CSE 246: Computer Arithmetic Algorithms and Hardware Design

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CSE 246 Computer Arithmetic Algorithms and
Hardware Design
Fall 2006 Lecture 11 Cordic, Log, Square,
Exponential Functions

• Instructor
• Prof. Chung-Kuan Cheng

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Project

• IEEE Computer Society Author Kit (5Pages)
• Introduction
• Statement of Problem
• Approaches
• Examples
• Experiments
• Conclusion
• 30 Minutes

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Cordic Algorithms

• Coordinate Rotations Digital Computer
• Rotate vector (x,y) to (x,y)

(x,y)
a
(x,y)
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Cordic Algorithms

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Cordic Algorithms

• Key Given cos a, sin a, tan a we can derive

i ai
0 45
1 26.6
2 14
3 7.1
4 3.6
5 1.8
6 0.9
7 0.4
8 0.2
9 0.1
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Cordic Algorithms (Example)

• Find

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Cordic Algorithms

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Logarithms Method 1

• Find

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Logarithms Method 1

• I.
• II.
• III. A table of

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Logarithms Method 1 (Example)

• Find ln(x), x 1.625
• 10.50.1251.625

1. 1 0 1 1.-1 _ -1 -1 0
-1 x 1 1 0 1 _ 0. 1 1 0 1
0.1 1 0 1 1.0 1 _ 0 1 1 0 1 x 0 1 1
0 1 _ 1.0 0 0 0 1
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Logarithms Method 1 (Example)

• -ln x (1.-1) ln(1.01) ln(1.0000-1)

1. 0 0 0 0 0 1 1. 0 0 0 0 0-1 1. 0 0 0 0 0 0 0 0
0 0
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Logarithms Method 2

• Let define
• Initially xlt2, ie. y00
• If

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Logarithms Method 2

• for i 1 to l do
• x x2
• if x 2
• then yi 1
• x x/2
• else yi 0

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Logarithms Method 2 (Example)

• Find ln2(x), x 1.11 (1.75)

x2 1.1 1 x 1.1 1 1 1 1
1 1 1 1 1 1 __ 1 1 0 0 0 1 y1 1
x2/2 1.1 0 0 0 1 x 1.1
0 0 0 1 1 1 0 0 0 1 1 1 0 0 0
1 1 1 0 0 0 1 _ 1 0.0 1 0 1 1 0 0 0
0 1 y2 1
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Logarithms Method 2 (Example)
(x2/2)2/2 1.00101100001 y3 0 ln2 1.11
0.110
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Squarer

• x3 x2 x1 x0
• X x3 x2 x1 x0
• x3x0 x2x0 x1x0 x0x0
• x3x1 x2x1 x1x1 x0x1
• x3x2 x2x2 x1x2 x0x2
• x3x3 x2x3 x1x3 x0x3 _
• x3x2 x3x1 x3x0 x2x0 x1x0 x0
• x3 x2x1 x1
• x2 _

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Exponentiation ex

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Exponentiation ex

• I.
• II.
• min
• max