1044 lines
24 KiB
JavaScript
1044 lines
24 KiB
JavaScript
'use strict';
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/**
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*
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* This class offers the possibility to calculate fractions.
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* You can pass a fraction in different formats. Either as array, as double, as string or as an integer.
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*
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* Array/Object form
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* [ 0 => <numerator>, 1 => <denominator> ]
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* { n => <numerator>, d => <denominator> }
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*
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* Integer form
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* - Single integer value as BigInt or Number
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*
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* Double form
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* - Single double value as Number
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*
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* String form
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* 123.456 - a simple double
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* 123/456 - a string fraction
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* 123.'456' - a double with repeating decimal places
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* 123.(456) - synonym
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* 123.45'6' - a double with repeating last place
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* 123.45(6) - synonym
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*
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* Example:
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* let f = new Fraction("9.4'31'");
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* f.mul([-4, 3]).div(4.9);
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*
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*/
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// Set Identity function to downgrade BigInt to Number if needed
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if (typeof BigInt === 'undefined') BigInt = function (n) { if (isNaN(n)) throw new Error(""); return n; };
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const C_ZERO = BigInt(0);
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const C_ONE = BigInt(1);
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const C_TWO = BigInt(2);
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const C_THREE = BigInt(3);
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const C_FIVE = BigInt(5);
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const C_TEN = BigInt(10);
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const MAX_INTEGER = BigInt(Number.MAX_SAFE_INTEGER);
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// Maximum search depth for cyclic rational numbers. 2000 should be more than enough.
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// Example: 1/7 = 0.(142857) has 6 repeating decimal places.
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// If MAX_CYCLE_LEN gets reduced, long cycles will not be detected and toString() only gets the first 10 digits
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const MAX_CYCLE_LEN = 2000;
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// Parsed data to avoid calling "new" all the time
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const P = {
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"s": C_ONE,
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"n": C_ZERO,
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"d": C_ONE
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};
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function assign(n, s) {
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try {
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n = BigInt(n);
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} catch (e) {
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throw InvalidParameter();
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}
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return n * s;
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}
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function ifloor(x) {
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return typeof x === 'bigint' ? x : Math.floor(x);
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}
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// Creates a new Fraction internally without the need of the bulky constructor
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function newFraction(n, d) {
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if (d === C_ZERO) {
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throw DivisionByZero();
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}
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const f = Object.create(Fraction.prototype);
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f["s"] = n < C_ZERO ? -C_ONE : C_ONE;
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n = n < C_ZERO ? -n : n;
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const a = gcd(n, d);
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f["n"] = n / a;
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f["d"] = d / a;
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return f;
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}
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const FACTORSTEPS = [C_TWO * C_TWO, C_TWO, C_TWO * C_TWO, C_TWO, C_TWO * C_TWO, C_TWO * C_THREE, C_TWO, C_TWO * C_THREE]; // repeats
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function factorize(n) {
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const factors = Object.create(null);
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if (n <= C_ONE) {
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factors[n] = C_ONE;
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return factors;
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}
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const add = (p) => { factors[p] = (factors[p] || C_ZERO) + C_ONE; };
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while (n % C_TWO === C_ZERO) { add(C_TWO); n /= C_TWO; }
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while (n % C_THREE === C_ZERO) { add(C_THREE); n /= C_THREE; }
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while (n % C_FIVE === C_ZERO) { add(C_FIVE); n /= C_FIVE; }
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// 30-wheel trial division: test only residues coprime to 2*3*5
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// Residue step pattern after 5: 7,11,13,17,19,23,29,31, ...
