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Expression.elm
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Expression.elm
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module Expression
( Expression(..)
, Environment
, evaluate
, evaluateExn
, parse
, derivative
, optimize
, toString
, variables
) where
{-| A type for simple mathematical expressions.
# Types
@docs Expression, Environment
# Parsing
@docs parse
# Evaluation
@docs evaluate, evaluateExn
# Differentiation
@docs derivative
-}
import Dict
import Set
import Maybe
import Native.ParseExpression
-- TODO: Add sin, cos, tan, cosh, sinh, tanh
type Expression
= Var String
| Constant Float
| Mul Expression Expression
| Add Expression Expression
| Exp Float Expression
| Pow Expression Float
| LogBase Float Expression
| Sin Expression
| Cos Expression
{-| An environment of values for variables to be used in evaluating an expression. -}
type alias Environment = Dict.Dict String Float
{-| Parse an expression from a string. -}
parse : String -> Result String Expression
parse = Native.ParseExpression.parse
toString : Expression -> String
toString =
let parenthesize x = "(" ++ x ++ ")"
in \expr ->
case expr of
Var x ->
x
Constant x ->
Basics.toString x
Add e1 e2 ->
parenthesize (toString e1 ++ " + " ++ toString e2)
Mul e1 e2 ->
parenthesize (toString e1 ++ " * " ++ toString e2)
Exp b e1 ->
parenthesize ((if b < 0 then parenthesize else identity) (Basics.toString b) ++ "^" ++ toString e1)
Pow e1 p ->
parenthesize (toString e1 ++ "^" ++ Basics.toString p)
LogBase b e1 ->
if b == e
then "log(" ++ toString e1 ++ ")"
else "log(" ++ Basics.toString b ++ "," ++ toString e1 ++ ")"
Sin e1 ->
"sin(" ++ toString e1 ++ ")"
Cos e1 ->
"cos(" ++ toString e1 ++ ")"
maybeMap2 : (a -> b -> c) -> Maybe a -> Maybe b -> Maybe c
maybeMap2 f ma mb =
case ma of
Just a ->
case mb of
Just b -> Just (f a b)
Nothing -> Nothing
Nothing -> Nothing
{-| Evaluate an expression in the given environment, returning `Nothing` if a variable is not found. -}
evaluate : Expression -> (Environment -> Maybe Float)
evaluate expr env =
case expr of
Var x ->
Dict.get x env
Constant c ->
Just c
Mul e1 e2 ->
maybeMap2 (*) (evaluate e1 env) (evaluate e2 env)
Add e1 e2 ->
maybeMap2 (+) (evaluate e1 env) (evaluate e2 env)
LogBase b e1 ->
Maybe.map (logBase b) (evaluate e1 env)
Exp b e1 ->
Maybe.map (\x -> b ^ x) (evaluate e1 env)
Pow e1 p ->
Maybe.map (\x -> x ^ p) (evaluate e1 env)
Sin e1 ->
Maybe.map sin (evaluate e1 env)
Cos e1 ->
Maybe.map cos (evaluate e1 env)
{-| Evaluate an expression in the given environment, throwing an exception if a variable is not found. -}
evaluateExn : Expression -> (Environment -> Float)
evaluateExn expr env =
case expr of
Var x ->
case Dict.get x env of { Just a -> a }
Constant c ->
c
Mul e1 e2 ->
evaluateExn e1 env * evaluateExn e2 env
Add e1 e2 ->
evaluateExn e1 env + evaluateExn e2 env
LogBase b e1 ->
logBase b (evaluateExn e1 env)
Exp b e1 ->
b ^ (evaluateExn e1 env)
Pow e1 p ->
(evaluateExn e1 env) ^ p
Sin e1 ->
sin (evaluateExn e1 env)
Cos e1 ->
cos (evaluateExn e1 env)
{-| Take the derivative of an expression with respect to the given variable. -}
derivative : String -> Expression -> Expression
derivative x expr =
case expr of
Var y ->
if y == x then Constant 1 else Constant 0
Constant _ -> Constant 0
Mul e1 e2 ->
Add (Mul e1 (derivative x e2)) (Mul (derivative x e1) e2)
Add e1 e2 ->
Add (derivative x e1) (derivative x e2)
LogBase b e1 ->
let d = derivative x e1 in
Mul d (Pow (Mul (Constant (logBase e b)) e1) (-1))
Exp b e1 ->
let d = derivative x e1 in
Mul (Constant (logBase e b)) (Mul d expr)
Pow e1 p ->
Mul (derivative x e1)
(Mul (Constant p) (Pow e1 (p - 1)))
Sin e1 ->
Mul (derivative x e1) (Cos e1)
Cos e1 ->
Mul (Constant (-1))
(Mul (derivative x e1) (Sin e1))
-- TODO: Do a proper optimization at some point with putting things into a normal
-- form.
