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King Fahd University of Petroleum and MineralsKing Fahd University of Petroleum and Minerals

Computer Engineering Department


College of ComputerScience And Engineering


Dr. A. Bouhraoua Arithmetic for Computers 1

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Lectures 3:Arithmetic for Computers – Application

to the MIPS Architecture

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Why Arithmetics• Microprocessors perform:

– Data Movement– Additions– Subtractions– Logical Operations– Comparisons

Arithmetic Operations

Understand Arithmetic for the Computer• Algorithms• Architectures• Circuits

Goal

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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 words0000 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

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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

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Number Representation• Numbers in computers

– Just a representation of real numbers – Real numbers have an infinite number of digits

153ten = …..00000……………………0000010011001two

= 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

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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

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Two’s Complement

What is the result of A – B when B >> A?Will Try to borrow from a string of leading 0s. Result will have a string of leading 1s.

Leading 0s: Positive numberLeading 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

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Negation and Sign Extension

• Two’s 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

• Two’s 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

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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

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Subtraction• Subtract A from B

• Add A to Two’s 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 >0, 0 or <0• Overflow can occur• MIPS instructions: sub, subi, subu, subiu

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Overflow Condition• Addition is operation between A and B (any sign)• Subtraction is addition of two’s complement• Overflow in addition/subtraction

– 33rd bit set with the value of result instead of the proper sign

Operation Operand A Operand B Result

A+B > 0 > 0 < 0

A+B < 0 < 0 > 0

A-B > 0 < 0 < 0

A-B < 0 > 0 > 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

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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 << sll

Shift Right >> slr

Bit-by-bit AND & and, andi

Bit-by-bit OR | or, ori

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