numbers and arithmetic · dr. shadrokh samavi 5 course learning objectives 1. explain the relative...

‹#›Dr. Shadrokh Samavi 1

Numbers and Arithmetic


Computer Arithmetic

Website: http://ece.iut.ac.ir/faculty/samavi/computer_arithmetic.htm

Text: Parhami, Behrooz, “Computer Arithmetic: Algorithms and Hardware Designs” Oxford University Press, 2000. http://www.ece.ucsb.edu/Faculty/Parhami/text_comp_arit.htm

Part 1: Number RepresentationPart 2: Addition/SubtractionPart 3: MultiplicationPart 4: DivisionPart 5: Real Arithmetic (Floating-Point)Part 6: Function EvaluationPart 7: Implementation Topics

Slides are intended to illustrate the content of Parhami’s book.


Part I Number Representation

1 Numbers and Arithmetic

2 Representing Signed Numbers

3 Redundant Number Systems

4 Residue Number Systems

Part II Addition/Subtraction

5 Basic Addition and Counting

6 Carry-Lookahead Adders

7 Variations in Fast Adders

8 Multioperand Addition

Part III Multiplication

9 Basic Multiplication Schemes

10 High-Radix Multipliers

11 Tree and Array Multipliers

12 Variations in Multipliers


Part IV Division

13 Basic Division Schemes

14 High-Radix Dividers

15 Variations in Dividers

16 Division by Convergence

Part V Real Arithmetic

17 Floating-Point Representations

18 Floating-Point Operations

19 Errors and Error Control

20 Precise and Certifiable Arithmetic

Part VI Function Evaluation

21 Square-Rooting Methods

22 The CORDIC Algorithms

23 Variations in Function Evaluation

24 Arithmetic by Table Lookup

Part VII Implementation Topics

25 High-Throughput Arithmetic

26 Low-Power Arithmetic

27 Fault-Tolerant Arithmetic

28 Past, Present, and Future


Course Learning Objectives

1. explain the relative merits of number systems usedby arithmetic circuits including both fixed- andfloating-point number systems.

2. demonstrate the use of key acceleration algorithmsand hardware for addition/subtraction, multiplication,and division, plus certain functions.

3. distinguish between the relative theoretical meritsof the different acceleration schemes.


4. identify the implementation limitations constrainingthe speed of acceleration schemes

5. evaluate, design, and optimize arithmetic circuitsfor low-power

6. evaluate, design, and optimize arithmetic circuitsfor precision



7. design, simulate, and evaluate an arithmetic circuitusing appropriate references including current journaland conference literature.

8. write a paper compatible with journal formatstandards on an arithmetic design.

9. make a professional presentation with strongtechnical content and audience interaction.



1.1 What Is Computer Arithmetic?


Pentium Division Bug (1994-95): Pentium’s radix-4 SRTalgorithm occasionally produced an incorrect quotient. Firstnoted in 1994 by T. Nicely who computed sums of reciprocals oftwin primes:

1/5 + 1/7 + 1/11 + 1/13 + . . . +1/p + 1/(p + 2) + . . .

Worst-case example of division error in Pentium:

4 195 8353 145 727

1.333 820 44...1.333 739 06...

c ==Correct quotient

double FLP value;accurate to only 14 bits(worse than single!)


Scope of computer arithmetic.


1.2 A Motivating Example


Patriot Missile battery once failed to intercept an incomingScud missile which killed 28. Reported cause: “softwareproblem” (inaccurate calculation of the time since boot).Specifics of the problem: time in tenths of second asmeasured by the system’s internal clock was multiplied by1/10 to get the time in seconds. Internal registers were24 bits wide

Patriot Missile


1/10 = 0.0001 1001 1001 1001 1001 100 (chopped to 24 b)Error ≅ 0.1100 1100 × 2–23 ≅ 9.5 × 10–8Error in 100-hr operation period:≅ 9.5 × 10–8 × 100 × 60 × 60 × 10 = 0.34 sDistance traveled by Scud = (0.34 s) × (1676 m/s) ≅ 570 mThis put the Scud outside the Patriot’s “range gate”.Ironically, the fact that the bad time calculation had beenimproved in some (but not all) code parts. It meant thatinaccuracies did not cancel out.