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for (let si = 0, p = C_TWO + C_FIVE; p * p <= n;) {
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while (n % p === C_ZERO) { add(p); n /= p; }
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p += FACTORSTEPS[si];
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si = (si + 1) & 7; // fast modulo 8
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}
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if (n > C_ONE) add(n);
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return factors;
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}
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const parse = function (p1, p2) {
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let n = C_ZERO, d = C_ONE, s = C_ONE;
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if (p1 === undefined || p1 === null) { // No argument
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/* void */
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} else if (p2 !== undefined) { // Two arguments
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if (typeof p1 === "bigint") {
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n = p1;
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} else if (isNaN(p1)) {
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throw InvalidParameter();
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} else if (p1 % 1 !== 0) {
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throw NonIntegerParameter();
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} else {
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n = BigInt(p1);
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}
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if (typeof p2 === "bigint") {
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d = p2;
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} else if (isNaN(p2)) {
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throw InvalidParameter();
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} else if (p2 % 1 !== 0) {
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throw NonIntegerParameter();
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} else {
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d = BigInt(p2);
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}
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s = n * d;
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} else if (typeof p1 === "object") {
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if ("d" in p1 && "n" in p1) {
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n = BigInt(p1["n"]);
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d = BigInt(p1["d"]);
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if ("s" in p1)
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n *= BigInt(p1["s"]);
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} else if (0 in p1) {
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n = BigInt(p1[0]);
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if (1 in p1)
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d = BigInt(p1[1]);
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} else if (typeof p1 === "bigint") {
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n = p1;
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} else {
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throw InvalidParameter();
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}
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s = n * d;
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} else if (typeof p1 === "number") {
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if (isNaN(p1)) {
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throw InvalidParameter();
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}
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if (p1 < 0) {
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s = -C_ONE;
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p1 = -p1;
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}
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if (p1 % 1 === 0) {
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n = BigInt(p1);
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} else {
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let z = 1;
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let A = 0, B = 1;
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let C = 1, D = 1;
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let N = 10000000;
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if (p1 >= 1) {
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z = 10 ** Math.floor(1 + Math.log10(p1));
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p1 /= z;
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}
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// Using Farey Sequences
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while (B <= N && D <= N) {
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let M = (A + C) / (B + D);
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if (p1 === M) {
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if (B + D <= N) {
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n = A + C;
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d = B + D;
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} else if (D > B) {
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n = C;
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d = D;
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} else {
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n = A;
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d = B;
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}
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break;
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} else {
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if (p1 > M) {
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A += C;
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B += D;
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} else {
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C += A;
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D += B;
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}
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if (B > N) {
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n = C;
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d = D;
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} else {
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n = A;
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d = B;
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}
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}
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}
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n = BigInt(n) * BigInt(z);
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d = BigInt(d);
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}
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} else if (typeof p1 === "string") {
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let ndx = 0;
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let v = C_ZERO, w = C_ZERO, x = C_ZERO, y = C_ONE, z = C_ONE;
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let match = p1.replace(/_/g, '').match(/\d+|./g);
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if (match === null)
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throw InvalidParameter();
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if (match[ndx] === '-') {// Check for minus sign at the beginning
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s = -C_ONE;
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ndx++;
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} else if (match[ndx] === '+') {// Check for plus sign at the beginning
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ndx++;
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}
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if (match.length === ndx + 1) { // Check if it's just a simple number "1234"
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w = assign(match[ndx++], s);
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} else if (match[ndx + 1] === '.' || match[ndx] === '.') { // Check if it's a decimal number
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if (match[ndx] !== '.') { // Handle 0.5 and .5
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v = assign(match[ndx++], s);
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}
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ndx++;
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// Check for decimal places
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if (ndx + 1 === match.length || match[ndx + 1] === '(' && match[ndx + 3] === ')' || match[ndx + 1] === "'" && match[ndx + 3] === "'") {
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w = assign(match[ndx], s);
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y = C_TEN ** BigInt(match[ndx].length);
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ndx++;
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}
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// Check for repeating places
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if (match[ndx] === '(' && match[ndx + 2] === ')' || match[ndx] === "'" && match[ndx + 2] === "'") {
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x = assign(match[ndx + 1], s);
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z = C_TEN ** BigInt(match[ndx + 1].length) - C_ONE;
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ndx += 3;
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}
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} else if (match[ndx + 1] === '/' || match[ndx + 1] === ':') { // Check for a simple fraction "123/456" or "123:456"
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w = assign(match[ndx], s);
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y = assign(match[ndx + 2], C_ONE);
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ndx += 3;
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} else if (match[ndx + 3] === '/' && match[ndx + 1] === ' ') { // Check for a complex fraction "123 1/2"
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v = assign(match[ndx], s);
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w = assign(match[ndx + 2], s);
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y = assign(match[ndx + 4], C_ONE);
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ndx += 5;
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}
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if (match.length <= ndx) { // Check for more tokens on the stack
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d = y * z;
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s = /* void */
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n = x + d * v + z * w;
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} else {
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throw InvalidParameter();
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}
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} else if (typeof p1 === "bigint") {
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n = p1;
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s = p1;
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d = C_ONE;
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} else {
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throw InvalidParameter();
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}
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if (d === C_ZERO) {
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throw DivisionByZero();
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}
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P["s"] = s < C_ZERO ? -C_ONE : C_ONE;
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P["n"] = n < C_ZERO ? -n : n;
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P["d"] = d < C_ZERO ? -d : d;
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};
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function modpow(b, e, m) {
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let r = C_ONE;
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for (; e > C_ZERO; b = (b * b) % m, e >>= C_ONE) {
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if (e & C_ONE) {
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r = (r * b) % m;
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}
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}
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return r;
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}
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function cycleLen(n, d) {
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for (; d % C_TWO === C_ZERO;
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d /= C_TWO) {
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}
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for (; d % C_FIVE === C_ZERO;
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d /= C_FIVE) {
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}
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if (d === C_ONE) // Catch non-cyclic numbers
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return C_ZERO;
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// If we would like to compute really large numbers quicker, we could make use of Fermat's little theorem:
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// 10^(d-1) % d == 1
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// However, we don't need such large numbers and MAX_CYCLE_LEN should be the capstone,
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// as we want to translate the numbers to strings.
