expression.hs

-- Expression type
data Expression = Constant Float
                | Variable String
                | Partial String String
                | BinOp String Expression Expression
                | Func String Expression
                deriving Show

-- -- Evaluation
eval e = simplify e

-- -- Pretty Printing
pretty (Variable a)  = a
pretty (Constant a)  = show a
pretty (Func a b)    = a ++ "(" ++ pretty b ++ ")"
pretty (Partial a b) = "d" ++ a ++ "/d" ++ b
pretty (BinOp a b c) = "(" ++ pretty b ++ " " ++ a ++ " " ++ pretty c ++ ")"

-- -- Differentiation
diff (Constant a) _  = Constant 0
diff (Variable a) v  | a == v = Constant 1
                     | a /= v = Partial a v
diff (BinOp a b c) v | a == "+" = BinOp "+" (diff b v) (diff c v)
                     | a == "*" = BinOp "+" (BinOp "*" (diff b v) c) (BinOp "*" b (diff c v))
                     | a == "/" = BinOp "/" (BinOp "+" (BinOp "*" (diff b v) c) (BinOp "*" (Constant (-1)) (BinOp "*" b (diff c v)))) (BinOp "*" c c)
diff (Func a b)   v  = BinOp "*" (diff b v) (Func (a ++ "'") b)

-- -- Simplification
-- Special Cases
simplify (BinOp "*" (Constant 0) _) = Constant 0
simplify (BinOp "*" _ (Constant 0)) = Constant 0
simplify (BinOp "*" (Constant 1) b) = simplify b
simplify (BinOp "*" a (Constant 1)) = simplify a
simplify (BinOp "*" (Constant a) (Constant b)) = Constant (a * b)

simplify (BinOp "+" (Constant 0) b) = simplify b
simplify (BinOp "+" a (Constant 0)) = simplify a
simplify (BinOp "+" (Constant a) (Constant b)) = Constant (a + b)

simplify (BinOp "/" a (Constant 1)) = simplify a
simplify (BinOp "/" (Constant a) (Constant b)) = Constant (a / b)

-- General Cases
simplify (BinOp a (Func b c) d)   = BinOp a (simplify (Func b c)) (simplify d)
simplify (BinOp a b (Func c d))   = BinOp a (simplify b) (simplify (Func c d))
simplify (BinOp a (Variable b) c) = BinOp a (Variable b) (simplify c)
simplify (BinOp a b (Variable c)) = BinOp a (simplify b) (Variable c)
simplify (BinOp a b c)            = simplify (BinOp a (simplify b) (simplify c))

simplify (Func a b)           = Func a (simplify b)

simplify a = a

-- Some example expressions
x     = Variable "x"
y     = Variable "y"
expr1 = BinOp "*" (BinOp "+" (Constant 1) (Constant 6.3)) (BinOp "*" (Constant 3) (Constant 2.4))
expr2 = BinOp "*" (Constant 10) (BinOp "+" (Constant 5) (Constant 1))
expr3 = BinOp "+" (BinOp "+" (BinOp "*" (Constant 5) (Constant 4.2)) (Constant 6.3)) (BinOp "*" (Constant 2) (Constant 3))
expr4 = BinOp "+" (Constant 5) (Constant 6)
fx    = Func "f" (BinOp "*" x x)