Using Calculator, with input a real number between 0 and 1 for example, how do I get the angle for which the sin equals that real number?

That is, how do I enter arcsin (or any other arc-trig) to determine the angle? Thanks.

Bob

Here is what its is doing if calculate 13.5 / 1539 it shows me 8.7719e-03

While that is technically a correct answer I think I want tit to display it in the format I have become accustomed to like .0087719

What do I need to do to get it from this format 8.7719e-03 to this format .0087719

Right now I have OSX 10.4.11, Calculator version 4.0.6 but I have the same problem with Leopard too

David

Additional information

For an advanced discussion of why this happens, see “What Every Computer Scientist Should Know About Floating-Point Arithmetic”

Interesting though that the computer scientists at Apple do not know this, and that they even point out what they should know, but don’t, on their help site.

sin 360

(degree mode)

the answer should be zero but the calc shows -2.449293598294706e-16

os x 10.3.9

There are exact number storing formats available, but they are much more cumbersome to use and coders of simple applications as calculators do not generally bother.

Sounds plausible, but that is setting the bar extremely low, don’t you think? I mean, the bar is pretty much just laying on the ground at that point.

I would think any calculator would use a number-storing format which could produce 100% accurate results on simple two-decimal arithmetic. When I think of rounding errors, I think more of legitimate repeating decimals like the square root of 2 or something.

If you want an explanation check this out.

It can be solved by using integer math rather than floating point. Windows had the same problem with its calculator, but it was upgraded in XP which fixed the problem.

Oh, and I am really really surprised your software “engineer” did not know this. It is really basic introductory computer science stuff.

And “precision” isn’t a misnomer at all.

Funny you should mention Netscape, because I actually thought of their frame sizing algorithm when I was thinking about this bug. I know it’s probably unrelated but it just reminded me of it. Basically, if you are using a frameset and you specify your frame dimensions in pixels, Netscape (at least up through version 4) would convert your pixels to a window size percentage, and then convert it back to pixels resulting in rounding errors all the time.

Kim and David,

Thanks for the Sun link. It looks like it explains floating points pretty thoroughly, but sheesh, 105 pages of engineer-speak. All I know is that even if this problem is legitimate, these errors should never make it to the consumer. As for our principal software engineer, I’m pretty sure his astonishment had more to do with not knowing how a calculator which had been around since 1984 could suddenly have this bug, as opposed to not knowing what floating point operations are. I mean, what could have possibly changed in Panther that would have introduced such a bug in the calculator? That is more the question…

In this case, the user entered two numbers with specified precision of 10^-2. The computer then spit out a number with an apparent precision of 10^-12. All the digits beyond .01 are made up. Nonsense. Artifacts. Subtracting (or adding) two numbers does not increase the number of fractional digits that are relevant (note that dividing does), so displaying those digits is incorrect. When presented with two numbers like the ones shown and asked to perform an addition or subtraction operation, the calculator should only show the answer with the same degree of precision. (Although a truly comprehensive floating point solution would also indicate the degree of error in the calculation — i.e., 9533.14 +/- 1×10^-14)

My closest analogy is in high school chemistry class when we were asked to weigh samples and perform calculations using the results. The calculations would be good to +/- 2 digits of accuracy (tolerance of about 1%), but my handy pocket calculator could spit out 8 digit calculation results. I’d hand in the paper with all those digits neatly written out at which point the teacher would patiently explain that all the digits past the first two or three were useless and not worth the ink in which they were written.

I can’t imagine it would add a great deal of code to have the calculator keep track of the number of significant digits entered and use it in calculating the number of significant digits to display on output.