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let rem = C_TEN % d;
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let t = 1;
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for (; rem !== C_ONE; t++) {
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rem = rem * C_TEN % d;
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if (t > MAX_CYCLE_LEN)
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return C_ZERO; // Returning 0 here means that we don't print it as a cyclic number. It's likely that the answer is `d-1`
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}
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return BigInt(t);
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}
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function cycleStart(n, d, len) {
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let rem1 = C_ONE;
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let rem2 = modpow(C_TEN, len, d);
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for (let t = 0; t < 300; t++) { // s < ~log10(Number.MAX_VALUE)
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// Solve 10^s == 10^(s+t) (mod d)
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if (rem1 === rem2)
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return BigInt(t);
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rem1 = rem1 * C_TEN % d;
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rem2 = rem2 * C_TEN % d;
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}
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return 0;
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}
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function gcd(a, b) {
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if (!a)
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return b;
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if (!b)
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return a;
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while (1) {
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a %= b;
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if (!a)
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return b;
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b %= a;
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if (!b)
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return a;
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}
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}
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/**
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* Module constructor
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*
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* @constructor
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* @param {number|Fraction=} a
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* @param {number=} b
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*/
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function Fraction(a, b) {
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parse(a, b);
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if (this instanceof Fraction) {
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a = gcd(P["d"], P["n"]); // Abuse a
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this["s"] = P["s"];
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this["n"] = P["n"] / a;
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this["d"] = P["d"] / a;
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} else {
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return newFraction(P['s'] * P['n'], P['d']);
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}
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}
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const DivisionByZero = function () { return new Error("Division by Zero"); };
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const InvalidParameter = function () { return new Error("Invalid argument"); };
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const NonIntegerParameter = function () { return new Error("Parameters must be integer"); };
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Fraction.prototype = {
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"s": C_ONE,
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"n": C_ZERO,
