Lectures 3:

1
COE 308

• Lectures 3
• Arithmetic for Computers Application to the
  MIPS Architecture

2
Why Arithmetics

• Microprocessors perform
• Data Movement
• Additions
• Subtractions
• Logical Operations
• Comparisons

Arithmetic Operations

• Understand Arithmetic for the Computer
• Algorithms
• Architectures
• Circuits

Goal
3
Binary Numbers
N dw-1Bw-1 diBi d1B1 d0B0

• Base B is Ten ? Decimal Number
• Base B is Two ? Binary Number

In MIPS, 32 bits words 0000 0000 0000 0000 0000
0000 0000 0011two 3ten 0000 0000 0000
0000 0000 0000 0001 0101two
21ten ....... 1111 1111 1111 1111 1111 1111 1111
1110two 4,294,967,294ten 1111 1111 1111 1111
1111 1111 1111 1111two 4,294,967,295ten
4
ACSII or Decimal vs Binary Numbers

• Attractive because Human- Friendly
  representation
• Non attractive to computers
• Understand only binary
• Binary operations easy to perform
• Does not know how to perform operations on ASCII
  or Decimal number.
• Need to convert ASCII/Decimal to binary before
  and after operation
• Used only for I/O (User Interface)

Use Binary Representation For Internal
Computations
Convert to/from ASCII for I/O Only
5
Number Representation

• Numbers in computers
• Just a representation of real numbers
• Real numbers have an infinite number of digits

153ten ..000000000010011001two
0000 0000 0000 0000 0000 0000 1001 1001two
in a 32 bits word
Add, Subtract, Multiply, Divide
Result bigger than number of available slots
(bits)
Overflow
6
Signed Numbers

• Subtraction of a big number from small number is
  a negative number

Need for Signed Numbers Representation
Intuitive Representation Sign Magnitude
Sign
Magnitude

• Many Shortcomings
• Where to put the Sign bit? Left? Right?
• Extra step to set the sign for addition and
  subtraction
• Positive and negative 0 leads to programming
  problems

7
Twos Complement
What is the result of A B when B gtgt A?
Will Try to borrow from a string of leading 0s.
Result will have a string of leading 1s.
Leading 0s Positive number Leading 1s Negative
number

• Imbalance between the number of negative numbers
  and the number of positive numbers
• Not a worry anymore.
• Used by ALL Computers today
• Easy recognition of positive/negative numbers
• Easy arithmetics

8
Negation and Sign Extension

• Twos complement negation
• Invert all bits of number one by one
• Add the value 1
• Coming from x x -1 (0 (-1) -1)
• Means x x 1
• Twos Complement Sign Extension
• Copy the Most Significant Bit to the left
• Coming from the fact that
• Positive numbers have an infinite number of 0s to
  the left
• Negative numbers have an infinite number of 1s to
  the left

9
Addition

• Straight Forward Operation
• Adding two number 6 and 7

0000 0000 0000 0000 0000 0000 0000 0110two
6ten 0000 0000 0000 0000 0000 0000 0000 0111two
7ten 0000 0000 0000 0000 0000 0000 0000 1101two
13ten

(0) 0 0
(0) 0 0
(1) 0 0
(1) 1 1
(0) 1 1
0 1


(0)0 (0)0 (0)0 (1)1 (1)1 (0)1

• Overflow can occur
• MIPS instructions add, addi, addu, addui

10
Subtraction

• Subtract A from B
• Add A to Twos Complement of B

0000 0000 0000 0000 0000 0000 0000 0111two
7ten 1111 1111 1111 1111 1111 1111 1111 1010two
-6ten 0000 0000 0000 0000 0000 0000 0000
0001two 1ten

• Result either gt0, 0 or lt0
• Overflow can occur
• MIPS instructions sub, subi, subu, subiu

11
Overflow Condition

• Addition is operation between A and B (any sign)
• Subtraction is addition of twos complement
• Overflow in addition/subtraction
• 33rd bit set with the value of result instead of
  the proper sign

Operation Operand A Operand B Result
AB gt 0 gt 0 lt 0
AB lt 0 lt 0 gt 0
A-B gt 0 lt 0 lt 0
A-B lt 0 gt 0 gt 0

• Add (add), add immediate (addi) and subtract
  (sub) cause exceptions on overflow
• Add unsigned(addu) , add immediate
  unsigned(addiu) and subtract unsigned (subu) do
  NOT cause exceptions on overflow

12
Logical Operations

• And Bit to bit logical AND
• Used to Clear specified bits
• Or Bit to Bit logical OR
• Used to Set specified bits
• Shift Operations
• Many uses Many algorithms, available in some
  high level languages like C/C
• SLL, SLR

Logical Operations C Operators MIPS Instruction
Shift Left ltlt sll
Shift Right gtgt slr
Bit-by-bit AND and, andi
Bit-by-bit OR or, ori