What is Rounding Error

1
What is Rounding Error?

• AiS Challenge
• STI 2003
• Richard Allen

2
What is Rounding Error?

• As I was going up the stair
• I met a man who wasnt there!
• He wasnt there again today!
• I wish, I wish hed stay away!

3
Floating-Point Numbers

• Nearly all computation on a digital computer is
  done in floating-point arithmetic (FPA).
• A floating-point number system (FPS) is a number
  system that uses a finite number of digits to
  represent the real number system that we use in
  exact computation.
• Floating point numbers are generally represented
  in binary in a computer.

4
A Floating-Point Number System

• Decimal example A FPS with 1-decimal digit
  ranging from -9.0 to 9.0 system contains only
  55 numbers.
• 0.0
• ?.01 ?.02 ?.03 ?.04 ?.05 ?.06 ?.07 ?.08 ?.09
• ?0.1 ?0.2 ?0.3 ?0.4 ?0.5 ?0.6 ?0.7 ?0.8 ?0.9
• ?1.0 ?2.0 ?3.0 ?4.0 ?5.0 ?6.0 ?7.0 ?8.0 ?9.0

5
Rounding Arithmetic

• If pi 3.14156... were represented in our
  1-digit FPS, it would be 3.0.
• In a 5-decimal system, pi could be represented by
• 3.1416 (rounding)
• 3.1415 (truncation or chopping)
• Rounding error is the difference between the
  result in exact arithmetic and the result in our
  floating-point system.

6
Some Interesting Implications of Rounding Error

• Check out the site http
• http//gala.univ-perp.fr/langlois/rounding_error
  .html
• for some interesting examples.
• Three of these examples follow
• The Patriot
• Down and out in Vancouver
• I shot an arrow in the air (The sad story of the
  Ariane 5)

7
The Patriot

• During the Gulf War, an Iraqi Scud got through
  the Patriot anti-missile system and hit a
  barracks, killing 28 people.
• To track the Scud, the system had to determine
  the interval between tracking times by
  subtracting two values of a timer.
• The times in tenths of a second were stored as
  integers (4.2 was stored as 42) in the onboard
  computer.

8
The Patriot (cont.)

• To compute the interval, timer values were
  con-verted to floating-point representation by
  multi-plying by 0.1.
• 0.1 has a non-terminating binary expansion
  con-sequently, the interval was computed with
  error - the larger the value, the larger the
  error.
• At the time of the incident, the missile battery
  had been operating for over 100 hrs, resulting in
  an error of 0.34 sec. in the timer causing the
  system to look in the wrong place for the
  incoming Scud.

9
Down and Out in Vancouver Stock Exchange

• In 1982 the Vancouver Stock Exchange introduced
  an index with a nominal value of 1000.000.
• After each transaction, the index was recomputed
  and truncated to three decimal places.
• After 22 months of transactions, the index was
  524.881.
• The true value was 1098.811.

10
The sad story of the Ariane 5

• On June 4, 1996, Ariane was launched at Kourou
  and all went well for 36 sec.
• At second 37, the rocket veered off course and
  self-destructed.
• The problem was in the Inertial Reference System.

11
The sad story of the Ariane 5

• The IRS tried to convert a 64-bit floating point
  number to a 16-bit integer.
• The number was too large, which triggered an
  error and send a diagnostic message to the
  On-Board Computer.
• The OBC interpreted the diagnostic word as flight
  data.
• Finis!

12
Solution of a Quadratic Equation

• The solution to the quadratic equation
• x2 bx c 0
• is
• x -b /-sqrt(b2 4c)/2
• Example b -0.125335, c 0.360551e-6,
  using 6-digit decimal arithmetic
• r1 1.25333e-1 (1.253321232e-1)
• r2 2.50000e-6 (2.876764477e-6)

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Quadratic Equation (cont.)

• r1 looks pretty good, but r2 does not. Why?
• Cancellation occurred in forming
• b sqrt(b2 4c)/2 0.125335 - 0.125330/2
  
  2.5e-6
• However, death occurred when we formed
• b2 4c 0.0157089 - .00000144220
• 0.0157075
• Cancellation and the addition/subtraction of
  floating-point numbers with differing magnitudes
  can cause problems.

14
Quadratic Equation the cure

• We must restore the information that we dont
  have about the coefficient c.
• A little algebra will tell us that if r1 and r1
  are real, distinct roots of x2 bx c 0,
  then
• r1r2 c
• So, we compute r1 as before and compute r2 as
• r2 c/r1 3.60551e-6 / 1.25332e-1
• 2.87677 (2.876764477e-6)