Trac #17953: symbolic function args prevent forced conversion of result to numeric

It is generally preferred that the returned result type matches the
argument type when calling a builtin function. For example this is as
expected:
{{{
sage: factorial(8).parent()
Integer Ring
sage: factorial(SR(8)).parent()
Symbolic Ring
sage: exp(0).parent()
Integer Ring
sage: exp(SR(0)).parent()
Symbolic Ring
}}}
but this is not:
{{{
from sage.symbolic.function import BuiltinFunction

class TestFunction(BuiltinFunction):
    def __init__(self):
        BuiltinFunction.__init__(self, "testfun", nargs=2)
    def _eval_(self, n, x, *args, **kwds):
        print (parent(n), parent(x))
        return SR(5)

sage: TestFunction()(SR(1),GF(2)(1))
(Integer Ring, Finite Field of size 2)
5
sage: type(_)
<type 'int'>
}}}
An explanation could be that `factorial` and `exp` derive from
`GinacFunction`.

URL: http://trac.sagemath.org/17953
Reported by: rws
Ticket author(s): Ralf Stephan
Reviewer(s): Marc Mezzarobba

Author: Release Manager

src/sage/functions/log.py

@@ -626,7 +626,7 @@ def _eval_(self, n, z):
             sage: parent(lambert_w(Integer(0)))
             Integer Ring
             sage: parent(lambert_w(e))
-            Integer Ring
+            Symbolic Ring
         """
         if not isinstance(z, Expression):
             if n == 0 and z == 0:

src/sage/functions/special.py

@@ -285,11 +285,10 @@ def _maxima_init_evaled_(self, *args):
         correctly (see :trac:`7557`)::
 
             sage: t = f(1.2+2*I*elliptic_kc(1-.5),.5)
-            sage: t._maxima_init_(maxima)  # abs tol 1e-13
-            '0.88771548861928029 - 1.7301614091485560e-15*%i'
+            sage: maxima(t) # abs tol 1e-13
+            0.88771548861928029 - 1.7301614091485560e-15*%i
             sage: t.n() # abs tol 1e-13
-            0.887715488619280 - 1.79195288804672e-15*I
-
+            0.887715488619280 - 1.73016140914856e-15*I
         """
         args_maxima = []
         for a in args:

src/sage/symbolic/function.pyx

@@ -357,12 +357,14 @@ cdef class Function(SageObject):
             array([ 0.        ,  0.84147098,  0.90929743,  0.14112001, -0.7568025 ])
 
         Symbolic functions evaluate non-exact input numerically, and return
-        symbolic expressions on exact input::
+        symbolic expressions on exact input, or if any input is symbolic::
 
             sage: arctan(1)
             1/4*pi
             sage: arctan(float(1))
             0.7853981633974483
+            sage: type(lambert_w(SR(0)))
+            <type 'sage.symbolic.expression.Expression'>
 
         Precision of the result depends on the precision of the input::
 
@@ -444,7 +446,7 @@ cdef class Function(SageObject):
 
         # if the given input is a symbolic expression, we don't convert it back
         # to a numeric type at the end
-        if len(args) == 1 and parent_c(args[0]) is SR:
+        if any(parent_c(arg) is SR for arg in args):
             symbolic_input = True
         else:
             symbolic_input = False