Number Systems and Circuits for Addition

1
Number Systems and Circuits for Addition
Integer Overflow

• Lecture 8
• Section 1.5
• Tue, Jan 30, 2007

2
Addition of Signed Integers

• Express negative numbers in twos complement
  form.
• Add, using the ordinary rules of addition.
• Pay special attention to two bits
• The carry-in bit of the last column.
• The carry-out bit of the last column.

3
Subtraction of Signed Integers

• Express negative numbers in twos complement
  form.
• Then take the twos complement of the subtrahend.
• Add, using the ordinary rules of addition.
• Pay special attention to two bits
• The carry-in bit of the last column.
• The carry-out bit of the last column.

4
Addition of Signed Integers

• Eight cases

a b a b c-in c-out Valid
P P P 0 0 yes
P P N 1 0 no
P N P 1 1 yes
P N N 0 0 yes
N P P 1 1 yes
N P N 0 0 yes
N N P 0 1 no
N N N 1 1 yes
5
Integer Overflow

• If the result is too large to fit in the allotted
  space, the condition is called overflow.
• Integer overflow occurs when
• The sum of two positive integers is too large.
• The sum of two negative integers is too large.

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Addition of Signed Integers

• How do we detect overflow?

a b a b c-in c-out Valid
P P P 0 0 yes
P P N 1 0 no
P N P 1 1 yes
P N N 0 0 yes
N P P 1 1 yes
N P N 0 0 yes
N N P 0 1 no
N N N 1 1 yes
7
Addition of Signed Integers

• How do we detect overflow?

a b a b c-in c-out Valid
P P P 0 0 yes
P P N 1 0 no
P N P 1 1 yes
P N N 0 0 yes
N P P 1 1 yes
N P N 0 0 yes
N N P 0 1 no
N N N 1 1 yes
8
Integer Overflow

• Integer overflow is detected when the carry-in
  bit does not match the carry-out bit in the
  high-order position.

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Case P P ? N An Example

• 80 90 01010000 01011010
• 10101010
• 01010110
• 86
• 80 90 256 86

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Case N N ? P An Example

• (80) (90) 10110000 10100110
• 01010110
• 86
• (80) (90) 256 86

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Case P N ? P An Example

• 90 (80) 01011010 10110000
• 00001010
• 10
• 90 (80) 10

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Case P N ? N An Example

• 80 (90) 01010000 10100110
• 11110110
• 00001010
• 10
• 80 (90) 10

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Six Flags over the PC
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Intel x86 Status Register
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Intel x86 Flags
a b a b CF SF OF
100 200 300 0 0 0
231-1 1 -231 0 1 1
200 -100 100 1 0 0
100 -200 -100 0 1 0
-100 200 100 1 0 0
-200 100 -100 0 1 0
-231 -1 231-1 1 0 1
-100 -200 -300 1 1 0
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The Eflags Program
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Intel x86 Condition Codes
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Intel x86 Puzzler

• Explain why the combinations
• CF 1, SF 1, OF 1
• CF 0, SF 0, OF 1
• never occur.