Nicely done, especially compared to the fits that HP went through trying to figure out how they could prove or disprove that all 11 digits of their calculation were correct.
40 years ago these calculators were looked upon with a bit of skepticism. Engineers that were used to seeing the log tables with their own eyes and hand-manipulating slide rules were being asked to trust the results coming out of these calculators.
And lives depended on it, really. If you were a civil engineer designing a bridge and you needed to be absolutely sure the numbers you were using were accurate to 7 places and totally correct, would you suddenly put all your faith in this small brown box with no way to examine the inner workings?
HP went through a lot to build that trust and it was rightly earned. HP's reputation for building solid accurate calculators kept that business going for decades to come.
Trusting the numbers coming out of a calculator was ALWAYS a bad idea, especially if lives depended on it. It wasn't just the calculator making a mistake - you could have made a mistake keying in the numbers.
This is dealt with by:
1. running the results through the inverse equations to verify that you get the inputs back again
2. calculating the results using an independent method to verify them
3. having a different group of people independently check your numbers
4. have your results pass a "reasonableness" test, i.e. do they make sense
5. put the resulting design on a test rig and verify the numbers experimentally
How do I know this? I worked on critical flight control systems for Boeing.
Any engineer who just punches numbers into a calculator and bets lives on the results ought to be fired.
These days, we can also have our calculators keep track of the units associated with a quantity, which helps greatly with sanity-checking. If the units come out wrong or mis-match anywhere in the calculation, you'll know immediately.
It's nice to have extra checks, but that changes nothing about what I said. If you are making designs that lives depend on, it is not HP's responsibility, it is YOURS and you should be using the techniques I outlined.
I'm sorry to be so blunt about this, but I feel strongly about it. I too often run into engineers that either argue that they can design a perfect system that is not subject to human error, or they try to shift responsibility onto other people or organizations.
BTW, I think every engineer should watch the series on the Smithsonian channel "Air Disasters". Each episode chronicles a particular disaster, and then painstakingly goes through what went wrong and how the problems were fixed. One can learn a lot about systems design from these case studies on how and why things go wrong.
In your opinion, if doing step 5 is cheap enough, do you still need 1-4? (That is, is it "okay"--in a pragmatic sense--to have sloppy thinking, if you can just throw numbers at the wall to see if they stick?)
A very old school surveying teacher (head of department as it happens) wasn't entirely convinced about calculators in the late eighties. I showed him how to get an extra decimal place of seconds of arc by subtracting the whole number of degrees.
The real sticking point was getting a real arctan2 function (returning the angle in the range [0,360]). He really liked some BBC Basic programs I hacked up. Fun times.
