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 wasn’t there! He wasn’t there again today! I wish, I wish he’d 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 x^2 +b*x + c = 0 is x = [-b +/-sqrt(b^2 – 4*c)]/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)

13. Quadratic Equation (cont.) r1 looks pretty good, but r2 does not. Why? Cancellation occurred in forming [b – sqrt(b^2 – 4*c)]/2 = [0.125335 - 0.125330]/2 = 2.5e-6 However, death occurred when we formed [b^2 – 4*c] = [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 don’t have about the coefficient c. A little algebra will tell us that if r1 and r1 are real, distinct roots of x^2 + b*x + c = 0, then r1*r2 = 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)