Number Systems and Arithmetic Jason Mars Thursday, January 24, 13 - - PowerPoint PPT Presentation

▶
number systems and arithmetic
SMART_READER_LITE
LIVE PREVIEW

Number Systems and Arithmetic Jason Mars Thursday, January 24, 13 - - PowerPoint PPT Presentation

Sep 28, 2022 575 likes •1.5k views

Number Systems and Arithmetic Jason Mars Thursday, January 24, 13 What do all those bits mean? bits (011011011100010 ....01) data instruction number text chars .............. R-format I-format ... integer floating point single precision

slide-1
SLIDE 1

Number Systems and Arithmetic

Jason Mars

Thursday, January 24, 13

slide-2
SLIDE 2

What do all those bits mean?

bits (011011011100010 ....01) instruction R-format I-format ... data number text chars .............. integer floating point signed unsigned single precision double precision ... ... ... ...

Thursday, January 24, 13

slide-3
SLIDE 3

Questions About Numbers

  • How do you represent
  • negative numbers?
  • fractions?
  • really large numbers?
  • really small numbers?
  • How do you
  • do arithmetic?
  • identify errors (e.g. overflow)?
  • What is an ALU and what does it look like?
  • ALU=arithmetic logic unit

Thursday, January 24, 13

slide-4
SLIDE 4

Introduction to Binary Numbers

  • Consider a 4 bit binary number
  • Examples of binary arithmetic

Binary Binary Decimal 0000 1 0001 2 0010 3 0011 Decimal 4 0100 5 0101 6 0110 7 0111 1 1 1 + 1 1 1 1 1 + 1 1

3 + 2 = 5 3 + 3 = 6

Thursday, January 24, 13

slide-5
SLIDE 5

Introduction to Binary Numbers

  • Consider a 4 bit binary number
  • Examples of binary arithmetic

Binary Binary Decimal 0000 1 0001 2 0010 3 0011 Decimal 4 0100 5 0101 6 0110 7 0111 1 1 1 + 1 1 1 1 1 + 1 1

3 + 2 = 5 3 + 3 = 6

1

Thursday, January 24, 13

slide-6
SLIDE 6

Introduction to Binary Numbers

  • Consider a 4 bit binary number
  • Examples of binary arithmetic

Binary Binary Decimal 0000 1 0001 2 0010 3 0011 Decimal 4 0100 5 0101 6 0110 7 0111 1 1 1 + 1 1 1 1 1 + 1 1

3 + 2 = 5 3 + 3 = 6

1

Thursday, January 24, 13

slide-7
SLIDE 7

Introduction to Binary Numbers

  • Consider a 4 bit binary number
  • Examples of binary arithmetic

Binary Binary Decimal 0000 1 0001 2 0010 3 0011 Decimal 4 0100 5 0101 6 0110 7 0111 1 1 1 + 1 1 1 1 1 + 1 1

3 + 2 = 5 3 + 3 = 6

1 1

Thursday, January 24, 13

slide-8
SLIDE 8

Introduction to Binary Numbers

  • Consider a 4 bit binary number
  • Examples of binary arithmetic

Binary Binary Decimal 0000 1 0001 2 0010 3 0011 Decimal 4 0100 5 0101 6 0110 7 0111 1 1 1 + 1 1 1 1 1 + 1 1

3 + 2 = 5 3 + 3 = 6

1 1

Thursday, January 24, 13

slide-9
SLIDE 9

Introduction to Binary Numbers

  • Consider a 4 bit binary number
  • Examples of binary arithmetic

Binary Binary Decimal 0000 1 0001 2 0010 3 0011 Decimal 4 0100 5 0101 6 0110 7 0111 1 1 1 + 1 1 1 1 1 + 1 1

3 + 2 = 5 3 + 3 = 6

1 1 1

Thursday, January 24, 13

slide-10
SLIDE 10

Introduction to Binary Numbers

  • Consider a 4 bit binary number
  • Examples of binary arithmetic

Binary Binary Decimal 0000 1 0001 2 0010 3 0011 Decimal 4 0100 5 0101 6 0110 7 0111 1 1 1 + 1 1 1 1 1 + 1 1

3 + 2 = 5 3 + 3 = 6

1 1 1 1

Thursday, January 24, 13

slide-11
SLIDE 11

Negative Numbers?

