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+/**
+ * @license Fraction.js v4.3.7 31/08/2023
+ * https://www.xarg.org/2014/03/rational-numbers-in-javascript/
+ *
+ * Copyright (c) 2023, Robert Eisele (robert@raw.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 => <numerator>, 1 => <denominator> ]
+ * [ n => <numerator>, 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 InvalidParameter();
+ }
+ return n * s;
+ }
+
+ // Creates a new Fraction internally without the need of the bulky constructor
+ function newFraction(n, d) {
+
+ if (d === 0) {
+ throw 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 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 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 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 InvalidParameter();
+ }
+
+ if (d === 0) {
+ throw 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']);
+ }
+ }
+
+ var DivisionByZero = function() { return new Error("Division by Zero"); };
+ var InvalidParameter = function() { return new Error("Invalid argument"); };
+ var NonIntegerParameter = function() { return 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 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);
+ },
+
+ /**
+ * Rounds a rational number to a multiple of another rational number
+ *
+ * Ex: new Fraction('0.9').roundTo("1/8") => 7 / 8
+ **/
+ "roundTo": function(a, b) {
+
+ /*
+ k * x/y ≤ a/b < (k+1) * x/y
+ ⇔ k ≤ a/b / (x/y) < (k+1)
+ ⇔ k = floor(a/b * y/x)
+ */
+
+ parse(a, b);
+
+ return newFraction(this['s'] * Math.round(this['n'] * P['d'] / (this['d'] * P['n'])) * P['n'], P['d']);
+ },
+
+ /**
+ * 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 (Math.abs(s['sub'](thisABS).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 exports === "object") {
+ Object.defineProperty(exports, "__esModule", { 'value': true });
+ exports['default'] = Fraction;
+ module['exports'] = Fraction;
+ } else {
+ root['Fraction'] = Fraction;
+ }
+
+})(this);
generated by cgit v1.2.3 (git 2.25.1) at 2025-12-06 14:48:59 +0000