On Sat, Nov 10, 2007 at 06:56:23PM +0100, Hans van Thiel wrote:
Thanks to all who've replied; Carl's explanation in particular was very interesting. So the precision, suggested by the many decimals in the 'show', is not the actual precision the user should 'count on'. If you take 1/60 of a degree to be approximately 0.0003 radians, you should not use sin for smaller values. In all cases the actual precision of sin appears to be 4 to 5 decimals, and results should be rounded to that before using them. Now I'm wondering about cos, tan and also the inverses, asin etc. :-)
What you're observing in these calculations is not so much the inaccuracy of sin as the inaccuracy of pi. Pi is an irrational number, so the Double constant pi is only an approximation, with an error probably around 3e-16 (1e-16 fractional error). Simple calculus tells you that the error in sin(pi) will thus be around:
sin(truepi+3e-16) ~ 0 + 3e-16
from which we conclude that the error in pi isn't really as bad as we'd naively expect.
For small epsilon, we can see that
sin((1+/-epsilon)*(truepi+3e-16)) ~ 0 +/- pi*epsilon + 3e-16.
which roughly explains your data quoted.
Results of sin should *not* be rounded before using them, unless you really want a less accurate answer.
Users who "count on" a particular precision, should not be using computers to do their arithmetic. A limitation in the technology, which derives from the floating point representation, should not become a norm, IMHO. Every engineering discipline uses error approximation and calculation of the cumulation/cancelation of those errors to estimate the reliability of
On Sat, 2007-11-10 at 11:14 -0800, David Roundy wrote: the total. That's if you use a slide rule, that's if you use a computer. There's no difference. If you mean that people should not do arithmetic on computers, if the results are vital, unless they understand the scope and limits of the tools they're using, I agree, of course.
The blog article on the accuracy of sine glossed over the relevant issue: that there's no point in getting better accuracy. It points out that sin(pi) only has five or six digits of accuracy, it means that the "correct" answer is about 1.225e-16, because "correct" is defined to mean the sine of the double which is nearest to the actual value of pi.
It's nice to be accurate, but the claim that for large arguments the computed value of sin is effectively "random" is pretty irrelevant, as the same thing can be said of the "true" value of sin, when defined as "the sine of the double closest to your desired x value" which is the closest we could possibly come to a "true" sin. If you're taking the sine of a large number, your code is broken, and no sin function is going to fix it.