/* * * decimal.js v9.0.1 * An arbitrary-precision Decimal type for JavaScript. * https://github.com/MikeMcl/decimal.js * Copyright (c) 2017 Michael Mclaughlin * MIT Licence * https://github.com/MikeMcl/decimal.js/LICENCE * */ // ----------------------------------- EDITABLE DEFAULTS ------------------------------------ // // The maximum exponent magnitude. // The limit on the value of `toExpNeg`, `toExpPos`, `minE` and `maxE`. var EXP_LIMIT = 9e15, // 0 to 9e15 // The limit on the value of `precision`, and on the value of the first argument to // `toDecimalPlaces`, `toExponential`, `toFixed`, `toPrecision` and `toSignificantDigits`. MAX_DIGITS = 1e9, // 0 to 1e9 // Base conversion alphabet. NUMERALS = '0123456789abcdef', // The natural logarithm of 10 (1025 digits). LN10 = '2.3025850929940456840179914546843642076011014886287729760333279009675726096773524802359972050895982983419677840422862486334095254650828067566662873690987816894829072083255546808437998948262331985283935053089653777326288461633662222876982198867465436674744042432743651550489343149393914796194044002221051017141748003688084012647080685567743216228355220114804663715659121373450747856947683463616792101806445070648000277502684916746550586856935673420670581136429224554405758925724208241314695689016758940256776311356919292033376587141660230105703089634572075440370847469940168269282808481184289314848524948644871927809676271275775397027668605952496716674183485704422507197965004714951050492214776567636938662976979522110718264549734772662425709429322582798502585509785265383207606726317164309505995087807523710333101197857547331541421808427543863591778117054309827482385045648019095610299291824318237525357709750539565187697510374970888692180205189339507238539205144634197265287286965110862571492198849978748873771345686209167058', // Pi (1025 digits). PI = '3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461284756482337867831652712019091456485669234603486104543266482133936072602491412737245870066063155881748815209209628292540917153643678925903600113305305488204665213841469519415116094330572703657595919530921861173819326117931051185480744623799627495673518857527248912279381830119491298336733624406566430860213949463952247371907021798609437027705392171762931767523846748184676694051320005681271452635608277857713427577896091736371787214684409012249534301465495853710507922796892589235420199561121290219608640344181598136297747713099605187072113499999983729780499510597317328160963185950244594553469083026425223082533446850352619311881710100031378387528865875332083814206171776691473035982534904287554687311595628638823537875937519577818577805321712268066130019278766111959092164201989380952572010654858632789', // The initial configuration properties of the Decimal constructor. DEFAULTS = { // These values must be integers within the stated ranges (inclusive). // Most of these values can be changed at run-time using the `Decimal.config` method. // The maximum number of significant digits of the result of a calculation or base conversion. // E.g. `Decimal.config({ precision: 20 });` precision: 20, // 1 to MAX_DIGITS // The rounding mode used when rounding to `precision`. // // ROUND_UP 0 Away from zero. // ROUND_DOWN 1 Towards zero. // ROUND_CEIL 2 Towards +Infinity. // ROUND_FLOOR 3 Towards -Infinity. // ROUND_HALF_UP 4 Towards nearest neighbour. If equidistant, up. // ROUND_HALF_DOWN 5 Towards nearest neighbour. If equidistant, down. // ROUND_HALF_EVEN 6 Towards nearest neighbour. If equidistant, towards even neighbour. // ROUND_HALF_CEIL 7 Towards nearest neighbour. If equidistant, towards +Infinity. // ROUND_HALF_FLOOR 8 Towards nearest neighbour. If equidistant, towards -Infinity. // // E.g. // `Decimal.rounding = 4;` // `Decimal.rounding = Decimal.ROUND_HALF_UP;` rounding: 4, // 0 to 8 // The modulo mode used when calculating the modulus: a mod n. // The quotient (q = a / n) is calculated according to the corresponding rounding mode. // The remainder (r) is calculated as: r = a - n * q. // // UP 0 The remainder is positive if the dividend is negative, else is negative. // DOWN 1 The remainder has the same sign as the dividend (JavaScript %). // FLOOR 3 The remainder has the same sign as the divisor (Python %). // HALF_EVEN 6 The IEEE 754 remainder function. // EUCLID 9 Euclidian division. q = sign(n) * floor(a / abs(n)). Always positive. // // Truncated division (1), floored division (3), the IEEE 754 remainder (6), and Euclidian // division (9) are commonly used for the modulus operation. The other rounding modes can also // be used, but they may not give useful results. modulo: 1, // 0 to 9 // The exponent value at and beneath which `toString` returns exponential notation. // JavaScript numbers: -7 toExpNeg: -7, // 0 to -EXP_LIMIT // The exponent value at and above which `toString` returns exponential notation. // JavaScript numbers: 21 toExpPos: 21, // 0 to EXP_LIMIT // The minimum exponent value, beneath which underflow to zero occurs. // JavaScript numbers: -324 (5e-324) minE: -EXP_LIMIT, // -1 to -EXP_LIMIT // The maximum exponent value, above which overflow to Infinity occurs. // JavaScript numbers: 308 (1.7976931348623157e+308) maxE: EXP_LIMIT, // 1 to EXP_LIMIT // Whether to use cryptographically-secure random number generation, if available. crypto: false // true/false }, // ----------------------------------- END OF EDITABLE DEFAULTS ------------------------------- // inexact, quadrant, external = true, decimalError = '[DecimalError] ', invalidArgument = decimalError + 'Invalid argument: ', precisionLimitExceeded = decimalError + 'Precision limit exceeded', cryptoUnavailable = decimalError + 'crypto unavailable', mathfloor = Math.floor, mathpow = Math.pow, isBinary = /^0b([01]+(\.[01]*)?|\.[01]+)(p[+-]?\d+)?$/i, isHex = /^0x([0-9a-f]+(\.[0-9a-f]*)?|\.[0-9a-f]+)(p[+-]?\d+)?$/i, isOctal = /^0o([0-7]+(\.[0-7]*)?|\.[0-7]+)(p[+-]?\d+)?$/i, isDecimal = /^(\d+(\.\d*)?|\.\d+)(e[+-]?\d+)?$/i, BASE = 1e7, LOG_BASE = 7, MAX_SAFE_INTEGER = 9007199254740991, LN10_PRECISION = LN10.length - 1, PI_PRECISION = PI.length - 1, // Decimal.prototype object P = { name: '[object Decimal]' }; // Decimal prototype methods /* * absoluteValue abs * ceil * comparedTo cmp * cosine cos * cubeRoot cbrt * decimalPlaces dp * dividedBy div * dividedToIntegerBy divToInt * equals eq * floor * greaterThan gt * greaterThanOrEqualTo gte * hyperbolicCosine cosh * hyperbolicSine sinh * hyperbolicTangent tanh * inverseCosine acos * inverseHyperbolicCosine acosh * inverseHyperbolicSine asinh * inverseHyperbolicTangent atanh * inverseSine asin * inverseTangent atan * isFinite * isInteger isInt * isNaN * isNegative isNeg * isPositive isPos * isZero * lessThan lt * lessThanOrEqualTo lte * logarithm log * [maximum] [max] * [minimum] [min] * minus sub * modulo mod * naturalExponential exp * naturalLogarithm ln * negated neg * plus add * precision sd * round * sine sin * squareRoot sqrt * tangent tan * times mul * toBinary * toDecimalPlaces toDP * toExponential * toFixed * toFraction * toHexadecimal toHex * toNearest * toNumber * toOctal * toPower pow * toPrecision * toSignificantDigits toSD * toString * truncated trunc * valueOf toJSON */ /* * Return a new Decimal whose value is the absolute value of this Decimal. * */ P.absoluteValue = P.abs = function () { var x = new this.constructor(this); if (x.s < 0) x.s = 1; return finalise(x); }; /* * Return a new Decimal whose value is the value of this Decimal rounded to a whole number in the * direction of positive Infinity. * */ P.ceil = function () { return finalise(new this.constructor(this), this.e + 1, 2); }; /* * Return * 1 if the value of this Decimal is greater than the value of `y`, * -1 if the value of this Decimal is less than the value of `y`, * 0 if they have the same value, * NaN if the value of either Decimal is NaN. * */ P.comparedTo = P.cmp = function (y) { var i, j, xdL, ydL, x = this, xd = x.d, yd = (y = new x.constructor(y)).d, xs = x.s, ys = y.s; // Either NaN or ±Infinity? if (!xd || !yd) { return !xs || !ys ? NaN : xs !== ys ? xs : xd === yd ? 0 : !xd ^ xs < 0 ? 1 : -1; } // Either zero? if (!xd[0] || !yd[0]) return xd[0] ? xs : yd[0] ? -ys : 0; // Signs differ? if (xs !== ys) return xs; // Compare exponents. if (x.e !== y.e) return x.e > y.e ^ xs < 0 ? 1 : -1; xdL = xd.length; ydL = yd.length; // Compare digit by digit. for (i = 0, j = xdL < ydL ? xdL : ydL; i < j; ++i) { if (xd[i] !== yd[i]) return xd[i] > yd[i] ^ xs < 0 ? 