1. Arithmetic Logical Unit
Be able to explain the organization of the
classical von Neumann machine and its
major functional components
Focus on ALU design issues

2. Arithmetic Where we've been: What's up ahead:
Performance (seconds, cycles, instructions)
Abstractions: Instruction Set Architecture Assembly Language and Machine Language
What's up ahead:
Implementing the Architecture
Focus on ALU 32
operation
result a b
ALU
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3. Numbers Many interpretations for bit strings Numbers
Binary numbers (base 2)
Octal and Hexadecimal representation
Decimal: n-1
Factors to consider:
numbers are finite (overflow)
fractions and real numbers
negative numbers
How do we represent negative numbers?
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4. Representations for Negative Numbers
Sign Magnitude One's Complement Two's Complement 000 = = = = = = = = = = = = = = = = = = = = = = = = -1
Which would you use?
Rotation measurement (shaft encoder) – which would you use?
What is the highest negative number that can be represented by an 8 bit number?
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5. Addition & Subtraction
Just like in grade school (carry/borrow 1s)
+   0110
Two's complement operations easy
subtraction using addition of negative numbers  1010
0001
Overflow (result too large for finite computer word):
e.g., adding two n-bit positive numbers yields negative number  0001 note that “overflow” is somewhat misleading, it does not mean a carry “overflowed”
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6. An exception (interrupt) occurs
Overflow - Exception
An exception (interrupt) occurs
Control jumps to predefined address for exception processing
Interrupted address is saved for possible resumption
Details based on software system / language
example: flight control vs. homework assignment
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7. Review: The Multiplexor
Selects one of the inputs to be the output, based on a control input
Lets build our ALU using a MUX: S C A B
note: we call this a 2-input mux
even though it has 3 inputs! 1
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8. Building a 32 bit ALU
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9. What about subtraction (a – b) ?
Two's complement approach: just negate b and add.
How do we negate?
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10. Multiplication More complicated than addition
accomplished via addition and shifting
How many bits in the result?
More time
More storage
Let's look at 3 versions based on grade school algorithm (multiplicand) __x_ (multiplier)
Negative numbers: convert and multiply
there are better techniques, we won’t look at them
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11. Multiplication: Implementation
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12. Multiplication Hardware – second version
Initially 0
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13. Final Version One partial-product addition per cycle.
OK if multiplication is not frequent.
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14. Neat Multiplication Trick
Simple example: * 9 =
1221 * 9 = 10989
1221 * (10 – 1) =
1221*10 – 1221*1 =
12210 – 1221 = 10989
More complex: * 99 =
1221 * 100 – 1221 * 1 =
– 1221 =
– 1221 =
Can we apply this to binary numbers?
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15. Neat Multiplication Trick – Binary Numbers
1010 * 011 = 10 * 3 = 30
1010 * (100 – 001) =
1010 * 100 – 1010 * 001 =
– 1010 =
(32 + 8) – (8 + 2) = 30
1010 * = 10 * 14 = 140
1010 * (10000 – 00010) =
– 1010*00010 =
– =
( ) – ( ) = 160 – 20 =
140
Identify runs of 1, and convert the multiply problem into a shifting problem + subtraction.
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16. Booth’s Algorithm
End of run (current bit = 0, bit to right = 1)
Middle of run
Beginning of run (current bit = 1, bit to right = 0)
Booth’s Algorithm:
00: Middle of string of 0s, so no operation
01: End of string of 1s, so add multiplicand to the left half of the partial product
10: Beginning of the 1s run, so subtract the multiplicand from the left half of the partial product
11: Middle of the 1s run, so no operation
What happens if the runs are short: ?
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17. Floating Point (a brief look)
We need a way to represent
numbers with fractions, e.g., = ´ (Do you recognize this?)
very small numbers, e.g., =1.0 ´ 10-9
very large numbers, e.g., ´ 109
Scientific notation – one non-zero digit to left of decimal point
Binary equivalent – one non-zero digit to left of binary point (1.0 ´ 2-1)
Representation:
sign, exponent, fraction: (–1)sign ´ fraction ´ 2exponent
leading bit is always 1, so this is implied. For example in is stored 011.
significand = 1 + fraction => significand 1 bit longer than fraction stored
more bits for fraction (significand) gives more accuracy
more bits for exponent increases range
IEEE 754 floating point standard:
single precision: 1 bit sign, 8 bit exponent, 23 bit fraction = 32 bits
what if exponent is negative?
bias the exponent by 127, so -1 is represented by =126
double precision: 1 bit sign, 11 bit exponent, 52 bit fraction = 64 bits
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18. Floating point addition
Floating point arithmetic deserves special attention
A simple floating point addition problem
= ´ ´ 10-1 (scientific notation)
assume 4 digit significand
To add we need the exponents to be the same
Align the decimal point
9.999 ´ ´ 101
Add the significands
Sum is ´101
Normalize to scientific notation
´101 = ´102
We started with 4 digit significand. Reduce to 4 digit significand.
result is ´102
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19. Pipelines for Vector Addition
Illustrate the potential advantages of pipelines
Pipelines for instruction execution discussed in Chapter 6
Objective: Compute C = A – B, where A and B are 7 x 1 vectors
i.e. C(i) = A(i) – B(i)
Each subtraction involves
Complement of B(i)
Increment to get two’s
complement
Add to A(i) R1
Complement R4
B(i) A(i) R3
Adder R2
Increment
Use pipeline for overlapping operations
Registers act as buffers storing intermediate values R5
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20. B(i) A(i) R2 Increment R5 R1 Complement R4 R3 Adder Clock Pulse # R16
B(6)
B(4)+1
A(4)
C(3) 7
B(7)
B(5)+1
A(5)
C(4) 8
B(6)+1
A(6)
C(5) 9
B(7)+1
A(7)
C(6) 10
C(7) R2
Increment R5 R1
Complement R4 R3
Adder
What if A, B are 8 element?
Advantage of pipeline?
Basic assumptions / requirements of good pipelines?
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21. Binary to Floating Point
Hexadecimal number: 24A60000
Bit pattern:
x x16-2+0x x x x16-6
Exponent: Sign = = -54
Mantissa: =
x 2-54
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22. Chapter 3 Assignments
3.3.1, 3.3.2, 3.3.3, 3.5.1, , , ,
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