Transcript CSI/CCE Computer Architecture

CS 447 – Computer Architecture
Lecture 3
Computer Arithmetic (2)
September 5, 2007
Karem Sakallah
[email protected]
www.qatar.cmu.edu/~msakr/15447-f07/
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Last Time
°Representation of finite-width
unsigned and signed integers
°Addition and subtraction of integers
°Multiplication of unsigned integers
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Today
°Review of multiplication of unsigned
integers
°Multiplication of signed integers
°Division of unsigned integers
°Representation of real numbers
°Addition, subtraction, multiplication,
and division of real numbers
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Multiplication
°Complex
°Work out partial product for each digit
°Take care with place value (column)
°Add partial products
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Multiplication Example
°
°
°
°
1011 Multiplicand (11 dec)
x 1101 Multiplier
(13 dec)
1011 Partial products
0000
Note: if multiplier bit is 1 copy
° 1011
multiplicand (place value)
° 1011
otherwise zero
° 10001111 Product (143 dec)
° Note: need double length result
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Unsigned Binary Multiplication
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Execution of Example
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Flowchart for Unsigned Binary Multiplication
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Multiplying Negative Numbers
°This does not work!
°Solution 1
• Convert to positive if required
• Multiply as above
• If signs were different, negate answer
°Solution 2
• Booth’s algorithm
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Observation
°Which of these two multiplications is
more difficult?
98,765 x 10,001
98,765 x 9,999
°Note:
98,765 x 10,001 = 98,765 x (10,000 + 1)
98,765 x 9,999 = 98,765 x (10,000 – 1)
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In Binary
Let Q = d d d 0 1 1 . . . 1 1 0 d d
QL = d d d 1 0 0 . . . 0 0 0 d d
QR = 0 0 0 0 0 0 . . . 0 1 0 0 0
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In Binary
Let Q = d d d 0 1 1 . . . 1 1 0 d d
QL = d d d 1 0 0 . . . 0 0 0 d d
QR = 0 0 0 0 0 0 . . . 0 1 0 0 0
Then Q = QL – QR
And M x Q = M x QL – M x QR
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A bit more explanation
P=MxQ
where M and Q are n-bit two’s complement
integers
e.g., Q = -23 q3 + 22 q2 + 21 q1 + 20 q0
which can be re-written as follows:
Q = 20(q-1–q0)+21(q0-q1)+22(q1-q2)+23(q2-q3)
leading to
P=
M x 20 x (q-1 - q0) +
M x 21 x (q0 - q1) +
M x 22 x (q1 - q2) +
M x 23 x (q2 - q3)
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Finally!
P=
M x 20 x (q-1 - q0) +
M x 21 x (q0 - q1) +
M x 22 x (q1 - q2) +
M x 23 x (q2 - q3)
qi qi-1 qi-1 – qi
Action
0
0
0
Noop
0
1
1
Add M
1
0
-1
Subtract M
1
1
0
Noop
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Example of Booth’s Algorithm
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Booth’s Algorithm
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Division
°More complex than multiplication
°Negative numbers are really bad!
°Based on long division
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Division of Unsigned Binary Integers
00001101
Quotient
1011 10010011
1011
001110
Partial
1011
Remainders
001111
1011
100
Dividend
Divisor
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Remainder
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Flowchart for Unsigned Binary Division
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Real Numbers
°Numbers with fractions
°Could be done in pure binary
• 1001.1010 = 24 + 20 +2-1 + 2-3 =9.625
°Where is the binary point?
°Fixed?
• Very limited
°Moving?
• How do you show where it is?
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Sign bit
Floating Point
Biased
Exponent
Significand or Mantissa
° +/- .significand x 2exponent
° Misnomer
° Point is actually fixed between sign bit
and body of mantissa
° Exponent indicates place value (point
position)
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Floating Point Examples
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Signs for Floating Point
°Mantissa is stored in two’s
complement
°Exponent is in excess or biased
notation
• e.g. Excess (bias) 128 means
• 8 bit exponent field
• Pure value range 0-255
• Subtract 128 to get correct value
• Range -128 to +127
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Normalization
°FP numbers are usually normalized
°i.e. exponent is adjusted so that
leading bit (MSB) of mantissa is 1
°Since it is always 1 there is no need to
store it
°(c.f. Scientific notation where numbers
are normalized to give a single digit
before the decimal point
°e.g. 3.123 x 103)
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FP Ranges
°For a 32 bit number
• 8 bit exponent
• +/- 2256  1.5 x 1077
°Accuracy
• The effect of changing lsb of mantissa
• 23 bit mantissa 2-23  1.2 x 10-7
• About 6 decimal places
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Expressible Numbers
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Density of Floating Point Numbers
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IEEE 754 Formats
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FP Arithmetic +/-
°Check for zeros
°Align significands (adjusting
exponents)
°Add or subtract significands
°Normalize result
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FP Addition & Subtraction Flowchart
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FP Arithmetic x/
° Check for zero
° Add/subtract exponents
° Multiply/divide significands (watch sign)
° Normalize
° Round
° All intermediate results should be in
double length storage
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Floating Point Multiplication
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Floating Point Division
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