ECE 8053 Introduction to Computer Arithmetic

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ECE 8053 Introduction to Computer Arithmetic
(Website http//www.ece.msstate.edu/classes/ece80
53/fall_2003/)
Course Text Content Part 1 Number
Representation Part 2 Addition/Subtraction Part
3 Multiplication Part 4 Division Part 5 Real
Arithmetic (Floating-Point) Part 6 Function
Evaluation Part 7 Implementation Topics
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Course Learning Objectives
Computer 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
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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
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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
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Importance of Computer Arithmetic
3.2 GHz Pentium has a clock cycle of 0.31 ns Can
one integer addition be done lt 0.31 ns in
execution stage of Pipeline?
What if you had to build a 32-bit adder ripple
carry and a gate delay was approximately 0.1 ns?
STEP 1 1101 1110 11011
-Note added from right to left. Why?
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Ripple-carry Structure
STEP 2 Design a circuit
x31
y31
x1
y1
x0
y0
c31
c2
c1
c00
c32
z31
z1
z0
Each box is a full-adder Implement it
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Full-Adder Implementation
xy
00
01
11
10
cin
0
0
1
0
1
1
1
0
1
0
xy
00
01
11
10
cin
0
0
0
1
0
1
0
1
1
1
cout cinx ciny x y
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Adder Circuit Analysis
STEP 3 Analysis
Critical Path is carry chain, 2 gate delays/bit 2
? 32 64 (64)(0.1ns) 6.4ns 6.4ns gtgt
0.31ns Must Use Faster Adder and/or Pipeline
More!!!!
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Number SystemsRoman Numeral System
Symbolic Digits

• RULES
• If symbol is repeated or lies to the right
  of another higher-valued symbol, value is
  additive
• XX101020
• CXX1001010120
• If symbol is repeated or lies to the left of
  a higher-valued symbol, value is subtractive
• XXC - (1010) 100 80
• XLVIII -(10) 50 5 3 48

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Weighted Positional Number System
Example Arabic Number System (first used by
Chinese)
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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

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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

• This is a radix-polynomial form

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Number System

• A Number System is defined if the following exist

• A digit set
• A radix or base value
• An addition operation
• A multiplication operation

• Example The binary number system

• Addition operator defined by addition table
• Multiplication operator defined by multiplication
  table

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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

smallest representable number
largest representable number
range of representable numbers-inclusive
(-exclusive interval bounds

• When ALU produces a result gtXmax or ltXmin,
  incorrect result occurs
• ALU should produce an error signal

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Example

• Assume 4-bit registers, unsigned binary numbers

Answer in register is 00112310
Overflow
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Machine Representations

• Most familiar number systems are
• nonredundant every value is uniquely
  represented by a radix polynomial
• weighted sequence of weights determines the
  value of the n-tuple formed from the digit
  set
• positional wi depends only on position i
• conventional number systemswhere ß is a
  constant. These are fixed-radix systems.

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Example
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
a coefficient of the weight value w282
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Fixed-Radix Systems
Register of length n can represent a number with
a fractional part and an integral part
k number of integral digits m number of
fractional digits n k m
radix point
A programmer can use an implied radix point in
any position
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Scaling Factors
Fixed point arithmetic can utilize scaling
factors to adjust radix point position
a scaling factor
no correction
must divide by a
must multiply by a
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Unit in the Last Position (ulp)

• Given w0r-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 fractions
• For integers

ulp 1
Example
98.67510
1 ulp 1?10-30.001
1 ulp is the smallest amount a fixed point number
may increase or decrease
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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
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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 system
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
Can Repeatedly Divide to Obtain Converted Value
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Radix Conversion Example
XI 34610 ?s10
?d 3
Fixed-point Decimal to Ternary Integer
Conversion, (arithmetic in old radix)
XI 1102113
Check by evaluating the radix polynomial
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Radix Conversion (fractional)
Consider the fractional part of the value in ?d
Fixed point system
Thus, PI is the desired digit We can repeatedly
multiply by the ?d value
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Radix Conversion Example
XI 0.29110 ?s10
?d 5
Fixed-point Decimal to Pentary Fractional
Conversion (arithmetic in old radix)
0.29110 is Finite Fraction for ?s10, but
infinite fraction for ?d 5