optimize : Expression -> Expression
optimize expr =
case expr of
Var _ ->
expr
Constant _ ->
expr
Mul e1 e2 ->
case (optimize e1, optimize e2) of
(Constant 0, _) ->
Constant 0
(_, Constant 0) ->
Constant 0
(Constant x, Constant y) ->
Constant (x * y)
(Constant 1, e2') ->
e2'
(e1', Constant 1) ->
e1'
(e1', e2') ->
Mul e1' e2'
Add e1 e2 ->
case (optimize e1, optimize e2) of
(Constant 0, e2') ->
e2'
(e1', Constant 0) ->
e1'
(Constant x, Constant y) ->
Constant (x + y)
(e1', e2') ->
Add e1' e2'
LogBase b e1 ->
case optimize e1 of
Constant x ->
Constant (logBase b x)
e1' ->
LogBase b e1'
Exp b e1 ->
case optimize e1 of
Constant x ->
Constant (b ^ x)
e1' ->
Exp b e1'
Pow e1 p ->
case optimize e1 of
Constant x ->
Constant (x ^ p)
e1' ->
Pow e1' p
Sin e1 ->
case optimize e1 of
Constant x ->
Constant (sin x)
e1' ->
Sin e1'
Cos e1 ->
case optimize e1 of
Constant x ->
Constant (cos x)
e1' ->
Cos e1'
variables : Expression -> Set.Set String
variables =
let
go acc expr =
case expr of
Var x -> Set.insert x acc
Constant _ -> acc
Mul e1 e2 -> go (go acc e1) e2
Add e1 e2 -> go (go acc e1) e2
Exp _ e1 -> go acc e1
Pow e1 _ -> go acc e1
LogBase _ e1 -> go acc e1
Sin e1 -> go acc e1
Cos e1 -> go acc e1
in
go Set.empty
{-
envName : String
envName = "__elmexpressionenvironment"
compileToString : Expression -> String
compileToString =
let parenthesize x = "(" ++ x ++ ")"
\e ->
case e of
Var x ->
"__elmexpressionenvironment" ++ "[" ++ toString x ++ "]"
Constant x ->
parenthesize (toString x)
Mul e1 e2 ->
parenthesize (compileToString e1 ++ "*" ++ compileToString e2)
Add e1 e2 ->
parenthesize (compileToString e1 ++ "+" ++ compileToString e2)
LogBase b e1 ->
parenthesize ("Math.log(" ++ compileToString e1 ++ ") / Math.log(" ++ toString b ++ ")")
Exp b e1 ->
"Math.pow(" ++ toString b ++ "," ++ compileToString e1 ++ ")"
Pow e1 p ->
"Math.pow(" ++ compileToString e1 ++ "," ++ toString p ++ ")"
-- Would be type:
-- Native.CompileExpression : String -> (Object -> Float)
compileExpressionString : String -> (Dict String Float -> Float)
compileExpressionString s =
-}