4.8 Sums, products and powers

Simple rational expressions are written down in GiNaC pretty much like in other CAS or like expressions involving numerical variables in C. The necessary operators +, -, * and / have been overloaded to achieve this goal. When you run the following code snippet, the constructor for an object of type mul is automatically called to hold the product of a and b and then the constructor for an object of type add is called to hold the sum of that mul object and the number one:

         ...
         symbol a("a"), b("b");
         ex MyTerm = 1+a*b;
         ...

For exponentiation, you have already seen the somewhat clumsy (though C-ish) statement pow(x,2); to represent x squared. This direct construction is necessary since we cannot safely overload the constructor ^ in C++ to construct a power object. If we did, it would have several counterintuitive and undesired effects:

• Due to C's operator precedence, 2*x^2 would be parsed as (2*x)^2.
• Due to the binding of the operator ^, x^a^b would result in (x^a)^b. This would be confusing since most (though not all) other CAS interpret this as x^(a^b).
• Also, expressions involving integer exponents are very frequently used, which makes it even more dangerous to overload ^ since it is then hard to distinguish between the semantics as exponentiation and the one for exclusive or. (It would be embarrassing to return 1 where one has requested 2^3.)

All effects are contrary to mathematical notation and differ from the way most other CAS handle exponentiation, therefore overloading ^ is ruled out for GiNaC's C++ part. The situation is different in ginsh, there the exponentiation-^ exists. (Also note that the other frequently used exponentiation operator ** does not exist at all in C++).

To be somewhat more precise, objects of the three classes described here, are all containers for other expressions. An object of class power is best viewed as a container with two slots, one for the basis, one for the exponent. All valid GiNaC expressions can be inserted. However, basic transformations like simplifying pow(pow(x,2),3) to x^6 automatically are only performed when this is mathematically possible. If we replace the outer exponent three in the example by some symbols a, the simplification is not safe and will not be performed, since a might be 1/2 and x negative.

Objects of type add and mul are containers with an arbitrary number of slots for expressions to be inserted. Again, simple and safe simplifications are carried out like transforming 3*x+4-x to 2*x+4.