Why Computers Suck At Math

antdude writes "This TechRadar article explains why computers suck at math, and how simple calculations can be a matter of life and death, like in the case of a Patriot defense system failing to take down a Scud missile attack: 'The calculation of where to look for confirmation of an incoming missile requires knowledge of the system time, which is stored as the number of 0.1-second ticks since the system was started up. Unfortunately, 0.1 seconds cannot be expressed accurately as a binary number, so when it's shoehorned into a 24-bit register — as used in the Patriot system — it's out by a tiny amount. But all these tiny amounts add up. At the time of the missile attack, the system had been running for about 100 hours, or 3,600,000 ticks to be more specific. Multiplying this count by the tiny error led to a total error of 0.3433 seconds, during which time the Scud missile would cover 687m. The radar looked in the wrong place to receive a confirmation and saw no target. Accordingly no missile was launched to intercept the incoming Scud — and 28 people paid with their lives.'"

Comments

Poor QA

[slifox]

It's pretty pathetic and negligent that software that controls explosive missles was not tested for over 100 hours of operation. That's a standard Quality Assurance procedure for even the simplest low-budget hardware...

It's also pretty pathetic that the system designers implemented a broken design and did not foresee this problem. High-resolution timekeeping has been accomplished pretty successfully already...

I wonder how much time and money was spent in research and development for this thing
It doesn't seem like we're getting a quality product for the likely huge sum that was paid for it...

Re:Poor QA

[dbIII]

Oh really? The problem with these systems is that they have never worked in anything other than rigged tests and are just silicon snake oil.
I remember having this same discussion where there was a story here about some sort of Israeli space lasers that could apparently even shoot down artillery shells. Only a few months after that a very large number of thirty year old rockets dumped at discount price by Iran for being obsolete came flying over the border from Lebanon. Since then a lot of even slower rockets came out of Gaza. The success rate of this amazing new space toy matches that of the Patriot - zero.

Re:Poor QA

[OeLeWaPpErKe]

Mod parent up ! This idiotic article blames computers for programmers using numerical approximation algorithms illadvisedly.

which is stored as the number of 0.1-second ticks since the system was started up. Unfortunately, 0.1 seconds cannot be expressed accurately as a binary number, so when it's shoehorned into a 24-bit register — as used in the Patriot system — it's out by a tiny amount. But all these tiny amounts add up. At the time of the missile attack, the system had been running for about 100 hours, or 3,600,000 ticks to be more specific. Multiplying this count by the tiny error led to a total error of 0.3433 seconds, during which time the Scud missile would cover 687m. The radar looked in the wrong place to receive a confirmation and saw no target. Accordingly no missile was launched to intercept the incoming Scud — and 28 people paid with their lives.'"

So in a system that should have clocks synchronized to less than a microsecond nobody bothered to run "ntpdate" even once in hundred days ? And surely the military has better clock synch than a stupid home pc ? This is stupidity, also known as "human error", causing those deaths. It's a case of "the correct answer to the wrong question".

What is always brought up as a "computer problem" is the crash in Paris of a jet due to infighting between the human pilot and the autopilot. Of course, there the ultimate mistake was the pilot's : he had forgotten to turn off the autopilot to land. It was set for cruising altitude (3km), and the pilot was trying to land. This resulted in ever more desperate attempts by the autopilot to get the plane to gain height, which eventually resulted in a total loss of lift for the plane, which naturally resulted in the plane hitting the ground nose-down and a big fireball. The computer did exactly as instructed, it's just that the pilot's (unintentionally given) instructions were stupid, and the fact that it took the pilot over 3 minutes to realize just how stupid he had been.

Re:Poor QA

[Hal_Porter]

There is a good GAO report on this.

This one?

http://www.fas.org/spp/starwars/gao/im92026.htm

Wow. People complain about the US government. Still look at the transparency. The GAO wrote a very readable report for the House Of Representatives and now we can all read it on the web. It's not unreasonable to think that the US's vast military superiority over everyone else on the planet is at least in part due to this sort of thing. I don't think any other government would do this - mistakes in the military would just get covered up as state secrets and anyone who tried to talk about them would get locked up or worse.

