CSC%204250%20Computer%20Architectures

1
CSC 4250Computer Architectures

• September 12, 2006Appendix H. Computer Arithmetic

2
Five Real Stories on Computer Arithmetic

1. Insufficient Accuracy
2. Truncation Errors
3. Numerical Overflow
4. Divide by Zero
5. 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 0.0001100110011001100
• 0.00011001100110011001100
• ? 23 bits ?
• Let x 0.00011001100110011001100
• Then
• 0.1 - x 0.1100110011001100 223
• 0.000110011001001 220
• Thus,
• (0.1 - x)/0.1 220 ? 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 3.6105 sec
• Accumulated error equals
• (3.6105) 220 0.3433 seconds
• gt 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
Company Price Market Cap
IBM 79.40 121.96B
MSFT 25.43 255.53B
GE 34.04 350.17B
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 SP 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 22211.25 577.5
• 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
  Orbiters 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
• Orbiters 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 xy is
  computed exactly and then rounded to q places, a
  second rounding to p places will not change the
  answer if q 2p2. 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 2p2
• 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 lt 2p2.

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

• v 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
• vmaxv (v0/ vmax )2 (vn/ vmax )2
• Replace vmax by an easier-to-handle quantity?