  • We would like a number system that provides
  • obvious representation of 0,1,2...
  • uses adder for addition
  • single value of 0
  • equal coverage of positive and negative numbers
  • easy detection of sign
  • easy negation

Thursday, January 24, 13

slide-12
SLIDE 12

Two’s Complement Representation

  • 2’s complement representation of negative numbers
  • Take the bitwise inverse and add 1
  • Biggest 4-bit Binary Number: 7

Smallest 4-bit Binary Number: -8

Decimal

  • 8
  • 7
  • 6
  • 5
  • 4
  • 3
  • 2
  • 1

1 2 3 4 5 6 7 Twos Complement Binary 1000 1001 1010 1011 1100 1101 1110 1111 0000 0001 0010 0011 0100 0101 0110 0111

Thursday, January 24, 13

slide-13
SLIDE 13

Two’s Complement Arithmetic (Subtraction)

2s Complement Binary 2s Complement Binary Decimal 0000 1 0001 2 0010 3 0011 1111 1110 1101 Decimal

  • 1
  • 2
  • 3

4 0100 5 0101 6 0110 7 0111 1100 1011 1010 1001

  • 4
  • 5
  • 6
  • 7

1000

  • 8

Examples: 7 - 6 = 7 + (- 6) = 1 3 - 5 = 3 + (- 5) = -2

1 1 1 1 1 + 1 1 1 1 1 +

Thursday, January 24, 13

slide-14
SLIDE 14

Two’s Complement Arithmetic (Subtraction)

2s Complement Binary 2s Complement Binary Decimal 0000 1 0001 2 0010 3 0011 1111 1110 1101 Decimal

  • 1
  • 2
  • 3

4 0100 5 0101 6 0110 7 0111 1100 1011 1010 1001

  • 4
  • 5
  • 6
  • 7

1000

  • 8

Examples: 7 - 6 = 7 + (- 6) = 1 3 - 5 = 3 + (- 5) = -2

1 1 1 1 1 + 1 1 1 1 1 + 1

Thursday, January 24, 13

slide-15
SLIDE 15

Two’s Complement Arithmetic (Subtraction)

2s Complement Binary 2s Complement Binary Decimal 0000 1 0001 2 0010 3 0011 1111 1110 1101 Decimal

  • 1
  • 2
  • 3

4 0100 5 0101 6 0110 7 0111 1100 1011 1010 1001

  • 4
  • 5
  • 6
  • 7

1000

  • 8

Examples: 7 - 6 = 7 + (- 6) = 1 3 - 5 = 3 + (- 5) = -2

1 1 1 1 1 + 1 1 1 1 1 + 1

Thursday, January 24, 13

slide-16
SLIDE 16

Two’s Complement Arithmetic (Subtraction)

2s Complement Binary 2s Complement Binary Decimal 0000 1 0001 2 0010 3 0011 1111 1110 1101 Decimal

  • 1
  • 2
  • 3

4 0100 5 0101 6 0110 7 0111 1100 1011 1010 1001

  • 4
  • 5
  • 6
  • 7

1000

  • 8

Examples: 7 - 6 = 7 + (- 6) = 1 3 - 5 = 3 + (- 5) = -2

1 1 1 1 1 + 1 1 1 1 1 + 1 1

Thursday, January 24, 13

slide-17
SLIDE 17

Two’s Complement Arithmetic (Subtraction)

2s Complement Binary 2s Complement Binary Decimal 0000 1 0001 2 0010 3 0011 1111 1110 1101 Decimal

  • 1
  • 2
  • 3

4 0100 5 0101 6 0110 7 0111 1100 1011 1010 1001

  • 4
  • 5
  • 6
  • 7

1000

  • 8

Examples: 7 - 6 = 7 + (- 6) = 1 3 - 5 = 3 + (- 5) = -2

1 1 1 1 1 + 1 1 1 1 1 + 1 1

Thursday, January 24, 13

slide-18
SLIDE 18

Two’s Complement Arithmetic (Subtraction)

2s Complement Binary 2s Complement Binary Decimal 0000 1 0001 2 0010 3 0011 1111 1110 1101 Decimal

  • 1
  • 2
  • 3

4 0100 5 0101 6 0110 7 0111 1100 1011 1010 1001

  • 4
  • 5
  • 6
  • 7

1000

  • 8

Examples: 7 - 6 = 7 + (- 6) = 1 3 - 5 = 3 + (- 5) = -2

1 1 1 1 1 + 1 1 1 1 1 + 1 1 1

Thursday, January 24, 13

slide-19
SLIDE 19

Two’s Complement Arithmetic (Subtraction)

2s Complement Binary 2s Complement Binary Decimal 0000 1 0001 2 0010 3 0011 1111 1110 1101 Decimal

  • 1
  • 2
  • 3

4 0100 5 0101 6 0110 7 0111 1100 1011 1010 1001

  • 4
  • 5
  • 6
  • 7

1000

  • 8

Examples: 7 - 6 = 7 + (- 6) = 1 3 - 5 = 3 + (- 5) = -2

1 1 1 1 1 + 1 1 1 1 1 + 1 1 1

Thursday, January 24, 13

slide-20
SLIDE 20

Two’s Complement Arithmetic (Subtraction)

