Wrong formulae for complex elementary functions

Wrong formulae for complex elementary functions

[Richard B. Kreckel]

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/>
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Re: Wrong formulae for complex elementary functions

[Michael Kerrisk]

Hi Andries,

Since you are the mathematician, can you comment?

Thanks,

Michael

On Fri, Nov 26, 2010 at 9:57 AM, Richard B. Kreckel <kreckel-HEL5OUoDxoc@public.gmane.org> wrote:
> 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/>
>



-- 
Michael Kerrisk
Linux man-pages maintainer; http://www.kernel.org/doc/man-pages/
Author of "The Linux Programming Interface"; http://man7.org/tlpi/
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Re: Wrong formulae for complex elementary functions

[Richard B. Kreckel]

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<kreckel-HEL5OUoDxoc@public.gmane.org>  wrote:
>> 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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Re: Wrong formulae for complex elementary functions

[Michael Kerrisk]

Hello Richard,

On Fri, Aug 5, 2011 at 11:34 PM, Richard B. Kreckel <kreckel-HEL5OUoDxoc@public.gmane.org> wrote:
> 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<kreckel-HEL5OUoDxoc@public.gmane.org>
>>  wrote:
>>>
>>> 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?

Sorry -- no, not yet.

> 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.

Now there's a good idea ;-). It would have been even better if you'd
sent me a program--I don't mess round with complex numbers every day
of the week...

> 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.

So, I eventually ironed out my complex math problems and came up with the below:

===
/* Link with "-lm" */

#include <complex.h>
#include <stdlib.h>
#include <unistd.h>
#include <stdio.h>



int
main(int argc, char *argv[])
{
    double complex z, c, f1, f2;
    double complex i = I;

    if (argc != 3) {
        fprintf(stderr, "Usage: %s <real> <imag>\n");
        exit(EXIT_FAILURE);
    }

    z = atof(argv[1]) + atof(argv[2]) * I;

    c = cacos(z);

    printf("cacos() = %6.3f %6.3f*i\n", creal(c), cimag(c));

    f1 = -i * clog(z + csqrt(z * z - 1));
    f2 = -i * clog(z + i * csqrt(1 - z * z));

    printf("f1      = %6.3f %6.3f*i\n", creal(f1), cimag(f1));
    printf("f2      = %6.3f %6.3f*i\n", creal(f2), cimag(f2));

    exit(EXIT_SUCCESS);
}
===

Some sample runs certainly seem to prove your point for that function at least:

$ ./cacos 1 1
cacos() =  0.905 -1.061*i
f1      =  0.905 -1.061*i
f2      =  0.905 -1.061*i
$ ./cacos -1 -1
cacos() =  2.237  1.061*i
f1      =  2.237  1.061*i
f2      =  2.237  1.061*i
$ ./cacos -1 1
cacos() =  2.237 -1.061*i
f1      = -2.237  1.061*i
f2      =  2.237 -1.061*i
$ ./cacos 1 -1
cacos() =  0.905  1.061*i
f1      = -0.905 -1.061*i
f2      =  0.905  1.061*i

The above is what you meant, right?

Thanks,

Michael

-- 
Michael Kerrisk
Linux man-pages maintainer; http://www.kernel.org/doc/man-pages/
Author of "The Linux Programming Interface"; http://man7.org/tlpi/
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Re: Wrong formulae for complex elementary functions

[Richard B. Kreckel]

Hi,

On 09/15/2011 06:31 AM, Michael Kerrisk wrote:
> The above is what you meant, right?

Yes, that is exactly what I meant.

It should work just like that for the other functions, too.

Let me know if there is anything else I can do.

   -richy.
-- 
Richard B. Kreckel
<http://www.ginac.de/~kreckel/>
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Re: Wrong formulae for complex elementary functions

[Andries E. Brouwer]

On Fri, Aug 05, 2011 at 11:34:54PM +0200, Richard B. Kreckel wrote:
> Hi everybody,
> 
> On 11/27/2010 08:37 AM, Michael Kerrisk wrote:
> >Hi Andries,
> >
> >Since you are the mathematician, can you comment?

