;;;;;
;;;;;
;;;;; Mathematical Functions
;;;;;
;;;;;

(define $abs
  (lambda [$x]
    (if (rational? x)
      (b.abs x)
      x)))

(define $neg
  (lambda [$x]
    (if (rational? x)
      (b.neg x)
      (* -1 x))))

(define $exp
  (lambda [$x]
    (if (float? x)
      (b.exp x)
      (if (term? x)
        (match x term-expr
          {[,0 1]
           [,1 e]
           [<mult $a ,(* i pi)> (** -1 a)]
           [_ (`exp x)]})
        (`exp x)))))

(define $log
  (lambda [$x]
    (if (float? x)
      (b.log x)
      (match x math-expr
        {[,1 0]
         [,e 1]
         [_ (`log x)]}))))

(define $cos
  (lambda [$x]
    (if (float? x)
      (b.cos x)
      (match x math-expr
        {[,0 1]
         [<term $n <cons ,π <nil>>> (** -1 (abs n))]
         [<div <mult _ ,π> ,2> 0]
         [_ (`cos x)]}))))

(define $sin
  (lambda [$x]
    (if (float? x)
      (b.sin x)
      (match x math-expr
        {[,0 0]
         [<mult _ ,π> 0]
         [<div <mult $n ,π> ,2> (** -1 (/ (- (abs n) 1) 2))]
         [_ (`sin x)]}))))

(define $tan
  (lambda [$x]
    (if (float? x)
      (b.tan x)
      (match x math-expr
        {[,0 0]
         [_ (`tan x)]}))))

(define $cosh
  (lambda [$x]
    (if (float? x)
      (b.cosh x)
      (match x math-expr
        {[,0 1]
         [_ (`cosh x)]}))))

(define $sinh
  (lambda [$x]
    (if (float? x)
      (b.sinh x)
      (match x math-expr
        {[,0 0]
         [_ (`sinh x)]}))))

(define $tanh
  (lambda [$x]
    (if (float? x)
      (b.tanh x)
      (match x math-expr
        {[,0 0]
         [_ (`tanh x)]}))))

(define $sinc
  (lambda [$x]
    (if (float? x)
      (if (eq? x 0.0)
        1.0
        (/ (b.sin x) x))
      (match x math-expr
        {[,0 1]
         [_ (/ (sin x) x)]}))))

(define $sigmoid
  (lambda [$z]
    (/ 1 (+ 1 (exp (* -1 z))))))

(define $kronecker-delta
  (cambda $js
    (if (all (eq? $ (car js)) (cdr js)) 1 0)))

(define $euler-totient-function
  (lambda [$n]
    (* n
       (product (map (lambda [$p] (- 1 (/ 1 p)))
                     (unique (p-f n)))))))

(define $ε
  (memoized-lambda [$n]
    (let {[[$es $os] (even-and-odd-permutations' n)]}
      (generate-tensor
        (cambda $is
          (if (member? is es)
            1
            (if (member? is os)
              -1
              0)))
        (take n (repeat1 n))))))

(define $ε'
  (memoized-lambda [$n $k]
    (let {[[$es $os] (even-and-odd-permutations' n)]}
      (generate-tensor
        (cambda $is
          (match (drop k is) (list integer)
            {[<join _ <cons $x <join _ <cons ?1#(lt? %1 x) _>>>> 0]
             [_ (if (member? is es)
                  1
                  (if (member? is os)
                    -1
                    0))]}))
        (take n (repeat1 n))))))