2s Complement Binary 2s Complement Binary Decimal 0000 1 0001 2 0010 3 0011 1111 1110 1101 Decimal

  • 1
  • 2
  • 3

4 0100 5 0101 6 0110 7 0111 1100 1011 1010 1001

  • 4
  • 5
  • 6
  • 7

1000

  • 8

Examples: 7 - 6 = 7 + (- 6) = 1 3 - 5 = 3 + (- 5) = -2

1 1 1 1 1 + 1 1 1 1 1 + 1 1 1

Thursday, January 24, 13

slide-21
SLIDE 21

Two’s Complement Arithmetic (Subtraction)

2s Complement Binary 2s Complement Binary Decimal 0000 1 0001 2 0010 3 0011 1111 1110 1101 Decimal

  • 1
  • 2
  • 3

4 0100 5 0101 6 0110 7 0111 1100 1011 1010 1001

  • 4
  • 5
  • 6
  • 7

1000

  • 8

Examples: 7 - 6 = 7 + (- 6) = 1 3 - 5 = 3 + (- 5) = -2

1 1 1 1 1 + 1 1 1 1 1 + 1 1 1 1

Thursday, January 24, 13

slide-22
SLIDE 22

Two’s Complement Arithmetic (Subtraction)

2s Complement Binary 2s Complement Binary Decimal 0000 1 0001 2 0010 3 0011 1111 1110 1101 Decimal

  • 1
  • 2
  • 3

4 0100 5 0101 6 0110 7 0111 1100 1011 1010 1001

  • 4
  • 5
  • 6
  • 7

1000

  • 8

Examples: 7 - 6 = 7 + (- 6) = 1 3 - 5 = 3 + (- 5) = -2

1 1 1 1 1 + 1 1 1 1 1 + 1 1 1 1 1

Thursday, January 24, 13

slide-23
SLIDE 23

Two’s Complement Arithmetic (Subtraction)

2s Complement Binary 2s Complement Binary Decimal 0000 1 0001 2 0010 3 0011 1111 1110 1101 Decimal

  • 1
  • 2
  • 3

4 0100 5 0101 6 0110 7 0111 1100 1011 1010 1001

  • 4
  • 5
  • 6
  • 7

1000

  • 8

Examples: 7 - 6 = 7 + (- 6) = 1 3 - 5 = 3 + (- 5) = -2

1 1 1 1 1 + 1 1 1 1 1 + 1 1 1 1 1 1

Thursday, January 24, 13

slide-24
SLIDE 24

Two’s Complement Arithmetic (Subtraction)

2s Complement Binary 2s Complement Binary Decimal 0000 1 0001 2 0010 3 0011 1111 1110 1101 Decimal

  • 1
  • 2
  • 3

4 0100 5 0101 6 0110 7 0111 1100 1011 1010 1001

  • 4
  • 5
  • 6
  • 7

1000

  • 8

Examples: 7 - 6 = 7 + (- 6) = 1 3 - 5 = 3 + (- 5) = -2

1 1 1 1 1 + 1 1 1 1 1 + 1 1 1 1 1 1 1

Thursday, January 24, 13

slide-25
SLIDE 25

Two’s Complement Arithmetic (Subtraction)

2s Complement Binary 2s Complement Binary Decimal 0000 1 0001 2 0010 3 0011 1111 1110 1101 Decimal

  • 1
  • 2
  • 3

4 0100 5 0101 6 0110 7 0111 1100 1011 1010 1001

  • 4
  • 5
  • 6
  • 7

1000

  • 8

Examples: 7 - 6 = 7 + (- 6) = 1 3 - 5 = 3 + (- 5) = -2

1 1 1 1 1 + 1 1 1 1 1 + 1 1 1 1 1 1 1 1

Thursday, January 24, 13

slide-26
SLIDE 26

Some Things We Want To Know About Our Number System

  • Negation
  • Sign extension
  • +3 => 0011, 00000011, 0000000000000011
  • -3 => 1101, 11111101, 1111111111111101
  • Overflow detection

0101 5 + 0110 6

Thursday, January 24, 13

slide-27
SLIDE 27

Some Things We Want To Know About Our Number System

  • Negation
  • Sign extension
  • +3 => 0011, 00000011, 0000000000000011
  • -3 => 1101, 11111101, 1111111111111101
  • Overflow detection

0101 5 + 0110 6

1

Thursday, January 24, 13

slide-28
SLIDE 28

Some Things We Want To Know About Our Number System

  • Negation
  • Sign extension
  • +3 => 0011, 00000011, 0000000000000011
  • -3 => 1101, 11111101, 1111111111111101
  • Overflow detection