1 : -1; } // Compare lengths. return xdL === ydL ? 0 : xdL > ydL ^ xs < 0 ? 1 : -1; }; /* * Return a new Decimal whose value is the cosine of the value in radians of this Decimal. * * Domain: [-Infinity, Infinity] * Range: [-1, 1] * * cos(0) = 1 * cos(-0) = 1 * cos(Infinity) = NaN * cos(-Infinity) = NaN * cos(NaN) = NaN * */ P.cosine = P.cos = function () { var pr, rm, x = this, Ctor = x.constructor; if (!x.d) return new Ctor(NaN); // cos(0) = cos(-0) = 1 if (!x.d[0]) return new Ctor(1); pr = Ctor.precision; rm = Ctor.rounding; Ctor.precision = pr + Math.max(x.e, x.sd()) + LOG_BASE; Ctor.rounding = 1; x = cosine(Ctor, toLessThanHalfPi(Ctor, x)); Ctor.precision = pr; Ctor.rounding = rm; return finalise(quadrant == 2 || quadrant == 3 ? x.neg() : x, pr, rm, true); }; /* * * Return a new Decimal whose value is the cube root of the value of this Decimal, rounded to * `precision` significant digits using rounding mode `rounding`. * * cbrt(0) = 0 * cbrt(-0) = -0 * cbrt(1) = 1 * cbrt(-1) = -1 * cbrt(N) = N * cbrt(-I) = -I * cbrt(I) = I * * Math.cbrt(x) = (x < 0 ? -Math.pow(-x, 1/3) : Math.pow(x, 1/3)) * */ P.cubeRoot = P.cbrt = function () { var e, m, n, r, rep, s, sd, t, t3, t3plusx, x = this, Ctor = x.constructor; if (!x.isFinite() || x.isZero()) return new Ctor(x); external = false; // Initial estimate. s = x.s * Math.pow(x.s * x, 1 / 3); // Math.cbrt underflow/overflow? // Pass x to Math.pow as integer, then adjust the exponent of the result. if (!s || Math.abs(s) == 1 / 0) { n = digitsToString(x.d); e = x.e; // Adjust n exponent so it is a multiple of 3 away from x exponent. if (s = (e - n.length + 1) % 3) n += (s == 1 || s == -2 ? '0' : '00'); s = Math.pow(n, 1 / 3); // Rarely, e may be one less than the result exponent value. e = mathfloor((e + 1) / 3) - (e % 3 == (e < 0 ? -1 : 2)); if (s == 1 / 0) { n = '5e' + e; } else { n = s.toExponential(); n = n.slice(0, n.indexOf('e') + 1) + e; } r = new Ctor(n); r.s = x.s; } else { r = new Ctor(s.toString()); } sd = (e = Ctor.precision) + 3; // Halley's method. // TODO? Compare Newton's method. for (;;) { t = r; t3 = t.times(t).times(t); t3plusx = t3.plus(x); r = divide(t3plusx.plus(x).times(t), t3plusx.plus(t3), sd + 2, 1); // TODO? Replace with for-loop and checkRoundingDigits. if (digitsToString(t.d).slice(0, sd) === (n = digitsToString(r.d)).slice(0, sd)) { n = n.slice(sd - 3, sd + 1); // The 4th rounding digit may be in error by -1 so if the 4 rounding digits are 9999 or 4999 // , i.e. approaching a rounding boundary, continue the iteration. if (n == '9999' || !rep && n == '4999') { // On the first iteration only, check to see if rounding up gives the exact result as the // nines may infinitely repeat. if (!rep) { finalise(t, e + 1, 0); if (t.times(t).times(t).eq(x)) { r = t; break; } } sd += 4; rep = 1; } else { // If the rounding digits are null, 0{0,4} or 50{0,3}, check for an exact result. // If not, then there are further digits and m will be truthy. if (!+n || !+n.slice(1) && n.charAt(0) == '5') { // Truncate to the first rounding digit. finalise(r, e + 1, 1); m = !r.times(r).times(r).eq(x); } break; } } } external = true; return finalise(r, e, Ctor.rounding, m); }; /* * Return the number of decimal places of the value of this Decimal. * */ P.decimalPlaces = P.dp = function () { var w, d = this.d, n = NaN; if (d) { w = d.length - 1; n = (w - mathfloor(this.e / LOG_BASE)) * LOG_BASE; // Subtract the number of trailing zeros of the last word. w = d[w]; if (w) for (; w % 10 == 0; w /= 10) n--; if (n < 0) n = 0; } return n; }; /* * n / 0 = I * n / N = N * n / I = 0 * 0 / n = 0 * 0 / 0 = N * 0 / N = N * 0 / I = 0 * N / n = N * N / 0 = N * N / N = N * N / I = N * I / n = I * I / 0 = I * I / N = N * I / I = N * * Return a new Decimal whose value is the value of this Decimal divided by `y`, rounded to * `precision` significant digits using rounding mode `rounding`. * */ P.dividedBy = P.div = function (y) { return divide(this, new this.constructor(y)); }; /* * Return a new Decimal whose value is the integer part of dividing the value of this Decimal * by the value of `y`, rounded to `precision` significant digits using rounding mode `rounding`. * */ P.dividedToIntegerBy = P.divToInt = function (y) { var x = this, Ctor = x.constructor; return finalise(divide(x, new Ctor(y), 0, 1, 1), Ctor.precision, Ctor.rounding); }; /* * Return true if the value of this Decimal is equal to the value of `y`, otherwise return false. * */ P.equals = P.eq = function (y) { return this.cmp(y) === 0; }; /* * Return a new Decimal whose value is the value of this Decimal rounded to a whole number in the * direction of negative Infinity. * */ P.floor = function () { return finalise(new this.constructor(this), this.e + 1, 3); }; /* * Return true if the value of this Decimal is greater than the value of `y`, otherwise return * false. * */ P.greaterThan = P.gt = function (y) { return this.cmp(y) > 0; }; /* * Return true if the value of this Decimal is greater than or equal to the value of `y`, * otherwise return false. * */ P.greaterThanOrEqualTo = P.gte = function (y) { var k = this.cmp(y); return k == 1 || k === 0; }; /* * Return a new Decimal whose value is the hyperbolic cosine of the value in radians of this * Decimal. * * Domain: [-Infinity, Infinity] * Range: [1, Infinity] * * cosh(x) = 1 + x^2/2! + x^4/4! + x^6/6! + ... * * cosh(0) = 1 * cosh(-0) = 1 * cosh(Infinity) = Infinity * cosh(-Infinity) = Infinity * cosh(NaN) = NaN * * x time taken (ms) result * 1000 9 9.8503555700852349694e+433 * 10000 25 4.4034091128314607936e+4342 * 100000 171 1.4033316802130615897e+43429 * 1000000 3817 1.5166076984010437725e+434294 * 10000000 abandoned after 2 minute wait * * TODO? Compare performance of cosh(x) = 0.5 * (exp(x) + exp(-x)) * */ P.hyperbolicCosine = P.cosh = function () { var k, n, pr, rm, len, x = this, Ctor = x.constructor, one = new Ctor(1); if (!x.isFinite()) return new Ctor(x.s ? 1 / 0 : NaN); if (x.isZero()) return one; pr = Ctor.precision; rm = Ctor.rounding; Ctor.precision = pr + Math.max(x.e, x.sd()) + 4; Ctor.rounding = 1; len = x.d.length; // Argument reduction: cos(4x) = 1 - 8cos^2(x) + 8cos^4(x) + 1 // i.e. cos(x) = 1 - cos^2(x/4)(8 - 8cos^2(x/4)) // Estimate the optimum number of times to use the argument reduction. // TODO? Estimation reused from cosine() and may not be optimal here. if (len < 32) { k = Math.ceil(len / 3); n = Math.pow(4, -k).toString(); } else { k = 16; n = '2.3283064365386962890625e-10'; } x = taylorSeries(Ctor, 1, x.times(n), new Ctor(1), true); // Reverse argument reduction var cosh2_x, i = k, d8 = new Ctor(8); for (; i--;) { cosh2_x = x.times(x); x = one.minus(cosh2_x.times(d8.minus(cosh2_x.times(d8)))); } return finalise(x, Ctor.precision = pr, Ctor.rounding = rm, true); }; /* * Return a new Decimal whose value is the hyperbolic sine of the value in radians of this * Decimal. * * Domain: [-Infinity, Infinity] * Range: [-Infinity, Infinity] * * sinh(x) = x + x^3/3! + x^5/5! + x^7/7! + ... * * sinh(0) = 0 * sinh(-0) = -0 * sinh(Infinity) = Infinity * sinh(-Infinity) = -Infinity * sinh(NaN) = NaN * * x time taken (ms) * 10 2 ms * 100 5 ms * 1000 14 ms * 10000 82 ms * 100000 886 ms 1.4033316802130615897e+43429 * 200000 2613 ms * 300000 5407 ms * 400000 8824 ms * 500000 13026 ms 8.7080643612718084129e+217146 * 1000000 48543 ms * * TODO? Compare performance of sinh(x) = 0.5 * (exp(x) - exp(-x)) * */ P.hyperbolicSine = P.sinh = function () { var k, pr, rm, len, x = this, Ctor = x.constructor; if (!x.isFinite() || x.isZero()) return new Ctor(x); pr = Ctor.precision; rm = Ctor.rounding; Ctor.precision = pr + Math.max(x.e, x.sd()) + 4; Ctor.rounding = 1; len = x.d.length; if (len < 3) { x = taylorSeries(Ctor, 2, x, x, true); } else { // Alternative argument reduction: sinh(3x) = sinh(x)(3 + 4sinh^2(x)) // i.e. sinh(x) = sinh(x/3)(3 + 4sinh^2(x/3)) // 3 multiplications and 1 addition // Argument reduction: sinh(5x) = sinh(x)(5 + sinh^2(x)(20 + 16sinh^2(x))) // i.e. sinh(x) = sinh(x/5)(5 + sinh^2(x/5)(20 + 16sinh^2(x/5))) // 4 multiplications and 2 additions // Estimate the optimum number of times to use the argument reduction. k = 1.4 * Math.sqrt(len); k = k > 16 ? 