[Haskell-cafe] Some More Sinus Results

[Hans van Thiel]

On Sat, 2007-11-10 at 11:14 -0800, David Roundy wrote:
> 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
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.