Transcript Lecture 6 Instruction Set Architectures

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
 2’s Complement
 BCD

0
-2n-1
0
to
to
to
2n - 1
2n-1 - 1
10n/4 - 1
But, what about?
very large numbers
9,349,398,989,787,762,244,859,087,678
 very small number
0.0000000000000000000000045691
 rationals
2/3
2
 Irrationals 2.71828…., 3.1415926…..,

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Conceptual Overview: Finite-precision
numbers
Negative Positive
underflow underflow
Negative
overflow
Expressible
negative
numbers
zero
……………….
Expressible
positive
numbers
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:


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Fixed point
Floating point
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Fixed-Point Data Format
4
2
1
1/2
……
S
Sign
1
0
0
Integer
0
1
1
.
0
0
1/4
0
1/8
0
1
0
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
number=400 000 000 000 000 000 000 000 000. 000
It is much more economical to represent as 4*1026

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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
s
exp
 MSB is sign bit: S=0/1
 exp field encodes E
 frac field encodes M
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frac
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IEEE 754 standard
Three formats: single/double/extended
precision (32,64,80 bits).
Single precision:
s
exp
1 bit
8 bits
frac
23 bits
Double precision: see page 192 in P&H 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: 1+127 = 128!
(Note 128-127 = +1!)
How about if the real exponent is -1 ?
Answer: -1+127 = 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
= 11011011011010000000000
(23 bits! With leading 1 hiding!)
frac
Exponent
E =
Exp =
1310, Bias =
12710
14010 = 10001100
Floating Point Representation:
Binary:01000110011011011011010000000000
exp
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frac
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Denormalized Values
Condition

exp = 000…0
Significand value M = 0.xxx…x

xxx…x: bits of frac
Cases

frac = 000…0
 Represents value 0
 Note that have distinct values +0 and –0

frac  000…0
 Numbers very close to 0.0
 Lose precision as get smaller
 “Gradual underflow”
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Special Values
Condition

exp = 111…1
 (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)


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Represents case when no numeric value can be
determined
E.g., sqrt(–1), 
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IEEE 754: Summary
Normalized:
±
0<exp<max
Any bit pattern
Denormalized:
±
0
Any nonzero bit pattern
zero:
±
0
0
Infinite:
±
11…1
0
NaN:
±
11…1
Any nonzero bit pattern
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IEEE754: Summary (Cont.)

-Normalized -Denorm
NaN
Negative
overflow
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+Denorm
+Normalized +
NaN
0 +0
Negative
underflow
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Positive
underflow
Positive
overflow
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FP Addition
Operands
(–1)s1 M1 2E1
(–1)s2 M2 2E2
 Assume E1 > E2
Exact Result: (–1)s M 2E


Exponent E: E1
Sign s, significand M: Result of signed align & add
E1–E2
(–1)s1 M1
(–1)s2 M2
+
(–1)s M
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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 = 1*23+1*22+0*21+1*20+0*2-1+1*2-2+1*2-3 = 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
37÷2 = 18 … 1
18÷2 = 9 … 0
9÷2 = 4 … 1
4÷2 = 2 … 0
2÷2=
1 … 0
1÷2 = 0 … 1
1001012
Can it always be converted into an accurate binary number?
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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 =
.75 * 2 =
.5 * 2 =
0.75
1.5
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?
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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


two’s 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).
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