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Contents

1 Introduction and Conventions _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 4

1.1 Outline _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 4

1.2 Motivation _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 4

1.3 Conventions _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 5

1.4 Recursive Function Evaluation _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 6

2 Arithmetic Operations _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 8

2.1 Overview _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 8

2.2 Implementation Techniques _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 9

3 Number Representations _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 10

3.1 Binary Number Systems (BNS) _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 10

3.2 Gray Numbers _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 13

3.3 Redundant Number Systems _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 14

3.4 Residue Number Systems (RNS) _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 16

3.5 Floating-Point Numbers _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 18

3.6 Logarithmic Number System _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 19

3.7 Antitetrational Number System _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 19

3.8 Composite Arithmetic _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 20

3.9 Round-Off Schemes _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 21

4 Addition _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 22

4.1 Overview _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 22

4.2 1-Bit Adders, (m, k)-Counters _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 23

4.3 Carry-Propagate Adders (CPA) _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 26

4.4 Carry-Save Adder (CSA) _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 45

4.5 Multi-Operand Adders _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 46

4.6 Sequential Adders _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 52

5 Simple / Addition-Based Operations _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 53

5.1 Complement and Subtraction _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 53

5.2 Increment / Decrement _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 54

5.3 Counting _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 58

5.4 Comparison, Coding, Detection _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 60

5.5 Shift, Extension, Saturation _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 64

5.6 Addition Flags _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 66

5.7 Arithmetic Logic Unit (ALU) _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 68

6 Multiplication _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 69

6.1 Multiplication Basics _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 69

6.2 Unsigned Array Multiplier _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 71

6.3 Signed Array Multipliers _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 72

6.4 Booth Recoding _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 73

6.5 Wallace Tree Addition _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 75

6.6 Multiplier Implementations _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 75

6.7 Composition from Smaller Multipliers _ _ _ _ _ _ _ _ _ _ _ _ _ 76

6.8 Squaring _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 76

7 Division / Square Root Extraction _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 77

7.1 Division Basics _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 77

7.2 Restoring Division _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 78

7.3 Non-Restoring Division _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 78

7.4 Signed Division _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 79

7.5 SRT Division _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 80

7.6 High-Radix Division _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 81

7.7 Division by Multiplication _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 81

7.8 Remainder / Modulus _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 82

7.9 Divider Implementations _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 83

7.10 Square Root Extraction _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 84

8 Elementary Functions _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 85

8.1 Algorithms _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 85

8.2 Integer Exponentiation _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 86

8.3 Integer Logarithm _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 87

9 VLSI Design Aspects _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 88

9.1 Design Levels _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 88

9.2 Synthesis _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 90

9.3 VHDL _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 91

9.4 Performance _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 93

9.5 Testability _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 95

Bibliography _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 96

Computer Arithmetic