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/**
 * @license Fraction.js v4.2.0 05/03/2022
 * https://www.xarg.org/2014/03/rational-numbers-in-javascript/
 *
 * Copyright (c) 2021, Robert Eisele (robert@xarg.org)
 * Dual licensed under the MIT or GPL Version 2 licenses.
 **/

/**
 *
 * This class offers the possibility to calculate fractions.
 * You can pass a fraction in different formats. Either as array, as double, as string or as an integer.
 *
 * Array/Object form
 * [ 0 => <nominator>, 1 => <denominator> ]
 * [ n => <nominator>, d => <denominator> ]
 *
 * Integer form
 * - Single integer value
 *
 * Double form
 * - Single double value
 *
 * String form
 * 123.456 - a simple double
 * 123/456 - a string fraction
 * 123.'456' - a double with repeating decimal places
 * 123.(456) - synonym
 * 123.45'6' - a double with repeating last place
 * 123.45(6) - synonym
 *
 * Example:
 *
 * var f = new Fraction("9.4'31'");
 * f.mul([-4, 3]).div(4.9);
 *
 */

(function(root) {

"use strict";

// Maximum search depth for cyclic rational numbers. 2000 should be more than enough.
  // Example: 1/7 = 0.(142857) has 6 repeating decimal places.
  // If MAX_CYCLE_LEN gets reduced, long cycles will not be detected and toString() only gets the first 10 digits
  var MAX_CYCLE_LEN = 2000;

// Parsed data to avoid calling "new" all the time
  var P = {
    "s": 1,
    "n": 0,
    "d": 1
  };

function assign(n, s) {

if (isNaN(n = parseInt(n, 10))) {
      throw Fraction['InvalidParameter'];
    }
    return n * s;
  }

// Creates a new Fraction internally without the need of the bulky constructor
  function newFraction(n, d) {

if (d === 0) {
      throw Fraction['DivisionByZero'];
    }

var f = Object.create(Fraction.prototype);
    f["s"] = n < 0 ? -1 : 1;

n = n < 0 ? -n : n;

var a = gcd(n, d);

f["n"] = n / a;
    f["d"] = d / a;
    return f;
  }

function factorize(num) {

var factors = {};

var n = num;
    var i = 2;
    var s = 4;

while (s <= n) {

while (n % i === 0) {
        n/= i;
        factors[i] = (factors[i] || 0) + 1;
      }
      s+= 1 + 2 * i++;
    }

if (n !== num) {
      if (n > 1)
        factors[n] = (factors[n] || 0) + 1;
    } else {
      factors[num] = (factors[num] || 0) + 1;
    }
    return factors;
  }

var parse = function(p1, p2) {

var n = 0, d = 1, s = 1;
    var v = 0, w = 0, x = 0, y = 1, z = 1;

var A = 0, B = 1;
    var C = 1, D = 1;

var N = 10000000;
    var M;

if (p1 === undefined || p1 === null) {
      /* void */
    } else if (p2 !== undefined) {
      n = p1;
      d = p2;
      s = n * d;

if (n % 1 !== 0 || d % 1 !== 0) {
        throw Fraction['NonIntegerParameter'];
      }

} else
      switch (typeof p1) {

case "object":
          {
            if ("d" in p1 && "n" in p1) {
              n = p1["n"];
              d = p1["d"];
              if ("s" in p1)
                n*= p1["s"];
            } else if (0 in p1) {
              n = p1[0];
              if (1 in p1)
                d = p1[1];
            } else {
              throw Fraction['InvalidParameter'];
            }
            s = n * d;
            break;
          }
        case "number":
          {
            if (p1 < 0) {
              s = p1;
              p1 = -p1;
            }

if (p1 % 1 === 0) {
              n = p1;
            } else if (p1 > 0) { // check for != 0, scale would become NaN (log(0)), which converges really slow

if (p1 >= 1) {
                z = Math.pow(10, Math.floor(1 + Math.log(p1) / Math.LN10));
                p1/= z;
              }

// Using Farey Sequences
              // http://www.johndcook.com/blog/2010/10/20/best-rational-approximation/

while (B <= N && D <= N) {
                M = (A + C) / (B + D);

if (p1 === M) {
                  if (B + D <= N) {
                    n = A + C;
                    d = B + D;
                  } else if (D > B) {
                    n = C;
                    d = D;
                  } else {
                    n = A;
                    d = B;
                  }
                  break;

