Re: Wrong formulae for complex elementary functions

Hi everybody,

On 11/27/2010 08:37 AM, Michael Kerrisk wrote:

Hi Andries,

Since you are the mathematician, can you comment?

Thanks,

Michael

On Fri, Nov 26, 2010 at 9:57 AM, Richard B. Kreckel[off-list ref]  wrote:

quoted

Hi!

The man pages for cacos, cacosf, cacosl, catan, catanf, catanl, cacosh,
cacoshf, cacoshl, catanh, catanhf, and catanhl contain wrong maths.

cacos, cacosf, cacosl:
  The formula given in the man page
    cacos(z) = -i clog(z + csqrt(z * z - 1))
  gives wrong results in second and fourth quadrant of complex plain.
  The formula
    cacos(z) = -i clog(z + I*csqrt(1 - z * z))
  gives correct results.

catan, catanf, catanl:
  The formula given in the man page
    catan(z) = 1 / 2i clog((1 + iz) / (1 - iz))
  gives wrong results on the negative imaginary axis beginning at -I
  (along one of the two branch cuts). Besides, the formula is written
  in an ambiguous way.
  The formula
    catan(z) = (clog(1 + iz) - clog(1 - iz)) / 2i
  gives correct results.

cacosh, cacoshf, cacoshl:
  The formula given in the man page
    cacosh(z) = (0.5) * clog((1 + z) / (1 - z))
  gives wrong results everywhere in the complex plain. (The formula
  seems to be copied from the one for catanh, where it is sometimes
  correct.)
  The formula
    cacosh(z) = 2 * clog(csqrt((z + 1)/2) + csqrt((z - 1)/2))
  gives correct results.

catanh, catanhf, catanhl:
  The formula given in the man page
    catanh(z) = 0.5 * clog((1 + z) / (1 - z))
  gives wrong results on the positive real axis beginning at 1 (along
  one of the two branch cuts).
  The formula
    catanh(z) = 0.5 * (clog(1 + z) - clog(1 - z))
  gives correct results.

I've also checked casin, casinf, casinl, casinh, casinhf, and casinhl and
the formulae given there
    casin(z) = -i clog(iz + csqrt(1 - z * z))
    casinh(z) = clog(z + csqrt(z * z + 1))
are actually correct.

I suspect that some of these errors are due to somebody trying to "simplify"
these formulae. Since this can ruin the behavior, I recommend to add a
comment to the upstream sources that warns against attempt of
simplification.

For the sake of reference, I am looking at manpages-dev 3.25-1 on
Debian/squeeze.

Bye!
  -richy.
--
Richard B. Kreckel
<http://www.ginac.de/~kreckel/>

This doesn't have seem to have been fixed in the Linux man pages, does it?

May I suggest a non-mathematical approach to moving forward?

Just write a little C program and compare the complex results of
a) the library implementation of catan, cacos, etc.,
b) the formula written in the man page, and
c) the formula I proposed above.
If you include a couple of values from all four quadrants, you'll see that my formula always agrees with the implementation while the one from the man page doesn't.

Is there anything else I can do to convince you that this should be fixed? If yes, pretty please, let me know! I would really appreciate getting this fixed.

Best wishes
  -richy.
-- 
Richard B. Kreckel
<http://www.ginac.de/~kreckel/>
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