diff --git a/fraction.js b/fraction.js new file mode 100644 index 00000000..fec06120 --- /dev/null +++ b/fraction.js @@ -0,0 +1,908 @@ +/** + * @license Fraction.js v4.1.2 23/05/2021 + * 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 => , 1 => ] + * [ n => , d => ] + * + * 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); + * + */ + + const memo = {}; + + let 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 createError(name) { + + function errorConstructor() { + var temp = Error.apply(this, arguments); + temp['name'] = this['name'] = name; + this['stack'] = temp['stack']; + this['message'] = temp['message']; + } + + /** + * Error constructor + * + * @constructor + */ + function IntermediateInheritor() { } + IntermediateInheritor.prototype = Error.prototype; + errorConstructor.prototype = new IntermediateInheritor(); + + return errorConstructor; + } + + var DivisionByZero = Fraction['DivisionByZero'] = createError('DivisionByZero'); + var InvalidParameter = Fraction['InvalidParameter'] = createError('InvalidParameter'); + + function assign(n, s) { + + if (isNaN(n = parseInt(n, 10))) { + throwInvalidParam(); + } + return n * s; + } + + function throwInvalidParam() { + throw new InvalidParameter(); + } + + 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; + } 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 { + throwInvalidParam(); + } + 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; + } + + const key = p1+'#'+p2 + const memoized = memo[key] + if(memoized) { + s = memoized.s; + n = memoized.n; + d = memoized.d; + break; + } + + // 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) + throwInvalidParam(); + + 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: + throwInvalidParam(); + } + + if (d === 0) { + throw new DivisionByZero(); + } + + P["s"] = s < 0 ? -1 : 1; + P["n"] = Math.abs(n); + P["d"] = Math.abs(d); + memo[p1+'#'+p2] = {s:P["s"],n:P["n"],d:P["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) { + + if (!(this instanceof Fraction)) { + return new Fraction(a, b); + } + + parse(a, b); + + a = gcd(P["d"], P["n"]); // Abuse variable a + + this["s"] = P["s"]; + this["n"] = P["n"] / a; + this["d"] = P["d"] / a; + } + + Fraction.prototype = { + + "s": 1, + "n": 0, + "d": 1, + + /** + * Calculates the absolute value + * + * Ex: new Fraction(-4).abs() => 4 + **/ + "abs": function() { + + return new Fraction(this["n"], this["d"]); + }, + + /** + * Inverts the sign of the current fraction + * + * Ex: new Fraction(-4).neg() => 4 + **/ + "neg": function() { + + return new Fraction(-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 new Fraction( + 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 new Fraction( + 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 new Fraction( + 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 new Fraction( + 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 new Fraction(this); + }, + + /** + * 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 new Fraction(this["s"] * this["n"] % this["d"], 1); + } + + parse(a, b); + if (0 === P["n"] && 0 === this["d"]) { + Fraction(0, 0); // 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 new Fraction( + 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 new Fraction(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 new Fraction; + } + return new Fraction(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 new Fraction(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 new Fraction(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 new Fraction(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 new Fraction(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 new Fraction(Math.pow(this['s'] * this["d"], P['n']), Math.pow(this["n"], P['n'])); + } else { + return new Fraction(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 new Fraction(d, n); + } + return new Fraction(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) { + + // First naive implementation, needs improvement + + if (isNaN(this['n']) || isNaN(this['d'])) { + return this; + } + + var cont = this['abs']()['toContinued'](); + + eps = eps || 0.001; + + function rec(a) { + if (a.length === 1) + return new Fraction(a[0]); + return rec(a.slice(1))['inverse']()['add'](a[0]); + } + + for (var i = 0; i < cont.length; i++) { + var tmp = rec(cont.slice(0, i + 1)); + if (tmp['sub'](this['abs']())['abs']().valueOf() < eps) { + return tmp['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() => "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 g; + 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'] === -1 ? "-" : ""; + + 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; + } + + + export default Fraction; \ No newline at end of file diff --git a/strudel.mjs b/strudel.mjs index 10f34360..9500102a 100644 --- a/strudel.mjs +++ b/strudel.mjs @@ -1,4 +1,4 @@ -import Fraction from 'fraction.js' +import Fraction from './fraction.js' import { compose } from 'ramda'; // will remove this as soon as compose is implemented here import { isNote, toMidi } from './util.mjs';