__math.scss

// Some code is my own
// Some code is coming from https://github.com/adambom/Sass-Math/blob/master/math.scss
// Some code is coming from http://thesassway.com/advanced/inverse-trigonometric-functions-with-sass
// Some code coming from http://thesassway.com/advanced/math-sequences-with-sass

@function power($base, $exponent) {
    $ret: 1;
    @if $exponent > 0 {
        @for $i from 1 through $exponent {
            $ret: $ret * $base;
        }
    } @else if $exponent < 0 {
        @for $i from $exponent to 0 {
            $ret: $ret / $base;
        }
    }
    @return $ret;
}

@function factorial ($x) {
    $ret: 1;
    @if $x > 0 {
        @while $x > 0 {
            $ret: $ret * $x;
            
            $x: $x - 1;
        }
    }
    @return $ret;
}

@function sin ($x) {
    $ret: 0;
    @for $n from 0 to 25 {
        $ret: $ret + power(-1, $n) * power($x, 2 * $n + 1) / factorial(2 * $n + 1);
    }

    @return $ret;
}

@function cos ($x) {
    $ret: 0;
    @for $n from 0 to 25 {
        $ret: $ret + power(-1, $n) * power($x, 2 * $n) / factorial(2 * $n);
    }
    @return $ret;
}

@function exp ($x) {
    $ret: 0;
    @for $n from 0 to 25 {
        $ret: $ret + power($x, $n) / factorial($n);
    }
    @return $ret;
}

@function ln($x) {
    $ret: 0;
    $n: 1;
    $dx: .001;
    @while $n <= $x {
        $ret: $ret + $dx / $n;
        $n: $n + $dx;
    }
    @return $ret;
}

@function sqrt($x) {    
    @return exp(0.5 * ln($x));
}

$default-threshold: pi()/180/10;

@function convert-angle($value, $unit-name) {
    $factors: (
        rad: 1rad,
        deg: 180deg/pi(),
        grad: 200grad/pi(),
        turn: .5turn/pi()
    );

    @if not unitless($value) {
        @warn '`#{$value}` should be unitless';
        @return false;
    }

    @if not map-has-key($factors, $unit-name) {
        @warn 'unit `#{$unit-name}` is not a valid unit - please make sure it is either `deg`, `rad`, `grad` or `turn`';
        @return false;
    }

    @return $value*map-get($factors, $unit-name);
}

@function asin($z, $unit-name: deg, $threshold: $default-threshold) {
    $sum: 0;
    $complement: false;
    $sign: $z/abs($z);
    $z: abs($z);

    @if $z > sin(pi()/4) {
        $complement: true;
        $z: sqrt(1 - pow($z, 2));
    }

    $term: $z;
    $i: 0;
    $k: 1;

    @while $term > $threshold {
        $sum: $sum + $term;

        $i: $i + 1;
        $k: $k*(2*$i - 1)/(2*$i);
        $j: 2*$i + 1;

        $term: $k*pow($z, $j)/$j;
    }
    @return convert-angle($sign*(if($complement, pi()/2 - $sum, $sum)), $unit-name);
}

@function acos($z, $unit-name: deg, $threshold: $default-threshold) {
    @return convert-angle(pi()/2, $unit-name) - asin($z, $unit-name, $threshold);
}

@function atan($z, $unit-name: deg, $threshold: $default-threshold) {
    @return asin($z/sqrt(1 + pow($z, 2)), $unit-name, $threshold);
}

@function roundfloat($v, $d) {
    $d : power(10,$d);
    @return round($v * $d) / $d;
}

@function fibonacci($n) {
    $fib: 0 1;
    @for $i from 1 through $n {
        $new: nth($fib, length($fib)) + nth($fib, length($fib) - 1);
        $fib: append($fib, $new);
    }
    @return $fib;
}

@function juggler($n) {
    $juggler: ($n);
    @while nth($juggler, length($juggler)) != 1 {
        $last    : nth($juggler, length($juggler));
        $x       : if($last % 2 == 0, 1/2, 3/2);
        $new     : pow($last, $x);
        $juggler : append($juggler, $new);
    }
    @return $juggler;
}