0101 5 + 0110 6

11

Thursday, January 24, 13

slide-29
SLIDE 29

Some Things We Want To Know About Our Number System

  • Negation
  • Sign extension
  • +3 => 0011, 00000011, 0000000000000011
  • -3 => 1101, 11111101, 1111111111111101
  • Overflow detection

0101 5 + 0110 6

011

Thursday, January 24, 13

slide-30
SLIDE 30

Some Things We Want To Know About Our Number System

  • Negation
  • Sign extension
  • +3 => 0011, 00000011, 0000000000000011
  • -3 => 1101, 11111101, 1111111111111101
  • Overflow detection

0101 5 + 0110 6

011 1

Thursday, January 24, 13

slide-31
SLIDE 31

Some Things We Want To Know About Our Number System

  • Negation
  • Sign extension
  • +3 => 0011, 00000011, 0000000000000011
  • -3 => 1101, 11111101, 1111111111111101
  • Overflow detection

0101 5 + 0110 6

011 1

  • 5

Thursday, January 24, 13

slide-32
SLIDE 32

Some Things We Want To Know About Our Number System

  • Negation
  • Sign extension
  • +3 => 0011, 00000011, 0000000000000011
  • -3 => 1101, 11111101, 1111111111111101
  • Overflow detection

0101 5 + 0110 6

011 1

  • 5

Thursday, January 24, 13

slide-33
SLIDE 33

Overflow Detection

  • How do we detect overflow?
  • XOR Carry In and Carry Out of MSB

1 1 1 1 1 + 1 1 1 1 1 1 1 1 + 1 1 1 1 1 7 3 1

  • 6
  • 4
  • 5

7 1 1 1 + 1 1 1 1 1 1 1 1 + 1 1 1 2 3 5

  • 4
  • 2
  • 6

1 1

Thursday, January 24, 13

slide-34
SLIDE 34

Overflow Detection

  • How do we detect overflow?
  • XOR Carry In and Carry Out of MSB

1 1 1 1 1 + 1 1 1 1 1 1 1 1 + 1 1 1 1 1 7 3 1

  • 6
  • 4
  • 5

7 1 1 1 + 1 1 1 1 1 1 1 1 + 1 1 1 2 3 5

  • 4
  • 2
  • 6

1 1

OK

Thursday, January 24, 13

slide-35
SLIDE 35

Overflow Detection

  • How do we detect overflow?
  • XOR Carry In and Carry Out of MSB

1 1 1 1 1 + 1 1 1 1 1 1 1 1 + 1 1 1 1 1 7 3 1

  • 6
  • 4
  • 5

7 1 1 1 + 1 1 1 1 1 1 1 1 + 1 1 1 2 3 5

  • 4
  • 2
  • 6

1 1

OK Not OK

Thursday, January 24, 13

slide-36
SLIDE 36

Arithmetic -- The Heart of Instruction Execution

32 32 32

  • peration

result a b

ALU

Instruction Fetch Instruction Decode Operand Fetch Execute Result Store Next Instruction

Thursday, January 24, 13

slide-37
SLIDE 37

Designing an Arithmetic Logic Unit

ALU N N N A B Result Overflow Zero 3 ALUop CarryOut ALU Control Lines (ALUop) Function 000 And 001 Or 010 Add 110 Subtract 111 Set-on-less-than