} else {

if (p1 > M) {
                    A+= C;
                    B+= D;
                  } else {
                    C+= A;
                    D+= B;
                  }

if (B > N) {
                    n = C;
                    d = D;
                  } else {
                    n = A;
                    d = B;
                  }
                }
              }
              n*= z;
            } else if (isNaN(p1) || isNaN(p2)) {
              d = n = NaN;
            }
            break;
          }
        case "string":
          {
            B = p1.match(/\d+|./g);

if (B === null)
              throw Fraction['InvalidParameter'];

if (B[A] === '-') {// Check for minus sign at the beginning
              s = -1;
              A++;
            } else if (B[A] === '+') {// Check for plus sign at the beginning
              A++;
            }

if (B.length === A + 1) { // Check if it's just a simple number "1234"
              w = assign(B[A++], s);
            } else if (B[A + 1] === '.' || B[A] === '.') { // Check if it's a decimal number

if (B[A] !== '.') { // Handle 0.5 and .5
                v = assign(B[A++], s);
              }
              A++;

// Check for decimal places
              if (A + 1 === B.length || B[A + 1] === '(' && B[A + 3] === ')' || B[A + 1] === "'" && B[A + 3] === "'") {
                w = assign(B[A], s);
                y = Math.pow(10, B[A].length);
                A++;
              }

// Check for repeating places
              if (B[A] === '(' && B[A + 2] === ')' || B[A] === "'" && B[A + 2] === "'") {
                x = assign(B[A + 1], s);
                z = Math.pow(10, B[A + 1].length) - 1;
                A+= 3;
              }

} else if (B[A + 1] === '/' || B[A + 1] === ':') { // Check for a simple fraction "123/456" or "123:456"
              w = assign(B[A], s);
              y = assign(B[A + 2], 1);
              A+= 3;
            } else if (B[A + 3] === '/' && B[A + 1] === ' ') { // Check for a complex fraction "123 1/2"
              v = assign(B[A], s);
              w = assign(B[A + 2], s);
              y = assign(B[A + 4], 1);
              A+= 5;
            }

if (B.length <= A) { // Check for more tokens on the stack
              d = y * z;
              s = /* void */
              n = x + d * v + z * w;
              break;
            }

/* Fall through on error */
          }
        default:
          throw Fraction['InvalidParameter'];
      }

if (d === 0) {
      throw Fraction['DivisionByZero'];
    }

P["s"] = s < 0 ? -1 : 1;
    P["n"] = Math.abs(n);
    P["d"] = Math.abs(d);
  };

function modpow(b, e, m) {

var r = 1;
    for (; e > 0; b = (b * b) % m, e >>= 1) {

if (e & 1) {
        r = (r * b) % m;
      }
    }
    return r;
  }

function cycleLen(n, d) {

for (; d % 2 === 0;
      d/= 2) {
    }

for (; d % 5 === 0;
      d/= 5) {
    }

if (d === 1) // Catch non-cyclic numbers
      return 0;

// If we would like to compute really large numbers quicker, we could make use of Fermat's little theorem:
    // 10^(d-1) % d == 1
    // However, we don't need such large numbers and MAX_CYCLE_LEN should be the capstone,
    // as we want to translate the numbers to strings.

var rem = 10 % d;
    var t = 1;

for (; rem !== 1; t++) {
      rem = rem * 10 % d;

if (t > MAX_CYCLE_LEN)
        return 0; // Returning 0 here means that we don't print it as a cyclic number. It's likely that the answer is `d-1`
    }
    return t;
  }

function cycleStart(n, d, len) {

var rem1 = 1;
    var rem2 = modpow(10, len, d);

for (var t = 0; t < 300; t++) { // s < ~log10(Number.MAX_VALUE)
      // Solve 10^s == 10^(s+t) (mod d)

if (rem1 === rem2)
        return t;

rem1 = rem1 * 10 % d;
      rem2 = rem2 * 10 % d;
    }
    return 0;
  }

function gcd(a, b) {

if (!a)
      return b;
    if (!b)
      return a;

while (1) {
      a%= b;
      if (!a)
        return b;
      b%= a;
      if (!b)
        return a;
    }
  };

/**
   * Module constructor
   *
   * @constructor
   * @param {number|Fraction=} a
   * @param {number=} b
   */
  function Fraction(a, b) {

parse(a, b);

if (this instanceof Fraction) {
      a = gcd(P["d"], P["n"]); // Abuse variable a
      this["s"] = P["s"];
      this["n"] = P["n"] / a;
      this["d"] = P["d"] / a;
    } else {
      return newFraction(P['s'] * P['n'], P['d']);
    }
  }