16 : k | 0; x = x.times(Math.pow(5, -k)); x = taylorSeries(Ctor, 2, x, x, true); // Reverse argument reduction var sinh2_x, d5 = new Ctor(5), d16 = new Ctor(16), d20 = new Ctor(20); for (; k--;) { sinh2_x = x.times(x); x = x.times(d5.plus(sinh2_x.times(d16.times(sinh2_x).plus(d20)))); } } Ctor.precision = pr; Ctor.rounding = rm; return finalise(x, pr, rm, true); }; /* * Return a new Decimal whose value is the hyperbolic tangent of the value in radians of this * Decimal. * * Domain: [-Infinity, Infinity] * Range: [-1, 1] * * tanh(x) = sinh(x) / cosh(x) * * tanh(0) = 0 * tanh(-0) = -0 * tanh(Infinity) = 1 * tanh(-Infinity) = -1 * tanh(NaN) = NaN * */ P.hyperbolicTangent = P.tanh = function () { var pr, rm, x = this, Ctor = x.constructor; if (!x.isFinite()) return new Ctor(x.s); if (x.isZero()) return new Ctor(x); pr = Ctor.precision; rm = Ctor.rounding; Ctor.precision = pr + 7; Ctor.rounding = 1; return divide(x.sinh(), x.cosh(), Ctor.precision = pr, Ctor.rounding = rm); }; /* * Return a new Decimal whose value is the arccosine (inverse cosine) in radians of the value of * this Decimal. * * Domain: [-1, 1] * Range: [0, pi] * * acos(x) = pi/2 - asin(x) * * acos(0) = pi/2 * acos(-0) = pi/2 * acos(1) = 0 * acos(-1) = pi * acos(1/2) = pi/3 * acos(-1/2) = 2*pi/3 * acos(|x| > 1) = NaN * acos(NaN) = NaN * */ P.inverseCosine = P.acos = function () { var halfPi, x = this, Ctor = x.constructor, k = x.abs().cmp(1), pr = Ctor.precision, rm = Ctor.rounding; if (k !== -1) { return k === 0 // |x| is 1 ? x.isNeg() ? getPi(Ctor, pr, rm) : new Ctor(0) // |x| > 1 or x is NaN : new Ctor(NaN); } if (x.isZero()) return getPi(Ctor, pr + 4, rm).times(0.5); // TODO? Special case acos(0.5) = pi/3 and acos(-0.5) = 2*pi/3 Ctor.precision = pr + 6; Ctor.rounding = 1; x = x.asin(); halfPi = getPi(Ctor, pr + 4, rm).times(0.5); Ctor.precision = pr; Ctor.rounding = rm; return halfPi.minus(x); }; /* * Return a new Decimal whose value is the inverse of the hyperbolic cosine in radians of the * value of this Decimal. * * Domain: [1, Infinity] * Range: [0, Infinity] * * acosh(x) = ln(x + sqrt(x^2 - 1)) * * acosh(x < 1) = NaN * acosh(NaN) = NaN * acosh(Infinity) = Infinity * acosh(-Infinity) = NaN * acosh(0) = NaN * acosh(-0) = NaN * acosh(1) = 0 * acosh(-1) = NaN * */ P.inverseHyperbolicCosine = P.acosh = function () { var pr, rm, x = this, Ctor = x.constructor; if (x.lte(1)) return new Ctor(x.eq(1) ? 0 : NaN); if (!x.isFinite()) return new Ctor(x); pr = Ctor.precision; rm = Ctor.rounding; Ctor.precision = pr + Math.max(Math.abs(x.e), x.sd()) + 4; Ctor.rounding = 1; external = false; x = x.times(x).minus(1).sqrt().plus(x); external = true; Ctor.precision = pr; Ctor.rounding = rm; return x.ln(); }; /* * Return a new Decimal whose value is the inverse of the hyperbolic sine in radians of the value * of this Decimal. * * Domain: [-Infinity, Infinity] * Range: [-Infinity, Infinity] * * asinh(x) = ln(x + sqrt(x^2 + 1)) * * asinh(NaN) = NaN * asinh(Infinity) = Infinity * asinh(-Infinity) = -Infinity * asinh(0) = 0 * asinh(-0) = -0 * */ P.inverseHyperbolicSine = P.asinh = function () { var pr, rm, x = this, Ctor = x.constructor; if (!x.isFinite() || x.isZero()) return new Ctor(x); pr = Ctor.precision; rm = Ctor.rounding; Ctor.precision = pr + 2 * Math.max(Math.abs(x.e), x.sd()) + 6; Ctor.rounding = 1; external = false; x = x.times(x).plus(1).sqrt().plus(x); external = true; Ctor.precision = pr; Ctor.rounding = rm; return x.ln(); }; /* * Return a new Decimal whose value is the inverse of the hyperbolic tangent in radians of the * value of this Decimal. * * Domain: [-1, 1] * Range: [-Infinity, Infinity] * * atanh(x) = 0.5 * ln((1 + x) / (1 - x)) * * atanh(|x| > 1) = NaN * atanh(NaN) = NaN * atanh(Infinity) = NaN * atanh(-Infinity) = NaN * atanh(0) = 0 * atanh(-0) = -0 * atanh(1) = Infinity * atanh(-1) = -Infinity * */ P.inverseHyperbolicTangent = P.atanh = function () { var pr, rm, wpr, xsd, x = this, Ctor = x.constructor; if (!x.isFinite()) return new Ctor(NaN); if (x.e >= 0) return new Ctor(x.abs().eq(1) ? x.s / 0 : x.isZero() ? x : NaN); pr = Ctor.precision; rm = Ctor.rounding; xsd = x.sd(); if (Math.max(xsd, pr) < 2 * -x.e - 1) return finalise(new Ctor(x), pr, rm, true); Ctor.precision = wpr = xsd - x.e; x = divide(x.plus(1), new Ctor(1).minus(x), wpr + pr, 1); Ctor.precision = pr + 4; Ctor.rounding = 1; x = x.ln(); Ctor.precision = pr; Ctor.rounding = rm; return x.times(0.5); }; /* * Return a new Decimal whose value is the arcsine (inverse sine) in radians of the value of this * Decimal. * * Domain: [-Infinity, Infinity] * Range: [-pi/2, pi/2] * * asin(x) = 2*atan(x/(1 + sqrt(1 - x^2))) * * asin(0) = 0 * asin(-0) = -0 * asin(1/2) = pi/6 * asin(-1/2) = -pi/6 * asin(1) = pi/2 * asin(-1) = -pi/2 * asin(|x| > 1) = NaN * asin(NaN) = NaN * * TODO? Compare performance of Taylor series. * */ P.inverseSine = P.asin = function () { var halfPi, k, pr, rm, x = this, Ctor = x.constructor; if (x.isZero()) return new Ctor(x); k = x.abs().cmp(1); pr = Ctor.precision; rm = Ctor.rounding; if (k !== -1) { // |x| is 1 if (k === 0) { halfPi = getPi(Ctor, pr + 4, rm).times(0.5); halfPi.s = x.s; return halfPi; } // |x| > 1 or x is NaN return new Ctor(NaN); } // TODO? Special case asin(1/2) = pi/6 and asin(-1/2) = -pi/6 Ctor.precision = pr + 6; Ctor.rounding = 1; x = x.div(new Ctor(1).minus(x.times(x)).sqrt().plus(1)).atan(); Ctor.precision = pr; Ctor.rounding = rm; return x.times(2); }; /* * Return a new Decimal whose value is the arctangent (inverse tangent) in radians of the value * of this Decimal. * * Domain: [-Infinity, Infinity] * Range: [-pi/2, pi/2] * * atan(x) = x - x^3/3 + x^5/5 - x^7/7 + ... * * atan(0) = 0 * atan(-0) = -0 * atan(1) = pi/4 * atan(-1) = -pi/4 * atan(Infinity) = pi/2 * atan(-Infinity) = -pi/2 * atan(NaN) = NaN * */ P.inverseTangent = P.atan = function () { var i, j, k, n, px, t, r, wpr, x2, x = this, Ctor = x.constructor, pr = Ctor.precision, rm = Ctor.rounding; if (!x.isFinite()) { if (!x.s) return new Ctor(NaN); if (pr + 4 <= PI_PRECISION) { r = getPi(Ctor, pr + 4, rm).times(0.5); r.s = x.s; return r; } } else if (x.isZero()) { return new Ctor(x); } else if (x.abs().eq(1) && pr + 4 <= PI_PRECISION) { r = getPi(Ctor, pr + 4, rm).times(0.25); r.s = x.s; return r; } Ctor.precision = wpr = pr + 10; Ctor.rounding = 1; // TODO? if (x >= 1 && pr <= PI_PRECISION) atan(x) = halfPi * x.s - atan(1 / x); // Argument reduction // Ensure |x| < 0.42 // atan(x) = 2 * atan(x / (1 + sqrt(1 + x^2))) k = Math.min(28, wpr / LOG_BASE + 2 | 0); for (i = k; i; --i) x = x.div(x.times(x).plus(1).sqrt().plus(1)); external = false; j = Math.ceil(wpr / LOG_BASE); n = 1; x2 = x.times(x); r = new Ctor(x); px = x; // atan(x) = x - x^3/3 + x^5/5 - x^7/7 + ... for (; i !== -1;) { px = px.times(x2); t = r.minus(px.div(n += 2)); px = px.times(x2); r = t.plus(px.div(n += 2)); if (r.d[j] !== void 0) for (i = j; r.d[i] === t.d[i] && i--;); } if (k) r = r.times(2 << (k - 1)); external = true; return finalise(r, Ctor.precision = pr, Ctor.rounding = rm, true); }; /* * Return true if the value of this Decimal is a finite number, otherwise return false. * */ P.isFinite = function () { return !!this.d; }; /* * Return true if the value of this Decimal is an integer, otherwise return false. * */ P.isInteger = P.isInt = function () { return !!this.d && mathfloor(this.e / LOG_BASE) > this.d.length - 2; }; /* * Return true if the value of this Decimal is NaN, otherwise return false. * */ P.isNaN = function () { return !this.s; }; /* * Return true if the value of this Decimal is negative, otherwise return false. * */ P.isNegative = P.isNeg = function () { return this.s < 0; }; /* * Return true if the value of this Decimal is positive, otherwise return false. * */ P.isPositive = P.isPos = function () { return this.s > 0; }; /* * Return true if the value of this Decimal is 0 or -0, otherwise return false. * */ P.isZero = function () { return !!this.d && this.d[0] === 0; }; /* * Return true if the value of this Decimal is less than `y`, otherwise return false. * */ P.lessThan = P.lt = function (y) { return this.cmp(y) < 0; }; /* * Return true if the value of this Decimal is less than or equal to `y`, otherwise return false. * */ P.lessThanOrEqualTo = P.lte = function (y) { return this.cmp(y) < 1; }; /* * Return the logarithm of the value of this Decimal to the specified base, rounded to `precision` * significant digits using rounding mode `rounding`. * * If no base is specified, return log[10](arg). * * log[base](arg) = ln(arg) / ln(base) * * The result will always be correctly rounded if the base of the log is 10, and 'almost always' * otherwise: * * Depending on the rounding mode, the result may be incorrectly rounded if the first fifteen * rounding digits are [49]99999999999999 or [50]00000000000000. In that case, the maximum error * between the result and the correctly rounded result will be one ulp (unit in the last place). * * log[-b](a) = NaN * log[0](a) = NaN * log[1](a) = NaN * log[NaN](a) = NaN * log[Infinity](a) = NaN * log[b](0) = -Infinity * log[b](-0) = -Infinity * log[b](-a) = NaN * log[b](1) = 0 * log[b](Infinity) = Infinity * log[b](NaN) = NaN * * [base] {number|string|Decimal} The base of the logarithm. * */ P.logarithm = P.log = function (base) { var isBase10, d, denominator, k, inf, num, sd, r, arg = this, Ctor = arg.constructor, pr = Ctor.precision, rm = Ctor.rounding, guard = 5; // Default base is 10. if (base == null) { base = new Ctor(10); isBase10 = true; } else { base = new Ctor(base); d = base.d; // Return NaN if base is negative, or non-finite, or is 0 or 1. if (base.s < 0 || !d || !d[0] || base.eq(1)) return new Ctor(NaN); isBase10 = base.eq(10); } d = arg.d; // Is arg negative, non-finite, 0 or 1? if (arg.s < 0 || !d || !d[0] || arg.eq(1)) { return new Ctor(d && !d[0] ? -1 / 0 : arg.s != 1 ? NaN : d ? 