Patriot Missile


Importance of Computer Arithmetic

3.2 GHz Pentium has a clock cycle of 0.31 ns. Can oneinteger addition be done < 0.31 ns in execution stageof Pipeline?

What if you had to build a 32-bit adder – ripple carryand a gate delay was approximately 0.1 ns?

11011110

11011-Note: added from right to left.

STEP 1


Ripple-carry Structure

STEP 2 – Design a circuit

– Each box is a full-adder

c32 z31 z1 z0

c31 c2 c1 c0=0

x31 y31 x1 y1 x0 y0


Full-Adder Implementation

x y cin z cout0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 1 1 0 0 1 0 1 0 1 0 1 1 1 0 0 1 1 1 1 1 1

xy

cin00 01 11 10

0

1

0 1 0 1

1 0 1 0

xycin

00 01 11 10

0

1

0 0 1 0

0 1 1 1

in in in in

in

z c x y c x y c x y c x y

c x y

( )out inc c x y xy cout = cinx + ciny + x y


Adder Circuit Analysis

STEP 3 – Analysis

Critical Path is carry chain, 2 gate delays/bit2 32 = 64(64)(0.1ns) = 6.4ns6.4ns >> 0.31nsMust Use Faster Adder and/or Pipeline More!!!!


Addition Paradigms

• right to left serial1

147865+30921178786

• right to left, parallel147865+30921177786001000178786


1.3 Numbers and Their Encodings


Numbers versus their representations (numerals)The number “twenty-seven” can be represented in differentways using numerals or numeration systems:||||| ||||| ||||| ||||| ||||| || sticks or unary code 27 radix-10 ordecimal code (27)ten 11011 radix-2 or binary code (11011)twoXXVII Roman numeralsEncoding of digit sets as binary strings: BCD example

Digit BCD representation0 0 0 0 01 0 0 0 12 0 0 1 03 0 0 1 14 0 1 0 05 0 1 0 16 0 1 1 07 0 1 1 18 1 0 0 09 1 0 0 1


Number Systems: “Roman” Numeral System

Symbolic Digitssymbol value

I 1 V 5 X 10 L 50 C 100 D 500 M 1000

RULES:• If symbol is repeated or lies to

the right of another higher-valuedsymbol, value is additive

XX=10+10=20CXX=100+10+10=120

• If symbol is repeated or lies to theleft of a higher-valued symbol,value is subtractive

XXC = - (10+10) + 100 = 80XLVIII = -(10) + 50 + 5 + 3 = 48


Weighted Positional Number System

Example: “Arabic” Number System

symbol (digit)

value (in 1’s position)

0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9


1.4 Fixed-Radix Positional Number Systems


Binary Number System

• n-ordered sequence:

• each xi{0,1} is a BInary digiT (BIT)

• magnitude of n is important• sequence is a short-hand notation• more precise definition is:

1 2 2 1 0n nx x x x x

• This is a radix-polynomial form


Number SystemA Number System is defined if the followings exist

Example: The binary number system 1.2.3. Addition operator defined by addition table4. Multiplication operator defined by multiplication table

1. A digit set2. A radix or base value3. An addition operation4. A multiplication operation

{0,1}ix 2

+ 0 1 0 0 1 1 1 10

0 10 0 01 0 1


Number System Observations• Cardinality of digit set (2) is equal to radix value• Addition operator XOR, Multiplication is AND

•How many integers exist?Mathematically, there are an infinite number, In computer, finite range due to register length

minX

maxX

min max[ , ]X X

smallest representable numberlargest representable number

range of representable numbers[-inclusive; (-exclusive interval bounds

•When ALU produces a result >Xmax or


Encoding of numbers in 4 bits


Negative Radix Number System


Example•Assume 4-bit registers, unsigned binary numbers

min 2 100000 0X max 2 101111 15X

min max 2 2[ , ] [0000 ,1111 ]X X

max 2 101 10000 16X

max 2 21 [0000 ,1111 ] (mod16)X X

1101 130110 6

1 0011 19

XY 19(mod16) 3

Answer in register is 00112=310Overflow


4 3 1 07 8 6 8 2 8 4 8X

876024X

Since octal is fixed-radix and positional,we can rewrite this value using shorthand notation

Note the importance of the use of 0 to serve as acoefficient of the weight value w2=82

