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William Stallings
Computer Organization
and Architecture
Chapter 9
Computer Arithmetic
Rev. by Luciano Gualà (2008)
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1
The von Neumann computer
Arithmetic and Logic Unit
Input
Output
Equipment
Main
Memory
Program Control Unit
Rev. by Luciano Gualà (2008)
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2
Control Lines
PC
IR
Address Lines
ALU
CPU Internal Bus
Registri
Data Lines
CPU Components:
Top Level View
AC
MBR
MAR
Control Unit
Control Signals
Rev. by Luciano Gualà (2008)
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3
Arithmetic & Logic Unit
• Does the calculations
• Almost everything else in the computer is there
to service this unit
• Handles integers
• May handle floating point (real) numbers
Rev. by Luciano Gualà (2008)
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4
ALU Inputs and Outputs
Rev. by Luciano Gualà (2008)
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Reminder
• Let A=an-1an-2…a1a0 be a (pure) binary number
• Its (decimal) value is:
n 1
A   2 ai
i
i 0
Example:
10010 = 24 + 21 = 16 + 2 = 18
Rev. by Luciano Gualà (2008)
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6
Integer Representation
• Only have 0 & 1 to represent everything
• Positive numbers stored in binary
 e.g. 41=00101001
 Binary digit = Bit
• No minus sign
• Negative integer representations:
 Sign-Magnitude
 Two’s complement
Rev. by Luciano Gualà (2008)
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Sign-Magnitude
• Left most bit is sign bit
• 0 means positive
• 1 means negative
+18 = 00010010
-18 = 10010010
• Problems
 Need to consider both sign and magnitude in
arithmetic
 Two representations of zero (+0 and -0)
Rev. by Luciano Gualà (2008)
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8
A better representation:
Two’s Complement
• Let A=an-1an-2…a1a0 be a n-length binary string
• It’s value is in Two’s Complement representation is:
A  2
n 1
n2
an 1   2 ai
i
i 0
if an-1=0 then A is a non-negative number
if an-1=1 then A is a negative number
Rev. by Luciano Gualà (2008)
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9
Two’s Complement: examples
-128 64 32 16
0
1
0
0
8
4
2
1
0
1
1
0
+4
+2
8
4
2
1
0
0
1
1
+2
+1
+64
-128 64 32 16
1
0
0
0
-128
-128 64 32 16
1
-128
0
0
0
= 70
8
4
2
1
1
0
0
0
+8
Rev. by Luciano Gualà (2008)
= -125
= -120
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Benefits
• Only one representation of zero
+0
+1
+2
+3
+4
+5
+6
+7
=
=
=
=
=
=
=
=
0000
0001
0010
0011
0100
0101
0110
0111
-1
-2
-3
-4
-5
-6
-7
-8
=
=
=
=
=
=
=
=
1111
1110
1101
1100
1011
1010
1001
1000
• Arithmetic works easily (see later)
Rev. by Luciano Gualà (2008)
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Geometric Depiction of Two’s
Complement Integers
Rev. by Luciano Gualà (2008)
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Range of Numbers
• 8 bit 2’s complement
+127 = 01111111 = 27 -1
-128 = 10000000 = -27
• 16 bit 2’s complement
+32767 = 01111111 11111111 = 215 - 1
-32768 = 10000000 00000000 = -215
Rev. by Luciano Gualà (2008)
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…when you cannot sleep…
…don’t count sheep in two’s complement…
Rev. by Luciano Gualà (2008)
8 - 14
2’s Complement representation for
negative numbers
•
To represent a negative number using the
“two’s complement” technique:
1. First decide how many bits are used for
representation
2. Then write the modulo of the negative number (in
pure binary)
3. Then, change each 0 in 1, each 1 in 0 (Boolean
Complement or “one’s complement”)
4. Finally, add 1 (as the result of Step 3 was a pure
binary number)
Rev. by Luciano Gualà (2008)
8 - 15
Examples
• E.g.: how to represent -3 with 4 bits:
 Start from +3 = 0011
 Boolean complement gives
 Add 1 to LSB gives -3
• Represent -20 with 8 bits:
 Start from +20 =
 Bolean complement gives
 Add 1
1100
1101
00010100
11101011
11101100
• Negation works in the same way, e.g. negation of -3 is
obtained by the “two’s complement” of -3:
 Representation of -3
 Boolean complement gives
 Add 1 to LSB gives -(-3)=+3
1101
0010
0011
Rev. by Luciano Gualà (2008)
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Negation Special Case 1
•
•
•
•
•
•
0=
0000
Bitwise NOT
1111 (Boolean complement)
Add 1 to LSB
+1
Result
1 0000
Carry is ignored, so:
- 0 = 0 OK !
Rev. by Luciano Gualà (2008)
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Negation Special Case 2
•
•
•
•
•
•
•
•
-8 =
1000
Bitwise NOT
0111 (Boolean complement)
Add 1 to LSB
+1
Result
1000
So:
-(-8) = -8 WRONG !
Monitor MSB (sign bit)
If it does not change during negation there is a
mistake (but for 0!)
