Arithmetic Logical Unit

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
• Performance (seconds, cycles, instructions)
• Abstractions Instruction Set Architecture
  Assembly Language and Machine Language
• What's up ahead
• Implementing the Architecture
• Focus on ALU

3
Numbers

• Many interpretations for bit strings
• Numbers
• Binary numbers (base 2)0000 0001 0010 0011 0100
  0101 0110 0111 1000 1001...
• Octal and Hexadecimal representation
• Decimal 0...2n-1
• Factors to consider
• numbers are finite (overflow)
• fractions and real numbers
• negative numbers
• How do we represent negative numbers?

4
Representations for Negative Numbers

• Sign Magnitude One's Complement
  Two's Complement000 0 000 0 000
  0001 1 001 1 001 1010 2 010
  2 010 2011 3 011 3 011 3100
  -0 100 -3 100 -4101 -1 101 -2
  101 -3110 -2 110 -1 110 -2111
  -3 111 -0 111 -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?

5
Addition Subtraction

• Just like in grade school (carry/borrow 1s)
  0111 0111
•  0110 - 0110
• 1101 0001
• Two's complement operations easy
• subtraction using addition of negative numbers
  0111  1010
• 0001
• Overflow (result too large for finite computer
  word)
• e.g., adding two n-bit positive numbers yields
  negative number 0111  0001 note that
  overflow is somewhat misleading, 1000 it
  does not mean a carry overflowed

6
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

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

note we call this a 2-input mux even
though it has 3 inputs!
0
1
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 0010 (multiplicand) __x_101
  1 (multiplier)
• Negative numbers convert and multiply
• there are better techniques, we wont look at them

11
Multiplication Implementation
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12
Multiplication Hardware second version
Initially 0
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 1221 9
• 1221 9 10989
• 1221 (10 1)
• 122110 12211
• 12210 1221 10989
• More complex 1221 99
• 1221 100 1221 1
• 122100 1221
• 122100 1221 120879
• Can we apply this to binary numbers?

15
Neat Multiplication Trick Binary Numbers

• 1010 011 10 3 30
• 1010 (100 001)
• 1010 100 1010 001
• 101000 1010
• (32 8) (8 2)
• 30
• 1010 01110 10 14 140
• 1010 (10000 00010)
• 10100000 101000010
• 10100000 10100
• (128 32) ( 16 4) 160 20
• 140
• Identify runs of 1, and convert the multiply
  problem into a shifting problem subtraction.

16
Booths Algorithm

• 0 1 1 1 1 0
• End of run (current bit 0, bit to right 1)
• Middle of run
• Beginning of run
  (current bit 1, bit to right 0)
• Booths 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 01001010101 ?

17
Floating Point (a brief look)

• We need a way to represent
• numbers with fractions, e.g., 3.14163.1416 100
  (Do you recognize this?)
• very small numbers, e.g., .0000000011.0 10-9
• very large numbers, e.g., 3.15576 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 1.011 is stored 011.
• significand 1 fraction gt 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
  -1127126
• double precision 1 bit sign, 11 bit exponent,
  52 bit fraction 64 bits

18
Floating point addition

• Floating point arithmetic deserves special
  attention
• A simple floating point addition problem
• 99.99 0.1610 9.999 101 1.610 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 0.01610 101
• Add the significands
• Sum is 10.015 101
• Normalize to scientific notation
• 10.015 101 1.0015 102
• We started with 4 digit significand. Reduce to 4
  digit significand.
• result is 1.002 102

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 twos
• complement
• Add to A(i)

• Use pipeline for overlapping operations
• Registers act as buffers storing intermediate
  values

20
B(i) A(i)
Clock Pulse R1 R2 R3 R4 R5
1 B(1)
2 B(2) B(1)
3 B(3) B(2) B(1)1 A(1)
4 B(4) B(3) B(2)1 A(2) C(1)
5 B(5) B(4) B(3)1 A(3) C(2)
6 B(6) B(5) B(4)1 A(4) C(3)
7 B(7) B(6) B(5)1 A(5) C(4)
8 B(7) B(6)1 A(6) C(5)
9 B(7)1 A(7) C(6)
10 C(7)
What if A, B are 8 element? Advantage of
pipeline? Basic assumptions / requirements of
good pipelines?
21
Binary to Floating Point

• Hexadecimal number 24A60000
• Bit pattern
• 0010 0100 1010 0110 0000 0000 0000 0000
• 0 01001001 0100 1100 0000 0000 0000 000
• 73 4x16-1 12x16-20x16-3 0x16-4
  0x16-5 0x16-6
• Exponent Sign 73 -127 -54
• Mantissa 1.25 .046875 1.29688
• 1.29688 x 2-54

8/29/2016
22
Chapter 3 Assignments

• 3.3.1, 3.3.2, 3.3.3, 3.5.1, 3.10.1, 3.10.2,
  3.10.3, 3.11.2