Fraction['DivisionByZero'] = new Error("Division by Zero");
  Fraction['InvalidParameter'] = new Error("Invalid argument");
  Fraction['NonIntegerParameter'] = new Error("Parameters must be integer");

Fraction.prototype = {

"s": 1,
    "n": 0,
    "d": 1,

/**
     * Calculates the absolute value
     *
     * Ex: new Fraction(-4).abs() => 4
     **/
    "abs": function() {

return newFraction(this["n"], this["d"]);
    },

/**
     * Inverts the sign of the current fraction
     *
     * Ex: new Fraction(-4).neg() => 4
     **/
    "neg": function() {

return newFraction(-this["s"] * this["n"], this["d"]);
    },

/**
     * Adds two rational numbers
     *
     * Ex: new Fraction({n: 2, d: 3}).add("14.9") => 467 / 30
     **/
    "add": function(a, b) {

parse(a, b);
      return newFraction(
        this["s"] * this["n"] * P["d"] + P["s"] * this["d"] * P["n"],
        this["d"] * P["d"]
      );
    },

/**
     * Subtracts two rational numbers
     *
     * Ex: new Fraction({n: 2, d: 3}).add("14.9") => -427 / 30
     **/
    "sub": function(a, b) {

parse(a, b);
      return newFraction(
        this["s"] * this["n"] * P["d"] - P["s"] * this["d"] * P["n"],
        this["d"] * P["d"]
      );
    },

/**
     * Multiplies two rational numbers
     *
     * Ex: new Fraction("-17.(345)").mul(3) => 5776 / 111
     **/
    "mul": function(a, b) {

parse(a, b);
      return newFraction(
        this["s"] * P["s"] * this["n"] * P["n"],
        this["d"] * P["d"]
      );
    },

/**
     * Divides two rational numbers
     *
     * Ex: new Fraction("-17.(345)").inverse().div(3)
     **/
    "div": function(a, b) {

parse(a, b);
      return newFraction(
        this["s"] * P["s"] * this["n"] * P["d"],
        this["d"] * P["n"]
      );
    },

/**
     * Clones the actual object
     *
     * Ex: new Fraction("-17.(345)").clone()
     **/
    "clone": function() {
      return newFraction(this['s'] * this['n'], this['d']);
    },

/**
     * Calculates the modulo of two rational numbers - a more precise fmod
     *
     * Ex: new Fraction('4.(3)').mod([7, 8]) => (13/3) % (7/8) = (5/6)
     **/
    "mod": function(a, b) {

if (isNaN(this['n']) || isNaN(this['d'])) {
        return new Fraction(NaN);
      }

if (a === undefined) {
        return newFraction(this["s"] * this["n"] % this["d"], 1);
      }

parse(a, b);
      if (0 === P["n"] && 0 === this["d"]) {
        throw Fraction['DivisionByZero'];
      }

/*
       * First silly attempt, kinda slow
       *
       return that["sub"]({
       "n": num["n"] * Math.floor((this.n / this.d) / (num.n / num.d)),
       "d": num["d"],
       "s": this["s"]
       });*/

/*
       * New attempt: a1 / b1 = a2 / b2 * q + r
       * => b2 * a1 = a2 * b1 * q + b1 * b2 * r
       * => (b2 * a1 % a2 * b1) / (b1 * b2)
       */
      return newFraction(
        this["s"] * (P["d"] * this["n"]) % (P["n"] * this["d"]),
        P["d"] * this["d"]
      );
    },

/**
     * Calculates the fractional gcd of two rational numbers
     *
     * Ex: new Fraction(5,8).gcd(3,7) => 1/56
     */
    "gcd": function(a, b) {

parse(a, b);

// gcd(a / b, c / d) = gcd(a, c) / lcm(b, d)

return newFraction(gcd(P["n"], this["n"]) * gcd(P["d"], this["d"]), P["d"] * this["d"]);
    },

/**
     * Calculates the fractional lcm of two rational numbers
     *
     * Ex: new Fraction(5,8).lcm(3,7) => 15
     */
    "lcm": function(a, b) {

parse(a, b);