Thursday, January 24, 13

slide-38
SLIDE 38

A One Bit ALU

  • This 1-bit ALU will perform AND, OR, and ADD

1 1 1 1 1 + 1 1 1

  • 4
  • 2
  • 6

1

Thursday, January 24, 13

slide-39
SLIDE 39

A One Bit ALU

  • This 1-bit ALU will perform AND, OR, and ADD

1 1 1 1 1 + 1 1 1

  • 4
  • 2
  • 6

1

Thursday, January 24, 13

slide-40
SLIDE 40

A One Bit ALU

  • This 1-bit ALU will perform AND, OR, and ADD

1 1 1 1 1 + 1 1 1

  • 4
  • 2
  • 6

1

1

Thursday, January 24, 13

slide-41
SLIDE 41

A One Bit ALU

  • This 1-bit ALU will perform AND, OR, and ADD

1 1 1 1 1 + 1 1 1

  • 4
  • 2
  • 6

1

1

Thursday, January 24, 13

slide-42
SLIDE 42

A One Bit ALU

  • This 1-bit ALU will perform AND, OR, and ADD

1 1 1 1 1 + 1 1 1

  • 4
  • 2
  • 6

1

1

Thursday, January 24, 13

slide-43
SLIDE 43

A One Bit ALU

  • This 1-bit ALU will perform AND, OR, and ADD

1 1 1 1 1 + 1 1 1

  • 4
  • 2
  • 6

1

Thursday, January 24, 13

slide-44
SLIDE 44

A One Bit ALU

  • This 1-bit ALU will perform AND, OR, and ADD

1 1 1 1 1 + 1 1 1

  • 4
  • 2
  • 6

1

1

Thursday, January 24, 13

slide-45
SLIDE 45

A One Bit ALU

  • This 1-bit ALU will perform AND, OR, and ADD

1 1 1 1 1 + 1 1 1

  • 4
  • 2
  • 6

1

1

Thursday, January 24, 13

slide-46
SLIDE 46

A One Bit ALU

  • This 1-bit ALU will perform AND, OR, and ADD

1 1 1 1 1 + 1 1 1

  • 4
  • 2
  • 6

1

1

Thursday, January 24, 13

slide-47
SLIDE 47

A One Bit ALU

  • This 1-bit ALU will perform AND, OR, and ADD

1 1 1 1 1 + 1 1 1

  • 4
  • 2
  • 6

1

1

Thursday, January 24, 13

slide-48
SLIDE 48

32 Bit ALU

1-bit ALU 32-bit ALU

Thursday, January 24, 13

slide-49
SLIDE 49

Relationship Between ISAs and ALUs

32 32 32

  • peration

result a b

ALU

Instruction Fetch Instruction Decode Operand Fetch Execute Result Store Next Instruction Thursday, January 24, 13

slide-50
SLIDE 50

Relationship Between ISAs and ALUs

  • After Decode
  • Implements All Arithmetic Execution
  • Inputs
  • Source Operands, Registers
  • Destination
  • Destination Operand

32 32 32

  • peration

result a b

ALU

Instruction Fetch Instruction Decode Operand Fetch Execute Result Store Next Instruction Thursday, January 24, 13

slide-51
SLIDE 51

Relationship Between ISAs and ALUs

  • After Decode
  • Implements All Arithmetic Execution
  • Inputs
  • Source Operands, Registers
  • Destination
  • Destination Operand
  • Will all come together with single cycle cpu, I promise ;-)

32 32 32

  • peration

result a b

ALU

Instruction Fetch Instruction Decode Operand Fetch Execute Result Store Next Instruction Thursday, January 24, 13

slide-52
SLIDE 52

The ALU’s Job - A Few Example Operations

Thursday, January 24, 13

slide-53
SLIDE 53

The ALU’s Job - A Few Example Operations

  • And, Or, Addition
  • add

Thursday, January 24, 13

slide-54
SLIDE 54

The ALU’s Job - A Few Example Operations

  • And, Or, Addition
  • add
  • Subtraction
  • sub

Thursday, January 24, 13

slide-55
SLIDE 55

The ALU’s Job - A Few Example Operations

  • And, Or, Addition
  • add
  • Subtraction
  • sub
  • Set-on-less-than
  • slt

Thursday, January 24, 13

slide-56
SLIDE 56

The ALU’s Job - A Few Example Operations

  • And, Or, Addition
  • add
  • Subtraction
  • sub
  • Set-on-less-than
  • slt
  • Branch-if-equal
  • beq

Thursday, January 24, 13

slide-57
SLIDE 57

Adding Subtraction

  • Keep in mind the following:
  • (A - B) is the same as: A + (-B)
  • 2’s Complement negate: Take the inverse of every bit and add 1
  • Bit-wise inverse of B is !B:
  • A - B = A + (-B) = A + (!B + 1) = A + !B + 1

Thursday, January 24, 13

slide-58
SLIDE 58

Adding Subtraction

  • Keep in mind the following:
  • (A - B) is the same as: A + (-B)
  • 2’s Complement negate: Take the inverse of every bit and add 1
  • Bit-wise inverse of B is !B:
  • A - B = A + (-B) = A + (!B + 1) = A + !B + 1

Thursday, January 24, 13

slide-59
SLIDE 59

Adding Subtraction

  • Keep in mind the following:
  • (A - B) is the same as: A + (-B)
  • 2’s Complement negate: Take the inverse of every bit and add 1
  • Bit-wise inverse of B is !B:
  • A - B = A + (-B) = A + (!B + 1) = A + !B + 1

Lets throw in NOR while we’re at it!

Thursday, January 24, 13

slide-60
SLIDE 60

Adding Subtraction

  • Keep in mind the following:
  • (A - B) is the same as: A + (-B)
  • 2’s Complement negate: Take the inverse of every bit and add 1
  • Bit-wise inverse of B is !B:
  • A - B = A + (-B) = A + (!B + 1) = A + !B + 1

Binvert a b CarryIn CarryOut Operation 1 2

  • Result

1 Ainvert 1

Lets throw in NOR while we’re at it!