0 : 1 / 0); } // The result will have a non-terminating decimal expansion if base is 10 and arg is not an // integer power of 10. if (isBase10) { if (d.length > 1) { inf = true; } else { for (k = d[0]; k % 10 === 0;) k /= 10; inf = k !== 1; } } external = false; sd = pr + guard; num = naturalLogarithm(arg, sd); denominator = isBase10 ? getLn10(Ctor, sd + 10) : naturalLogarithm(base, sd); // The result will have 5 rounding digits. r = divide(num, denominator, sd, 1); // If at a rounding boundary, i.e. the result's rounding digits are [49]9999 or [50]0000, // calculate 10 further digits. // // If the result is known to have an infinite decimal expansion, repeat this until it is clear // that the result is above or below the boundary. Otherwise, if after calculating the 10 // further digits, the last 14 are nines, round up and assume the result is exact. // Also assume the result is exact if the last 14 are zero. // // Example of a result that will be incorrectly rounded: // log[1048576](4503599627370502) = 2.60000000000000009610279511444746... // The above result correctly rounded using ROUND_CEIL to 1 decimal place should be 2.7, but it // will be given as 2.6 as there are 15 zeros immediately after the requested decimal place, so // the exact result would be assumed to be 2.6, which rounded using ROUND_CEIL to 1 decimal // place is still 2.6. if (checkRoundingDigits(r.d, k = pr, rm)) { do { sd += 10; num = naturalLogarithm(arg, sd); denominator = isBase10 ? getLn10(Ctor, sd + 10) : naturalLogarithm(base, sd); r = divide(num, denominator, sd, 1); if (!inf) { // Check for 14 nines from the 2nd rounding digit, as the first may be 4. if (+digitsToString(r.d).slice(k + 1, k + 15) + 1 == 1e14) { r = finalise(r, pr + 1, 0); } break; } } while (checkRoundingDigits(r.d, k += 10, rm)); } external = true; return finalise(r, pr, rm); }; /* * Return a new Decimal whose value is the maximum of the arguments and the value of this Decimal. * * arguments {number|string|Decimal} * P.max = function () { Array.prototype.push.call(arguments, this); return maxOrMin(this.constructor, arguments, 'lt'); }; */ /* * Return a new Decimal whose value is the minimum of the arguments and the value of this Decimal. * * arguments {number|string|Decimal} * P.min = function () { Array.prototype.push.call(arguments, this); return maxOrMin(this.constructor, arguments, 'gt'); }; */ /* * n - 0 = n * n - N = N * n - I = -I * 0 - n = -n * 0 - 0 = 0 * 0 - N = N * 0 - I = -I * N - n = N * N - 0 = N * N - N = N * N - I = N * I - n = I * I - 0 = I * I - N = N * I - I = N * * Return a new Decimal whose value is the value of this Decimal minus `y`, rounded to `precision` * significant digits using rounding mode `rounding`. * */ P.minus = P.sub = function (y) { var d, e, i, j, k, len, pr, rm, xd, xe, xLTy, yd, x = this, Ctor = x.constructor; y = new Ctor(y); // If either is not finite... if (!x.d || !y.d) { // Return NaN if either is NaN. if (!x.s || !y.s) y = new Ctor(NaN); // Return y negated if x is finite and y is ±Infinity. else if (x.d) y.s = -y.s; // Return x if y is finite and x is ±Infinity. // Return x if both are ±Infinity with different signs. // Return NaN if both are ±Infinity with the same sign. else y = new Ctor(y.d || x.s !== y.s ? x : NaN); return y; } // If signs differ... if (x.s != y.s) { y.s = -y.s; return x.plus(y); } xd = x.d; yd = y.d; pr = Ctor.precision; rm = Ctor.rounding; // If either is zero... if (!xd[0] || !yd[0]) { // Return y negated if x is zero and y is non-zero. if (yd[0]) y.s = -y.s; // Return x if y is zero and x is non-zero. else if (xd[0]) y = new Ctor(x); // Return zero if both are zero. // From IEEE 754 (2008) 6.3: 0 - 0 = -0 - -0 = -0 when rounding to -Infinity. else return new Ctor(rm === 3 ? -0 : 0); return external ? finalise(y, pr, rm) : y; } // x and y are finite, non-zero numbers with the same sign. // Calculate base 1e7 exponents. e = mathfloor(y.e / LOG_BASE); xe = mathfloor(x.e / LOG_BASE); xd = xd.slice(); k = xe - e; // If base 1e7 exponents differ... if (k) { xLTy = k < 0; if (xLTy) { d = xd; k = -k; len = yd.length; } else { d = yd; e = xe; len = xd.length; } // Numbers with massively different exponents would result in a very high number of // zeros needing to be prepended, but this can be avoided while still ensuring correct // rounding by limiting the number of zeros to `Math.ceil(pr / LOG_BASE) + 2`. i = Math.max(Math.ceil(pr / LOG_BASE), len) + 2; if (k > i) { k = i; d.length = 1; } // Prepend zeros to equalise exponents. d.reverse(); for (i = k; i--;) d.push(0); d.reverse(); // Base 1e7 exponents equal. } else { // Check digits to determine which is the bigger number. i = xd.length; len = yd.length; xLTy = i < len; if (xLTy) len = i; for (i = 0; i < len; i++) { if (xd[i] != yd[i]) { xLTy = xd[i] < yd[i]; break; } } k = 0; } if (xLTy) { d = xd; xd = yd; yd = d; y.s = -y.s; } len = xd.length; // Append zeros to `xd` if shorter. // Don't add zeros to `yd` if shorter as subtraction only needs to start at `yd` length. for (i = yd.length - len; i > 0; --i) xd[len++] = 0; // Subtract yd from xd. for (i = yd.length; i > k;) { if (xd[--i] < yd[i]) { for (j = i; j && xd[--j] === 0;) xd[j] = BASE - 1; --xd[j]; xd[i] += BASE; } xd[i] -= yd[i]; } // Remove trailing zeros. for (; xd[--len] === 0;) xd.pop(); // Remove leading zeros and adjust exponent accordingly. for (; xd[0] === 0; xd.shift()) --e; // Zero? if (!xd[0]) return new Ctor(rm === 3 ? -0 : 0); y.d = xd; y.e = getBase10Exponent(xd, e); return external ? finalise(y, pr, rm) : y; }; /* * n % 0 = N * n % N = N * n % I = n * 0 % n = 0 * -0 % n = -0 * 0 % 0 = N * 0 % N = N * 0 % I = 0 * N % n = N * N % 0 = N * N % N = N * N % I = N * I % n = N * I % 0 = N * I % N = N * I % I = N * * Return a new Decimal whose value is the value of this Decimal modulo `y`, rounded to * `precision` significant digits using rounding mode `rounding`. * * The result depends on the modulo mode. * */ P.modulo = P.mod = function (y) { var q, x = this, Ctor = x.constructor; y = new Ctor(y); // Return NaN if x is ±Infinity or NaN, or y is NaN or ±0. if (!x.d || !y.s || y.d && !y.d[0]) return new Ctor(NaN); // Return x if y is ±Infinity or x is ±0. if (!y.d || x.d && !x.d[0]) { return finalise(new Ctor(x), Ctor.precision, Ctor.rounding); } // Prevent rounding of intermediate calculations. external = false; if (Ctor.modulo == 9) { // Euclidian division: q = sign(y) * floor(x / abs(y)) // result = x - q * y where 0 <= result < abs(y) q = divide(x, y.abs(), 0, 3, 1); q.s *= y.s; } else { q = divide(x, y, 0, Ctor.modulo, 1); } q = q.times(y); external = true; return x.minus(q); }; /* * Return a new Decimal whose value is the natural exponential of the value of this Decimal, * i.e. the base e raised to the power the value of this Decimal, rounded to `precision` * significant digits using rounding mode `rounding`. * */ P.naturalExponential = P.exp = function () { return naturalExponential(this); }; /* * Return a new Decimal whose value is the natural logarithm of the value of this Decimal, * rounded to `precision` significant digits using rounding mode `rounding`. * */ P.naturalLogarithm = P.ln = function () { return naturalLogarithm(this); }; /* * Return a new Decimal whose value is the value of this Decimal negated, i.e. as if multiplied by * -1. * */ P.negated = P.neg = function () { var x = new this.constructor(this); x.s = -x.s; return finalise(x); }; /* * n + 0 = n * n + N = N * n + I = I * 0 + n = n * 0 + 0 = 0 * 0 + N = N * 0 + I = I * N + n = N * N + 0 = N * N + N = N * N + I = N * I + n = I * I + 0 = I * I + N = N * I + I = I * * Return a new Decimal whose value is the value of this Decimal plus `y`, rounded to `precision` * significant digits using rounding mode `rounding`. * */ P.plus = P.add = function (y) { var carry, d, e, i, k, len, pr, rm, xd, yd, x = this, Ctor = x.constructor; y = new Ctor(y); // If either is not finite... if (!x.d || !y.d) { // Return NaN if either is NaN. if (!x.s || !y.s) y = new Ctor(NaN); // Return x if y is finite and x is ±Infinity. // Return x if both are ±Infinity with the same sign. // Return NaN if both are ±Infinity with different signs. // Return y if x is finite and y is ±Infinity. else if (!x.d) y = new Ctor(y.d || x.s === y.s ? x : NaN); return y; } // If signs differ... if (x.s != y.s) { y.s = -y.s; return x.minus(y); } xd = x.d; yd = y.d; pr = Ctor.precision; rm = Ctor.rounding; // If either is zero... if (!xd[0] || !yd[0]) { // Return x if y is zero. // Return y if y is non-zero. if (!yd[0]) y = new Ctor(x); return external ? finalise(y, pr, rm) : y; } // x and y are finite, non-zero numbers with the same sign. // Calculate base 1e7 exponents. k = mathfloor(x.e / LOG_BASE); e = mathfloor(y.e / LOG_BASE); xd = xd.slice(); i = k - e; // If base 1e7 exponents differ... if (i) { if (i < 0) { d = xd; i = -i; len = yd.length; } else { d = yd; e = k; len = xd.length; } // Limit number of zeros prepended to max(ceil(pr / LOG_BASE), len) + 1. k = Math.ceil(pr / LOG_BASE); len = k > len ? k + 1 : len + 1; if (i > len) { i = len; d.length = 1; } // Prepend zeros to equalise exponents. Note: Faster to use reverse then do unshifts. d.reverse(); for (; i--;) d.push(0); d.reverse(); } len = xd.length; i = yd.length; // If yd is longer than xd, swap xd and yd so xd points to the longer array. if (len - i < 0) { i = len; d = yd; yd = xd; xd = d; } // Only start adding at yd.length - 1 as the further digits of xd can be left as they are. for (carry = 0; i;) { carry = (xd[--i] = xd[i] + yd[i] + carry) / BASE | 0; xd[i] %= BASE; } if (carry) { xd.unshift(carry); ++e; } // Remove trailing zeros. // No need to check for zero, as +x + +y != 0 && -x + -y != 0 for (len = xd.length; xd[--len] == 0;) xd.pop(); y.d = xd; y.e = getBase10Exponent(xd, e); return external ? finalise(y, pr, rm) : y; }; /* * Return the number of significant digits of the value of this Decimal. * * [z] {boolean|number} Whether to count integer-part trailing zeros: true, false, 1 or 0. * */ P.precision = P.sd = function (z) { var k, x = this; if (z !== void 0 && z !== !!z && z !== 1 && z !== 0) throw Error(invalidArgument + z); if (x.d) { k = getPrecision(x.d); if (z && x.e + 1 > k) k = x.e + 1; } else { k = NaN; } return k; }; /* * Return a new Decimal whose value is the value of this Decimal rounded to a whole number using * rounding mode `rounding`. * */ P.round = function () { var x = this, Ctor = x.constructor; return finalise(new Ctor(x), x.e + 1, Ctor.rounding); }; /* * Return a new Decimal whose value is the sine of the value in radians of this Decimal. * * Domain: [-Infinity, Infinity] * Range: [-1, 1] * * sin(x) = x - x^3/3! + x^5/5! - ... * * sin(0) = 0 * sin(-0) = -0 * sin(Infinity) = NaN * sin(-Infinity) = NaN * sin(NaN) = NaN * */ P.sine = P.sin = function () { var pr, rm, x = this, Ctor = x.constructor; if (!x.isFinite()) return new Ctor(NaN); if (x.isZero()) return new Ctor(x); pr = Ctor.precision; rm = Ctor.rounding; Ctor.precision = pr + Math.max(x.e, x.sd()) + LOG_BASE; Ctor.rounding = 1; x = sine(Ctor, toLessThanHalfPi(Ctor, x)); Ctor.precision = pr; Ctor.rounding = rm; return finalise(quadrant > 2 ? x.neg() : x, pr, rm, true); }; /* * Return a new Decimal whose value is the square root of this Decimal, rounded to `precision` * significant digits using rounding mode `rounding`. * * sqrt(-n) = N * sqrt(N) = N * sqrt(-I) = N * sqrt(I) = I * sqrt(0) = 0 * sqrt(-0) = -0 * */ P.squareRoot = P.sqrt = function () { var m, n, sd, r, rep, t, x = this, d = x.d, e = x.e, s = x.s, Ctor = x.constructor; // Negative/NaN/Infinity/zero? if (s !== 1 || !d || !d[0]) { return new Ctor(!s || s < 0 && (!d || d[0]) ? NaN : d ? x : 1 / 0); } external = false; // Initial estimate. s = Math.sqrt(+x); // Math.sqrt underflow/overflow? // Pass x to Math.sqrt as integer, then adjust the exponent of the result. if (s == 0 || s == 1 / 0) { n = digitsToString(d); if ((n.length + e) % 2 == 0) n += '0'; s = Math.sqrt(n); e = mathfloor((e + 1) / 2) - (e < 0 || e % 2); if (s == 1 / 0) { n = '1e' + e; } else { n = s.toExponential(); n = n.slice(0, n.indexOf('e') + 1) + e; } r = new Ctor(n); } else { r = new Ctor(s.toString()); } sd = (e = Ctor.precision) + 3; // Newton-Raphson iteration. for (;;) { t = r; r = t.plus(divide(x, t, sd + 2, 1)).times(0.5); // TODO? Replace with for-loop and checkRoundingDigits. if (digitsToString(t.d).slice(0, sd) === (n = digitsToString(r.d)).slice(0, sd)) { n = n.slice(sd - 3, sd + 1); // The 4th rounding digit may be in error by -1 so if the 4 rounding digits are 9999 or // 4999, i.e. approaching a rounding boundary, continue the iteration. if (n == '9999' || !rep && n == '4999') { // On the first iteration only, check to see if rounding up gives the exact result as the // nines may infinitely repeat. if (!rep) { finalise(t, e + 1, 0); if (t.times(t).eq(x)) { r = t; break; } } sd += 4; rep = 1; } else { // If the rounding digits are null, 0{0,4} or 50{0,3}, check for an exact result. // If not, then there are further digits and m will be truthy. if (!+n || !+n.slice(1) && n.charAt(0) == '5') { // Truncate to the first rounding digit. finalise(r, e + 1, 1); m = !r.times(r).eq(x); } break; } } } external = true; return finalise(r, e, Ctor.rounding, m); }; /* * Return a new Decimal whose value is the tangent of the value in radians of this Decimal. * * Domain: [-Infinity, Infinity] * Range: [-Infinity, Infinity] * * tan(0) = 0 * tan(-0) = -0 * tan(Infinity) = NaN * tan(-Infinity) = NaN * tan(NaN) = NaN * */ P.tangent = P.tan = function () { var pr, rm, x = this, Ctor = x.constructor; if (!x.isFinite()) return new Ctor(NaN); if (x.isZero()) return new Ctor(x); pr = Ctor.precision; rm = Ctor.rounding; Ctor.precision = pr + 10; Ctor.rounding = 1; x = x.sin(); x.s = 1; x = divide(x, new Ctor(1).minus(x.times(x)).sqrt(), pr + 10, 0); Ctor.precision = pr; Ctor.rounding = rm; return finalise(quadrant == 2 || quadrant == 4 ? x.neg() : x, pr, rm, true); }; /* * n * 0 = 0 * n * N = N * n * I = I * 0 * n = 0 * 0 * 0 = 0 * 0 * N = N * 0 * I = N * N * n = N * N * 0 = N * N * N = N * N * I = N * I * n = I * I * 0 = N * I * N = N * I * I = I * * Return a new Decimal whose value is this Decimal times `y`, rounded to `precision` significant * digits using rounding mode `rounding`. * */ P.times = P.mul = function (y) { var carry, e, i, k, r, rL, t, xdL, ydL, x = this, Ctor = x.constructor, xd = x.d, yd = (y = new Ctor(y)).d; y.s *= x.s; // If either is NaN, ±Infinity or ±0... if (!xd || !xd[0] || !yd || !yd[0]) { return new Ctor(!y.s || xd && !xd[0] && !yd || yd && !yd[0] && !xd // Return NaN if either is NaN. // Return NaN if x is ±0 and y is ±Infinity, or y is ±0 and x is ±Infinity. ? NaN // Return ±Infinity if either is ±Infinity. // Return ±0 if either is ±0. : !xd || !yd ? y.s / 0 : y.s * 0); } e = mathfloor(x.e / LOG_BASE) + mathfloor(y.e / LOG_BASE); xdL = xd.length; ydL = yd.length; // Ensure xd points to the longer array. if (xdL < ydL) { r = xd; xd = yd; yd = r; rL = xdL; xdL = ydL; ydL = rL; } // Initialise the result array with zeros. r = []; rL = xdL + ydL; for (i = rL; i--;) r.push(0); // Multiply! for (i = ydL; --i >= 0;) { carry = 0; for (k = xdL + i; k > i;) { t = r[k] + yd[i] * xd[k - i - 1] + carry; r[k--] = t % BASE | 0; carry = t / BASE | 0; } r[k] = (r[k] + carry) % BASE | 0; } // Remove trailing zeros. for (; !r[--rL];) r.pop(); if (carry) ++e; else r.shift(); y.d = r; y.e = getBase10Exponent(r, e); return external ? finalise(y, Ctor.precision, Ctor.rounding) : y; }; /* * Return a string representing the value of this Decimal in base 2, round to `sd` significant * digits using rounding mode `rm`. * * If the optional `sd` argument is present then return binary exponential notation. * * [sd] {number} Significant digits. Integer, 1 to MAX_DIGITS inclusive. * [rm] {number} Rounding mode. Integer, 0 to 