// This file:
// Copyright (c) 2013 Takeru Suzuki
// Licensed under the MIT license.
// https://github.com/terkel/mathsass

// Constants
$E: 2.718281828459045;
$PI: 3.141592653589793;
$LN2: 0.6931471805599453;
$SQRT2: 1.4142135623730951;

@function error($message) {
  @warn "#{_error("The direction used does not exist")}";
  @return null;
}

// Returns the factorial of a non-negative integer.
// @param {Number} $x A non-negative integer.
// @return {Number}
// @example
//     fact(0) // 1
//     fact(8) // 40320
@function fact($x) {
  @if $x < 0 or $x != floor($x) {
    @warn "Argument for `fact()` must be a positive integer.";
    @return null;
  }
  $ret: 1;
  @while $x > 0 {
    $ret: $ret * $x;
    $x: $x - 1;
  }
  @return $ret;
}

// Returns a two-element list containing the normalized fraction and exponent of number.
// @param {Number} $x
// @return {List} fraction, exponent
@function frexp($x) {
  $exp: 0;
  @if $x < 0 {
    $x: $x * -1;
  }
  @if $x < 0.5 {
    @while $x < 0.5 {
      $x: $x * 2;
      $exp: $exp - 1;
    }
  } @else if $x >= 1 {
    @while $x >= 1 {
      $x: $x / 2;
      $exp: $exp + 1;
    }
  }
  @return $x, $exp;
}

// Returns $x * 2^$exp
// @param {Number} $x
// @param {Number} $exp
@function ldexp($x, $exp) {
  $b: if($exp >= 0, 2, 1 / 2);
  @if $exp < 0 {
    $exp: $exp * -1;
  }
  @while $exp > 0 {
    @if $exp % 2 == 1 {
      $x: $x * $b;
    }
    $b: $b * $b;
    $exp: floor($exp * 0.5);
  }
  @return $x;
}

// Returns the natural logarithm of a number.
// @param {Number} $x
// @example
//     log(2)  // 0.69315
//     log(10) // 2.30259
@function log($x) {
  @if $x <= 0 {
    @return 0 / 0;
  }
  $k: nth(frexp($x / $SQRT2), 2);
  $x: $x / ldexp(1, $k);
  $x: ($x - 1) / ($x + 1);
  $x2: $x * $x;
  $i: 1;
  $s: $x;
  $sp: null;
  @while $sp != $s {
    $x: $x * $x2;
    $i: $i + 2;
    $sp: $s;
    $s: $s + $x / $i;
  }
  @return $LN2 * $k + 2 * $s;
}

@function ipow($base, $exp) {
  @if $exp != floor($exp) {
    @return error("Exponent for `ipow()` must be an integer.");
  }
  $r: 1;
  $s: 0;
  @if $exp < 0 {
    $exp: $exp * -1;
    $s: 1;
  }
  @while $exp > 0 {
    @if $exp % 2 == 1 {
      $r: $r * $base;
    }
    $exp: floor($exp * 0.5);
    $base: $base * $base;
  }
  @return if($s != 0, 1 / $r, $r);
}

// Returns E^x, where x is the argument, and E is Euler's constant, the base of the natural logarithms.
// @param {Number} $x
// @example
//     exp(1)  // 2.71828
//     exp(-1) // 0.36788
@function exp($x) {
  $ret: 0;
  @for $n from 0 to 24 {
    $ret: $ret + ipow($x, $n) / fact($n);
  }
  @return $ret;
}

// Returns base to the exponent power.
// @param {Number} $base The base number
// @param {Number} $exp The exponent to which to raise base
// @return {Number}
// @example
//     pow(4, 2)   // 16
//     pow(4, -2)  // 0.0625
//     pow(4, 0.2) // 1.31951
@function pow($base, $exp) {
  @if $exp == floor($exp) {
    @return ipow($base, $exp);
  } @else {
    @return exp(log($base) * $exp);
  }
}

// Returns the square root of a number.
// @param {Number} $x
// @example
//     sqrt(2) // 1.41421
//     sqrt(5) // 2.23607
@function sqrt($x) {
  @if $x < 0 {
    @return error("Argument for `sqrt()` must be a positive number.");
  }
  $ret: 1;
  @for $i from 1 through 24 {
    $ret: $ret - (pow($ret, 2) - $x) / (2 * $ret);
  }
  @return $ret;
}