Topic 3d Representation of Real Numbers

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Topic 3d Representation of Real Numbers

• Introduction to Computer Systems Engineering
• (CPEG 323)

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Recap

• What can be represented in n bits?
• Unsigned 0 to 2n - 1
• 2s Complement -2n-1 to 2n-1 - 1
• BCD 0 to 10n/4 - 1
• But, what about?
• very large numbers 9,349,398,989,787,762,244,859,0
  87,678
• very small number 0.0000000000000000000000045691
• rationals 2/3
• Irrationals 2.71828., 3.1415926..,

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Conceptual Overview Finite-precision numbers
Positive underflow
Negative underflow
zero
Expressible negative numbers
Expressible positive numbers
Negative overflow
Positive overflow

• Is there any underflow region in integer
  representation?
• Between two adjacent integer numbers, there is
  no other integer.

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Representing Real Numbers

• How to represent fractional parts to the right of
  the decimal'' point?
• A number like 0.12 (i.e., (1/2)10) not
  represented well by integers 0 or 1!
• Two ways to represent real numbers better
• Fixed point
• Floating point

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Fixed-Point Data Format
4
2
1
1/2
1/4
1/8

0
S
1
0
0
1
1
.
0
0
0
0
1
0
Sign
Integer
Fractional
hypothetical binary point
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Fixed Point

• Pros
• Add two reals just by adding the integers
• Much less logic than floating point
• Faster
• Often used in digital signal processing
• Cons
• The range of numbers is narrow
• number400 000 000 000 000 000 000 000 000. 000
• It is much more economical to represent as 41026

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Recall Scientific Notation
decimal point
exponent
-23
6.02 x 10
Significand
radix (base)

• Issues
• Arithmetic (, -, , / )
• Representation, Normal form
• Range and Precision(Determined by?)
• Rounding and errors
• Exceptions (e.g., divide by zero, overflow,
  underflow)
• Properties

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Scientific Notation Normalized

• 12.35 x 10-9 ?
• 1.235 x 10-8 ?
• scientific notation has a single digit to the
  left of the decimal point
• Normalized scientific notation such a single
  digit must be non-zero.

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Floating Point Representation

• Numerical Form (1)s M 2E
• Sign bit s determines whether number is negative
  or positive
• Significand M normally a fractional value in
  range 1.0,2.0).
• Exponent E weights value by power of two
• Encoding

• MSB is sign bit S0/1
• exp field encodes E
• frac field encodes M

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IEEE 754 standard

• Three formats single/double/extended precision
  (32,64,80 bits).
• Single precision

Double precision see page 192 in PH book
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IEEE 754 Standard Floating Point Representation

• the representation
• (1)s (1 Fraction) 2E-bias
• where E is the exponent part in
  the notation
• Sign bit s determines whether number is negative
  or positive
• Fraction is normally a fractional value in range
  between 0 and 1
• A leading 1 added to the fraction is implicit
• Exponent using a biased notation
• For single precision the bias is 127

That is, when you convert the notation to the
number it represents, the exponential part used
should be reading E out from the IEEE 754
notation, and calculated as E-127.
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Advantage of using the biased notation for
exponents

• Under the single-precision IEEE 754 standard
  bias 127
• If the real exponent is 1, what is the biased
  exponent ? Answer 1127 128! (Note 128-127
  1!)
• How about if the real exponent is -1 ?

• Answer -1127 126! (Note 126-127 -1!)

How about if the real exponent is -127 ?
E -127 -127, then E ?
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Observations

• Using biased notation, the exponential part in
  the single-precision IEEE 754 notation itself
  never get nagative.

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Normalized Encoding Example

• Value Float F 15213.010
• 1521310 1.1101101101101 X 213
• Significand
• M 1.1101101101101
• frac 11011011011010000000000
• (23 bits! With leading 1 hiding!)
• Exponent
• E 1310, Bias 12710
• Exp 14010 10001100

Floating Point Representation Binary010001100110
11011011010000000000
exp
frac
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Denormalized Values

• Condition
•  exp 0000
• Significand value M 0.xxxx
• xxxx bits of frac
• Cases
• frac 0000
• Represents value 0
• Note that have distinct values 0 and 0
• frac ? 0000
• Numbers very close to 0.0
• Lose precision as get smaller
• Gradual underflow

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Special Values

• Condition
•  exp 1111
• ??(infinity)
• Operation that overflows
• Both positive and negative
• E.g., 1.0/0.0 ?1.0/?0.0 ?, 1.0/?0.0 ??
• Not-a-Number (NaN)
• Represents case when no numeric value can be
  determined
• E.g., sqrt(1), ?????

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IEEE 754 Summary
0ltexpltmax Any bit pattern
Normalized
0 Any nonzero bit pattern
Denormalized
zero
0 0
Infinite
111 0
NaN
111 Any nonzero bit pattern
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IEEE754 Summary (Cont.)
?
??
Normalized
-Normalized
Denorm
-Denorm
NaN
?0
NaN
0
Negative overflow
Positive underflow
Positive overflow
Negative underflow
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FP Addition

• Operands
• (1)s1 M1 2E1
• (1)s2 M2 2E2
• Assume E1 gt E2
• Exact Result (1)s M 2E
• Exponent E E1
• Sign s, significand M Result of signed align
  add

(1)s1 M1
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Decimal Number Conversion

• Convert Binary to Decimal (base 2 to base 10)
• x x x x x. d d d d d d 2

• 24 23 22 21 20 2-1 2-2 2-3
  2-4 2-5 2-6

1101.0112 12312202112002-112-212-3
13.37510
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Decimal Number Conversion (Cont.)

• Convert Decimal Integer to binary Integer
• divide the decimal value by 2 and then write
  down the remainder from bottom to top (Divide 2
  and Get the Remainders)
• 3710 ?2

Quotient reminder 372
18 1 182 9 0 92
4 1 42 2 0
22 1 0 12
0 1
1001012
Can it always be converted into an accurate
binary number?
YES!
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Decimal Number Conversion (Cont.)

• Convert Decimal Fraction to Binary Fraction
• multiply the decimal value by 2 and then write
  down the integer number from top to bottom
  (Multiply 2 and Get the Integers) .
• 0.37510 ?2

.375 2 0.75 .75 2
1.5 .5 2 1.0
Where to put the binary point?
Prior to the First number!
0.0112
Can it always be converted into an accurate
binary number?
NO!
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Decimal Number Conversion (Cont.)

• Convert Decimal Number to Binary Binary
• Put Together Integer Part . Fraction Part
• 37.37510 100101.011

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Summary

• Computer arithmetic is constrained by limited
  precision
• Bit patterns have no inherent meaning but
  standards do exist
• twos complement
• IEEE 754 floating point
• OPcode determines meaning of the bit patterns
• Performance and accuracy are important so there
  are many complexities in real machines (i.e.,
  algorithms and implementation).