This is an exercise in the use of types with multiple constructors.
The derivative of an expression is a concept from calculus. The derivative of x (with respect to x) is 1, and the derivative of expression x^2 is 2x. If you are unfamiliar with derivatives, the general rules that are needed here are given below.
Symbolic mathematics packages such as Mathematica and Maple will take derivatives for you. They do so symbolically, in the same way as you do when you are doing it by hand.
For this assignment, you will write a program to compute the derivative of an expression. Do so by modifying the starting point file. It computes derivatives of expressions involving only a variable called xx, constants and addition. The variable is called xx just because it is probably a bad idea to have a global constant called x.
Your assignment is to extend the given program so that it can handle subtraction, multiplication and constant powers.
The rules for taking the derivative a' of an expression a (with respect to x) are as follows. Here, c is a constant and x is the independent variable, called xx in the program.
| c' | = | 0 |
| x' | = | 1 |
| (a+b)' | = | a' + b' |
| (a - b)' | = | a' - b' |
| (ab)' | = | ab' + a'b |
| (ac)' | = | cac-1a' |
The derivative rules can lead to some rather silly looking expressions. If you apply them exactly as stated, you find that the derivative of 2+3*x is 0 + 3*1 + 0*x. It is nice to do some simplifications. A simplification function is provided. You should extend the simplifier as well, to get simplifications involving subtractions, multiplication and constant powers. Do at least the following simplifications.
| y-0 | = | y |
| 0*y | = | 0 |
| y*0 | = | 0 |
| 1*y | = | y |
| y*1 | = | y |
| y^1 | = | y |
Also perform constant arithmetic. For example, your expression should not contain subexpressions that contain only constants, except for simple constants. Expression 3*4 should be simplified to 12.
Note. The program is broken into two parts. deriv.ast implements expressions and derivatives. File testderiv.ast contains a tester. Extend it to do more tests. Be sure to test your new code!
Note. You are only supposed to deal with constant powers. The equation given for the derivative of a power does not work when c is a general expression. You will lose points if your program uses that rule inappropriately.
Note. Internal documentation is important in any program, and it is important that it be correct. When you make modifications, keep the documentation up to date. Failure to do so will cause you to lose points.
Note. Pay attention to what you are writing. I have received submissions for this assignment containing "facts" such as 0*x = x, x0 = x, x0 = 0, 0-x = x and assorted other goodies. (By the way, 00 is undefined. It is not terribly important how your program handles that little problem. If you treat x0 as 1, that is ok.)
Note. To compute a real number to a real power, use operator ^~. It computes powers using logarithms and exponentials. (x^~y = exp(y ln(x)).)
Submit both of your programs, the derivative finder and the tester, in the usual way, using the handin program. Be sure to indicate that the assignment number is 5.