8 inclusive. * */ P.toBinary = function (sd, rm) { return toStringBinary(this, 2, sd, rm); }; /* * Return a new Decimal whose value is the value of this Decimal rounded to a maximum of `dp` * decimal places using rounding mode `rm` or `rounding` if `rm` is omitted. * * If `dp` is omitted, return a new Decimal whose value is the value of this Decimal. * * [dp] {number} Decimal places. Integer, 0 to MAX_DIGITS inclusive. * [rm] {number} Rounding mode. Integer, 0 to 8 inclusive. * */ P.toDecimalPlaces = P.toDP = function (dp, rm) { var x = this, Ctor = x.constructor; x = new Ctor(x); if (dp === void 0) return x; checkInt32(dp, 0, MAX_DIGITS); if (rm === void 0) rm = Ctor.rounding; else checkInt32(rm, 0, 8); return finalise(x, dp + x.e + 1, rm); }; /* * Return a string representing the value of this Decimal in exponential notation rounded to * `dp` fixed decimal places using rounding mode `rounding`. * * [dp] {number} Decimal places. Integer, 0 to MAX_DIGITS inclusive. * [rm] {number} Rounding mode. Integer, 0 to 8 inclusive. * */ P.toExponential = function (dp, rm) { var str, x = this, Ctor = x.constructor; if (dp === void 0) { str = finiteToString(x, true); } else { checkInt32(dp, 0, MAX_DIGITS); if (rm === void 0) rm = Ctor.rounding; else checkInt32(rm, 0, 8); x = finalise(new Ctor(x), dp + 1, rm); str = finiteToString(x, true, dp + 1); } return x.isNeg() && !x.isZero() ? '-' + str : str; }; /* * Return a string representing the value of this Decimal in normal (fixed-point) notation to * `dp` fixed decimal places and rounded using rounding mode `rm` or `rounding` if `rm` is * omitted. * * As with JavaScript numbers, (-0).toFixed(0) is '0', but e.g. (-0.00001).toFixed(0) is '-0'. * * [dp] {number} Decimal places. Integer, 0 to MAX_DIGITS inclusive. * [rm] {number} Rounding mode. Integer, 0 to 8 inclusive. * * (-0).toFixed(0) is '0', but (-0.1).toFixed(0) is '-0'. * (-0).toFixed(1) is '0.0', but (-0.01).toFixed(1) is '-0.0'. * (-0).toFixed(3) is '0.000'. * (-0.5).toFixed(0) is '-0'. * */ P.toFixed = function (dp, rm) { var str, y, x = this, Ctor = x.constructor; if (dp === void 0) { str = finiteToString(x); } else { checkInt32(dp, 0, MAX_DIGITS); if (rm === void 0) rm = Ctor.rounding; else checkInt32(rm, 0, 8); y = finalise(new Ctor(x), dp + x.e + 1, rm); str = finiteToString(y, false, dp + y.e + 1); } // To determine whether to add the minus sign look at the value before it was rounded, // i.e. look at `x` rather than `y`. return x.isNeg() && !x.isZero() ? '-' + str : str; }; /* * Return an array representing the value of this Decimal as a simple fraction with an integer * numerator and an integer denominator. * * The denominator will be a positive non-zero value less than or equal to the specified maximum * denominator. If a maximum denominator is not specified, the denominator will be the lowest * value necessary to represent the number exactly. * * [maxD] {number|string|Decimal} Maximum denominator. Integer >= 1 and < Infinity. * */ P.toFraction = function (maxD) { var d, d0, d1, d2, e, k, n, n0, n1, pr, q, r, x = this, xd = x.d, Ctor = x.constructor; if (!xd) return new Ctor(x); n1 = d0 = new Ctor(1); d1 = n0 = new Ctor(0); d = new Ctor(d1); e = d.e = getPrecision(xd) - x.e - 1; k = e % LOG_BASE; d.d[0] = mathpow(10, k < 0 ? LOG_BASE + k : k); if (maxD == null) { // d is 10**e, the minimum max-denominator needed. maxD = e > 0 ? d : n1; } else { n = new Ctor(maxD); if (!n.isInt() || n.lt(n1)) throw Error(invalidArgument + n); maxD = n.gt(d) ? (e > 0 ? d : n1) : n; } external = false; n = new Ctor(digitsToString(xd)); pr = Ctor.precision; Ctor.precision = e = xd.length * LOG_BASE * 2; for (;;) { q = divide(n, d, 0, 1, 1); d2 = d0.plus(q.times(d1)); if (d2.cmp(maxD) == 1) break; d0 = d1; d1 = d2; d2 = n1; n1 = n0.plus(q.times(d2)); n0 = d2; d2 = d; d = n.minus(q.times(d2)); n = d2; } d2 = divide(maxD.minus(d0), d1, 0, 1, 1); n0 = n0.plus(d2.times(n1)); d0 = d0.plus(d2.times(d1)); n0.s = n1.s = x.s; // Determine which fraction is closer to x, n0/d0 or n1/d1? r = divide(n1, d1, e, 1).minus(x).abs().cmp(divide(n0, d0, e, 1).minus(x).abs()) < 1 ? [n1, d1] : [n0, d0]; Ctor.precision = pr; external = true; return r; }; /* * Return a string representing the value of this Decimal in base 16, round to `sd` significant * digits using rounding mode `rm`. * * If the optional `sd` argument is present then return binary exponential notation. * * [sd] {number} Significant digits. Integer, 1 to MAX_DIGITS inclusive. * [rm] {number} Rounding mode. Integer, 0 to 8 inclusive. * */ P.toHexadecimal = P.toHex = function (sd, rm) { return toStringBinary(this, 16, sd, rm); }; /* * Returns a new Decimal whose value is the nearest multiple of `y` in the direction of rounding * mode `rm`, or `Decimal.rounding` if `rm` is omitted, to the value of this Decimal. * * The return value will always have the same sign as this Decimal, unless either this Decimal * or `y` is NaN, in which case the return value will be also be NaN. * * The return value is not affected by the value of `precision`. * * y {number|string|Decimal} The magnitude to round to a multiple of. * [rm] {number} Rounding mode. Integer, 0 to 8 inclusive. * * 'toNearest() rounding mode not an integer: {rm}' * 'toNearest() rounding mode out of range: {rm}' * */ P.toNearest = function (y, rm) { var x = this, Ctor = x.constructor; x = new Ctor(x); if (y == null) { // If x is not finite, return x. if (!x.d) return x; y = new Ctor(1); rm = Ctor.rounding; } else { y = new Ctor(y); if (rm === void 0) { rm = Ctor.rounding; } else { checkInt32(rm, 0, 8); } // If x is not finite, return x if y is not NaN, else NaN. if (!x.d) return y.s ? x : y; // If y is not finite, return Infinity with the sign of x if y is Infinity, else NaN. if (!y.d) { if (y.s) y.s = x.s; return y; } } // If y is not zero, calculate the nearest multiple of y to x. if (y.d[0]) { external = false; x = divide(x, y, 0, rm, 1).times(y); external = true; finalise(x); // If y is zero, return zero with the sign of x. } else { y.s = x.s; x = y; } return x; }; /* * Return the value of this Decimal converted to a number primitive. * Zero keeps its sign. * */ P.toNumber = function () { return +this; }; /* * Return a string representing the value of this Decimal in base 8, round to `sd` significant * digits using rounding mode `rm`. * * If the optional `sd` argument is present then return binary exponential notation. * * [sd] {number} Significant digits. Integer, 1 to MAX_DIGITS inclusive. * [rm] {number} Rounding mode. Integer, 0 to 8 inclusive. * */ P.toOctal = function (sd, rm) { return toStringBinary(this, 8, sd, rm); }; /* * Return a new Decimal whose value is the value of this Decimal raised to the power `y`, rounded * to `precision` significant digits using rounding mode `rounding`. * * ECMAScript compliant. * * pow(x, NaN) = NaN * pow(x, ±0) = 1 * pow(NaN, non-zero) = NaN * pow(abs(x) > 1, +Infinity) = +Infinity * pow(abs(x) > 1, -Infinity) = +0 * pow(abs(x) == 1, ±Infinity) = NaN * pow(abs(x) < 1, +Infinity) = +0 * pow(abs(x) < 1, -Infinity) = +Infinity * pow(+Infinity, y > 0) = +Infinity * pow(+Infinity, y < 0) = +0 * pow(-Infinity, odd integer > 0) = -Infinity * pow(-Infinity, even integer > 0) = +Infinity * pow(-Infinity, odd integer < 0) = -0 * pow(-Infinity, even integer < 0) = +0 * pow(+0, y > 0) = +0 * pow(+0, y < 0) = +Infinity * pow(-0, odd integer > 0) = -0 * pow(-0, even integer > 0) = +0 * pow(-0, odd integer < 0) = -Infinity * pow(-0, even integer < 0) = +Infinity * pow(finite x < 0, finite non-integer) = NaN * * For non-integer or very large exponents pow(x, y) is calculated using * * x^y = exp(y*ln(x)) * * Assuming the first 15 rounding digits are each equally likely to be any digit 0-9, the * probability of an incorrectly rounded result * P([49]9{14} | [50]0{14}) = 2 * 0.2 * 10^-14 = 4e-15 = 1/2.5e+14 * i.e. 1 in 250,000,000,000,000 * * If a result is incorrectly rounded the maximum error will be 1 ulp (unit in last place). * * y {number|string|Decimal} The power to which to raise this Decimal. * */ P.toPower = P.pow = function (y) { var e, k, pr, r, rm, s, x = this, Ctor = x.constructor, yn = +(y = new Ctor(y)); // Either ±Infinity, NaN or ±0? if (!x.d || !y.d || !x.d[0] || !y.d[0]) return new Ctor(mathpow(+x, yn)); x = new Ctor(x); if (x.eq(1)) return x; pr = Ctor.precision; rm = Ctor.rounding; if (y.eq(1)) return finalise(x, pr, rm); // y exponent e = mathfloor(y.e / LOG_BASE); // If y is a