Date: Sun, 12 Aug 2012 23:02:44 -0000 From: Stephen Montgomery-Smith <stephen@missouri.edu> To: Peter Jeremy <peter@rulingia.com> Cc: Diane Bruce <db@db.net>, Bruce Evans <brde@optusnet.com.au>, John Baldwin <jhb@freebsd.org>, David Chisnall <theraven@freebsd.org>, Bruce Evans <bde@freebsd.org>, Steve Kargl <sgk@troutmask.apl.washington.edu>, David Schultz <das@freebsd.org>, Warner Losh <imp@bsdimp.com> Subject: Re: Use of C99 extra long double math functions after r236148 Message-ID: <500DAD41.5030104@missouri.edu> Resent-Message-ID: <20120812230235.GC20453@server.rulingia.com> In-Reply-To: <20120722121219.GC73662@server.rulingia.com> References: <20120717084457.U3890@besplex.bde.org> <5004A5C7.1040405@missouri.edu> <5004DEA9.1050001@missouri.edu> <20120717040118.GA86840@troutmask.apl.washington.edu> <20120717042125.GF66913@server.rulingia.com> <20120717043848.GB87001@troutmask.apl.washington.edu> <20120717225328.GA86902@server.rulingia.com> <20120717232740.GA95026@troutmask.apl.washington.edu> <20120718001337.GA87817@server.rulingia.com> <20120718123627.D1575@besplex.bde.org> <20120722121219.GC73662@server.rulingia.com>
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I just realized that catan(z) = reverse( catanh(reverse (z))), just like
casin relates to casinh (remember reverse(x+I*y) = y+I*x). This is a
consequence of catan and catanh being odd functions, as well as the
standard relation catan(z) = -I*catanh(I*z).
So I would modify Peter's code by taking out the minus signs.
Maybe it would make a difference if the answers involved -0.
On 07/22/12 07:12, Peter Jeremy wrote:
> /*
> * Arc-tangent of a complex argument z = x + I*y.
> *
> * catan(z) = reverse( catanh(reverse (z)))
> *
> */
> double complex
> catan(double complex z)
> {
> double complex r;
>
> r = catanh(cpack(cimag(z), creal(z)));
> return (cpack(cimag(r), creal(r)));
> }
>
I am attaching code that computes all six arc-trig-hyp functions. Peter
made a remark that catanh(z) would be hard to compute when |z|=1.
Fortunately that is not the case, because when |z|=1, the imaginary part
of catanh(z) is plus or minus PI/4. So we won't face the same problems
that clog(z) has.
I feel that I am done with these functions for now. I tried to change
my comments to conform to the style given to me by Bruce. However
spacing inside mathematical expressions is something where I am
inconsistent.
The functions still need a lot of work to handle -0, infs and NaNs
correctly. I will leave that to you guys, because you seem so much
better at it than me. I still don't understand why the proper test is
"if (x!=x) return(x+x)" rather than "if (isnan(x)) return(NAN)".
However, I think you might find that a lot of the handling when the
input or output is infinity works without any changes. For example,
catanh(1) seems to produce the correct answer, and maybe even
catanh(1-0*I) will work better than expected.
I am also attaching the test code. I run it like this, so that only
ULPs greater than 3 appear:
./test3 | perl -lne '@a=split " ",$_;print if $a[2]>3 || $a[3]>3'
and get outputs like this:
21987 atanh 3.13643 0.299452 0.443282 0.0665108 0.473296 0.0824781
67013 atan 0.377411 3.03922 -0.0170315 -0.442191 -0.0211662 -0.474753
70044 acosh 0.883474 3.23108 1.06353 0.107343 0.433696 0.242278
70044 acos 3.23108 0.883474 1.06353 0.107343 0.242278 -0.433696
96124 atan 0.509279 3.21631 -0.0346851 -0.461121 -0.0440054 -0.497841
The first example is the count of which example I am trying. The third
and fourth entries are the ULPs of the real and imaginary parts of the
answer. The 5th and 6th entries are the real and imaginary parts of the
input, and the 7th and 8th entries are the real and imaginary parts of
the answer.
I use the unuran port to generate data where the x and y values of the
inputs are normally distributed N(0,1). As you can see from the parts I
commented out, I tried many variations (mostly to check edge cases:
close to zero, very large, or close to branch cuts). In particular, if
you want random data on the unit disk, use h=hypot(x,y); x/=h; y/=h;
My next project will be to get clog(z) to work well when |z|=1.
Stephen
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