Hi everybody,

Now that I see this discussion again:
Mathematically speaking there is no unique correct definitions.
The function is multiple-valued and some arbitrary agreement
fixes one of the values.

Richard's point of view is that the actual behavior of the library functions
should be documented. Not a bad idea, but perhaps one should document what
the standard says, and then file a bug report in case the implementation
does something else.

I lost most of my old docs, but just rediscovered a copy of
ISO-C-FDIS.1999-04.pdf, I don't know whether that still is the
appropriate source to consult.

(i)
It says in 7.3.1: cacos: branch cuts outside [-1,1] along the real axis
returns arccos z where Re arccos z lies in [0,pi].
So, if z = cos w then w = arccos z, but because cos (w + 2pi) = cos w
and cos (-w) = cos w, we can pick w with real part in [0,pi].

It says in 7.3.7.2: clog: branch cut along the negative real axis
returns log z where Im log z lies in [-I*pi,I*pi].

It says in 7.3.8.3: csqrt: branch cut along the negative real axis
returns sqrt z where Im sqrt z >= 0.

We want to check whether it is true that
cacos(z) = -I* clog(z + I*csqrt(1 - z * z))

Suppose w = cacos(z). Then w is defined by cos w = z and Re w in [0,pi].
The question is whether iw = log(cos w + i sqrt(sin^2 w)).
If sin w has nonnegative imaginary part, then sqrt(sin^2 w) = sin w
and the question is whether iw = log(e^{iw}). This logarithm is
chosen with imaginary part in [-pi,pi] but w has imaginary part
in [0,pi], so this equality holds.
If sin w has negative imaginary part, then sqrt(sin^2 w) = -sin w
and the question is whether iw = log(e^{-iw}), and again this holds.

I agree with Richard's equation.

(ii)
Claim: catan(z) = (clog(1 + iz) - clog(1 - iz)) / 2i

7.3.5.3: catan: branch cuts outside [-i,i] along the imaginary axis
returns atan z where Re atan z lies in [-pi/2, pi/2].

Suppose w = catan(z). Then w is defined by tan w = z and Re w in [-pi/2, pi/2].
Up to a multiple of 2pi*i the RHS is
(1/2i)log((cos w + i sin w)/(cos w - i sin w)) = (1/2i) log e^{2iw},
looks good. The two multiples of 2pi*i partially cancel, so after
dividing by 2i this is in [-pi/2, pi/2].

I agree with Richard's equation.

(iii)
Claim: catanh(z) = (clog(1 + z) - clog(1 - z))/2

7.3.6.3: catanh: branch cuts outside [-1,1] along the real axis
returns atanh z where Im atanh z lies in [-pi/2,pi/2].

Suppose w = catanh(z).
Then w is defined by tanh w = z and Re w in [-pi/2,pi/2].
Up to a multiple of 2pi*i the RHS is correct, and just like before
we get the right value.

I agree with Richard's equation.

So, if I make no mistake, Richard is correct on all counts.

Andries
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Re: Wrong formulae for complex elementary functions

[Michael Kerrisk]

Hi Andries!

Nice to hear from you again!