Rev. by Luciano Gualà (2008)
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Conversion Between Lengths
• Positive number: pack with leading zeroes
+18 =
00010010
+18 = 00000000 00010010
• Negative number: pack with leading ones
-18 =
11101110
-18 = 11111111 11101110
• i.e. pack with MSB (sign bit)
Rev. by Luciano Gualà (2008)
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Addition and Subtraction
• Addition: standard
• Overflow: when the result needs more bits to be
represented
• Monitor MSB: if it changes there may be an overflow
 When Pos + Pos or Neg + Neg the sign bit should not change: if
it does there is an overflow
• Subtraction: take two’s complement of subtrahend and
add to minuend
 i.e.:
a - b = a + (-b)
• So we only need addition and complement circuits
Rev. by Luciano Gualà (2008)
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Addition: examples
1001 + -7
0101 = 5
1110
-2
1100 + -4
0100 = 4
0
1 0000
1100 + -4
1111 = -1
-5
1 1011
0011 +
0100 =
0111
0101 +
0100 =
1001
1001 + -7
1010 = -6
overflow
1 0011
3
4
7
5
4
overflow
Rev. by Luciano Gualà (2008)
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Hardware for Addition and
Subtraction
Rev. by Luciano Gualà (2008)
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Real Numbers
• …very informally, numbers with the
fractional point
• Actually, only finite precision real numbers
• we need to code the binary point
• we need to code the binary point position
• Solution: floating point numbers
Rev. by Luciano Gualà (2008)
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Reminder
• Let A=an-1an-2…a1a0 ,a-1,a-2,…,a-m be a binary number
• Its value is:
n 1
A   2 ai 
i
i 0
1
2 a
j
j  m
j
Example:
1001,1011 = 23 + 20 +2-1 + 2-3 + 2-4 =9,6875
2-1
2-2
2-3
2-4
2-5
0,5
0,250
0,125
0,0625
0,03125
Rev. by Luciano Gualà (2008)
8 - 24
Real Numbers
• Where is the binary point?
 Fixed?
• Very limited
 Moving?
• How do you show where it is?
• Example:
976.000.000.000.000
= 9,76 * 1014
0,0000000000000976
= 9,76 * 10-14
Rev. by Luciano Gualà (2008)
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Sign bit
Floating Point
Biased
Exponent
Significand or Mantissa
• Represents the value
+/- .<significand> * <base><exponent>
• “Floating” refers to the fact that the true
position of the point “floats” according to the
value of the exponent
Rev. by Luciano Gualà (2008)
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Normalization
(10,0101)2= 10,0101* 20 = 10010,1 * 2-3
= 1001010 * 2-5
= 0,00100101 * 24
• Floating Point numbers are usually normalized
 exponent is adjusted so that there is a single bit equal to 1
before the binary point
• Similar to scientific notation for decimal numbers where
numbers are normalized to give a single digit before the
decimal point
3123 = 3.123 x 103
Rev. by Luciano Gualà (2008)
8 - 27
Normalization
• A normalized number ( 0) has the following form:
+/- 1,bbb…b * 2 +/- E
where
b  {0,1}
• The base (or radix) for the exponent is suppose to be 2
(so it is not represented)
• Since MSB of the mantissa is always 1 there is no need
to represent it
Rev. by Luciano Gualà (2008)
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Typical Representation of
Floating Point with 32 bits
• Mantissa uses 23 bits to store a 24 bits pure binary
number in the interval [1,2)
• Sign is stored in the first bit
• Exponent value is represented in excess or biased
notation with 8 bits
• Excess (bias) 127 means




8 bit exponent field
Nominal exponent value has range 0-255
Subtract 127 (= 28-1-1) to get correct exponent value
Real range of exponent value is -127 to +128
Rev. by Luciano Gualà (2008)
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excess (bias) notation
• Two parameters are
specified:
 the number of bits n
 the bias value K
(usually, K=2n -1 -1)
• the string consisting
of the number i in
pure binary
represents the value
i–K
n = 4 K=7
0000 = -7
0001 = -6
0010 = -5
0011 = -4
0100 = -3
0101 = -2
0110 = -1
0111 = 0
Rev. by Luciano Gualà (2008)
1000 =
1001 =
1010 =
1011 =
1100 =
1101 =
1110 =
1111 =
1
2
3
4
5
6
7
8
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Floating Point Examples
147 = 127+20
107 = 127-20
Rev. by Luciano Gualà (2008)
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Expressible Numbers
Notice:
we are
representing
232 different
numbers in
both cases!!
-(2-2-23)*2128
-2-127
2-127
Rev. by Luciano Gualà (2008)
(2-2-23)*2128
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density
• Precision decreases with the increase of modulo
(representation is denser for smaller numbers)
• How many numbers can we represent in the interval
[2,4)?
• And in the range [4,8)?
• they are the same: 223
Note: Finite precision: (10.0/3.0)*3.0 may be different from 10.0
Rev. by Luciano Gualà (2008)
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IEEE 754
•
•
•
•
Standard for floating point storage
32 and 64 bit standards
8 and 11 bit exponent respectively
Extended formats (both mantissa and exponent)
for intermediate results
Rev. by Luciano Gualà (2008)
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IEEE 754 Formats
Rev. by Luciano Gualà (2008)
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Special cases (single format)
• exponent [-126,127]: normalized numbers
• Some strings are interpreted as special strings
• Two representations for zero (posit. and negat.)
 +/- 0 = 0/1 00000000 00000000000000000000000
• Able to represent infinity (posit. and negat.)
 +/-  = 0/1 11111111 00000000000000000000000
• Non-normalized numbers:
 0/1 00000000 (string different to all 0s)
 exponent=-126
 form of significand: 0.bbbb..
• NAN = Not A Number
 Result of an operation which has no solution
 Propagated as NAN through subsequent operations
 representation: 0/1 11111111 (string different to all 0s)
Rev. by Luciano Gualà (2008)
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FP Arithmetic: addition and
subtraction
•
•
•
•
Check for zero
Align significands (adjusting exponents)
Add or subtract significands
Normalize result
Rev. by Luciano Gualà (2008)
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FP Arithmetic: multiplication
and division
•
•
•
•
•
•
Check for zero
Add/subtract exponents
Multiply/divide significands (watch sign)
Normalize
Round
All intermediate results should be in double
length storage
Rev. by Luciano Gualà (2008)
8 - 38