CSE 246: Computer Arithmetic Algorithms and Hardware Design

CSE 246: Computer Arithmetic Algorithms and Hardware Design Fall 2006 Lecture 1: Introduction and Numbers Instructor: Prof. Chung-Kuan Cheng

Agenda • Administration • Motivation • Lecture 1: Numbers

Administration • Textbook: Computer Arithmetic Algorithms and Hardware Designs, Behrooz Parhami, Oxford • Recommended: Art of Computer Programming, Volume 2, Seminumerical Algorithms (3rd Edition), Donald E. Knuth • Numerical Computing with IEEE Floating Point Arithmetic, Michael L. Overton, SIAM • Computer Arithmetic Algorithms, Israel Koren, A K Peters, Natick, Massachusetts • Digital Arithmetic, Milos D. Ercegovac and Tomas Lang, Morgan Kaufmann • CMOS VLSI Design, Neil H.E. Weste and David Harris, Addison Wesley • Principles and Practices of Interconnection Networks, William James Dally and Brian Towles, Morgan Kaufmann • In addition: set of papers to read

Administration • No classes on the following days • Tu 10/17 BIBE • Tu 10/24 EPEP • Tu 11/7 ICCAD

Administration • Grading: • Homework – 20% • Midterm – 35% • Project • Report – 25% • Presentation – 20% • Midterm: Thursday, 10/2/06

Administration • Potential project samples: • Interconnect and switch modules • Data path components using FPGAs, nano technologies • Low power logic styles • Reconfigurable blocks • Low power data path components • Low power/reliable coding 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 paradigms • Interconnect effects: clock, control, bus, signal • Low power designs • Wider bit width • Reliability centric designs • FPGAs and nano technologies

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

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)

Numbers (Radix Conversion) • Repeatedly divide the integer by (R)r • Repeatedly multiply the fraction by (R)r • Example • (201.31)10=(13001.123)5 (xk-1, …, x0 . X-1, … , x-l)r=(XK-1, …, X0 . X-1, …, X-L)R

Numbers • Avoid Division (Montgomory System) • Simplify Mod operation • mod r-1, mod r+1 Example: 29110 mod 9 = 2+9+1 mod 9 = 12 mod 9 = 3 29110 mod 11 = 2-9+1mod 11 = -6 mod 11 = 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