From: Hubert HOLIN (Hubert.Holin_at_[hidden])
Date: 2001-05-25 17:49:08
Paris (U.E.), le 26/05/2001
--- In boost_at_y..., Peter Schmitteckert (boost) <boost_at_s...> wrote:
> Salut,
>
> first of all I would see the library in boost, but I suggest that problems
Thanks!
> with accuracy/overflows should be revised, or at least be mentioned.
>
> Special Functions:
> ==============
>
> There are calculations of the following form
>
> if (abs(x) <= numeric_limits<T>::epsilon())
> {
> return(static_cast<T>(1));
> }
> else
> {
> return(sin(x)/x);
> }
>
> I would like to suggest to use something similar to
>
> const T eps = sqrt(sqrt(numeric_limits<T>::epsilon()));
> if (abs(x) <= eps) // or "<" ? which is better for performance?
> {
> return(static_cast<T>(1) - x*x/6);
> } else
> {
> return(sin(x)/x);
> }
>
> e.g. using t.C with g++ 2.9.3 one can see a small inaccuracy with long
> double:
> #include <iostream.h>
> #include <math.h>
>
> using namespace std;
>
> int main()
> {
>
> cout << sizeof(double) << '\t' << sizeof(long double) << endl << endl;
>
> cout.precision(20);
>
> cout << "double: " << endl;
> for( double x = 1e-7; x >1e-8; x *=0.8 )
> {
> cout << x << '\t' << sin(x) << '\t' << sin(x) / x << '\t' <<
> double(1.0) - x*x/6.0
> << '\t' << double(1.0) - x*x/6.0 + x*x*x*x/120.0 - x*x*x*x*x*x /
> 5040. <<endl;
> }
>
> cout << endl << "long double: " << endl;
>
> for( long double x = 1e-7; x >1e-8; x *=0.8 )
> {
> cout << x << '\t' << sin(x) << '\t' << sin(x) / x << '\t' << 1.0 -
> x*x/6.0
> << '\t' << 1.0 - x*x/6.0 + x*x*x*x/120.0 - x*x*x*x*x*x / 5040.
> <<endl;
> }
>
> return 0;
> }
>
> Same applies to sinhc, similar to atanh().
Two remarks here:
1) could you compare the discrepancy to numeric_limits<T>::epsilon()
for small values of the argument.
2) this is, as noted in the docs, a "stop-gap" library, we will have
to fine-tune it for various hardware/software combinations... so this
situation will probably require some extensive list of ifdefs.
> By the way, why is no-one using the correct name:
> arctan == arcus tangens; artanh = area tangens hyperbolicus
Histerical raisins ;-)
> Inaddition, what is the deeper meening of sinc_pi, it behaves
> like sinc. It confused me quite some time.
"sinc" is sometimes the name of a one-parameter family of
functions, of which "sinc_pi" is the most famous (this is the approach
we have taken here, see docs). At other times (f.i. MATLAB), "sinc"
just designates what I called here "sinc_pi". Just one more unfortunate
ambiguity of mathematics...
> Quaternions:
> ===========
[SNIP]
(for this part see another post, will be done).
Looks like I forgot to turn my escapes into ordinary white spaces
for this update, sorry...
Hubert Holin
Hubert.Holin_at_[hidden]