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_2002/)
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
1.7GHz Pentium has a clock cycle of 0.59 ns 1
integer addition lt 0.59 ns in execution stage of
P/L
Assume a 4-bit adder ripple carry, 0.2 ns gate
delay
STEP 1 1101 1110 11011
-Note added from right to left. Why?
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Ripple-carry Structure
STEP 2 Design a circuit
x3
y3
x2
y2
x1
y1
x0
y0
c4
c3
c3
c3
c00
z4
z3
z2
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
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Adder Circuit Analysis
STEP 3 Analysis
Critical Path is 3 gates 3?412 (12)(0.2ns)2.4ns
2.4ns gt 0.59ns Must Use Faster Adder!!!!
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Roman 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
  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 nkm
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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Scaling Factor Example
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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 ulp

Example
98.67510
1 ulp 1?10-30.001
1 ulp is the smallest amount a fixed point number
may increase or decrease