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8/3/2019 Part 07 - Computer Arithmetic
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University of Pune
S.E. I.T.
Subject code: 214442
Part 07: Computer Arithmetic
Computer Organization
UNIT I
8/3/2019 Part 07 - Computer Arithmetic
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Fixed point numbers
Floating point numbers
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1 1 0 1 - Number
0 0 1 0 - 1¶s Complement
1 0 1 1 - Number
0 1 0 0 - 1¶s Complement
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1 1 0 1 - Number
0 0 1 0 - 1¶s Complement
+ 1
---------------
0 0 1 1 - 2¶s complement
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0 0 1 0 0 0 (+8)
0 0 1 0 0 1 (+9)
0 1 0 0 0 1 (+17)
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A ddition of 28 and 15:
1 1 1
0 0 1 1 1 0 0 (+28)
0 0 0 1 1 1 1 (+15)
0 1 0 1 0 1 1 (+43)
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A ddition of 28 and -15:x (10000) ² 1·s Complement of 15
0 1 1 1 0 0 (+28)
1 1 0 0 0 0 (-15)
1 0 0 1 1 0 0
1
0 0 1 1 0 1 (13)
A dd end-around carry
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A ddition of -28 and 15:x (100011) ² 1·s Complement of 28
1 0 0 0 1 1 (-28)
0 0 1 1 1 1 (+15)
1 1 0 0 1 0 (-13)
Result is in 1·s complement form
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A ddition of -28 and -15:x (10000) ² 1·s Complement of 15
x (100011) ² 1·s Complement of 28
1 1 0 0 0 1 1 (-28)
1 1 1 0 0 0 0 (-15)
1 1 0 1 0 0 1 1
1
1 0 1 0 1 0 0 (-43)
A dd end-around carry
Result is in 1·s complement form
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A ddition of 28 and 15:
1 1 1
0 0 1 1 1 0 0 (+28)
0 0 0 1 1 1 1 (+15)
0 1 0 1 0 1 1 (+43)
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A ddition of 28 and -15:x (10001) ² 2·s Complement of 15
0 1 1 1 0 0 (+28)
1 1 0 0 0 1 (-15)
1 0 0 1 1 0 1 (+13)
Ignore the carry
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A ddition of -28 and 15:x (100100) ² 2·s Complement of 28
1 1 0 0 1 0 0 (-28)
0 0 0 1 1 1 1 (+15)
1 1 1 0 0 1 1 (-13)
Result is in 2·s complement form
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A ddition of -28 and -15:x (10001) ² 2·s Complement of 15
x (100100) ² 2·s Complement of 28
1 1 0 0 1 0 0 (-28)
1 1 1 0 0 0 1 (-15)
1 1 0 1 0 1 0 1 (-43)
Ignore end-around carry
Result is in 2·s complement form
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Paper and Pencil Example:Multiplicand 1100
2= 12
Multiplier × 11012= 13
1100
0000
1100
1100
Product 100111002= 156
m-bit multiplicand × n-bit multiplier = (m+n)-bit product
A ccomplished via shifting and addition
Consumes more time and more chip area
Binary multiplication is easy
0 × multiplicand = 0
1 × multiplicand = multiplicand
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Start
Multiplier[0]?
1a. Product = Product + Multiplicand
2. Shift the Multiplicand Left 1 bit
3. Shift the Multiplier Right 1 bit
32nd Repetition?
Done
= 0= 1
No
Yes
Initialize
P
roduct =0
Multiplicand is zero extended
Multiplicand
64-bit ALU
Control
Multiplier
Product
64 bits
64 bits
write
shift left
add
shift right
Multiplier[0]32 bits
64 bits
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Consider: 11002 × 11012 ,Product = 100111002
4-bit multiplicand and multiplier are used in this
example
Multiplicand is zero extended because it is unsigned
000101100000SLL Multiplicand and SRL Multiplier
00001100Multiplier[0] = 0 => Do Nothing
001100110000S
LL Multiplicand andS
RL Multiplier
00011000 0110SLL Multiplicand and SRL Multiplier
000011000000SLL Multiplicand and SRL Multiplier
00001100Multiplier[0] = 1 => ADD +
00111100Multiplier[0] = 1 => ADD +
10011100Multiplier[0] = 1 => ADD +
2
00001100 000000001101Initialize0
1
3
4
Multiplicand ProductMultiplier Iteration