// lcm(a / b, c / d) = lcm(a, c) / gcd(b, d)

if (P["n"] === 0 && this["n"] === 0) {
        return newFraction(0, 1);
      }
      return newFraction(P["n"] * this["n"], gcd(P["n"], this["n"]) * gcd(P["d"], this["d"]));
    },

/**
     * Calculates the ceil of a rational number
     *
     * Ex: new Fraction('4.(3)').ceil() => (5 / 1)
     **/
    "ceil": function(places) {

places = Math.pow(10, places || 0);

if (isNaN(this["n"]) || isNaN(this["d"])) {
        return new Fraction(NaN);
      }
      return newFraction(Math.ceil(places * this["s"] * this["n"] / this["d"]), places);
    },

/**
     * Calculates the floor of a rational number
     *
     * Ex: new Fraction('4.(3)').floor() => (4 / 1)
     **/
    "floor": function(places) {

places = Math.pow(10, places || 0);

if (isNaN(this["n"]) || isNaN(this["d"])) {
        return new Fraction(NaN);
      }
      return newFraction(Math.floor(places * this["s"] * this["n"] / this["d"]), places);
    },

/**
     * Rounds a rational numbers
     *
     * Ex: new Fraction('4.(3)').round() => (4 / 1)
     **/
    "round": function(places) {

places = Math.pow(10, places || 0);

if (isNaN(this["n"]) || isNaN(this["d"])) {
        return new Fraction(NaN);
      }
      return newFraction(Math.round(places * this["s"] * this["n"] / this["d"]), places);
    },

/**
     * Gets the inverse of the fraction, means numerator and denominator are exchanged
     *
     * Ex: new Fraction([-3, 4]).inverse() => -4 / 3
     **/
    "inverse": function() {

return newFraction(this["s"] * this["d"], this["n"]);
    },

/**
     * Calculates the fraction to some rational exponent, if possible
     *
     * Ex: new Fraction(-1,2).pow(-3) => -8
     */
    "pow": function(a, b) {

parse(a, b);

// Trivial case when exp is an integer

if (P['d'] === 1) {

if (P['s'] < 0) {
          return newFraction(Math.pow(this['s'] * this["d"], P['n']), Math.pow(this["n"], P['n']));
        } else {
          return newFraction(Math.pow(this['s'] * this["n"], P['n']), Math.pow(this["d"], P['n']));
        }
      }

// Negative roots become complex
      //     (-a/b)^(c/d) = x
      // <=> (-1)^(c/d) * (a/b)^(c/d) = x
      // <=> (cos(pi) + i*sin(pi))^(c/d) * (a/b)^(c/d) = x         # rotate 1 by 180°
      // <=> (cos(c*pi/d) + i*sin(c*pi/d)) * (a/b)^(c/d) = x       # DeMoivre's formula in Q ( https://proofwiki.org/wiki/De_Moivre%27s_Formula/Rational_Index )
      // From which follows that only for c=0 the root is non-complex. c/d is a reduced fraction, so that sin(c/dpi)=0 occurs for d=1, which is handled by our trivial case.
      if (this['s'] < 0) return null;

// Now prime factor n and d
      var N = factorize(this['n']);
      var D = factorize(this['d']);

// Exponentiate and take root for n and d individually
      var n = 1;
      var d = 1;
      for (var k in N) {
        if (k === '1') continue;
        if (k === '0') {
          n = 0;
          break;
        }
        N[k]*= P['n'];

if (N[k] % P['d'] === 0) {
          N[k]/= P['d'];
        } else return null;
        n*= Math.pow(k, N[k]);
      }

for (var k in D) {
        if (k === '1') continue;
        D[k]*= P['n'];

if (D[k] % P['d'] === 0) {
          D[k]/= P['d'];
        } else return null;
        d*= Math.pow(k, D[k]);
      }

if (P['s'] < 0) {
        return newFraction(d, n);
      }
      return newFraction(n, d);
    },

/**
     * Check if two rational numbers are the same
     *
     * Ex: new Fraction(19.6).equals([98, 5]);
     **/
    "equals": function(a, b) {

parse(a, b);
      return this["s"] * this["n"] * P["d"] === P["s"] * P["n"] * this["d"]; // Same as compare() === 0
    },

/**
     * Check if two rational numbers are the same
     *
     * Ex: new Fraction(19.6).equals([98, 5]);
     **/
    "compare": function(a, b) {

parse(a, b);
      var t = (this["s"] * this["n"] * P["d"] - P["s"] * P["n"] * this["d"]);
      return (0 < t) - (t < 0);
    },

