Arithmetic

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Arithmetic

• CPSC 321 Computer Architecture
• Andreas Klappenecker

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Signed Numbers (3bits)
sign magnitude ones complement twos complement
0002 0 0002 0 0002 0
0012 1 0012 1 0012 1
0102 2 0102 2 0102 2
0112 3 0112 3 0112 3
1002 -0 1002 -3 1002 -4
1012 -1 1012 -2 1012 -3
1102 -2 1102 -1 1102 -2
1112 -3 1112 -0 1112 -1
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Twos complement

• The unsigned sum of an n-bit number with its
  negative yields?
• Example with 3 bits
• 0112 3
• 1012 -3
• 10002 2n gt negate(x) 2n-x
• 1012 -22 20

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Negating Signed Integers

• Suppose that you have an n-bit integer in twos
  complement representation
• Negate positive to negative
• 00112 3 ? 11012 -3
• invert bits and add 1
• Negate negative to positive
• Trick invert bits and add 1
• -3 11012 ? 00102 ? 00112

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Addition Subtraction

• Just like in grade school (carry/borrow 1s)
  0111 0111 0110  0110 - 0110 - 0101
• 1101 0001 0001
• Two's complement operations easy
• subtraction using addition of negative numbers
  0111 7  1010 -6 0001

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Detecting Overflow
Operation Operand A Operand B Overflow if result
AB gt0 gt0 lt0
AB lt0 lt0 gt0
A-B gt0 lt0 lt0
A-B lt0 gt0 gt0
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MIPS instructions

• lb loads a byte and stores the sign-extended
  version in a word.
• lbu loads a byte and stores it in a word
• Which of these two is typically used to process
  characters?

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Logic Gates AND

• AND
• a b c
• 0 0 0
• 0 1 0
• 1 0 0
• 1 1 1

a
c
b
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Logic Gates OR

• OR
• a b c
• 0 0 0
• 0 1 1
• 1 0 1
• 1 1 1

a
c
b
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An ALU (arithmetic logic unit)

• Let's build an ALU to support the andi and ori
  instructions
• Selection of operation 0 and, 1 or
• we'll just build a 1 bit ALU, and use 32 of
  them
• Possible Implementation (sum-of-products)

a
b
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The Multiplexor

• Selects one of the inputs to be the output,
  based on a control input
• Build (and/or) ALU using a MUX

note it is called a 2-input mux even
though it has 3 inputs!
0
1
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Different Implementations

• Not easy to decide the best way to build
  something
• Don't want too many inputs to a single gate
• for our purposes, ease of comprehension is
  important
• Dont want to have to go through too many gates
• Let's look at a 1-bit ALU for addition

cout a b a cin b cin sum a xor b xor cin
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Different Implementations

• How could we build a 1-bit ALU for add, and, and
  or?
• How could we build a 32-bit ALU?

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Building a 32 bit ALU
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What about subtraction (a b) ?

• Two's complement approach just negate b and
  add.
• How do we negate?
• A solution