Thursday, January 24, 13

slide-61
SLIDE 61

Adding Set-on-less-than

  • We are mostly there!
  • A < B = (A - B) < 0
  • If true, set LSB to 1, all others 0
  • Else, set all bits to 0

Thursday, January 24, 13

slide-62
SLIDE 62

Adding Set-on-less-than

  • We are mostly there!
  • A < B = (A - B) < 0
  • If true, set LSB to 1, all others 0
  • Else, set all bits to 0

Binvert a b CarryIn CarryOut Operation 1 2

  • Result

1 Ainvert 1 3 Less

Thursday, January 24, 13

slide-63
SLIDE 63

Adding Set-on-less-than

  • We are mostly there!
  • A < B = (A - B) < 0
  • If true, set LSB to 1, all others 0
  • Else, set all bits to 0

Binvert a b CarryIn CarryOut Operation 1 2

  • Result

1 Ainvert 1 3 Less

Binvert a b CarryIn Operation 1 2

  • Result

1 3 Less Overflow detection Set Overflow Ainvert 1

MSB

Thursday, January 24, 13

slide-64
SLIDE 64

Adding Set-on-less-than

  • We are mostly there!
  • A < B = (A - B) < 0
  • If true, set LSB to 1, all others 0
  • Else, set all bits to 0

1 1 1 1 1 + 1 1 1 1 1

What about overflow?

Binvert a b CarryIn CarryOut Operation 1 2

  • Result

1 Ainvert 1 3 Less

Binvert a b CarryIn Operation 1 2

  • Result

1 3 Less Overflow detection Set Overflow Ainvert 1

MSB

Thursday, January 24, 13

slide-65
SLIDE 65

Overflow Detection

  • When is Overflow Possible?
  • Positive + Negative = Overflow Impossible
  • Negative + Positive = Overflow Impossible
  • Positive + Positive = Can Overflow
  • Negative + Negative = Can Overflow

Decimal

  • 8
  • 7
  • 6
  • 5
  • 4
  • 3
  • 2
  • 1

1 2 3 4 5 6 7 Twos Complement Binary 1000 1001 1010 1011 1100 1101 1110 1111 0000 0001 0010 0011 0100 0101 0110 0111

Thursday, January 24, 13

slide-66
SLIDE 66

Overflow Detection

  • When is Overflow Possible?
  • Positive + Negative = Overflow Impossible
  • Negative + Positive = Overflow Impossible
  • Positive + Positive = Can Overflow
  • Negative + Negative = Can Overflow

Decimal

  • 8
  • 7
  • 6
  • 5
  • 4
  • 3
  • 2
  • 1

1 2 3 4 5 6 7 Twos Complement Binary 1000 1001 1010 1011 1100 1101 1110 1111 0000 0001 0010 0011 0100 0101 0110 0111

  • A Problem for SLT ( -7 < 6 ) -->

Thursday, January 24, 13

slide-67
SLIDE 67

Overflow Detection

  • When is Overflow Possible?
  • Positive + Negative = Overflow Impossible
  • Negative + Positive = Overflow Impossible
  • Positive + Positive = Can Overflow
  • Negative + Negative = Can Overflow

Decimal

  • 8
  • 7
  • 6
  • 5
  • 4
  • 3
  • 2
  • 1

1 2 3 4 5 6 7 Twos Complement Binary 1000 1001 1010 1011 1100 1101 1110 1111 0000 0001 0010 0011 0100 0101 0110 0111

  • A Problem for SLT ( -7 < 6 ) -->
  • -7 + (-6) = 3 -->

Thursday, January 24, 13

slide-68
SLIDE 68

Overflow Detection

  • When is Overflow Possible?
  • Positive + Negative = Overflow Impossible
  • Negative + Positive = Overflow Impossible
  • Positive + Positive = Can Overflow
  • Negative + Negative = Can Overflow

Decimal

  • 8
  • 7
  • 6
  • 5
  • 4
  • 3
  • 2
  • 1

1 2 3 4 5 6 7 Twos Complement Binary 1000 1001 1010 1011 1100 1101 1110 1111 0000 0001 0010 0011 0100 0101 0110 0111

  • A Problem for SLT ( -7 < 6 ) -->
  • -7 + (-6) = 3 -->
  • MSB=0 -->

Thursday, January 24, 13

slide-69
SLIDE 69

Overflow Detection

  • When is Overflow Possible?
  • Positive + Negative = Overflow Impossible
  • Negative + Positive = Overflow Impossible
  • Positive + Positive = Can Overflow
  • Negative + Negative = Can Overflow

Decimal

  • 8
  • 7
  • 6
  • 5
  • 4
  • 3
  • 2
  • 1

1 2 3 4 5 6 7 Twos Complement Binary 1000 1001 1010 1011 1100 1101 1110 1111 0000 0001 0010 0011 0100 0101 0110 0111

  • A Problem for SLT ( -7 < 6 ) -->
  • -7 + (-6) = 3 -->
  • MSB=0 -->
  • -7 > 6!!

