add is_exact method to SR

Author: mforets

src/sage/symbolic/expression.pyx

@@ -2293,6 +2293,61 @@ cdef class Expression(CommutativeRingElement):
         """
         return is_a_relational(self._gobj)
 
+    def is_exact(self):
+        """
+        Return True if this expression only contains exact numerical coefficients.
+
+        EXAMPLES::
+
+            sage: x, y = var('x, y')
+            sage: (x+y-1).is_exact()
+            True
+            sage: (x+y-1.9).is_exact()
+            False
+            sage: x.is_exact()
+            True
+            sage: pi.is_exact()
+            True
+            sage: (sqrt(x-y) - 2*x + 1).is_exact()
+            True
+            sage: ((x-y)^0.5 - 2*x + 1).is_exact()
+            False
+
+        TESTS::
+
+            sage: (sin(x*cos(2*x*pi)) - 10*y^3 - 1/(x+4)).is_exact()
+            True
+            sage: (sin(x*cos(2.0*x*pi)) - 10*y^3 - 1/(x+4)).is_exact()
+            False
+            sage: SR(42).is_exact()
+            True
+            sage: SR(42.01).is_exact()
+            False
+            sage: SR(I).is_exact()
+            True
+            sage: (x-I).is_exact()
+            True
+            sage: (x-CC(0,1)).is_exact()
+            False
+        """
+        # generator over all numerical elements in the subexpression tree of expr
+        def numelems_gen(expr):
+            if expr.is_numeric():
+                yield expr
+            elif expr.operator() is not None:
+                for op in expr.operands():
+                    if op.is_numeric():
+                        yield op
+                    else:
+                        for opp in numelems_gen(op):
+                            yield opp
+        # stop at the first inexact number in the subexpression tree of self,
+        # and if there is no such element, then self is exact
+        for nelem in numelems_gen(self):
+            if not nelem.pyobject().base_ring().is_exact():
+                return False
+        return True
+
     cpdef bint is_infinity(self):
         """
         Return True if self is an infinite expression.