ece 8053 introduction to computer arithmetic (website: course & text content: part 1: number...
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ECE 8053 Introduction to Computer Arithmetic
(Website: http://www.ece.msstate.edu/classes/ece8053/fall_2003/)
Course & Text Content:Part 1: Number RepresentationPart 2: Addition/SubtractionPart 3: MultiplicationPart 4: DivisionPart 5: Real Arithmetic (Floating-Point)Part 6: Function EvaluationPart 7: Implementation Topics
Course Learning ObjectivesComputer Arithmetic students will be able to ...
1. explain the relative merits of number systems used by arithmetic circuits including both fixed- and floating-point number systems
2. demonstrate the use of key acceleration algorithms and hardware for addition/subtraction, multiplication, and division, plus certain functions
3. distinguish between the relative theoretical merits of the different acceleration schemes
Course Learning Objectives(Continued)
4. identify the implementation limitations constraining the speed of acceleration schemes
5. evaluate, design, and optimize arithmetic circuits for low-power
6. evaluate, design, and optimize arithmetic circuits for precision
Course Learning Objectives(Continued)
7. design, simulate, and evaluate an arithmetic circuit using appropriate references including current journal and conference literature
8. write a paper compatible with journal format standards on an arithmetic design
9. make a professional presentation with strong technical content and audience interaction
Importance of Computer Arithmetic3.2 GHz Pentium has a clock cycle of 0.31 ns
Can one integer addition be done < 0.31 ns in execution stage ofPipeline?
What if you had to build a 32-bit adder – ripple carry and a gate delay was approximately 0.1 ns?
STEP 11101
1110 11011
-Note: added from right to left. Why?
Ripple-carry Structure
STEP 2 – Design a circuit
c32 z31 z1 z0
c31 c2 c1 c0=0
x31 y31 x1 y1 x0 y0
– Each box is a full-adder– Implement it
Full-Adder Implementationx y cin z cout 0 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
cin
00 01 11 10
0
1
0 1 0 1
1 0 1 0
xy
cin
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!!!!
Number SystemsRoman 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-valued symbol, value is additive XX=10+10=20 CXX=100+10+10=120• If symbol is repeated or lies to the left of a higher-valued symbol, value is subtractive XXC = - (10+10) + 100 = 80 XLVIII = -(10) + 50 + 5 + 3 = 48
Weighted Positional Number System
Example: Arabic Number System (first used by Chinese)
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
Addition Paradigms• right to left serial 1 147865 +30921 178786
• right to left, parallel 147865 +30921 177786 001000 178786
•random 461325 147865 +30921 177786 001000 178786
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
11 2 2 1 0
1 2 2 1 00
2 2 2 2 2 2n
n n in n i
i
x x x x x x
• This is a radix-polynomial form
Number System• A Number System is defined if the following 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 1 0 0 0 1 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 number
largest representable number
range of representable numbers[-inclusive; (-exclusive interval bounds
•When ALU produces a result >Xmax or <Xmin, incorrect result occurs
•ALU should produce an error signal
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 13
0110 6
1 0011 19
X
Y 19(mod16) 3
Answer in register is 00112=310
Overflow
Machine Representations• Most familiar number systems are:1. nonredundant – every value is uniquely represented
by a radix polynomial2. 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
Example
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
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 part
k – 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 FactorsFixed point arithmetic can utilize scaling factors to adjust radix point position
a – scaling factor
2
( )aX aY a X Y
aX aY a XY
aX X
aY Y
no correction
must divide by a
must multiply by a
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
Example98.67510
1 ulp = 110-3=0.001
1 ulp is the smallest amount a fixed point number may increase or decrease
ulp = 1
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
Radix Conversion 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 x
x x x x x
x
1 2 2 1
0
{[( ) ] }k d k d dQ x x x x
Desired 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 ExampleXI = 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
1
1 12 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 x
X P P
P x
P x x
Radix Conversion ExampleXI = 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 )
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