1. CS267 L13 Floating Point.1 Demmel Sp 1999 CS 267 Applications of Parallel Computers Lecture 13: Floating Point Arithmetic James Demmel http://www.cs.berkeley.edu/~demmel/cs267_Spr99

2. CS267 L13 Floating Point.2 Demmel Sp 1999 Outline °History °IEEE Floating point formats °Error analysis °Ex: Polynomial evaluation °Extra precision °Cray arithmetic - a pathological example °Exception Handling °Using Exception Handling to go faster °Heterogeneous computing

3. CS267 L13 Floating Point.3 Demmel Sp 1999 History °In the beginning was darkness… °ACM Turing Award 1989 to W. Kahan for design of the IEEE Floating Point Standard

4. CS267 L13 Floating Point.4 Demmel Sp 1999 IEEE Floating Point Formats °Scientific notation in decimal or binary °List of formats, sizes ranges °Definitions (macheps) °What is an exception?

5. CS267 L13 Floating Point.5 Demmel Sp 1999 Error Analysis °Basic error formula °Example: adding n numbers °Def of Numerical stability Ex: log(1-x)/x? (too complicated)

6. CS267 L13 Floating Point.6 Demmel Sp 1999 Example: polynomial evaluation °Picture from ma221 text, error bounds

7. CS267 L13 Floating Point.7 Demmel Sp 1999 Simulating extra precision °Priest/Shewchuk

8. CS267 L13 Floating Point.8 Demmel Sp 1999 Cray Arithmetic °History °Impact: D&C code? °Impact: x/sqrt(x^2+y^2)

9. CS267 L13 Floating Point.9 Demmel Sp 1999 Exception Handling °List 5 cases, responses

10. CS267 L13 Floating Point.10 Demmel Sp 1999 Using Exception Handling to Design Faster Algorithms °Sturm Sequence loops °comparison of triangular solve

11. CS267 L13 Floating Point.11 Demmel Sp 1999 Hazards of Heterogeneous Computing °Subnormals °Multiply-accumulate °Cray/IEEE °Compiler differences °Repeatability/debugging/nonassociativity