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"d": C_ONE,
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/**
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* Calculates the absolute value
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*
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* Ex: new Fraction(-4).abs() => 4
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**/
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"abs": function () {
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return newFraction(this["n"], this["d"]);
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},
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/**
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* Inverts the sign of the current fraction
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*
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* Ex: new Fraction(-4).neg() => 4
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**/
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"neg": function () {
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return newFraction(-this["s"] * this["n"], this["d"]);
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},
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/**
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* Adds two rational numbers
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*
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* Ex: new Fraction({n: 2, d: 3}).add("14.9") => 467 / 30
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**/
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"add": function (a, b) {
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parse(a, b);
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return newFraction(
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this["s"] * this["n"] * P["d"] + P["s"] * this["d"] * P["n"],
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this["d"] * P["d"]
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);
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},
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/**
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* Subtracts two rational numbers
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*
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* Ex: new Fraction({n: 2, d: 3}).add("14.9") => -427 / 30
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**/
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"sub": function (a, b) {
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parse(a, b);
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return newFraction(
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this["s"] * this["n"] * P["d"] - P["s"] * this["d"] * P["n"],
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this["d"] * P["d"]
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);
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},
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/**
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* Multiplies two rational numbers
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*
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* Ex: new Fraction("-17.(345)").mul(3) => 5776 / 111
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**/
|
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"mul": function (a, b) {
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parse(a, b);
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return newFraction(
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this["s"] * P["s"] * this["n"] * P["n"],
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this["d"] * P["d"]
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);
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},
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/**
|
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* Divides two rational numbers
|
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*
|
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* Ex: new Fraction("-17.(345)").inverse().div(3)
|
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**/
|
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"div": function (a, b) {
|
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|
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parse(a, b);
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return newFraction(
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this["s"] * P["s"] * this["n"] * P["d"],
|
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this["d"] * P["n"]
|
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);
|
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},
|
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|
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/**
|
||
* Clones the actual object
|
||
*
|
||
* Ex: new Fraction("-17.(345)").clone()
|
||
**/
|
||
"clone": function () {
|
||
return newFraction(this['s'] * this['n'], this['d']);
|
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},
|
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|
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/**
|
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* Calculates the modulo of two rational numbers - a more precise fmod
|
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*
|
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* Ex: new Fraction('4.(3)').mod([7, 8]) => (13/3) % (7/8) = (5/6)
|
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* Ex: new Fraction(20, 10).mod().equals(0) ? "is Integer"
|
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**/
|
||
"mod": function (a, b) {
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|
||
if (a === undefined) {
|
||
return newFraction(this["s"] * this["n"] % this["d"], C_ONE);
|
||
}
|
||
|
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parse(a, b);
|
||
if (C_ZERO === P["n"] * this["d"]) {
|
||
throw DivisionByZero();
|
||
}
|
||
|
||
/**
|
||
* I derived the rational modulo similar to the modulo for integers
|
||
*
|
||
* https://raw.org/book/analysis/rational-numbers/
|
||
*
|
||
* n1/d1 = (n2/d2) * q + r, where 0 ≤ r < n2/d2
|
||
* => d2 * n1 = n2 * d1 * q + d1 * d2 * r
|
||
* => r = (d2 * n1 - n2 * d1 * q) / (d1 * d2)
|
||
* = (d2 * n1 - n2 * d1 * floor((d2 * n1) / (n2 * d1))) / (d1 * d2)
|
||
* = ((d2 * n1) % (n2 * d1)) / (d1 * d2)
|
||
*/
|
||
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);
|
||
|
||
// https://raw.org/book/analysis/rational-numbers/
|
||
// 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);
|
||
|
||
// https://raw.org/book/analysis/rational-numbers/
|
||
// lcm(a / b, c / d) = lcm(a, c) / gcd(b, d)
|
||
|
||
if (P["n"] === C_ZERO && this["n"] === C_ZERO) {
|
||
return newFraction(C_ZERO, C_ONE);
|
||
}
|
||
return newFraction(P["n"] * this["n"], gcd(P["n"], this["n"]) * gcd(P["d"], this["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 integer exponent
|
||
*
|
||
* 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'] === C_ONE) {
|
||
|
||
if (P['s'] < C_ZERO) {
|
||
return newFraction((this['s'] * this["d"]) ** P['n'], this["n"] ** P['n']);
|
||
} else {
|
||
return newFraction((this['s'] * this["n"]) ** P['n'], 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
|
||
// ⇔ (cos(c*pi/d) + i*sin(c*pi/d)) * (a/b)^(c/d) = x # DeMoivre's formula
|
||
// From which follows that only for c=0 the root is non-complex
|
||
if (this['s'] < C_ZERO) return null;
|
||
|
||
// Now prime factor n and d
|
||
let N = factorize(this['n']);
|
||
let D = factorize(this['d']);
|
||
|
||
// Exponentiate and take root for n and d individually
|
||
let n = C_ONE;
|
||
let d = C_ONE;
|
||
for (let k in N) {
|
||
if (k === '1') continue;
|
||
if (k === '0') {
|
||
n = C_ZERO;
|
||
break;
|
||
}
|
||
N[k] *= P['n'];
|
||
|
||
if (N[k] % P['d'] === C_ZERO) {
|
||
N[k] /= P['d'];
|
||
} else return null;
|
||
n *= BigInt(k) ** N[k];
|
||
}
|
||
|
||
for (let k in D) {
|
||
if (k === '1') continue;
|
||
D[k] *= P['n'];
|
||
|
||
if (D[k] % P['d'] === C_ZERO) {
|
||
D[k] /= P['d'];
|
||
} else return null;
|
||
d *= BigInt(k) ** D[k];
|
||
}
|
||
|
||
if (P['s'] < C_ZERO) {
|
||
return newFraction(d, n);
|
||
}
|
||
return newFraction(n, d);
|
||
},
|
||
|
||
/**
|
||
* Calculates the logarithm of a fraction to a given rational base
|
||
*
|
||
* Ex: new Fraction(27, 8).log(9, 4) => 3/2
|
||
*/
|
||
"log": function (a, b) {
|
||
|
||
parse(a, b);
|
||
|
||
if (this['s'] <= C_ZERO || P['s'] <= C_ZERO) return null;
|
||
|
||
const allPrimes = Object.create(null);
|
||
|
||
const baseFactors = factorize(P['n']);
|
||
const T1 = factorize(P['d']);
|
||
|
||
const numberFactors = factorize(this['n']);
