Integer Arithmetic

1
Topics

• Integer Arithmetic
• Floating Point Representation
• Floating Point Arithmetic

2
2s Complement Integers
3
2s Complement Addition/Subtraction
A B ? A
What is Overflow? How is it identified?
4
Unsigned Integer Multiplication
5
Unsigned Integer Multiplication
Q x M ? AQ
6
Unsigned Integer Multiplication Flow Diagram
7
2s Comp MultiplicationBooths Algorithm
Add one extra bit to the Q register
Q-1
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2s Comp MultiplicationBooths Algorithm
Q-1
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2s Comp MultiplicationBooths Algorithm
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Booth (7) x (3)

• A Q M
• 3 7
• 0000 0011 0 0111
• --------------------
• 1001 0011 0 0111 A lt- (A - M) 1st
• 1100 1001 1 0111 Shift
• --------------------
• 2nd
• 1110 0100 1 0111 Shift
• --------------------
• 0101 0100 1 0111 A lt- (A M) 3rd
• 0010 1010 0 0111 Shift
• --------------------
• 4th
• 0001 0101 0 0111 Shift
• --------------------

11
Booth (7) x (-3)

• A Q M
• -3 7
• 0000 1101 0 0111
• --------------------
• 1001 1101 0 0111 A lt- (A - M) 1st
• 1100 1110 1 0111 Shift
• --------------------
• 0011 1110 1 0111 A lt- (A M) 2nd
• 0001 1111 0 0111 Shift
• --------------------
• 1010 1111 0 0111 A lt- (A - M) 3rd
• 1101 0111 1 0111 Shift
• --------------------
• 4th
• 1110 1011 1 0111 Shift
• --------------------

12
Booth (-7) x (3)

• A Q M
• 3 -7
• 0000 0011 0 1001
• --------------------
• 0111 0011 0 1001 A lt- (A - M) 1st
• 0011 1001 1 1001 Shift
• --------------------
• 2nd
• 0001 1100 1 1001 Shift
• --------------------
• 1010 1100 1 1001 A lt- (A M) 3rd
• 1101 0110 0 1001 Shift
• --------------------
• 4th
• 1110 1011 0 1001 Shift
• --------------------

13
Booth (-7) x (-3)

• A Q M
• -3 -7
• 0000 1101 0 1001
• --------------------
• 0111 1101 0 1001 A lt- (A - M) 1st
• 0011 1110 1 1001 Shift
• --------------------
• 1100 1110 1 1001 A lt- (A M) 2nd
• 1110 0111 0 1001 Shift
• --------------------
• 0101 0111 0 1001 A lt- (A - M) 3rd
• 0010 1011 1 1001 Shift
• --------------------
• 4th
• 0001 0101 1 1001 Shift
• --------------------

14
Unsigned Integer Division
15
Unsigned Integer Division
16
Unsigned Integer Division
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Unsigned Integer Division
Divisor ? M Dividend ? Q Quotient in
Q Remainder in A
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What about 2s Comp Division ?

• Can we do something like Booths Algorithm?
• Why might we use Booths Algorithm for
  multiplication but not for Division?

19
Single Precision Floating Point NumbersIEEE
Standard

• 32 bit Single Precision Floating Point Numbers
  are stored as
• S EEEEEEEE FFFFFFFFFFFFFFFFFFFFFFF
• S Sign 1 bit
• E Exponent 8 bits
• F Fraction 23 bits
  
• The value V
• If E255 and F is nonzero, then V NaN
  ("Not a Number")
• If E255 and F is zero and S is 1, then V -
  Infinity
• If E255 and F is zero and S is 0, then V
  Infinity
• If 0ltElt255 then V (-1)S 2 (E-127)
  (1.F) (exponent range -127 to 128)
• If E0 and F is nonzero, then V (-1)S 2
  (-126) (0.F) ("unnormalized" values)
• If E0 and F is zero and S is 1, then V - 0
• If E0 and F is zero and S is 0, then V 0

Significand
Note 255 decimal 11111111 in binary (8 bits)
20
FP Examples
21
Double Precision Floating Point NumbersIEEE
Standard

• 64 bit Double Precision Floating Point Numbers
  are stored as
• S EEEEEEEEEEE FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF
  FFFFFFFFFFFFFFFFFF
• S Sign 1 bit
• E Exponent 11 bits
• F Fraction 52 bits
  
• The value V
• If E2047 and F is nonzero, then V NaN
  ("Not a Number")
• If E2047 and F is zero and S is 1, then V -
  Infinity
• If E2047 and F is zero and S is 0, then V
  Infinity
• If 0ltElt2047 then V (-1)S 2 (E-1023)
  (1.F) (exponent range -1023 to 1024)
• If E0 and F is nonzero, then V (-1)S 2
  (-1022) (0.F) ("unnormalized" values)
• If E0 and F is zero and S is 1, then V - 0
• If E0 and F is zero and S is 0, then V 0

Significand
Note 2047 decimal 11111111111 in binary (11
bits)
22
32 bit 2s Complement Integer Numbers
All the Integers from -2,147,483,648 to
2,147,483,647, i.e.
- 2 Gig to 2 Gig-1
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32 bit FP Numbers
24
Density of 32 bit FP Numbers
Note ONLY 232 FP numbers are representable
There are only 232 distinct combinations of
bits in 32 bits !
25
The Added Denormalized FP Numbers
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Floating Point Addition / Subtraction

• Steps
• Check for zero
• Align the significands (fractions)
• Add or Subtract the significands
• Normalize the Result
• Bad Results
• Exponent Overflow
• Exponent Underflow
• Significand Overflow
• Significand Underflow

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Floating Point Addition/Subraction
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Floating Point Multiplication

• Check for zero
• Multiply significands
• Add exponents
• Normalize
• Overflow/underflow?

29
Floating Point Multiplication
30
Floating Point Division

• Check for zero
• Divide significands
• Subract exponents
• Normalize
• Overflow/underflow?

31
Floating Point Division