On Thu, Sep 15, 2011 at 6:24 PM, Andries E. Brouwer
<Andries.Brouwer-rh8NL+sEX9E@public.gmane.org> wrote:
> On Fri, Aug 05, 2011 at 11:34:54PM +0200, Richard B. Kreckel wrote:
>> Hi everybody,
>>
>> On 11/27/2010 08:37 AM, Michael Kerrisk wrote:
>> >Hi Andries,
>> >
>> >Since you are the mathematician, can you comment?
>
> Hi everybody,
>
> Now that I see this discussion again:
> Mathematically speaking there is no unique correct definitions.
> The function is multiple-valued and some arbitrary agreement
> fixes one of the values.
>
> Richard's point of view is that the actual behavior of the library functions
> should be documented. Not a bad idea, but perhaps one should document what
> the standard says, and then file a bug report in case the implementation
> does something else.
>
> I lost most of my old docs, but just rediscovered a copy of
> ISO-C-FDIS.1999-04.pdf, I don't know whether that still is the
> appropriate source to consult.
>
> (i)
> It says in 7.3.1: cacos: branch cuts outside [-1,1] along the real axis
> returns arccos z where Re arccos z lies in [0,pi].
> So, if z = cos w then w = arccos z, but because cos (w + 2pi) = cos w
> and cos (-w) = cos w, we can pick w with real part in [0,pi].
>
> It says in 7.3.7.2: clog: branch cut along the negative real axis
> returns log z where Im log z lies in [-I*pi,I*pi].
>
> It says in 7.3.8.3: csqrt: branch cut along the negative real axis
> returns sqrt z where Im sqrt z >= 0.
>
> We want to check whether it is true that
> cacos(z) = -I* clog(z + I*csqrt(1 - z * z))
>
> Suppose w = cacos(z). Then w is defined by cos w = z and Re w in [0,pi].
> The question is whether iw = log(cos w + i sqrt(sin^2 w)).
> If sin w has nonnegative imaginary part, then sqrt(sin^2 w) = sin w
> and the question is whether iw = log(e^{iw}). This logarithm is
> chosen with imaginary part in [-pi,pi] but w has imaginary part
> in [0,pi], so this equality holds.
> If sin w has negative imaginary part, then sqrt(sin^2 w) = -sin w
> and the question is whether iw = log(e^{-iw}), and again this holds.
>
> I agree with Richard's equation.
>
> (ii)
> Claim: catan(z) = (clog(1 + iz) - clog(1 - iz)) / 2i
>
> 7.3.5.3: catan: branch cuts outside [-i,i] along the imaginary axis
> returns atan z where Re atan z lies in [-pi/2, pi/2].
>
> Suppose w = catan(z). Then w is defined by tan w = z and Re w in [-pi/2, pi/2].
> Up to a multiple of 2pi*i the RHS is
> (1/2i)log((cos w + i sin w)/(cos w - i sin w)) = (1/2i) log e^{2iw},
> looks good. The two multiples of 2pi*i partially cancel, so after
> dividing by 2i this is in [-pi/2, pi/2].
>
> I agree with Richard's equation.
>
> (iii)
> Claim: catanh(z) = (clog(1 + z) - clog(1 - z))/2
>
> 7.3.6.3: catanh: branch cuts outside [-1,1] along the real axis
> returns atanh z where Im atanh z lies in [-pi/2,pi/2].
>
> Suppose w = catanh(z).
> Then w is defined by tanh w = z and Re w in [-pi/2,pi/2].
> Up to a multiple of 2pi*i the RHS is correct, and just like before
> we get the right value.
>
> I agree with Richard's equation.
>
> So, if I make no mistake, Richard is correct on all counts.

Thanks very much for the above analysis. I'm happy to have a second
mathematician confirm what the first one says,

Best regards,

Michael


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Author of "The Linux Programming Interface"; http://man7.org/tlpi/
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Re: Wrong formulae for complex elementary functions

[Michael Kerrisk]