"simplify": function(eps) {

if (isNaN(this['n']) || isNaN(this['d'])) {
        return this;
      }

eps = eps || 0.001;

var thisABS = this['abs']();
      var cont = thisABS['toContinued']();

for (var i = 1; i < cont.length; i++) {

var s = newFraction(cont[i - 1], 1);
        for (var k = i - 2; k >= 0; k--) {
          s = s['inverse']()['add'](cont[k]);
        }

if (s['sub'](thisABS)['abs']().valueOf() < eps) {
          return s['mul'](this['s']);
        }
      }
      return this;
    },

/**
     * Check if two rational numbers are divisible
     *
     * Ex: new Fraction(19.6).divisible(1.5);
     */
    "divisible": function(a, b) {

parse(a, b);
      return !(!(P["n"] * this["d"]) || ((this["n"] * P["d"]) % (P["n"] * this["d"])));
    },

/**
     * Returns a decimal representation of the fraction
     *
     * Ex: new Fraction("100.'91823'").valueOf() => 100.91823918239183
     **/
    'valueOf': function() {

return this["s"] * this["n"] / this["d"];
    },

/**
     * Returns a string-fraction representation of a Fraction object
     *
     * Ex: new Fraction("1.'3'").toFraction(true) => "4 1/3"
     **/
    'toFraction': function(excludeWhole) {

var whole, str = "";
      var n = this["n"];
      var d = this["d"];
      if (this["s"] < 0) {
        str+= '-';
      }

if (d === 1) {
        str+= n;
      } else {

if (excludeWhole && (whole = Math.floor(n / d)) > 0) {
          str+= whole;
          str+= " ";
          n%= d;
        }

str+= n;
        str+= '/';
        str+= d;
      }
      return str;
    },

/**
     * Returns a latex representation of a Fraction object
     *
     * Ex: new Fraction("1.'3'").toLatex() => "\frac{4}{3}"
     **/
    'toLatex': function(excludeWhole) {

var whole, str = "";
      var n = this["n"];
      var d = this["d"];
      if (this["s"] < 0) {
        str+= '-';
      }

if (d === 1) {
        str+= n;
      } else {

if (excludeWhole && (whole = Math.floor(n / d)) > 0) {
          str+= whole;
          n%= d;
        }

str+= "\\frac{";
        str+= n;
        str+= '}{';
        str+= d;
        str+= '}';
      }
      return str;
    },

/**
     * Returns an array of continued fraction elements
     *
     * Ex: new Fraction("7/8").toContinued() => [0,1,7]
     */
    'toContinued': function() {

var t;
      var a = this['n'];
      var b = this['d'];
      var res = [];

if (isNaN(a) || isNaN(b)) {
        return res;
      }

do {
        res.push(Math.floor(a / b));
        t = a % b;
        a = b;
        b = t;
      } while (a !== 1);

return res;
    },

/**
     * Creates a string representation of a fraction with all digits
     *
     * Ex: new Fraction("100.'91823'").toString() => "100.(91823)"
     **/
    'toString': function(dec) {

var N = this["n"];
      var D = this["d"];

if (isNaN(N) || isNaN(D)) {
        return "NaN";
      }

dec = dec || 15; // 15 = decimal places when no repetation

var cycLen = cycleLen(N, D); // Cycle length
      var cycOff = cycleStart(N, D, cycLen); // Cycle start

var str = this['s'] < 0 ? "-" : "";

str+= N / D | 0;

N%= D;
      N*= 10;

if (N)
        str+= ".";

if (cycLen) {

for (var i = cycOff; i--;) {
          str+= N / D | 0;
          N%= D;
          N*= 10;
        }
        str+= "(";
        for (var i = cycLen; i--;) {
          str+= N / D | 0;
          N%= D;
          N*= 10;
        }
        str+= ")";
      } else {
        for (var i = dec; N && i--;) {
          str+= N / D | 0;
          N%= D;
          N*= 10;
        }
      }
      return str;
    }
  };

if (typeof define === "function" && define["amd"]) {
    define([], function() {
      return Fraction;
    });
  } else if (typeof exports === "object") {
    Object.defineProperty(Fraction, "__esModule", { 'value': true });
    Fraction['default'] = Fraction;
    Fraction['Fraction'] = Fraction;
    module['exports'] = Fraction;
  } else {
    root['Fraction'] = Fraction;
  }

})(this);