CSC 4250 Computer Architectures

1. CSC 4250Computer Architectures September 12, 2006Appendix H. Computer Arithmetic

2. Five Real Stories on Computer Arithmetic • Insufficient Accuracy • Truncation Errors • Numerical Overflow • Divide by Zero • Wrong Units of Measurement

3. 1. Insufficient Accuracy • First Iraq War • Feb 25, 1991; Dhahran, Saudi Arabia • Iraqi Scud Missile • American Patriot Missile failed to intercept • 28 American soldiers died in the attack: • 20% of the 148 total deaths

4. Fixed Point Data • Patriot system contains an internal clock, a counter that is incremented every 0.1 seconds • Time in seconds is determined by multiplying the counter value by 0.1 (24 bit approximation)

5. Approximation Base 10 Base 2 0.1 = 0.000110011001100[1100]… ≈ 0.00011001100110011001100 ← 23 bits → • Let x = 0.00011001100110011001100 • Then 0.1 − x = 0.110011001100[1100]…× 2–23 = 0.00011001100[1001]…× 2–20 • Thus, (0.1 − x)/0.1 = 2–20 ← relative error

6. Patriot System • Mobile system • Designed to avoid detection • Should run for only a few hours at each site • Ran for 100 hours in Dhahran: 100 hrs = 360,000 sec = 3.6×105 sec • Accumulated error equals (3.6×105) × 2–20 = 0.3433 seconds > Clock step size

7. Software System Upgrade • Software (assembly language) written in 1970s • Modified to cope with high speed of missiles → Clock time more accurately converted • Not all function calls replaced by new code • Patriot likely confused by different clock times • Patriot did not track the Scud → No Patriot missile was fired

8. 2. Truncation Errors • Dow Jones Industrial Average equals a multiple of the total prices of the 30 underlying stocks • Present multiple is about 8.00 • If IBM ($79.40, closing price on 9/7) goes up $1, then DJI average goes up 8.00 points • If Microsoft ($25.43, closing price on 9/7) goes up $1, then DJI average goes up 8.00 points • If GE ($34.04, closing price on 9/7) goes up $1, then DJIA goes up 8.00 points • Is DJIA a good measure of the market? • What happens to DJIA when IBM goes up 10%? MSFT? • Faster to update index than to compute it from definition

9. Market Cap

10. Effect on DJIA • If the market cap of IBM goes up $12.196B, the DJIA will go up ?? points • If the market cap of MSFT goes up $25.553B, the DJIA will go up ?? points • If the market cap of GE goes up $35.017B, the DJIA will go up ?? points • The S&P 500 index is generally regarded as a better measure of the market because it takes market cap into account (it is float weighted)

11. Truncation Errors (2) • Vancouver Stock Exchange introduced new stock index in 1982 • Updated and truncated (three decimal digits) after each transaction • After 22 months, index fell from 1000 to 525 ─ 524.881, to be precise • Correct value is 1098.811 ─ based on underlying stocks

12. Truncation Error (3) • Assume 2500 transactions a day • Average error per transaction is 0.0005 • Average error per day is 1.25 • Assume 21 trade days a month • Total estimated error is 22×21×1.25 = 577.5 • Correct value is 1098.811 → Estimated wrong value = 521.311 • Observed index value = 528.881

13. 3. Numerical Overflow • European Space Agency • Ariane 5 rocket; June 4, 1996 • $500 million communication satellites on board • Off course and exploded 37 seconds after liftoff • Overflow when 64-bit floating-point number converted to 16-bit signed integer • Value measured horizontal velocity of rocket • 16 bit integer sufficient for Ariane 4 • Part of Ariane 4 software reused in Ariane 5

14. Divide by Zero • US Navy missile cruiser USS Yorktown • September 1997; off coast of Virginia • A zero was entered by mistake into data field of Remote Database Manager Program • A divide by zero → Database overflow → Propulsion system shut down • Ship dead in water for 2 hours 45 minutes

15. 5. Wrong Units of Measurement • Mars Climate Orbiter; $125 million • September 23, 1999 • Lockheed Martin built & operated the orbiter • LM engineers gave navigation commands for the Orbiter’s thruster in English units • NASA use metric units • Navigation information did not transfer to JPL • Orbiter got to within 60km of Mars, 100 km closer than planned • Orbiter’s propulsion system overheated and became disabled

16. Four Rounding Rules • Round to even: 1.25 → 1.2; 1.35 → 1.4; −1.35 → −1.4 • Round toward zero (= truncate): 1.25 → 1.2; 1.35 → 1.3; −1.35 → −1.3 • Round toward plus infinity: 1.25 → 1.3; 1.35 → 1.4; −1.35 → −1.3 • Round toward minus infinity: 1.25 → 1.2; 1.35 → 1.3; −1.35 → −1.4

17. Important Rule on Accuracy (p. H-35) • If x and y have p-bit significands, and x+y is computed exactly and then rounded to q places, a second rounding to p places will not change the answer if q ≥ 2p+2. This is true not only for addition, but also for multiplication, division, and square root.

18. IEEE Arithmetic • Double precision, q = 53 • Single precision, p = 24 • So, q ≥ 2p+2 • Accurate single precision can be implemented by computing in double precision, and then rounded to single precision

19. Double Rounding (p. H-34) • Want to compute 1.9 × 0.66 • Exact result is 1.254 • Say extended precision is 3 digits • Rounded to extended precision, the result is 1.25 • Further rounded to single precision, the result is 1.2 • Correctly rounded result is 1.3, obtainable by directly rounding once • What went wrong? • We have: q = 3; p = 2; q < 2p+2.

20. Objective • Computed result should be accurate to last bit (as if calculated in exact arithmetic, and then rounded correctly).

21. Cray-1 • Computation not accurate • Cray division: a/b = a × (1/b) • Take b = 10 (Recall Patriot System). • Last three bits could be wrong on Cray divides • Read p. H-64 and p. H-65

22. Floating Point Operations Not Associative • We have ( 3.14 + 1010 ) −1010 = 0 • But 3.14 + (1010 −1010 ) = 3.14 • Also, (1020 × 1020 ) ×10−20 = Overflow 1020 × ( 1020 ×10−20 ) = 1020

23. Euclidean Length of Vector • √[ v02 + v12 + v22 + … + vn2 ] • May overflow or underflow with vi2, even though final result is a normal number. • Try vi = 1040, or vi = 10−40

24. Better Way to Compute Euclidean Length • Let vmax = max │vi│ • Then compute vmax√[ (v0/ vmax )2 + … + (vn/ vmax )2 ] • Replace vmax by an easier-to-handle quantity?