;; Solution to cubic equations by Sigh_

;;

;; Solves equations of the type ax^3 + bx^2 +cx + d = 0 (a != 0)

;;

;; Usage:

;; $cubic(a,b,c,d)

;; or

;; /cubic a b c d

alias cubic {
  var %i = $calc((3*$3 /$1 -$2 ^2/$1 ^2)/3),%x = $calc((2*$2 ^3 /$1 ^3-9*$2 $&
    *$3 /$1 ^2+27*$4 /$1)/27),%n = $calc(%x ^2 /4+%i ^3/27),%r = $iif($isid,return,echo -a)
  if (%n > 0) {
    var %s = $cbrt($calc($sqrt(%n)-%x /2)),%u = $cbrt($calc(-$sqrt(%n)-%x /2)), $&
      %p = $x($calc($2 /-3/$1 -(%s +%u)/2)),%t = $iif($calc(3^0.5*(%s -%u)/2) != 1,$x($v1))
    %r $x($calc(%s +%u -$2 /3/$1)) , %p + %t $+ i , %p - %t $+ i
    return
  }
  var %k = $calc((%x ^2/4-%n)^0.5),%j = -1 * $cbrt(%k),%t = $acos($calc(%x $&
    /-2/%k)) / 3,%m = $cos(%t),%b = $sqrt(3) * $sin(%t),%p = $calc($2 /-3/$1))
  %r $x($calc(-2*%j *%m -$2 /3/$1)) , $x($calc(%j *(%m +%b)+%p)) , $x($calc(%j *(%m -%b)+%p))
}

alias -l cbrt return $iif($1 < 0,-) $+ $calc($abs($1)^(1/3))

alias -l x return $round($1,3)