Horner form of symbolic expressions

src/sage/calculus/wester.py

@@ -617,9 +617,8 @@
     sage: # (YES) Convert the above to Horner's form.
     sage: #      Verify(Horner(p, x), ((((a[5]*x+a[4])*x
     sage: #        +a[3])*x+a[2])*x+a[1])*x);
-    sage: # We use the trick of evaluating the algebraic poly at a symbolic variable:
     sage: restore('x')
-    sage: p(x)
+    sage: SR(p).horner(x)
     ((((a4*x + a3)*x + a2)*x + a1)*x + a0)*x
 
 ::

src/sage/symbolic/expression.pyx

@@ -6081,6 +6081,41 @@ cdef class Expression(CommutativeRingElement):
             sig_off()
         return new_Expression_from_GEx(self._parent, x)
 
+    def horner(self, x):
+        """
+        Rewrite this expression as a polynomial in Horner form in ``x``.
+
+        EXAMPLES::
+
+            sage: add((i+1)*x^i for i in range(5)).horner(x)
+            (((5*x + 4)*x + 3)*x + 2)*x + 1
+
+            sage: x, y, z = SR.var('x,y,z')
+            sage: (x^5 + y*cos(x) + z^3 + (x + y)^2 + y^x).horner(x)
+            z^3 + ((x^3 + 1)*x + 2*y)*x + y^2 + y*cos(x) + y^x
+
+            sage: expr = sin(5*x).expand_trig(); expr
+            5*cos(x)^4*sin(x) - 10*cos(x)^2*sin(x)^3 + sin(x)^5
+            sage: expr.horner(sin(x))
+            (5*cos(x)^4 - (10*cos(x)^2 - sin(x)^2)*sin(x)^2)*sin(x)
+            sage: expr.horner(cos(x))
+            sin(x)^5 + 5*(cos(x)^2*sin(x) - 2*sin(x)^3)*cos(x)^2
+
+        TESTS::
+
+            sage: SR(0).horner(x), SR(1).horner(x), x.horner(x)
+            (0, 1, x)
+            sage: (x^(1/3)).horner(x)
+            Traceback (most recent call last):
+            ...
+            ValueError: Cannot return dense coefficient list with noninteger exponents.
+        """
+        coef = self.coefficients(x, sparse=False)
+        res = coef[-1]
+        for c in reversed(coef[:-1]):
+            res = res*x + c
+        return res
+
     def collect_common_factors(self):
         """
         This function does not perform a full factorization but only