CSE 246: Computer Arithmetic Algorithms and Hardware Design

CSE 246: Computer Arithmetic Algorithms and Hardware Design Winter 2005 Lecture 1: Numbers Instructor: Prof. Chung-Kuan Cheng

Agenda • Administrative • Motivation • Lecture 1: Numbers

Administrative • Textbook: Computer Arithmetic Algorithms, Israel Koren, 2nd Edition, Published by AK Peters Natick, Massachusetts • Recommended: Art of Computer Programming, Volume 2, Seminumerical Algorithms (3rd Edition), Donald E. Knuth • In addition: set of papers to read

Administrative • Grading: • Homework – 20% • Midterm – 35% • Project • Report – 25% • Presentation – 20% • Midterm: Tuesday, Feb. 8th • Homework 1: • Due: 1/18/05

Administrative • Potential project samples: • Design interconnect and switch modules • Use FPGAs, nano technologies for add/sub • Design reconfigurable blocks • Design Low power adder, multiplier • Invent Low power/reliable number systems

Motivation Why do we care about arithmetic algorithms and hardware design? • Classic problems – well defined • Advancements will have a huge impact • Solutions will be widely used • New paradigm • Interconnect dominated: clock, control, bus, signal • Power driven • Reliability centric • FPGAs

Motivation • Should a new business focus on building market or new technology? • New technology: a market will be built around new technology

Motivation • What if we had a 10GHz chip that was 2 cm x 2 cm? • It takes 2 clock cycles (time of flight) to get from one end of the chip to the other • How would the clock be distributed? • What if the electrical input is 1 Volt/100 Watts? How do we get 100 Amps through the chip?

Topics • Numbers • Binary numbers, negative numbers, redundant numbers, residual numbers • Addition/Subtraction • Prefix adders (zero deficiency) • Multiplication/Division • Floating point operations • Functions: (sqrt),log, exp, CORDIC • Optimization, analysis, fault tolerance

Other Topics • Potential focus on the following topics: • Power reduction • Interconnect • FPGAs

Goals/Background • Why do you want to take this class? What would you like to learn? • Fulfill course requirement • Hardware • Software • Work • Research • Curiosity

Numbers • Special Symbols • Symbols used to represent a value • Roman Numerals 1 = I 100 = C 5 = V 500 = D 10 = X 1000 = M 50 = L For example: 2004 = MMIV

Numbers • Position Symbols • The value depends on the position of the number • For example: 125 = 100 + 20 + 5 One 100, Two 10s, and Five 1s Another example: 1 hour, 3 minutes • Positional systems includes radixes: 2, -2, 2, 2j (imaginary)

Numbers • Summation of positional numbers • Given: • Value is: (where y is the base) • For example: • Consider • Note that position systems provide a complete range of numbers (e.g. –2 to 5)

Signed Numbers • Biased numbers • Signed Bit • Complementary representation • Positive number: x (mod p) • Negative number: (M-x) (mod p) (Note: mod p is added implicitly) • One’s complement Two’s complement M=2n-1 M=2n Flip each bit Flip each bit + 1 • Two’s complement can be used for subtraction

Signed Numbers • Two’s complement subtraction: • (M-x+M-y) mod M = M-(x+y) • Two’s complement conversion: • Positive number: • To negative:

Signed Numbers • Two’s complement Proof as follows: Which leads to: Example:

Next time • Talk about redundant numbers

CSE 246: Computer Arithmetic Algorithms and Hardware Design