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 CSE 246

Agenda o o o Administration Motivation Lecture 1: Numbers CSE 246 2

Administration o o o o Textbook: Computer Arithmetic Algorithms and Hardware Designs, Behrooz Parhami, Oxford Recommended: Art of Computer Programming, Volume 2, Seminumerical Algorithms (3 rd 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 CSE 246 3

Administration o No classes on the following days n n n CSE 246 Tu 10/17 BIBE Tu 10/24 EPEP Tu 11/7 ICCAD 4

Administration o o Grading: n Homework – 20% n Midterm – 35% n Project o Report – 25% o Presentation – 20% Midterm: Thursday, 10/2/06 CSE 246 5

Administration o Potential project samples: n n n CSE 246 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 6

Agenda o o o Administration Motivation Lecture 1: Numbers CSE 246 7

Motivation Why do we care about arithmetic algorithms and hardware design? o Classic problems – well defined o Advancements will have a huge impact n o Solutions will be widely used New paradigms n n n CSE 246 Interconnect effects: clock, control, bus, signal Low power designs Wider bit width Reliability centric designs FPGAs and nano technologies 8

Motivation o o Should a new business focus on building market or new technology? New technology: a market will be built around new technology CSE 246 9

Topics o Numbers n o Addition/Subtraction n o o Binary numbers, negative numbers, redundant numbers, residual numbers Prefix adders (zero deficiency) Multiplication/Division Floating point operations Functions: (sqrt), log, exp, CORDIC Optimization, analysis, fault tolerance CSE 246 10

Other Topics o Potential focus on the following topics: n n n CSE 246 Power reduction Interconnect FPGAs 11

Goals/Background o Why do you want to take this class? What would you like to learn? n n n CSE 246 Fulfill course requirement Hardware Software Work Research Curiosity 12

Agenda o o o Administration Motivation Lecture 1: Numbers CSE 246 13

Numbers o Special Symbols n n 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 CSE 246 14

Numbers o Position Symbols n The value depends on the position of the number For example: 125 = 100 + 20 + 5 One 100, Two 10 s, and Five 1 s Another example: 1 hour, 3 minutes o n CSE 246 Positional systems includes radixes: 2, -2, 2, 2 j (imaginary) 15

Numbers o Summation of positional numbers n Given: Value is: n For example: n o Consider 4 -2 1 0 0 1 1 CSE 246 (where y is the base) 0 1 0 1 0 0 1 1 0 1 -2 -1 4 5 2 3 o o Note that position systems provide a complete range of numbers (e. g. – 2 to 5) 16

Numbers (Radix Conversion) (xk-1, …, x 0. X-1, … , x-l)r=(XK-1, …, X 0. X-1, …, X-L)R o o o Repeatedly divide the integer by (R)r Repeatedly multiply the fraction by (R)r Example n CSE 246 (201. 31)10=(13001. 123)5 17

Numbers o o Avoid Division (Montgomory System) Simplify Mod operation n mod r-1, mod r+1 Example: 29110 mod 9 = 2+9+1 mod 9 = 12 mod 9 = 3 29110 mod 11 = 2 -9+1 mod 11 = -6 mod 11 = 5 CSE 246 18

Signed Numbers o o o Biased numbers Signed Bit Complementary representation n Positive number: x (mod p) Negative number: (M-x) (mod p) (Note: mod p is added implicitly) n One’s complement n 0 0 CSE 246 0 1 1 1 n 0 1 -1 -0 Two’s complement M=2 n-1 0 0 0 1 Flip each bit 1 1 0 1 -2 -1 M=2 n Flip each bit + 1 Two’s complement can be used for subtraction 19

Signed Numbers o Two’s complement subtraction: n o (M-x+M-y) mod M = M-(x+y) Two’s complement conversion: n n CSE 246 Positive number: To negative: 20

Signed Numbers o Two’s complement Example: 0 0 1 1 0 1 0 1 Proof as follows: 0 1 2 3 -4 -3 -2 -1 Which leads to: CSE 246 21

Next time o Talk about redundant numbers CSE 246 22