small integer use the 'exponentiation by squaring' algorithm. if (e >= y.d.length - 1 && (k = yn < 0 ? -yn : yn) <= MAX_SAFE_INTEGER) { r = intPow(Ctor, x, k, pr); return y.s < 0 ? new Ctor(1).div(r) : finalise(r, pr, rm); } s = x.s; // if x is negative if (s < 0) { // if y is not an integer if (e < y.d.length - 1) return new Ctor(NaN); // Result is positive if x is negative and the last digit of integer y is even. if ((y.d[e] & 1) == 0) s = 1; // if x.eq(-1) if (x.e == 0 && x.d[0] == 1 && x.d.length == 1) { x.s = s; return x; } } // Estimate result exponent. // x^y = 10^e, where e = y * log10(x) // log10(x) = log10(x_significand) + x_exponent // log10(x_significand) = ln(x_significand) / ln(10) k = mathpow(+x, yn); e = k == 0 || !isFinite(k) ? mathfloor(yn * (Math.log('0.' + digitsToString(x.d)) / Math.LN10 + x.e + 1)) : new Ctor(k + '').e; // Exponent estimate may be incorrect e.g. x: 0.999999999999999999, y: 2.29, e: 0, r.e: -1. // Overflow/underflow? if (e > Ctor.maxE + 1 || e < Ctor.minE - 1) return new Ctor(e > 0 ? s / 0 : 0); external = false; Ctor.rounding = x.s = 1; // Estimate the extra guard digits needed to ensure five correct rounding digits from // naturalLogarithm(x). Example of failure without these extra digits (precision: 10): // new Decimal(2.32456).pow('2087987436534566.46411') // should be 1.162377823e+764914905173815, but is 1.162355823e+764914905173815 k = Math.min(12, (e + '').length); // r = x^y = exp(y*ln(x)) r = naturalExponential(y.times(naturalLogarithm(x, pr + k)), pr); // r may be Infinity, e.g. (0.9999999999999999).pow(-1e+40) if (r.d) { // Truncate to the required precision plus five rounding digits. r = finalise(r, pr + 5, 1); // If the rounding digits are [49]9999 or [50]0000 increase the precision by 10 and recalculate // the result. if (checkRoundingDigits(r.d, pr, rm)) { e = pr + 10; // Truncate to the increased precision plus five rounding digits. r = finalise(naturalExponential(y.times(naturalLogarithm(x, e + k)), e), e + 5, 1); // Check for 14 nines from the 2nd rounding digit (the first rounding digit may be 4 or 9). if (+digitsToString(r.d).slice(pr + 1, pr + 15) + 1 == 1e14) { r = finalise(r, pr + 1, 0); } } } r.s = s; external = true; Ctor.rounding = rm; return finalise(r, pr, rm); }; /* * Return a string representing the value of this Decimal rounded to `sd` significant digits * using rounding mode `rounding`. * * Return exponential notation if `sd` is less than the number of digits necessary to represent * the integer part of the value in normal notation. * * [sd] {number} Significant digits. Integer, 1 to MAX_DIGITS inclusive. * [rm] {number} Rounding mode. Integer, 0 to 8 inclusive. * */ P.toPrecision = function (sd, rm) { var str, x = this, Ctor = x.constructor; if (sd === void 0) { str = finiteToString(x, x.e <= Ctor.toExpNeg || x.e >= Ctor.toExpPos); } else { checkInt32(sd, 1, MAX_DIGITS); if (rm === void 0) rm = Ctor.rounding; else checkInt32(rm, 0, 8); x = finalise(new Ctor(x), sd, rm); str = finiteToString(x, sd <= x.e || x.e <= Ctor.toExpNeg, sd); } return x.isNeg() && !x.isZero() ? '-' + str : str; }; /* * Return a new Decimal whose value is the value of this Decimal rounded to a maximum of `sd` * significant digits using rounding mode `rm`, or to `precision` and `rounding` respectively if * omitted. * * [sd] {number} Significant digits. Integer, 1 to MAX_DIGITS inclusive. * [rm] {number} Rounding mode. Integer, 0 to 8 inclusive. * * 'toSD() digits out of range: {sd}' * 'toSD() digits not an integer: {sd}' * 'toSD() rounding mode not an integer: {rm}' * 'toSD() rounding mode out of range: {rm}' * */ P.toSignificantDigits = P.toSD = function (sd, rm) { var x = this, Ctor = x.constructor; if (sd === void 0) { sd = Ctor.precision; rm = Ctor.rounding; } else { checkInt32(sd, 1, MAX_DIGITS); if (rm === void 0) rm = Ctor.rounding; else checkInt32(rm, 0, 8); } return finalise(new Ctor(x), sd, rm); }; /* * Return a string representing the value of this Decimal. * * Return exponential notation if this Decimal has a positive exponent equal to or greater than * `toExpPos`, or a negative exponent equal to or less than `toExpNeg`. * */ P.toString = function () { var x = this, Ctor = x.constructor, str = finiteToString(x, x.e <= Ctor.toExpNeg || x.e >= Ctor.toExpPos); return x.isNeg() && !x.isZero() ? '-' + str : str; }; /* * Return a new Decimal whose value is the value of this Decimal truncated to a whole number. * */ P.truncated = P.trunc = function () { return finalise(new this.constructor(this), this.e + 1, 1); }; /* * Return a string representing the value of this Decimal. * Unlike `toString`, negative zero will include the minus sign. * */ P.valueOf = P.toJSON = function () { var x = this, Ctor = x.constructor, str = finiteToString(x, x.e <= Ctor.toExpNeg || x.e >= Ctor.toExpPos); return x.isNeg() ? '-' + str : str; }; /* // Add aliases to match BigDecimal method names. // P.add = P.plus; P.subtract = P.minus; P.multiply = P.times; P.divide = P.div; P.remainder = P.mod; P.compareTo = P.cmp; P.negate = P.neg; */ // Helper functions for Decimal.prototype (P) and/or Decimal methods, and their callers. /* * digitsToString P.cubeRoot, P.logarithm, P.squareRoot, P.toFraction, P.toPower, * finiteToString, naturalExponential, naturalLogarithm * checkInt32 P.toDecimalPlaces, P.toExponential, P.toFixed, P.toNearest, * P.toPrecision, P.toSignificantDigits, toStringBinary, random * checkRoundingDigits P.logarithm, P.toPower, naturalExponential, naturalLogarithm * convertBase toStringBinary, parseOther * cos P.cos * divide P.atanh, P.cubeRoot, P.dividedBy, P.dividedToIntegerBy, * P.logarithm, P.modulo, P.squareRoot, P.tan, P.tanh, P.toFraction, * P.toNearest, toStringBinary, naturalExponential, naturalLogarithm, * taylorSeries, atan2, parseOther * finalise P.absoluteValue, P.atan, P.atanh, P.ceil, P.cos, P.cosh, * P.cubeRoot, P.dividedToIntegerBy, P.floor, P.logarithm, P.minus, * P.modulo, P.negated, P.plus, P.round, P.sin, P.sinh, P.squareRoot, * P.tan, P.times, P.toDecimalPlaces, P.toExponential, P.toFixed, * P.toNearest, P.toPower, P.toPrecision, P.toSignificantDigits, * P.truncated, divide, getLn10, getPi, naturalExponential, * naturalLogarithm, ceil, floor, round, trunc * finiteToString P.toExponential, P.toFixed, P.toPrecision, P.toString, P.valueOf, * toStringBinary * getBase10Exponent P.minus, P.plus, P.times, parseOther * getLn10 P.logarithm, naturalLogarithm * getPi P.acos, P.asin, P.atan, toLessThanHalfPi, atan2 * getPrecision P.precision, P.toFraction * getZeroString digitsToString, finiteToString * intPow P.toPower, parseOther * isOdd toLessThanHalfPi * maxOrMin max, min * naturalExponential P.naturalExponential, P.toPower * naturalLogarithm P.acosh, P.asinh, P.atanh, P.logarithm, P.naturalLogarithm, * P.toPower, naturalExponential * nonFiniteToString finiteToString, toStringBinary * parseDecimal Decimal * parseOther Decimal * sin P.sin * taylorSeries P.cosh, P.sinh, cos, sin * toLessThanHalfPi P.cos, P.sin * toStringBinary P.toBinary, P.toHexadecimal, P.toOctal * truncate intPow * * Throws: P.logarithm, P.precision, P.toFraction, checkInt32, getLn10, getPi, * naturalLogarithm, config, parseOther, random, Decimal */ function digitsToString(d) { var i, k, ws, indexOfLastWord = d.length - 1, str = '', w = d[0]; if (indexOfLastWord > 0) { str += w; for (i = 1; i < indexOfLastWord; i++) { ws = d[i] + ''; k = LOG_BASE - ws.length; if (k) str += getZeroString(k); str += ws; } w = d[i]; ws = w + ''; k = LOG_BASE - ws.length; if (k) str += getZeroString(k); } else if (w === 0) { return '0'; } // Remove trailing zeros of last w. for (; w % 10 === 0;) w /= 10; return str + w; } function checkInt32(i, min, max) { if (i !