|
||
const T2 = factorize(this['d']);
|
||
|
||
for (const prime in T1) {
|
||
baseFactors[prime] = (baseFactors[prime] || C_ZERO) - T1[prime];
|
||
}
|
||
for (const prime in T2) {
|
||
numberFactors[prime] = (numberFactors[prime] || C_ZERO) - T2[prime];
|
||
}
|
||
|
||
for (const prime in baseFactors) {
|
||
if (prime === '1') continue;
|
||
allPrimes[prime] = true;
|
||
}
|
||
for (const prime in numberFactors) {
|
||
if (prime === '1') continue;
|
||
allPrimes[prime] = true;
|
||
}
|
||
|
||
let retN = null;
|
||
let retD = null;
|
||
|
||
// Iterate over all unique primes to determine if a consistent ratio exists
|
||
for (const prime in allPrimes) {
|
||
|
||
const baseExponent = baseFactors[prime] || C_ZERO;
|
||
const numberExponent = numberFactors[prime] || C_ZERO;
|
||
|
||
if (baseExponent === C_ZERO) {
|
||
if (numberExponent !== C_ZERO) {
|
||
return null; // Logarithm cannot be expressed as a rational number
|
||
}
|
||
continue; // Skip this prime since both exponents are zero
|
||
}
|
||
|
||
// Calculate the ratio of exponents for this prime
|
||
let curN = numberExponent;
|
||
let curD = baseExponent;
|
||
|
||
// Simplify the current ratio
|
||
const gcdValue = gcd(curN, curD);
|
||
curN /= gcdValue;
|
||
curD /= gcdValue;
|
||
|
||
// Check if this is the first ratio; otherwise, ensure ratios are consistent
|
||
if (retN === null && retD === null) {
|
||
retN = curN;
|
||
retD = curD;
|
||
} else if (curN * retD !== retN * curD) {
|
||
return null; // Ratios do not match, logarithm cannot be rational
|
||
}
|
||
}
|
||
|
||
return retN !== null && retD !== null
|
||
? newFraction(retN, retD)
|
||
: null;
|
||
},
|
||
|
||
/**
|
||
* 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"];
|
||
},
|
||
|
||
/**
|
||
* Check if this rational number is less than another
|
||
*
|
||
* Ex: new Fraction(19.6).lt([98, 5]);
|
||
**/
|
||
"lt": function (a, b) {
|
||
|
||
parse(a, b);
|
||
return this["s"] * this["n"] * P["d"] < P["s"] * P["n"] * this["d"];
|
||
},
|
||
|
||
/**
|
||
* Check if this rational number is less than or equal another
|
||
*
|
||
* Ex: new Fraction(19.6).lt([98, 5]);
|
||
**/
|
||
"lte": function (a, b) {
|
||
|
||
parse(a, b);
|
||
return this["s"] * this["n"] * P["d"] <= P["s"] * P["n"] * this["d"];
|
||
},
|
||
|
||
/**
|
||
* Check if this rational number is greater than another
|
||
*
|
||
* Ex: new Fraction(19.6).lt([98, 5]);
|
||
**/
|
||
"gt": function (a, b) {
|
||
|
||
parse(a, b);
|
||
return this["s"] * this["n"] * P["d"] > P["s"] * P["n"] * this["d"];
|
||
},
|
||
|
||
/**
|
||
* Check if this rational number is greater than or equal another
|
||
*
|
||
* Ex: new Fraction(19.6).lt([98, 5]);
|
||
**/
|
||
"gte": function (a, b) {
|
||
|
||
parse(a, b);
|
||
return this["s"] * this["n"] * P["d"] >= P["s"] * P["n"] * this["d"];
|
||
},
|
||
|
||
/**
|
||
* Compare two rational numbers
|
||
* < 0 iff this < that
|
||
* > 0 iff this > that
|
||
* = 0 iff this = that
|
||
*
|
||
* Ex: new Fraction(19.6).compare([98, 5]);
|
||
**/
|
||
"compare": function (a, b) {
|
||
|
||
parse(a, b);
|
||
let t = this["s"] * this["n"] * P["d"] - P["s"] * P["n"] * this["d"];
|
||
|
||
return (C_ZERO < t) - (t < C_ZERO);
|
||
},
|
||
|
||
/**
|
||
* Calculates the ceil of a rational number
|
||
*
|
||
* Ex: new Fraction('4.(3)').ceil() => (5 / 1)
|
||
**/
|
||
"ceil": function (places) {
|
||
|
||
places = C_TEN ** BigInt(places || 0);
|
||
|
||
return newFraction(ifloor(this["s"] * places * this["n"] / this["d"]) +
|
||
(places * this["n"] % this["d"] > C_ZERO && this["s"] >= C_ZERO ? C_ONE : C_ZERO),
|
||
places);
|
||
},
|
||
|
||
/**
|
||
* Calculates the floor of a rational number
|
||
*
|
||
* Ex: new Fraction('4.(3)').floor() => (4 / 1)
|
||
**/
|
||
"floor": function (places) {
|
||
|
||
places = C_TEN ** BigInt(places || 0);
|
||
|
||
return newFraction(ifloor(this["s"] * places * this["n"] / this["d"]) -
|
||
(places * this["n"] % this["d"] > C_ZERO && this["s"] < C_ZERO ? C_ONE : C_ZERO),
|
||
places);
|
||
},
|
||
|
||
/**
|
||
* Rounds a rational numbers
|
||
*
|
||
* Ex: new Fraction('4.(3)').round() => (4 / 1)
|
||
**/
|
||
"round": function (places) {
|
||
|
||
places = C_TEN ** BigInt(places || 0);
|
||
|
||
/* Derivation:
|
||
|
||
s >= 0:
|
||
round(n / d) = ifloor(n / d) + (n % d) / d >= 0.5 ? 1 : 0
|
||
= ifloor(n / d) + 2(n % d) >= d ? 1 : 0
|
||
s < 0:
|
||
round(n / d) =-ifloor(n / d) - (n % d) / d > 0.5 ? 1 : 0
|
||
=-ifloor(n / d) - 2(n % d) > d ? 1 : 0
|
||
|
||
=>:
|
||
|
||
round(s * n / d) = s * ifloor(n / d) + s * (C + 2(n % d) > d ? 1 : 0)
|
||
where C = s >= 0 ? 1 : 0, to fix the >= for the positve case.
|
||
*/
|
||
|
||
return newFraction(ifloor(this["s"] * places * this["n"] / this["d"]) +
|
||
this["s"] * ((this["s"] >= C_ZERO ? C_ONE : C_ZERO) + C_TWO * (places * this["n"] % this["d"]) > this["d"] ? C_ONE : C_ZERO),
|
||
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)
|
||
⇔ k = floor((a * y) / (b * x))
|
||
*/
|
||
|
||
parse(a, b);
|
||
|
||
const n = this['n'] * P['d'];
|
||
const d = this['d'] * P['n'];
|
||
const r = n % d;
|
||
|
||
// round(n / d) = ifloor(n / d) + 2(n % d) >= d ? 1 : 0
|
||
let k = ifloor(n / d);
|
||
if (r + r >= d) {
|
||
k++;
|
||
}
|
||
return newFraction(this['s'] * k * P['n'], P['d']);
|
||
},
|
||
|
||
/**
|
||
* Check if two rational numbers are divisible
|
||
*
|
||
* Ex: new Fraction(19.6).divisible(1.5);
|
||
*/
|
||
"divisible": function (a, b) {
|
||
|
||
parse(a, b);
|
||
if (P['n'] === C_ZERO) return false;
|
||
return (this['n'] * P['d']) % (P['n'] * this['d']) === C_ZERO;
|
||
},
|
||
|
||
/**
|
||
* Returns a decimal representation of the fraction
|
||
*
|
||
* Ex: new Fraction("100.'91823'").valueOf() => 100.91823918239183
|
||
**/
|
||
'valueOf': function () {
|
||
//if (this['n'] <= MAX_INTEGER && this['d'] <= MAX_INTEGER) {
|
||
return Number(this['s'] * this['n']) / Number(this['d']);
|
||
//}
|
||
},
|
||
|
||
/**
|
||
* Creates a string representation of a fraction with all digits
|
||
*
|
||
* Ex: new Fraction("100.'91823'").toString() => "100.(91823)"
|
||
**/
|
||
'toString': function (dec = 15) {
|
||
|
||
let N = this["n"];
|
||
let D = this["d"];
|
||
|
||
let cycLen = cycleLen(N, D); // Cycle length
|
||
let cycOff = cycleStart(N, D, cycLen); // Cycle start
|
||
|
||
let str = this['s'] < C_ZERO ? "-" : "";
|
||
|
||
// Append integer part
|
||
str += ifloor(N / D);
|
||
|
||
N %= D;
|
||
N *= C_TEN;
|
||
|
||
if (N)
|
||
str += ".";
|
||
|
||
if (cycLen) {
|
||
|
||
for (let i = cycOff; i--;) {
|
||
str += ifloor(N / D);
|
||
N %= D;
|
||
N *= C_TEN;
|
||
}
|
||
str += "(";
|
||
for (let i = cycLen; i--;) {
|
||
str += ifloor(N / D);
|
||
N %= D;
|
||
N *= C_TEN;
|
||
}
|
||
str += ")";
|
||
} else {
|
||
for (let i = dec; N && i--;) {
|
||
str += ifloor(N / D);
|
||
N %= D;
|
||
N *= C_TEN;
|
||
}
|
||
}
|
||
return str;
|
||
},
|
||
|
||
/**
|
||
* Returns a string-fraction representation of a Fraction object
|
||
*
|
||
* Ex: new Fraction("1.'3'").toFraction() => "4 1/3"
|
||
**/
|
||
'toFraction': function (showMixed = false) {
|
||
|
||
let n = this["n"];
|
||
let d = this["d"];
|
||
let str = this['s'] < C_ZERO ? "-" : "";
|
||
|
||
if (d === C_ONE) {
|
||
str += n;
|
||
} else {
|
||
const whole = ifloor(n / d);
|
||
if (showMixed && whole > C_ZERO) {
|
||
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 (showMixed = false) {
|
||
|
||
let n = this["n"];
|
||
let d = this["d"];
|
||
let str = this['s'] < C_ZERO ? "-" : "";
|
||
|
||
if (d === C_ONE) {
|
||
str += n;
|
||
} else {
|
||
const whole = ifloor(n / d);
|
||
if (showMixed && whole > C_ZERO) {
|
||
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 () {
|
||
|
||
let a = this['n'];
|
||
let b = this['d'];
|
||
const res = [];
|
||
|
||
while (b) {
|
||
res.push(ifloor(a / b));
|
||
const t = a % b;
|
||
a = b;
|
||
b = t;
|
||
}
|
||
return res;
|
||
},
|
||
|
||
"simplify": function (eps = 1e-3) {
|
||
|
||
// Continued fractions give best approximations for a max denominator,
|
||
// generally outperforming mediants in denominator–accuracy trade-offs.
|
||
// Semiconvergents can further reduce the denominator within tolerance.
|
||
|
||
const ieps = BigInt(Math.ceil(1 / eps));
|
||
|
||
const thisABS = this['abs']();
|
||
const cont = thisABS['toContinued']();
|
||
|
||
for (let i = 1; i < cont.length; i++) {
|
||
|
||
let s = newFraction(cont[i - 1], C_ONE);
|
||
for (let k = i - 2; k >= 0; k--) {
|
||
s = s['inverse']()['add'](cont[k]);
|
||
}
|
||
|
||
let t = s['sub'](thisABS);
|
||
if (t['n'] * ieps < t['d']) { // More robust than Math.abs(t.valueOf()) < eps
|
||
return s['mul'](this['s']);
|
||
}
|
||
}
|
||
return this;
|
||
}
|
||
};
|
||
export {
|
||
Fraction as default, Fraction
|
||
};
|