Whilst Ngt10 divide by 10 and add 1
  to the exponent
• Else whilst Nlt1 multiply by 10 and decrement the
  exponent

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Add floating point

• Denormalise smaller number so that exponents
  equal
• Perform addition
• Renormalise
• Eg 949.75 52.0 1002.75
• 9.49750 E02 ? 9.49750 E02
• 5.20000 E01 ? 0.52000 E02
• 10.02750 E02 ? 1.00275 E03

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Note loss of accuracy

• Compare Octave which uses floating point numbers
  with Mathematica which uses full precision
  arithmetic
• Octave floating point gives only 5 figure accuracy

Mathematica 5! Out1120 10! Out23628800 50! O
ut330414093201713378043612608166064768844377641
568960512000000000000
Octave fact(5) ans
120 fact(10) ans 3628800 fact(50) ans
3.0414e64
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Loss of precison continued

• When there is a big difference between the
  numbers the addition is lost with floating point

Octave 325000000 108 ans 3.2500D08
Mathematica In1 325000000
108 Out1 325000108
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IEEE floating point numbers
Institution of Electrical and Electronic Engineers
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Single Precision
E
F
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Definition

• N-1s x 1.F x 2E-128
• Example 1
• 3.25
• In fixed point binary 11.01
• 1.101 x 21
• In IEEE format this is
• s0 E129, F10100 thus in IEEE it is
• S E F
• 01000 00011010 0000 0000 0000 0000 000

Delete this bit
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Example 2

• -0.375 -3/8
• In fixed point binary -0.011
• -11 x 1.1 x 2-2
• In IEEE format this is
• s1 E126, F1000 thus in IEEE it is
• S E F
• 10111 11101000 0000 0000 0000 0000 000

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Range

• IEEE32 1.17 1038 to 3.40 1038
• IEEE64 2.23 10308 to 1.79 10308
• 80bit 3.37 104932 to 1.18 104932

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