Re:Poor QA

[Shinobi]

To be honest, from working in two specialist fields(HPC system level programming and embedded applications(particularly sensor stuff), I've experienced that CompSci grads are more likely than CompEng or EE grads to make errors like this. A large part of it is simply that CompSci nowadays is too high-level and abstract, many of them don't know very much about how computers ACTUALLY work other than as a theoretical model.

A common remark is "Why should I need to know that, the compiler will take care of it better than I will anyway", completely forgetting that the compiler is only as smart as the programmer who coded it is. So you can get what I ran into with an odd appliance based around the SH-4 processor I was hired to fix some performance problems with. It ran fixed point integer and decimal math, and was ported over from ARM. But it only reached about 25% of maximum theoretical performance, while the ARM reached around 80%. Turns out GCC was at fault, using a generic method that wasn't suitable for the Super-H architecture. And the CompSci had no clue about such things.

Re:Poor QA

[Alef]

I don't think any other government would do this - mistakes in the military would just get covered up as state secrets and anyone who tried to talk about them would get locked up or worse.

Eh. Forgive me, but do you have any basis whatsoever for this claim, or are you just being arrogant?

Re:Poor QA

[Jeremi]

The computer did exactly as instructed, it's just that the pilot's (unintentionally given) instructions were stupid, and the fact that it took the pilot over 3 minutes to realize just how stupid he had been.

Sounds like a user interface problem to me. Given the potential consequences of that particular user error, the fact that the autopilot was still engaged should have been made more obvious to the pilot. (e.g. when the plane computer sees that a struggle is going on between the autopilot and the manual controls, it should prompt a loud, un-maskable synthesized voice shouting "THE AUTOPILOT IS ENGAGED, YOU IDIOT!")

Re:Poor QA

[Anonymous Coward]

So in a system that should have clocks synchronized to less than a microsecond nobody bothered to run "ntpdate" even once in hundred days ?

Do you want to be the one to explain to the generals why their stand-alone, truck-based mobile air protection system needs a hard-line network connection to work?

The real idiocy is here:

Unfortunately, 0.1 seconds cannot be expressed accurately as a binary number, so when it's shoehorned into a 24-bit register

Taken charitably, the article writer has oversimplified to the point of obscuring the point. It's perfectly possible to represent a 0.1-second tick in a 24-bit register. There's an overflow about once every 19 days. The problem is doing calculations *with* that number, and that takes knowing what the hell you're doing. Given the problem the system designers were trying to solve with Patriot, this should not have been a problem.

And surely the military has better clock synch than a stupid home pc ?

You'd be surprised how hard clock accuracy is to get right, *especially* under military conditions. A drift of 0.3433 seconds over 100 hours works out as an accuracy of 1 part in a million, give or take. Besides, the problem here wasn't clock drift, so it's a irrelevant.

Re:Poor QA

[OeLeWaPpErKe]

You missed the third option, which is for the motivation behind the firing of rockets to be removed.

http://www.youtube.com/watch?v=iNrCMdFoZqQ

So who do we allow to settle there ?

The "kingdom of Egypt" (the state of the Farao's) ? (exterminated to the last man by muslims)
The Hittite Emptre ? (exterminated by the Greeks, Romans, Persians)
The kingdom of Israel ?
The Assyrian Empire ?

Which of these do we restore ? (note that the palestinians, or to be more exact, the arabs only come into play about 4500 years after the Assyrian Empire)

Which do we restore ? And why do they have more rights than all the others who conquered that piece of land ?

Note the obvious truth : the Jews controlled Israel about 4300 years before the arabs even left their tiny province ...

What if some Greek starts firing rockets at the Arabs ? Will you tell them to leave ? He has at least as much right to Israel as they do ? What if the Jews start firing rockets into Jordan (territory that was part of the kingdom of Israel) ?

And of course, you shouldn't count out yourself. You're an Indo-European living in America. It seems hypocritical in the extreme to tell others to leave conquered lands. Your province of origin is northwestern Iran, every other place on this earth indoeuropeans live (including Europe), is obviously conquered from someone else.

