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This section goes beyond the approach presented so far and shows how to develop a UI application that can accept algebraic expressions as input and simplify, differentiate, and render them graphically. Figure 12-2 shows the project structure. Note in particular that you need to add library references to FSharp.PowerPack and System.Windows.Forms. Furthermore, you add the lexer ExprLexer.fsl and the parser ExprParser.fsy to the project so you can quickly edit them if necessary. You can manually include a reference to the PowerPack targets in the project file to automatically generate code for the parser and lexer definitions, or you can manually generate these files. You do the latter for this example, but in both cases you need to add the generated files (ExprParser.fs and ExprLexer.fs) to the project. The main Expr type that represents algebraic expressions is contained in Expr.fs. Although you can use the expression constructors defined in this type to create expression values on the fly, the most convenient method for embedding complex expressions into this representation is by parsing them.

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Figure 12-2. The SymbolicDifferentiation project in Visual Studio 2010 Armed with the ability to encode and parse algebraic expressions, you place the derivation and simplification logic in its own module and file (ExprUtil.fs). The last piece is rendering (in VisualExpr.fs). Finally, a simple UI client in Main.fs completes the application.

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Listing 12-3 shows the definition of the abstract syntax representation of expressions using a single Expr type. Expressions contain numbers, variables, negation, sums, differences, products, fractions, exponents, basic trigonometric functions (sin x, cos x), and ex. Let s look at this abstract syntax design more closely. In 9, you saw that choosing an abstract syntax often involves design choices, and that these choices often relate to the roles the abstract syntax representation should serve. In this case, you use the abstract syntax to compute symbolic derivatives and simplifications (using techniques similar to those earlier in this chapter) and also to graphically visualize the resulting expressions in a way that is pleasant for the human user. For this reason, you don t use an entirely minimalistic abstract syntax (for example, by replacing quotients with an inverse node), because it s helpful to maintain some additional structure in the input. Here, you represent sums and differences not as binary terms (as you do for products and quotients) but instead as a list of expression terms. The Sub term also carries the minuend, the term that is to be reduced, separately. As a result, you have to apply different strategies when simplifying them. Listing 12-3. Expr.fs: The Core Expression Type for the Visual Symbolic Differentiation Application namespace Symbolic.Expressions type Expr = | Num of | Var of | Neg of | Add of | Sub of | Prod of | Frac of | Pow of | Sin of | Cos of | Exp of decimal string Expr Expr list Expr * Expr list Expr * Expr Expr * Expr Expr * decimal Expr Expr Expr

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