The symbolic types of Mathemagix should understood to be mainly syntactic. For instance, we systematically use the rules

			(1)
			(2)

which can be incorrect when , and are complex numbers, but which are correct in a suitable differential algebra. It should be noticed that equality testing is syntactic: we do not require that equality in all algebraic models can be detected syntactically (even though our normal forms should do as good as a job as possible).

Some of the variants for symbolic expressions implement additional heuristic methods for zero-testing. This sometimes gives rise to massive simplifications, but may lead to incorrect results. More precisely, the following kinds of incorrectness can be encountered:

1. The semantics if coherent in itself but not compatible with the syntactic semantics described above. For instance, we may use evaluation at points for heuristic zero-tests, for which the rules (?) and (?) may be violated.

2. When using a strict ball evaluation policy, we may safely detect inequalities (modulo the above remark). However, many expressions, such as , typically evaluate to a ball containing the whole space, and will be assimilated to zero by a zero-test.

3. When replacing subexpressions such as by new variables, a positive answer to a zero-test usually means that the expression is really zero. However, it gets easier to miss certain identities, since only part of the algebraic/analytic structure is preserved.

Even though the symbolic types do not rely on the evaluator, function application (apply) is not a completely trivial operation. First of all, we have implemented arithmetic on functions:

The standard functions , , , etc. also receive special treatment. For instance, the expression ('exp + 1)(x) evaluates to e^x + 1.

The internal representation and the printed form do not need to coincide. For instance, sums can be kept in a rather expanded form; when printing them, some factoring may take place in order to enhance readability. Of course, such factoring would be to expensive during computations, but may be have a minor cost when printing.