So when will you give the good example ?

Re:Poor QA

[danlip]

The computer did exactly as instructed, it's just that the pilot's (unintentionally given) instructions were stupid, and the fact that it took the pilot over 3 minutes to realize just how stupid he had been.

Sounds like a user interface problem to me. Given the potential consequences of that particular user error, the fact that the autopilot was still engaged should have been made more obvious to the pilot. (e.g. when the plane computer sees that a struggle is going on between the autopilot and the manual controls, it should prompt a loud, un-maskable synthesized voice shouting "THE AUTOPILOT IS ENGAGED, YOU IDIOT!")

Or if the pilot is pushing hard on the stick the autopilot should disengage (with loud alarms).
If I tap on the breaks in my car the cruise control disengages, it does not fight me.
- Dan

Curse of binary floating point

[Carewolf]

Use decimal floating point or simple swich to fixed point. Fixed point not used as often as it should, and many developers don't know how difficult ordinary floiting point really is.

Re:Curse of binary floating point

[noidentity]

Unfortunately, 0.1 seconds cannot be expressed accurately as a binary number, so when it's shoehorned into a 24-bit register -- as used in the Patriot system -- it's out by a tiny amount.

Sorry, 0.1 seconds can be represented EXACTLY in such a system. It doesn't even need floating-point. Here is how such a system could represent the durations of 0.1 seconds, 25.7 seconds, and 123.4 seconds: 1, 257, and 1234. So like you say, fixed-point works here. No need for anything beyond integers in this case.

Fixed point numbers?

[Big_Mamma]

Use fixed point numbers? You know, in financial apps, you never store things as floating points, use cents or 1/1000th dollars instead!

Computers don't suck at math, those programmers do. You can get any precision mathematics on even 8 bit processors, most of the time compilers will figure out everything for you just fine. If you really have to use 24 bits counters with 0.1s precision, you *know* that your timer will wrap around every 466 hours, just issue a warning to reboot every 10 days or auto reboot when it overflows.

Stupid article, too

[hellfire]

Translation: computers are only as smart as the people programming them... and there's plenty of stupid people out there.

We knew this. This is no great revelation. So why is this news?

What?!

[jointm1k]

of 0.1-second ticks since the system was started up. Unfortunately, 0.1 seconds cannot be expressed accurately as a binary number, so when it's shoehorned into a 24-bit register

All they had to do is use integers, where a value of 1 represents 0.1 s.

Re:"User error"?

[betterunixthanunix]

So they designed a system that accumulated rounding errors over time, and their solution was to ask the system's users to reboot the system every so often? Somehow, that does not add to my sympathy for these programmers...

don't blame the computer for bad programming

[frovingslosh]

It is absurd to blame the computer (or worse, all computers) for what is bad programming. Computers can store a 1/10 of a second perfectly accurately, as long as it is stored in a variable that counts tenths of seconds rather than seconds. It can easily be stored as an integer that way, avoiding any floating point rounding errors.

There certainly are cases of bad math in computers, particularly Intel computers. But this isn't such an example. This is just a lazy and stupid programmer who didn't understand what he was really doing who should take the blame for the failure that killed people, not the computer.

Re:retrospective technological excuses

[david duncan scott]

Regardless, what isn't possible is is to design a system that can accurately track and shoot down missiles in flight. As the Patriot defence system so patently demonstrated.

You're right. Just as the failure of Samuel Langley's aircraft demonstrated that man would never fly, the failure of an anti-aircraft missile to destroy only half of the ballistic missiles (targets moving at what, twice the speed of the targets it was designed to destroy?) demonstrates that ABM's will never work.

Re:"User error"?

[Joce640k]

I'm calling "Horsepoo" on the whole story.

a) If they knew enough about it to put "reboot every 36 hours" in the manual they knew enough to fix it.

b) According to the summary, 36 hours would still be a complete miss (a third of 687 meters is still 229)

c) A fixed point integer (32 bits) can mark tenths of seconds with complete accuracy for over 13 years.

d) Leaving aside a,b and c, the story still doesn't make any sense. The system would start the calculation the moment it saw the missile, not 100 hours before it appeared on the radar.