Example


Fixed-Radix Systems

1 2 1 01 2 1 0

11

1

k kk k

km i

m ii m

X x x x x

x x x

Register of length n can represent a number with a fractional part and an integral partk – number of integral digitsm – number of fractional digitsn = k + m

1 2 1 0 1.k k mx x x x x x radix point

A programmer can use an implied radix point in any position


Scaling Factors

Fixed point arithmetic can utilize scaling factors to adjust radix point position

a – scaling factor

2

( )aX aY a X YaX aY a XY

aX XaY Y


Unit in the Last Position (ulp)

• Given w0=r-m and n, the position of the radix point is determined

• Simpler to disregard position of the radix point in fixed point by using unit in least (significant) position ulp

For fractionsFor integers

mulp r

Example98.675101 ulp = 110-3=0.001

1 ulp is the smallest amount a fixed point number may increase or decrease

ulp = 1


1.5 Number Radix Conversion


Converting whole part w: (105)ten = (?)fiveRepeatedly divide by five Quotient Remainder

105 0 21 1 4 4 0

Therefore, (105)ten = (410)five

Converting fractional part v: (105.486)ten = (410.?) Repeatedly multiply by five Whole Part Fraction

.486 2 .430 2 .150 0 .750 3 .750 3 .750

Therefore, (105.486)ten ≅ (410.22033)five = w.v

Radix Conversion


Horner’s rule used to convert (22033)five to decimal.

Radix Conversion


Horner’s rule used to convert (0.22033)five to decimal.

Radix Conversion


Radix Conversion

Given a number in old radix r, conversion to the new radix R representation

- can be accomplished doing the arithmetic in the old or new radix

- old and new representations may not be exactly equal


Given a value X represented in source system with radix s, represent the same number in a destination system with radix d

Consider the integral part of the number, XI, in the d system1 2 1 0

1 2 1 0

1 2 2 1 0{[( ) ] }0

k kI k d k d d d

k d k d d d

i d

X x x x xx x x x x

x

1 2 2 1

0

{[( ) ] }k d k d dQ x x x xDesired LSD x

Can Repeatedly Divide to Obtain Converted Value

Consider the integral part of the number, XI, in the d system

If XI is divided by d , we obtain x0 as a remainder and quotient

Radix Conversion


Radix Conversion Example

XI = 34610 s=10 d =3

5 4 3 2 1 0

10 10

1 3 1 3 0 3 2 3 1 3 1 3[243 81 18 3 1] 346

Fixed-point Decimal to Ternary Integer Conversion, (arithmetic in old radix)

Check by evaluating the radix polynomial

XI = 1102113


Radix Conversion (fractional)

Consider the fractional part of the value in d Fixed point system

Thus, PI is the desired digitWe can repeatedly multiply by the d value

1 2 ( 1)1 2 ( 1)

1 1 11 2 3

11 1

2 3

{ [ ( )]}

[ ( )]

m mF d d m d m d

d d d

d F I F

I

F d d

X x x x x

x x xX P PP x

P x x


XI = 0.29110 s=10 d =5

0.291 5 1.455 1

Fixed-point Decimal to Pentary Fractional Conversion(arithmetic in old radix)

0.29110 is Finite Fraction for s=10, but infinite fraction for d =5

0.455 5 2.275 2 0.275 5 1.375 1 0.375 5 1.875 1 0.875 5 4.375 4 0.375 5 1.875 1 0.875 5 4.375 4

10 50.291 (0.12114141414 )

Radix Conversion Example


1.6 Classes of Number Representations


1. Integers (fixed-point), unsigned & signed

2. Signed-magnitude, biased, complement

3. Signed-digit (redundant numbers )

4. Residue number system: (RNS)

5. Real numbers, floating-point

6. Real numbers, exact

Classes of Number Representations


Machine RepresentationsMost familiar number systems are:

1. Non-redundant – every value is uniquely represented by a radix polynomial

2. Weighted – sequence of weights

determines the value of the n-tuple formed from the digit set

3. Positional – wi depends only on position i4. Conventional number systems

where ß is a constant. These are fixed-radix systems.

1 2 2 1 0, , , , ,n nw w w w w

1 2 2 1 0, , , , ,n nx x x x x 1

0

n

i ii

X x w

iiw

numbers and arithmetic · dr. shadrokh samavi 5 course learning objectives 1. explain the relative...

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