Thursday, January 24, 13

slide-70
SLIDE 70

Overflow Detection

  • When is Overflow Possible?
  • Positive + Negative = Overflow Impossible
  • Negative + Positive = Overflow Impossible
  • Positive + Positive = Can Overflow
  • Negative + Negative = Can Overflow

1 1 1 1 1 + 1 1 1 1 1 1 1 1 + 1 1 1 1 1 7 3 1

  • 6
  • 4
  • 5

7 1 1 1 + 1 1 1 1 1 1 1 1 + 1 1 1 2 3 5

  • 4
  • 2
  • 6

1 1 Decimal

  • 8
  • 7
  • 6
  • 5
  • 4
  • 3
  • 2
  • 1

1 2 3 4 5 6 7 Twos Complement Binary 1000 1001 1010 1011 1100 1101 1110 1111 0000 0001 0010 0011 0100 0101 0110 0111

  • A Problem for SLT ( -7 < 6 ) -->
  • -7 + (-6) = 3 -->
  • MSB=0 -->
  • -7 > 6!!

Thursday, January 24, 13

slide-71
SLIDE 71

Overflow Detection

  • When is Overflow Possible?
  • Positive + Negative = Overflow Impossible
  • Negative + Positive = Overflow Impossible
  • Positive + Positive = Can Overflow
  • Negative + Negative = Can Overflow

1 1 1 1 1 + 1 1 1 1 1 1 1 1 + 1 1 1 1 1 7 3 1

  • 6
  • 4
  • 5

7 1 1 1 + 1 1 1 1 1 1 1 1 + 1 1 1 2 3 5

  • 4
  • 2
  • 6

1 1

OK

Decimal

  • 8
  • 7
  • 6
  • 5
  • 4
  • 3
  • 2
  • 1

1 2 3 4 5 6 7 Twos Complement Binary 1000 1001 1010 1011 1100 1101 1110 1111 0000 0001 0010 0011 0100 0101 0110 0111

  • A Problem for SLT ( -7 < 6 ) -->
  • -7 + (-6) = 3 -->
  • MSB=0 -->
  • -7 > 6!!

Thursday, January 24, 13

slide-72
SLIDE 72

Overflow Detection

  • When is Overflow Possible?
  • Positive + Negative = Overflow Impossible
  • Negative + Positive = Overflow Impossible
  • Positive + Positive = Can Overflow
  • Negative + Negative = Can Overflow

1 1 1 1 1 + 1 1 1 1 1 1 1 1 + 1 1 1 1 1 7 3 1

  • 6
  • 4
  • 5

7 1 1 1 + 1 1 1 1 1 1 1 1 + 1 1 1 2 3 5

  • 4
  • 2
  • 6

1 1

OK

Not OK

Decimal

  • 8
  • 7
  • 6
  • 5
  • 4
  • 3
  • 2
  • 1

1 2 3 4 5 6 7 Twos Complement Binary 1000 1001 1010 1011 1100 1101 1110 1111 0000 0001 0010 0011 0100 0101 0110 0111

  • A Problem for SLT ( -7 < 6 ) -->
  • -7 + (-6) = 3 -->
  • MSB=0 -->
  • -7 > 6!!

Thursday, January 24, 13

slide-73
SLIDE 73

Overflow Detection Logic

  • Carry into MSB ! = Carry out of MSB
  • For a N-bit ALU: Overflow = CarryIn[N - 1] XOR CarryOut[N - 1]

A0 B0 1-bit ALU Result0 CarryIn0 CarryOut0 A1 B1 1-bit ALU Result1 CarryIn1 CarryOut1 A2 B2 1-bit ALU Result2 CarryIn2 A3 B3 1-bit ALU CarryIn3 CarryOut3 Result3 CarryOut2 X Y X XOR Y 1 1 1 1 1 1

Thursday, January 24, 13

slide-74
SLIDE 74

Overflow Detection Logic

  • Carry into MSB ! = Carry out of MSB
  • For a N-bit ALU: Overflow = CarryIn[N - 1] XOR CarryOut[N - 1]

A0 B0 1-bit ALU Result0 CarryIn0 CarryOut0 A1 B1 1-bit ALU Result1 CarryIn1 CarryOut1 A2 B2 1-bit ALU Result2 CarryIn2 A3 B3 1-bit ALU CarryIn3 CarryOut3 Result3 CarryOut2 X Y X XOR Y 1 1 1 1 1 1 Overflow

Thursday, January 24, 13

slide-75
SLIDE 75

Adding Branch-if-equal

Thursday, January 24, 13

slide-76
SLIDE 76

Adding Branch-if-equal

  • Subtract to the rescue

again!

