type sym_const = Pi | E

type expr =
  | Const of float
  | SymConst of sym_const
  | Var of string
  | Add of expr * expr
  | Sub of expr * expr
  | Mul of expr * expr
  | Div of expr * expr
  | Pow of expr * expr
  | Neg of expr
  | Sin of expr
  | Cos of expr
  | Tan of expr
  | Sinh of expr
  | Cosh of expr
  | Tanh of expr
  | Asin of expr
  | Acos of expr
  | Atan of expr
  | Atan2 of expr * expr
  | Exp of expr
  | Ln of expr
  | Log of expr * expr
  | Sqrt of expr
  | Abs of expr

let rec to_string = function
  | Const f ->
      if Float.is_integer f then
        string_of_int (int_of_float f)
      else
        string_of_float f
  | SymConst Pi -> "π"
  | SymConst E -> "e"
  | Var s -> s
  | Add (e1, e2) -> to_string_add e1 ^ " + " ^ to_string_add e2
  | Sub (e1, e2) -> to_string e1 ^ " - " ^ to_string_paren e2
  | Mul (e1, e2) -> to_string_mul e1 ^ "*" ^ to_string_mul e2
  | Div (e1, e2) -> to_string_div e1 ^ "/" ^ to_string_paren e2
  | Pow (e1, e2) -> to_string_atom e1 ^ "^" ^ to_string_pow_exp e2
  | Neg e -> "-" ^ to_string_atom e
  | Sin e -> "sin(" ^ to_string e ^ ")"
  | Cos e -> "cos(" ^ to_string e ^ ")"
  | Tan e -> "tan(" ^ to_string e ^ ")"
  | Sinh e -> "sinh(" ^ to_string e ^ ")"
  | Cosh e -> "cosh(" ^ to_string e ^ ")"
  | Tanh e -> "tanh(" ^ to_string e ^ ")"
  | Asin e -> "asin(" ^ to_string e ^ ")"
  | Acos e -> "acos(" ^ to_string e ^ ")"
  | Atan e -> "atan(" ^ to_string e ^ ")"
  | Atan2 (e1, e2) -> "atan2(" ^ to_string e1 ^ ", " ^ to_string e2 ^ ")"
  | Exp e -> "e^" ^ to_string_pow_exp e
  | Ln e -> "ln(" ^ to_string e ^ ")"
  | Log (base_e, arg) -> "log(" ^ to_string base_e ^ ", " ^ to_string arg ^ ")"
  | Sqrt e -> "sqrt(" ^ to_string e ^ ")"
  | Abs e -> "abs(" ^ to_string e ^ ")"

and to_string_atom = function
  | (Const _ | SymConst _ | Var _ | Sin _ | Cos _ | Tan _ | Sinh _ | Cosh _ | Tanh _
    | Asin _ | Acos _ | Atan _ | Atan2 _ | Exp _ | Ln _ | Log _ | Sqrt _ | Abs _) as e -> to_string e
  | e -> "(" ^ to_string e ^ ")"

and to_string_mul = function
  | (Const _ | SymConst _ | Var _ | Pow _ | Sin _ | Cos _ | Tan _ | Sinh _ | Cosh _ | Tanh _
    | Asin _ | Acos _ | Atan _ | Atan2 _ | Exp _ | Ln _ | Log _ | Sqrt _ | Abs _ | Mul _) as e -> to_string e
  | e -> "(" ^ to_string e ^ ")"

and to_string_div = function
  | (Const _ | SymConst _ | Var _ | Pow _ | Sin _ | Cos _ | Tan _ | Sinh _ | Cosh _ | Tanh _
    | Asin _ | Acos _ | Atan _ | Atan2 _ | Exp _ | Ln _ | Log _ | Sqrt _ | Abs _ | Mul _) as e -> to_string e
  | e -> "(" ^ to_string e ^ ")"

and to_string_add = function
  | (Add _ | Sub _) as e -> to_string e
  | e -> to_string e

and to_string_pow_exp = function
  | (Const _ | SymConst _ | Var _ | Pow _) as e -> to_string e
  | e -> "(" ^ to_string e ^ ")"

and to_string_paren = function
  | (Add _ | Sub _) as e -> "(" ^ to_string e ^ ")"
  | e -> to_string e