Hello Richard,

On Thu, Sep 15, 2011 at 6:31 AM, Michael Kerrisk <mtk.manpages-Re5JQEeQqe8@public.gmane.orgm> wrote:
> Hello Richard,
>
> On Fri, Aug 5, 2011 at 11:34 PM, Richard B. Kreckel <kreckel-HEL5OUoDxoc@public.gmane.org> wrote:
>> 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<kreckel@ginac.de>
>>>  wrote:
>>>>
>>>> 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?
>
> Sorry -- no, not yet.
>
>> 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.
>
> Now there's a good idea ;-). It would have been even better if you'd
> sent me a program--I don't mess round with complex numbers every day
> of the week...
>
>> 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.
>
> So, I eventually ironed out my complex math problems and came up with the below:
>
> ===
> /* Link with "-lm" */
>
> #include <complex.h>
> #include <stdlib.h>
> #include <unistd.h>
> #include <stdio.h>
>
>
>
> int
> main(int argc, char *argv[])
> {
>    double complex z, c, f1, f2;
>    double complex i = I;
>
>    if (argc != 3) {
>        fprintf(stderr, "Usage: %s <real> <imag>\n");
>        exit(EXIT_FAILURE);
>    }
>
>    z = atof(argv[1]) + atof(argv[2]) * I;
>
>    c = cacos(z);
>
>    printf("cacos() = %6.3f %6.3f*i\n", creal(c), cimag(c));
>
>    f1 = -i * clog(z + csqrt(z * z - 1));
>    f2 = -i * clog(z + i * csqrt(1 - z * z));
>
>    printf("f1      = %6.3f %6.3f*i\n", creal(f1), cimag(f1));
>    printf("f2      = %6.3f %6.3f*i\n", creal(f2), cimag(f2));
>
>    exit(EXIT_SUCCESS);
> }
> ===
>
> Some sample runs certainly seem to prove your point for that function at least:
>
> $ ./cacos 1 1
> cacos() =  0.905 -1.061*i
> f1      =  0.905 -1.061*i
> f2      =  0.905 -1.061*i
> $ ./cacos -1 -1
> cacos() =  2.237  1.061*i
> f1      =  2.237  1.061*i
> f2      =  2.237  1.061*i
> $ ./cacos -1 1
> cacos() =  2.237 -1.061*i
> f1      = -2.237  1.061*i
> f2      =  2.237 -1.061*i
> $ ./cacos 1 -1
> cacos() =  0.905  1.061*i
> f1      = -0.905 -1.061*i
> f2      =  0.905  1.061*i
>
> The above is what you meant, right?
>
> Thanks,
>
> Michael

For man-pages-3.33, I have applied the changes below. They should be
consistent with what you proposed (and Andries confirmed). Could you
please carefully check...

I think it's not so necessary to add a warning to the page source.
Nowadays we have version control and logs, so the reasoning should be
easy to find.

Thanks,

Michael


diff --git a/man3/cacos.3 b/man3/cacos.3
index b483a0c..1ae6ff5 100644
--- a/man3/cacos.3
+++ b/man3/cacos.3
@@ -27,7 +27,7 @@ is chosen in the interval [0,pi].
 One has:
 .nf

-    cacos(z) = \-i clog(z + csqrt(z * z \- 1))
+    cacos(z) = \-i * clog(z + i * csqrt(1 \- z * z))
 .fi
 .SH VERSIONS
 These functions first appeared in glibc in version 2.1.
diff --git a/man3/cacosh.3 b/man3/cacosh.3
index 7bf60c9..bf3d99b 100644
--- a/man3/cacosh.3
+++ b/man3/cacosh.3
@@ -30,7 +30,7 @@ is chosen nonnegative.
 One has:
 .nf

-    cacosh(z) = (0.5) * clog((1 + z) / (1 \- z))
+    cacosh(z) = 2 * clog(csqrt((z + 1) / 2) + csqrt((z \- 1) / 2))
 .fi
 .SH VERSIONS
 These functions first appeared in glibc in version 2.1.
diff --git a/man3/catan.3 b/man3/catan.3
index 30692bb..fe8c39e 100644
--- a/man3/catan.3
+++ b/man3/catan.3
@@ -25,7 +25,7 @@ The real part of y is chosen in the interval [\-pi/2,pi/2].
 One has:
 .nf

-    catan(z) = 1 / 2i clog((1 + iz) / (1 \- iz))
+    catan(z) = (clog(1 + i * z) \- clog(1 \- i * z)) / (2 * i)
 .fi
 .SH VERSIONS
 These functions first appeared in glibc in version 2.1.
diff --git a/man3/catanh.3 b/man3/catanh.3
index e396021..4f2f230 100644
--- a/man3/catanh.3
+++ b/man3/catanh.3
@@ -27,7 +27,7 @@ is chosen in the interval [\-pi/2,pi/2].
 One has:
 .nf

-    catanh(z) = 0.5 * clog((1 + z) / (1 \- z))
+    catanh(z) = 0.5 * (clog(1 + z) \- clog(1 \- z))
 .fi
 .SH VERSIONS
 These functions first appeared in glibc in version 2.1.



-- 
Michael Kerrisk
Linux man-pages maintainer; http://www.kernel.org/doc/man-pages/
Author of "The Linux Programming Interface"; http://man7.org/tlpi/
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Re: Wrong formulae for complex elementary functions

[Richard B. Kreckel]

Hi!

On 09/15/2011 08:38 PM, Michael Kerrisk wrote:
> For man-pages-3.33, I have applied the changes below. They should be
> consistent with what you proposed (and Andries confirmed). Could you
> please carefully check...

Yes, the patch looks correct. Thanks a lot for applying it.

(I would have loved to try it but couldn't find the git repository for 
man-pages.)

> I think it's not so necessary to add a warning to the page source.
> Nowadays we have version control and logs, so the reasoning should be
> easy to find.

Certainly. Out of curiousity: Does the version control of man-pages date 
back enough to tell when and why the formulas got written the wrong way?

Bye,
   -richy.
-- 
Richard B. Kreckel
<http://www.ginac.de/~kreckel/>
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Re: Wrong formulae for complex elementary functions

[Michael Kerrisk]

Hi Richard,

On Thu, Sep 15, 2011 at 11:42 PM, Richard B. Kreckel <kreckel-HEL5OUoDxoc@public.gmane.org> wrote:
> Hi!
>
> On 09/15/2011 08:38 PM, Michael Kerrisk wrote:
>>
>> For man-pages-3.33, I have applied the changes below. They should be
>> consistent with what you proposed (and Andries confirmed). Could you
>> please carefully check...
>
> Yes, the patch looks correct. Thanks a lot for applying it.
>
> (I would have loved to try it but couldn't find the git repository for
> man-pages.)

Well, with the current state of affairs
(http://lwn.net/Articles/457142/), no one can find it...

>> I think it's not so necessary to add a warning to the page source.
>> Nowadays we have version control and logs, so the reasoning should be
>> easy to find.
>
> Certainly. Out of curiousity: Does the version control of man-pages date
> back enough to tell when and why the formulas got written the wrong way?

Unfortunately not. Version control dates back to the start of my watch
(Nov 2004), and those pages were added prior to that. Looking in old
tarballs, it seems the the error was there in the initial versions of
these pages.

Cheers,

Michael


-- 
Michael Kerrisk
Linux man-pages maintainer; http://www.kernel.org/doc/man-pages/
Author of "The Linux Programming Interface"; http://man7.org/tlpi/
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Re: Wrong formulae for complex elementary functions

[D. Barbier]

On 2011/9/16 Michael Kerrisk wrote:
> On Thu, Sep 15, 2011 at 11:42 PM, Richard B. Kreckel wrote:
[...]
>> (I would have loved to try it but couldn't find the git repository for
>> man-pages.)
>
> Well, with the current state of affairs
> (http://lwn.net/Articles/457142/), no one can find it...
[...]

Hello Michael,

Could you please push your repository somewhere so that we can start
translating your changes?  No need for a full remote repository, a
tarball of your .git directory would be fine.
Thanks.

Denis
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