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CHAPTER 9 INTRODUCING LANGUAGE-ORIENTED PROGRAMMING
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Introducing Active Patterns
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Pattern matching is one of the key techniques provided in F# for decomposing abstract syntax trees and other abstract representations of languages. So far in this book, all the examples of pattern matching have been directly over the core representations of data structures: for example, directly matching on the structure of lists, options, records, and discriminated unions. But pattern matching in F# is also extensible that is, you can define new ways of matching over existing types. You do this through a mechanism called active patterns. This book covers only the basics of active patterns. However, they can be indispensable, because they can let you continue to use pattern matching with your types even after you hide their representations. Active patterns also let you use pattern matching with .NET object types. The following section covers active patterns and how they work.
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In high-school math courses, you were probably taught that you can view complex numbers in two ways: as rectangular coordinates x + yi or as polar coordinates of a phase r and magnitude . In most computer systems, complex numbers are stored in the first format, although often the second format is more useful. Wouldn t it be nice if you could look at complex numbers through either lens You could do this by explicitly converting from one form to another when needed, but it would be better to have your programming language look after the transformations needed to do this for you. Active patterns let you do exactly that. First, here is a standard definition of complex numbers: [<Struct>] type Complex(r: float, i: float) = static member Polar(mag,phase) = Complex(mag * cos phase, mag * sin phase) member x.Magnitude = sqrt(r*r + i*i) member x.Phase = atan2 i r member x.RealPart = r member x.ImaginaryPart = i Here is a pattern that lets you view complex numbers as rectangular coordinates: let (|Rect|) (x:Complex) = (x.RealPart, x.ImaginaryPart) And here is an active pattern to help you view complex numbers in polar coordinates: let (|Polar|) (x:Complex) = (x.Magnitude, x.Phase) The key thing to note is that these definitions let you use Rect and Polar as tags in pattern matching. For example, you can now write the following to define addition and multiplication over complex numbers: let addViaRect a b = match a, b with | Rect (ar, ai), Rect (br, bi) -> Complex (ar+br, ai+bi) let mulViaRect a b = match a, b with | Rect (ar, ai), Rect (br, bi) -> Complex (ar*br - ai*bi, ai*br + bi*ar)
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CHAPTER 9 INTRODUCING LANGUAGE-ORIENTED PROGRAMMING
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As it happens, multiplication on complex numbers is easier to express using polar coordinates, implemented as follows: let mulViaPolar a b = match a, b with | Polar (m, p), Polar (n, q) -> Complex.Polar (m*n, p+q) Here is an example of using the (|Rect|) and (|Polar|) active patterns directly on some complex numbers via the pattern tags Rect and Polar. You first make the complex number 3+4i: > let c = Complex (3.0, 4.0);; val c : complex > c;; val it : complex = 3.0r+4.0i > match c with | Rect (x, y) -> printfn "x = %g, y = %g" x y;; x = 3, y = 4 val it : unit = () match c with | Polar (x, y) -> printfn "x = %g, y = %g" x y;; x = 5.0, y = 0.927295 val it : unit = () > addViaRect c c;; val it : complex = 6.0r+8.0i > mulViaRect c c;; val it : complex = -7.0r+24.0i > mulViaPolar c c;; val it : complex = -7.0r+24.0i As you may expect, you get the same results if you multiply via rectangular or polar coordinates. However, the execution paths are quite different. Let s look closely at the definition of mulViaRect. The important lines are in bold here: let mulViaRect a b = match a, b with | Rect (ar, ai), Rect (br, bi) -> Complex (ar*br - ai*bi, ai*br + bi*ar) When F# needs to match the values a and b against the patterns Rect (ar, ai) and Rect (br, bi), it doesn t look at the contents of a and b directly. Instead, it runs a function as part of pattern matching (which is why they re called active patterns). In this case, the function executed is (|Rect|), which produces a pair as its result. The elements of the pair are then bound to the variables ar and ai. Likewise, in the definition of mulViaPolar, the matching is performed partly by running the function (|Polar|). The functions (|Rect|) and (|Polar|) are allowed to do anything, as long as each ultimately produces a pair of results. Here are the types of (|Rect|) and (|Polar|): >
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