1. Detecting and Exploiting Narrow Bitwidth Computations Mihai Budiu Carnegie Mellon University mihaib@cs.cmu.edu joint work with Seth Copen Goldstein

2. 10/02/99Mihai Budiu -- Narrowing bitwidths - - SOCS2 2 Motivation: applications Media processing Digital Signal Processing FFT

3. 10/02/99Mihai Budiu -- Narrowing bitwidths - - SOCS2 3 “MMX” CPU support for narrow widths Reconfigurable hardware Motivation: hardware + ++++ (a & 0xf) | (b & 0x18) ba

4. 10/02/99Mihai Budiu -- Narrowing bitwidths - - SOCS2 4 No programming language support No compiler support Motivation int a; long b; int a; a = (a >> 16) & 0xf0;

5. 10/02/99Mihai Budiu -- Narrowing bitwidths - - SOCS2 5 Types The type of an integer value describes: sign (signed or unsigned) width in bits information about each bit –bit is constant ( 0 or 1 ) –bit is don’t care ( x ) –cannot predict bit -- don’t know ( u )

6. 10/02/99Mihai Budiu -- Narrowing bitwidths - - SOCS2 6 Why? You don’t have to implement: –don’t care bits –constant bits This may enable you to exploit hardware more efficiently, increasing performance

7. 10/02/99Mihai Budiu -- Narrowing bitwidths - - SOCS2 7 The Type Inference 1. Sign inference 2. Width inference (forward) 3. Width inference (backward) 4. Constant propagation (forward) 5. Don’t care discovery (backward)

8. 10/02/99Mihai Budiu -- Narrowing bitwidths - - SOCS2 8 The Type Inference Algorithms (from 10000 feet) Forms of abstract interpretation I/O values are assigned types conservatively Iterate until a fixed-point is reached “Data-flow at the bit level” Low complexity O(n^2 w).

9. 10/02/99Mihai Budiu -- Narrowing bitwidths - - SOCS2 9 u0uuu + u00uu u001u Constant Propagation

10. 10/02/99Mihai Budiu -- Narrowing bitwidths - - SOCS2 10 Don’t Care Propagation + xxuuu xxuux xxuuu lsb out bit from lsb in bit msb out bit from all in bits In Out

11. 10/02/99Mihai Budiu -- Narrowing bitwidths - - SOCS2 11 Mixing + xxuux xx0uu xx01u The mixing phase just takes bitwise the best of the information provided by the two propagations. u00uu u001u xxuuu xxuux u0uuu

12. 10/02/99Mihai Budiu -- Narrowing bitwidths - - SOCS2 12 Circuit Reduction for Reconfigurable Hardware

13. 10/02/99Mihai Budiu -- Narrowing bitwidths - - SOCS2 13 Circuit Reduction for Reconfigurable Hardware -- after synthesis --

14. 10/02/99Mihai Budiu -- Narrowing bitwidths - - SOCS2 14 C Program Size Reduction G721 from Mediabench

15. 10/02/99Mihai Budiu -- Narrowing bitwidths - - SOCS2 15 Conclusions Finding useless bits in a computation as a type inference problem Efficient algorithms exhibited (data-flow analysis) Big performance gain for reconfigurable hardware benchmarks (up to 95%) Effective even for C programs (e.g. 7% dynamic size reduction)