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BITS Pilani presentation

D. PowarLecturer,

BITS-Pilani, Hyderabad Campus

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BITS PilaniHyderabad Campus

CS F342

Computer Architecture

Lecture -11

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Last lecture review

Computer Arithmetic

9/19/2013 3

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Division

Unsigned division

Signed division

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Division of Unsigned Binary Integers

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Flowchart for Unsigned Binary Division 

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Bits of dividend are examined from L to R until the set of

bits examined represents a number greater than or equal

to the divisor

 Until this event occurs, 0s are placed in the quotient

 When the event occurs, a 1 is placed in the quotient and the

divisor is subtracted from the partial dividend

 This continues in cyclic pattern

 The process stops when all the bits of dividend are exhausted

More complex than multiplication

Algorithm:

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Needs operations like Shift, subtract, restore

Positive numbers: Q = 0111 (Dividend)

M = 0011 (Divisor)

Examples:

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Example: Restoring Twos Complement Division (7/3) 

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Signed division

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When performing (7)÷(-3), remainder is 0001,

But, when performing (-7)÷(3) and (-7)÷(-3) , remainder is 1111.

In general

The remainder is defined by

D=Q*V+R,

i.e., R = D - Q*V

Where

D = dividend

Q = quotient

V = divisor

R = remainder

How signed division is Different??

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1. Convert the operands into unsigned values

2. Compute unsigned division

3. Check the sign’s of reminder and quotient 

4. If negative (reminder or  quotient), take 2’s compliment of the same 

Division Algorithm for Signed Integers

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Example : (7/-3) 

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14

MIPS Division

• Use HI/LO registers for result –  HI: 32-bit remainder

 –  LO: 32-bit quotient

• Instructions– div rs, rt / divu rs, rt

Ex: div $s0,$t0

 –  No overflow or divide-by-0 checking• Software must perform checks if required

 –  Use mfhi, mflo to access result

Ex: mfhi $t0 $t0 <- reminder 

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Floating points arithmetic 

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Numbers with fractions

Could be done in pure binary

1001.1010 = 24 + 20 + 2-3 +2-1 =9.625

We need a way to represent

 – Numbers with fractions, e.g., 3.1416

 – Very small numbers,• Mass of electron: 9.1 X 10-28 gm

 – Very large numbers,• Distance (pluto – sun) : 5.9 X 1012 m

Floating points representation 

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m x r em : mantissa

r : radix [ can be 2,8,16,10]

e : exponent 

The number 537.25 is represented in a registerwith m = 53725 and e = 3

Interpreted as

Point is actually fixed between sign bit and bodyof mantissa

Exponent indicates place value (point position)

Floating Point 

.53725x103 

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 Exponent can be represented in Signedmagnitude, Signed 2’s complement, Signed 1’s

complement or biased form.

Biased representation: In this representation the sign bit is removed from being

a separate entity.

Bias is a +ve number that is added to each exponent as

the floating point number is formed, so that internally allexponent are +ve.

Biased representation is used in many computers.

Exponent representation:

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 Consider an exponent that ranges from -50 to 49.

Internally, it is represented by two digits (without sign) by

adding to it a bias of 50.

The exponent register contains the no. e+50, where ‘e’ is

the actual exponent.

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To simplify operations on floating-point numbers, it istypically required that they have to be normalized.

 A normalized number is one in which the most

significant digit of the significand is nonzero 

Floating Point numbers are usually normalizedi.e. exponent is adjusted so that leading bit(MSB) of mantissa is non zero.

Ex: .0001580x102 (Normalized number is : 0.1580x10-1)

Ex: 0.000123 (Normalized number is : 0.123x10-3),here ".123" is called a normalized mantissa, "-3"is called the exponent .

Normalization 

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For binary numbers, a normalized number is thereforeone in which the most significant bit of the significand is

one.

The typical convention is that there is one bit to the left of

the radix point. Thus, a normalized nonzero number isone in the form:

Because the most significant bit is always one, it is

unnecessary to store this bit; rather, it is implicit

Zero cannot be normalized 

Normalization

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The leftmost bit stores the sign of the number (1:negative, 0:positive)

IEEE-754 Floating Point Numbers

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Bias equals, where k: no: of bits in exponent

The range of true exponent values are: -126 to 127

The final portion is significand /mantissa

Single precision

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Examples: 32 bit (SP)