Thursday, January 24, 13

slide-77
SLIDE 77

Adding Branch-if-equal

  • Subtract to the rescue

again!

  • If (A - B = 0) --> A = B

Thursday, January 24, 13

slide-78
SLIDE 78

Adding Branch-if-equal

  • Subtract to the rescue

again!

  • If (A - B = 0) --> A = B
  • Zero Detection Logic is just
  • ne BIG NOR gate (for

equality test)

  • Any non-zero input to the

NOR gate will cause its

  • utput to be zero

Thursday, January 24, 13

slide-79
SLIDE 79

Adding Branch-if-equal

  • Subtract to the rescue

again!

  • If (A - B = 0) --> A = B
  • Zero Detection Logic is just
  • ne BIG NOR gate (for

equality test)

  • Any non-zero input to the

NOR gate will cause its

  • utput to be zero

. . . a0 Operation CarryIn ALU0 Less CarryOut b0 a1 CarryIn ALU1 Less CarryOut b1 Result0 Result1 a2 CarryIn ALU2 Less CarryOut b2 a31 CarryIn ALU31 Less b31 Result2 Result31 . . . . . . . . . Bnegate . . . Ainvert Overflow . . . Set CarryIn . . . . . . Zero Thursday, January 24, 13

slide-80
SLIDE 80

Full ALU

what signals accomplish: Binvert CIn Oper add? sub? and?

  • r?

beq? slt?

ALU a ALU operation b CarryOut Zero Result Overflow Thursday, January 24, 13

slide-81
SLIDE 81

Full ALU

what signals accomplish: Binvert CIn Oper add? sub? and?

  • r?

beq? slt?

2

ALU a ALU operation b CarryOut Zero Result Overflow Thursday, January 24, 13

slide-82
SLIDE 82

Full ALU

what signals accomplish: Binvert CIn Oper add? sub? and?

  • r?

beq? slt?

2 1 1 2

ALU a ALU operation b CarryOut Zero Result Overflow Thursday, January 24, 13

slide-83
SLIDE 83

Full ALU

what signals accomplish: Binvert CIn Oper add? sub? and?

  • r?

beq? slt?

2 1 1 2

ALU a ALU operation b CarryOut Zero Result Overflow Thursday, January 24, 13

slide-84
SLIDE 84

Full ALU

what signals accomplish: Binvert CIn Oper add? sub? and?

  • r?

beq? slt?

2 1 1 2 1

ALU a ALU operation b CarryOut Zero Result Overflow Thursday, January 24, 13

slide-85
SLIDE 85

Full ALU

what signals accomplish: Binvert CIn Oper add? sub? and?

  • r?

beq? slt?

2 1 1 2 1 1 1 2

ALU a ALU operation b CarryOut Zero Result Overflow Thursday, January 24, 13

slide-86
SLIDE 86

Full ALU

what signals accomplish: Binvert CIn Oper add? sub? and?

  • r?

beq? slt?

2 1 1 2 1 1 1 2 1 1 3

ALU a ALU operation b CarryOut Zero Result Overflow Thursday, January 24, 13

slide-87
SLIDE 87

The Disadvantage of Ripple Carry

  • The adder we just built is called a

“Ripple Carry Adder”

  • The carry bit may have to

propagate from LSB to MSB

  • Worst case delay for an N-bit

RC adder: 2N-gate delay A0 B0 1-bit ALU Result0 CarryOut0 A1 B1 1-bit ALU Result1 CarryIn1 CarryOut1 A2 B2 1-bit ALU Result2 CarryIn2 A3 B3 1-bit ALU Result3 CarryIn3 CarryOut3 CarryOut2 CarryIn0

a b CarryIn CarryOut

Thursday, January 24, 13

slide-88
SLIDE 88

The Disadvantage of Ripple Carry

  • The adder we just built is called a

“Ripple Carry Adder”

  • The carry bit may have to

propagate from LSB to MSB

  • Worst case delay for an N-bit

RC adder: 2N-gate delay A0 B0 1-bit ALU Result0 CarryOut0 A1 B1 1-bit ALU Result1 CarryIn1 CarryOut1 A2 B2 1-bit ALU Result2 CarryIn2 A3 B3 1-bit ALU Result3 CarryIn3 CarryOut3 CarryOut2 CarryIn0

a b CarryIn CarryOut

Faster Adders Possible: See C.6

Thursday, January 24, 13

slide-89
SLIDE 89

Teaser!! The ALU in Perspective

Thursday, January 24, 13

slide-90
SLIDE 90

Teaser!! The ALU in Perspective

Thursday, January 24, 13

slide-91
SLIDE 91

Teaser!! The ALU in Perspective Single Cycle CPU Next Time!!

Thursday, January 24, 13