From: Brian D Williams (talon57@well.com)
Date: Thu Mar 25 1999 - 07:41:50 MST
I wrote:
>Actually this is incorrect, the military initially claimed
>something like a 90% plus success rate, then an MIT prof claimed
>a 0% rate. The revised rate was near 60%.
This is related:
Computer Arithmetic Tragedies
disasters caused by computer arithmetic errors
Patriot Missile Failure
On February 25, 1991, during the Gulf War, an American Patriot
Missile battery in Dharan, Saudi Arabia, failed to intercept an
incoming Iraqi Scud missile. The Scud struck an American Army
barracks and killed 28 soliders. A report of the General
Accounting office, GAO/IMTEC-92-26, entitled Patriot Missile
Defense: Software Problem Led to System Failure at Dhahran, Saudi
Arabia reported on the cause of the failure. It turns out that the
cause was an inaccurate calculation of the time since boot due to
computer arithmetic errors. Specifically, the time in tenths of
second as measured by the system's internal clock was multiplied by
1/10 to produce the time in seconds. This calculation was
performed using a 24 bit fixed point register. In particular, the
value 1/10, which has a non-terminating binary expansion, was
chopped at 24 bits after the radix point. The small chopping
error, when multiplied by the large number giving the time in
tenths of a second, lead to a significant error. Indeed, the
Patriot battery had been up around 100 hours, and an easy
calculation shows that the resulting time error due to the
magnified chopping error was about 0.34 seconds. (The number 1/10
equals 1/24+1/25+1/28+1/29+1/212+1/213+....
In other words, the binary expansion of 1/10 is
0.0001100110011001100110011001100.... Now the 24 bit register in
the Patriot stored instead 0.00011001100110011001100 introducing an
error of 0.0000000000000000000000011001100... binary, or about
0.000000095 decimal. Multiplying by the number of tenths of a
second in 100 hours gives
0.000000095×100×60×60×10=0.34.)
A Scud travels at about 1,676 meters per second, and so travels
more than half a kilometer in this time. This was far enough that
the incoming Scud was outside the "range gate" that the Patriot
tracked. Ironically, the fact that the bad time calculation had
been improved in some parts of the code, but not all, contributed
to the problem, since it meant that the inaccuracies did not
cancel.
The following paragraph is excerpted from the GAO report.
The range gate's prediction of where the Scud will next appear is
a function of the Scud's known velocity and the time of the last
radar detection. Velocity is a real number that can be expressed as
a whole number and a decimal (e.g., 3750.2563...miles per hour).
Time is kept continuously by the system's internal clock in tenths
of seconds but is expressed as an integer or whole number (e.g.,
32, 33, 34...). The longer the system has been running, the
larger the number representing time. To predict where the Scud
will next appear, both time and velocity must be expressed as real
numbers. Because of the way the Patriot computer performs its
calculations and the fact that its registers are only 24 bits long,
the conversion of time from an integer to a real number cannot be
any more precise than 24 bits. This conversion results in a loss
of precision causing a less accurate time calculation. The effect
of this inaccuracy on the range gate's calculation is directly
proportional to the target's velocity and the length of the the
system has been running. Consequently, performing the conversion
after the Patriot has been running continuously for extended
periods causes the range gate to shift away from the center of the
target, making it less likely that the target, in this case a
Scud, will be successfully intercepted.
Brian
Member, Extropy Institute
www.extropy.org
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