Trac #17556: Move simplify_log() from simplify_full() to simplify_real()

The `simplify_log` function assumes that its argument is real; otherwise
the log contraction operation is invalid. For example,

{{{
sage: x,y = SR.var('x,y')
sage: assume(y, 'complex')
sage: f = log(x*y) - (log(x) + log(y))
sage: f(x=-1, y=i)
-2*I*pi
sage: f.simplify_log()
0
}}}

Now that we have a `simplify_real()` method, why not move
`simplify_log()` there instead?

In the process, the newer `simplify_rectform()` can be added to
`simplify_full()`.

I'm working on a patch for this.

URL: http://trac.sagemath.org/17556
Reported by: mjo
Ticket author(s): Michael Orlitzky
Reviewer(s): Ralf Stephan

src/doc/de/thematische_anleitungen/sage_gymnasium.rst

@@ -386,16 +386,22 @@ wir dies mit der ``simplify_full()`` Funktion::
     sage: (sin(x)^2 + cos(x)^2).simplify_full()
     1
 
-Dabei werden auch Additionstheoreme für trigonometrische Funktionen und manche
-Logarithmengesetze eingesetzt::
+Dabei werden auch Additionstheoreme für trigonometrische Funktionen eingesetzt::
 
     sage: var('x, y, z')
     (x, y, z)
-    sage: (sin(x + y)/(log(x) + log(y))).simplify_full()
-    (cos(y)*sin(x) + cos(x)*sin(y))/log(x*y)
+    sage: sin(x + y).simplify_full()
+    cos(y)*sin(x) + cos(x)*sin(y)
     sage: (sin(x)^2 + cos(x)^2).simplify_full()
     1
 
+Mit der verwandten Funktion ``simplify_real()`` werden auch Additionstheoreme
+bei Logarithmen angewandt, die nur mit reellen Werten erlaubt sind::
+
+    sage: x, y = var('x, y')
+    sage: (log(x) + log(y)).simplify_real()
+    log(x*y)
+
 Faktorisieren und ausmultiplizieren
 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 

src/sage/modules/vector_symbolic_dense.py

@@ -17,7 +17,7 @@
     sage: type(u)
     <class 'sage.modules.vector_symbolic_dense.Vector_symbolic_dense'>
     sage: u.simplify_full()
-    (1, log(6*y^2))
+    (1, log(3*y) + log(2*y))
 
 TESTS:
 

src/sage/symbolic/expression.pyx

@@ -8239,13 +8239,28 @@ cdef class Expression(CommutativeRingElement):
             1/2*log(2*t) - 1/2*log(t)
             sage: forget()
 
+        Complex logs are not contracted, :trac:`17556`::
+
+            sage: x,y = SR.var('x,y')
+            sage: assume(y, 'complex')
+            sage: f = log(x*y) - (log(x) + log(y))
+            sage: f.simplify_full()
+            log(x*y) - log(x) - log(y)
+            sage: forget()
+
+        The simplifications from :meth:`simplify_rectform` are
+        performed, :trac:`17556`::
+
+            sage: f = ( e^(I*x) - e^(-I*x) ) / ( I*e^(I*x) + I*e^(-I*x) )
+            sage: f.simplify_full()
+            sin(x)/cos(x)
+
         """
         x = self
         x = x.simplify_factorial()
+        x = x.simplify_rectform()
         x = x.simplify_trig()
         x = x.simplify_rational()
-        x = x.simplify_log('one')
-        x = x.simplify_rational()
         return x
 
     full_simplify = simplify_full
@@ -8395,7 +8410,8 @@ cdef class Expression(CommutativeRingElement):
     def simplify_real(self):
         r"""
         Simplify the given expression over the real numbers. This allows
-        the simplification of `\sqrt{x^{2}}` into `\left|x\right|`.
+        the simplification of `\sqrt{x^{2}}` into `\left|x\right|` and
+        the contraction of `\log(x) + \log(y)` into `\log(xy)`.
 
         INPUT:
 
@@ -8412,6 +8428,13 @@ cdef class Expression(CommutativeRingElement):
             sage: f.simplify_real()
             abs(x)
 
+        ::
+
+            sage: y = SR.var('y')
+            sage: f = log(x) + 2*log(y)
+            sage: f.simplify_real()
+            log(x*y^2)
+
         TESTS:
 
         We set the Maxima ``domain`` variable to 'real' before we call
@@ -8477,7 +8500,9 @@ cdef class Expression(CommutativeRingElement):
         for v in self.variables():
             assume(v, 'real');
 
-        result = self.simplify();
+        # This will round trip through Maxima, essentially performing
+        # self.simplify() in the process.
+        result = self.simplify_log()
 
         # Set the domain back to what it was before we were called.
         maxima.eval('domain: %s$' % original_domain)
@@ -9021,8 +9046,6 @@ cdef class Expression(CommutativeRingElement):
             log((sqrt(2) + 1)*(sqrt(2) - 1))
             sage: _.simplify_rational()
             0
-            sage: log_expr.simplify_full()   # applies both simplify_log and simplify_rational
-            0
 
         We should use the current simplification domain rather than
         set it to 'real' explicitly (:trac:`12780`)::