== ~~i || i < min || i > max) { throw Error(invalidArgument + i); } } /* * Check 5 rounding digits if `repeating` is null, 4 otherwise. * `repeating == null` if caller is `log` or `pow`, * `repeating != null` if caller is `naturalLogarithm` or `naturalExponential`. */ function checkRoundingDigits(d, i, rm, repeating) { var di, k, r, rd; // Get the length of the first word of the array d. for (k = d[0]; k >= 10; k /= 10) --i; // Is the rounding digit in the first word of d? if (--i < 0) { i += LOG_BASE; di = 0; } else { di = Math.ceil((i + 1) / LOG_BASE); i %= LOG_BASE; } // i is the index (0 - 6) of the rounding digit. // E.g. if within the word 3487563 the first rounding digit is 5, // then i = 4, k = 1000, rd = 3487563 % 1000 = 563 k = mathpow(10, LOG_BASE - i); rd = d[di] % k | 0; if (repeating == null) { if (i < 3) { if (i == 0) rd = rd / 100 | 0; else if (i == 1) rd = rd / 10 | 0; r = rm < 4 && rd == 99999 || rm > 3 && rd == 49999 || rd == 50000 || rd == 0; } else { r = (rm < 4 && rd + 1 == k || rm > 3 && rd + 1 == k / 2) && (d[di + 1] / k / 100 | 0) == mathpow(10, i - 2) - 1 || (rd == k / 2 || rd == 0) && (d[di + 1] / k / 100 | 0) == 0; } } else { if (i < 4) { if (i == 0) rd = rd / 1000 | 0; else if (i == 1) rd = rd / 100 | 0; else if (i == 2) rd = rd / 10 | 0; r = (repeating || rm < 4) && rd == 9999 || !repeating && rm > 3 && rd == 4999; } else { r = ((repeating || rm < 4) && rd + 1 == k || (!repeating && rm > 3) && rd + 1 == k / 2) && (d[di + 1] / k / 1000 | 0) == mathpow(10, i - 3) - 1; } } return r; } // Convert string of `baseIn` to an array of numbers of `baseOut`. // Eg. convertBase('255', 10, 16) returns [15, 15]. // Eg. convertBase('ff', 16, 10) returns [2, 5, 5]. function convertBase(str, baseIn, baseOut) { var j, arr = [0], arrL, i = 0, strL = str.length; for (; i < strL;) { for (arrL = arr.length; arrL--;) arr[arrL] *= baseIn; arr[0] += NUMERALS.indexOf(str.charAt(i++)); for (j = 0; j < arr.length; j++) { if (arr[j] > baseOut - 1) { if (arr[j + 1] === void 0) arr[j + 1] = 0; arr[j + 1] += arr[j] / baseOut | 0; arr[j] %= baseOut; } } } return arr.reverse(); } /* * cos(x) = 1 - x^2/2! + x^4/4! - ... * |x| < pi/2 * */ function cosine(Ctor, x) { var k, y, len = x.d.length; // Argument reduction: cos(4x) = 8*(cos^4(x) - cos^2(x)) + 1 // i.e. cos(x) = 8*(cos^4(x/4) - cos^2(x/4)) + 1 // Estimate the optimum number of times to use the argument reduction. if (len < 32) { k = Math.ceil(len / 3); y = Math.pow(4, -k).toString(); } else { k = 16; y = '2.3283064365386962890625e-10'; } Ctor.precision += k; x = taylorSeries(Ctor, 1, x.times(y), new Ctor(1)); // Reverse argument reduction for (var i = k; i--;) { var cos2x = x.times(x); x = cos2x.times(cos2x).minus(cos2x).times(8).plus(1); } Ctor.precision -= k; return x; } /* * Perform division in the specified base. */ var divide = (function () { // Assumes non-zero x and k, and hence non-zero result. function multiplyInteger(x, k, base) { var temp, carry = 0, i = x.length; for (x = x.slice(); i--;) { temp = x[i] * k + carry; x[i] = temp % base | 0; carry = temp / base | 0; } if (carry) x.unshift(carry); return x; } function compare(a, b, aL, bL) { var i, r; if (aL != bL) { r = aL > bL ? 1 : -1; } else { for (i = r = 0; i < aL; i++) { if (a[i] != b[i]) { r = a[i] > b[i] ? 1 : -1; break; } } } return r; } function subtract(a, b, aL, base) { var i = 0; // Subtract b from a. for (; aL--;) { a[aL] -= i; i = a[aL] < b[aL] ? 1 : 0; a[aL] = i * base + a[aL] - b[aL]; } // Remove leading zeros. for (; !a[0] && a.length > 1;) a.shift(); } return function (x, y, pr, rm, dp, base) { var cmp, e, i, k, logBase, more, prod, prodL, q, qd, rem, remL, rem0, sd, t, xi, xL, yd0, yL, yz, Ctor = x.constructor, sign = x.s == y.s ? 1 : -1, xd = x.d, yd = y.d; // Either NaN, Infinity or 0? if (!xd || !xd[0] || !yd || !yd[0]) { return new Ctor(// Return NaN if either NaN, or both Infinity or 0. !x.s || !y.s || (xd ? yd && xd[0] == yd[0] : !yd) ? NaN : // Return ±0 if x is 0 or y is ±Infinity, or return ±Infinity as y is 0. xd && xd[0] == 0 || !yd ? sign * 0 : sign / 0); } if (base) { logBase = 1; e = x.e - y.e; } else { base = BASE; logBase = LOG_BASE; e = mathfloor(x.e / logBase) - mathfloor(y.e / logBase); } yL = yd.length; xL = xd.length; q = new Ctor(sign); qd = q.d = []; // Result exponent may be one less than e. // The digit array of a Decimal from toStringBinary may have trailing zeros. for (i = 0; yd[i] == (xd[i] || 0); i++); if (yd[i] > (xd[i] || 0)) e--; if (pr == null) { sd = pr = Ctor.precision; rm = Ctor.rounding; } else if (dp) { sd = pr + (x.e - y.e) + 1; } else { sd = pr; } if (sd < 0) { qd.push(1); more = true; } else { // Convert precision in number of base 10 digits to base 1e7 digits. sd = sd / logBase + 2 | 0; i = 0; // divisor < 1e7 if (yL == 1) { k = 0; yd = yd[0]; sd++; // k is the carry. for (; (i < xL || k) && sd--; i++) { t = k * base + (xd[i] || 0); qd[i] = t / yd | 0; k = t % yd | 0; } more = k || i < xL; // divisor >= 1e7 } else { // Normalise xd and yd so highest order digit of yd is >= base/2 k = base / (yd[0] + 1) | 0; if (k > 1) { yd = multiplyInteger(yd, k, base); xd = multiplyInteger(xd, k, base); yL = yd.length; xL = xd.length; } xi = yL; rem = xd.slice(0, yL); remL = rem.length; // Add zeros to make remainder as long as divisor. for (; remL < yL;) rem[remL++] = 0; yz = yd.slice(); yz.unshift(0); yd0 = yd[0]; if (yd[1] >= base / 2) ++yd0; do { k = 0; // Compare divisor and remainder. cmp = compare(yd, rem, yL, remL); // If divisor < remainder. if (cmp < 0) { // Calculate trial digit, k. rem0 = rem[0]; if (yL != remL) rem0 = rem0 * base + (rem[1] || 0); // k will be how many times the divisor goes into the current remainder. k = rem0 / yd0 | 0; // Algorithm: // 1. product = divisor * trial digit (k) // 2. if product > remainder: product -= divisor, k-- // 3. remainder -= product // 4. if product was < remainder at 2: // 5. compare new remainder and divisor // 6. If remainder > divisor: remainder -= divisor, k++ if (k > 1) { if (k >= base) k = base - 1; // product = divisor * trial digit. prod = multiplyInteger(yd, k, base); prodL = prod.length; remL = rem.length; // Compare product and remainder. cmp = compare(prod, rem, prodL, remL); // product > remainder. if (cmp == 1) { k--; // Subtract divisor from product. subtract(prod, yL < prodL ? yz : yd, prodL, base); } } else { // cmp is -1. // If k is 0, there is no need to compare yd and rem again below, so change cmp to 1 // to avoid it. If k is 1 there is a need to compare yd and rem again below. if (k == 0) cmp = k = 1; prod = yd.slice(); } prodL = prod.length; if (prodL < remL) prod.unshift(0); // Subtract product from remainder. subtract(rem, prod, remL, base); // If product was < previous remainder. if (cmp == -1) { remL = rem.length; // Compare divisor and new remainder. cmp = compare(yd, rem, yL, remL); // If divisor < new remainder, subtract divisor from remainder. if (cmp < 1) { k++; // Subtract divisor from remainder. subtract(rem, yL < remL ? yz : yd, remL, base); } } remL = rem.length; } else if (cmp === 0) { k++; rem = [0]; } // if cmp === 1, k will be 0 // Add the next digit, k, to the result array. qd[i++] = k; // Update the remainder. if (cmp && rem[0]) { rem[remL++] = xd[xi] || 0; } else { rem = [xd[xi]]; remL = 1; } } while ((xi++ < xL || rem[0] !== void 0) && sd--); more = rem[0] !== void 0; } // Leading zero? if (!qd[0]) qd.shift(); } // logBase is 1 when divide is being used for base conversion. if (logBase == 1) { q.e = e; inexact = more; } else { // To calculate q.e, first get the number of digits of qd[0]. for (i = 1, k = qd[0]; k >= 10; k /= 10) i++; q.e = i + e * logBase - 1; finalise(q, dp ? pr + q.e + 1 : pr, rm, more); } return q; }; })(); /* * Round `x` to `sd` significant digits using rounding mode `rm`. * Check for over/under-flow. */ function finalise(x, sd, rm, isTruncated) { var digits, i, j, k, rd, roundUp, w, xd, xdi, Ctor = x.constructor; // Don't round if sd is null or undefined. out: if (sd != null) { xd = x.d; // Infinity/NaN. if (!xd) return x; // rd: the rounding digit, i.e. the digit after the digit that may be rounded up. // w: the word of xd containing rd, a base 1e7 number. // xdi: the index of w within xd. // digits: the number of digits of w. // i: what would be the index of rd within w if all the numbers were 7 digits long (i.e. if // they had leading zeros) // j: if > 0, the actual index of rd within w (if < 0, rd is a leading zero). // Get the length of the first word of the digits array xd. for (digits = 1, k = xd[0]; k >= 10; k /= 10) digits++; i = sd - digits; // Is the rounding digit in the first word of xd? if (i < 0) { i += LOG_BASE; j = sd; w = xd[xdi = 0]; // Get the rounding digit at index j of w. rd = w / mathpow(10, digits - j - 1) % 10 | 0; } else { xdi = Math.ceil((i + 1) / LOG_BASE); k = xd.length; if (xdi >= k) { if (isTruncated) { // Needed by `naturalExponential`, `naturalLogarithm` and `squareRoot`. for (; k++ <= xdi;) xd.push(0); w = rd = 0; digits = 1; i %= LOG_BASE; j = i - LOG_BASE + 1; } else { break out; } } else { w = k = xd[xdi]; // Get the number of digits of w. for (digits = 1; k >= 10; k /= 10) digits++; // Get the index of rd within w. i %= LOG_BASE; // