use Maxima's trigrat() in symbolic simplify

[rws]

The ask page ​http://ask.sagemath.org/question/11365/simplify-trigonometric-expression/ showed that trigrat() is not used. What a waste.

[rws]

Please review.

---

New commits:

​3db6f89

17065: do trigrat after trigexpand; doctest

[nbruin]

Quite some doctest failures on the buildbot. Some of them are probably not due this particular change, but others most probably are. A few are simply *better* answers now (that's just a matter of changing the expected output), but there are also some that are much *worse*. So the change is not uniformly an improvement.

ratsimp claims "canonical form", so that's definitely attractive. However, do we know if its rewrites always apply across the whole domain? We'd have to document if it deviates.

[rws]

Since setting "needs_work" makes the buildbot link disappear, and I'm too stupid to find the results otherwise (I tried), I cannot sort this out without help, sorry.

[kcrisman]

This can even lead to disaster, I think:

sage: F = (4*cos(9*pi/180)^2 - 3)*(4*cos(27*pi/180)^2 - 3)
sage: F.maxima_methods().trigrat()
sqrt(5) + 2*cos(7/10*pi) - 2*cos(1/10*pi) - 2*I*sin(3/5*pi) +
2*I*sin(2/5*pi) + 1

I really am not interested in seeing I turn up in such a simplification, regardless of the "correct" answer.

[nbruin]

Replying to rws:

Since setting "needs_work" makes the buildbot link disappear, and I'm too stupid to find the results otherwise (I tried), I cannot sort this out without help, sorry.

Oh, that's silly. I'm setting it back to "needs review" to make the report visible for now. Hopefully we can have this fixed.

[rws]

Thanks, the link is now here: ​http://build.sagedev.org/trac/builders/trac_builder/builds/1046 so the ticket can be set to the right status.

[rws]

Replying to kcrisman:

This can even lead to disaster, I think:

sage: F = (4*cos(9*pi/180)^2 - 3)*(4*cos(27*pi/180)^2 - 3)
sage: F.maxima_methods().trigrat()
sqrt(5) + 2*cos(7/10*pi) - 2*cos(1/10*pi) - 2*I*sin(3/5*pi) +
2*I*sin(2/5*pi) + 1

I really am not interested in seeing I turn up in such a simplification, regardless of the "correct" answer.

While this one could be resolved by adding a trigreduce to the queue:

sage: F = (4*cos(9*pi/180)^2 - 3)*(4*cos(27*pi/180)^2 - 3)
sage: F.maxima_methods().trigrat()
sqrt(5) + 2*cos(7/10*pi) - 2*cos(1/10*pi) - 2*I*sin(3/5*pi) +
2*I*sin(2/5*pi) + 1
sage: _.maxima_methods().trigreduce()
sqrt(5) - 2*cos(3/10*pi) - 2*cos(1/10*pi) + 1

I cannot see how to fix the doctest failure:

File "src/doc/de/thematische_anleitungen/sage_gymnasium.rst", line 394, in doc.de.thematische_anleitungen.sage_gymnasium
Failed example:
    (sin(x + y)/(log(x) + log(y))).simplify_full()
Expected:
    (cos(y)*sin(x) + cos(x)*sin(y))/log(x*y)
Got:
    ((-I*arctan2(0, x) - I*arctan2(0, y))*sin(x + y) + log(abs(x)*abs(y))*sin(x + y))/(arctan2(0, x)^2 + 2*arctan2(0, x)*arctan2(0, y) + arctan2(0, y)^2 + log(abs(x))^2 + 2*log(abs(x))*log(abs(y)) + log(abs(y))^2)

the minimal case being

sage: log(x).maxima_methods().trigrat()
I*arctan2(0, x) + log(abs(x))

[rws]

On the other hand, replacing trigrat with trigsimp/trigexpand/trigreduce will fail in this case:

sage: sin(1/8*pi)*sin(3/8*pi)*sin(5/8*pi)*sin(7/8*pi)
sin(7/8*pi)*sin(5/8*pi)*sin(3/8*pi)*sin(1/8*pi)
sage: _.maxima_methods().trigreduce()
TypeError                                 Traceback (most recent call last)
/home/ralf/sage/local/lib/python2.7/site-packages/sage/interfaces/interface.pyc in __init__(self, parent, value, is_name, name)
    624                 self._name = parent._create(value, name=name)
    625             except (TypeError, RuntimeError, ValueError) as x:
--> 626                 raise TypeError(x)
    627 
    628     def _latex_(self):

TypeError: ECL says: Unrecognised output from sp1sintcos.

[nbruin]

Replying to rws:

On the other hand, replacing trigrat with trigsimp/trigexpand/trigreduce will fail in this case: [...]

Yes, I noticed that one too. I have confirmed that the same problem arises in Maxima 5.34.1 on SBCL, so it's probably a maxima problem. This is now: ​https://sourceforge.net/p/maxima/bugs/2818/

[rws]

While some of the mentioned problems have been fixed in recent Maxima, this no longer works:

sage: F = (4*cos(9*pi/180)^2 - 3)*(4*cos(27*pi/180)^2 - 3)
sage: F.maxima_methods().trigrat()
sqrt(5) + 2*cos(7/10*pi) - 2*cos(1/10*pi) - 2*I*sin(3/5*pi) +
2*I*sin(2/5*pi) + 1
sage: _.maxima_methods().trigreduce()
sqrt(5) - 2*cos(3/10*pi) - 2*cos(1/10*pi) + 1