Transcript Lecture 6 Instruction Set Architectures

Topic 3
Number Representations
and Computer Arithmetics
Introduction to Computer
Systems Engineering
(CPEG 323)
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Outline
Introduction
Number representation and 2’s
complement
ALU design
Integer multiplication/division
Real Numbers
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Recap
Review
 Steps in executing an instruction?
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Fields of an instruction?
Performance?
Base of the operands?
What's up ahead:
 Number representation
 Arithmetic algorithms
 Logic implementation
 Instructions
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Decoded opcode
a
32
ALU
result
32
b
32
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Reading
Up to now, you should have read
(or are reading)

Patterson and Hennessy, chapters 12 and appendix A
New material
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Patterson and Hennessy, chapter 3
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Numbers Representation
It may get complicated:

How to represent negative numbers?
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What is the range of numbers?

What if a number is out of the range?

How about rational and irrational numbers?
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How does hardware add/sub/div/mul these
numbers?
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Bits and Numbers
Bits are just bits: NO inherent meaning
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conventions define relationship between
bits and numbers
How does the hardware know which
convention is to use?
Add $s1, $s2, $s3
Addu $s1, $s2, $s3
What is the sign of $s1, $s2, and $s3?
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Number Representations
Unsigned integers
Signed integers
BCD (Binary Coded Decimal)
Fixed point
Floating point
Other types of data:
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Characters (ASCII, Unicode)
Pixels (graphics)
Groups of bits
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Unsigned Integers
Why unsigned integers?


Memory access, PC, SP, RA
In C, unsigned int
How to represent?


Number your fingers!
Radix number system
 Decimal: 10 different symbols: 0 1 2 3 4 5 6 7 8 9
number=
n 1
i
(
d
*
1
0
)
 i
i 0
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Unsigned Integers(Radix number
system)
In general, k-radix number system
number=
n1
 (d
i 0
i
* ki )
 How many different di?
 What is the biggest number?
 What is the smallest one?
Special cases:


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2-radix (binary)
8 (octal)
16 (hexademical)
 How many different symbols do thy have?
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Unsigned Integers(Radix number
system, Cont.)
How to convert binary to octal and
hexadecimal?
Example:
010100=(0 *22+1 *21+0*20)*23+(1*22+0*21+0*20)
=2*81+ 4 *80= 248
9
4
8
B
6
Hexadecimal: 1
Binary: 000110010100100010110110
Octal: 0 6 2 4 4 2 6 6
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Unsigned Integers(BCD: binary coded decimal)
Representation: 12710
 How many bits necessary for
BCD? For binary?
 Storage space used ?
Addition
 How to determine a carry:
“If ai+ bi >10” vs. “If ai+ bi >2”
Which one is more efficient for
computer, decimal or binary?
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BCD
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010 1111
Number
0
1
2
3
4
5
6
7
8
9
No use
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Signed Integers
How to distinguish a negative number
from a positive number ?
How to do so efficiently ?
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Signed Integers (Sign and magnitude)
-12: -(sign) 12(absolute value)
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A separate sign bit
A magnitude
For hardware:
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Where to put the sign bit? Right or Left?
How to know the sign of addition? (e.g.
one extra step ?)
What is the sign of 0? (positive and
negative 0 ?)
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