Now ... at the speed of a scud missile (mach 5 if google serves me), it may be that an accuracy of 1/10th second isn't enough to compute the trajectory accurately enough to intercept it. At that speed you might need 10,000th second resolution or whatever. *That* would be believable (but unlikely - the designers would have to be complete idiots).

The rest of the article? Yawn. It's the same old recycled story we've been seeing since the 1970s (those of us who are old enough).

Re:And this is why...

[SpinyNorman]

It's the reporting that's garbage. It makes no sense at all. A system tracking missiles travelling at Mach 3 is keeping track of time to 0.1 sec accuracy?! Do you really believe that? Wanna buy a bridge?

0.1 sec at Mach 3 is 100m, so you'd have a hope in hell of ever hitting a 3m long target.

The problem isn't the people working for the defence company, who are hard-core PhDs with some very serious domain knowledge. The problem is people like yourself who are so math illiterate as not to be able to fact check a piece-of-shit story!

READ THE GD ARTICLE

[ToasterMonkey]

FTFA:
"So computers might suck at maths, but there's always a solution available to circumvent their inherent weaknesses. And in that case, it's probably more accurate to say that computer programmers suck at maths - or at least some of them do."

Thank you, come again.

So in a system that should have clocks synchronized to less than a microsecond nobody bothered to run "ntpdate" even once in hundred days ?

Yes, obviously they just needed to ssh into their patriot missile air defense system, edit a few lines in /etc/inet/ntp.conf and svcadm restart ntp.

The obvious problem in the article, if you read it, is computer's finite precision, and how it is dealt with. By 'computer', the author could have easily included the system libraries that are actually doing all the rounding and overflows instead of implementing arbitrary precision in software.

Everyone defending the way 'computers' is used in this article, and conflating it with 'processor' is a complete idiot.

Re:READ THE GD ARTICLE

[Jane Q. Public]

I'm obviously not a hardware designer? That's funny. I am not the cluless one here. How about some simple math? Maybe you would learn something.

A 24-bit register, with clock ticks every 0.1 second, would overflow in less than 20 days. And if the clock ticks were faster, then it would overflow even sooner. No wonder they recommended rebooting the system every few days.

Of course I do not recommend an infinitely large register. Simply one that is large enough for the job at hand. This one obviously isn't. Further, a 0.1-second resolution clock is obviously not adequate to a job requiring this kind of precision.

If the hardware clock is off (not overflowed but INACCURATE, which was the real situation here), no amount of software tweaking will properly fix the problem. The article did not state but implied -- incorrectly -- that the clock register was accumulating rounding errors; that is not the case. Nobody makes system clocks that way, nor did they in the 90s or even the 80s. The system clock is nothing but a counter that is incremented every clock tick. The actual problem was that the clock ticks were not sufficiently precise, so over time the count was off. Math libraries and rounding errors played no part whatsoever in that error.

Finally, I would like to point out that today's standard PC-type system clocks are large enough that they won't overflow for 100 years or so; that is the obvious and proper solution to the overflow problem. The problem of clock ticks that are sufficiently precise for timing of missile navigation, as far as I know, has not been addressed on standard PCs, however, and they do not try to correct for that in software because the adequate precision in the clock simply does not exist. It would amount to tilting at windmills. Keeping a count in software of the number of times the register overflows is also NOT an appropriate solution for a system clock, nor is any software tweak, because software by definition is volatile while the hardware clock is not. In other words, nobody does it that way, dude, because it's just plain the wrong answer.

As for your final comment, most Unix programmers know what epoch time is, when it started (00:00:00 UTC on 1 January 1970 according to ISO 8601), and when that date will roll over in the counter (approximately 65 YEARS later, so it isn't much of an issue). Nobody is arguing that we should make a missile system that needs to last, unmodified, for over 65 years. But proper hardware design in the first place, which was certainly possible at that time using ASICs if not straight-up custom chips, would have eliminated the problem.

Paid with their lives

[sochdot]

I'd just like to point out here that the 28 people were not killed by the failure of the intercept system. They were killed by the nice folks who launched the missile in the first place.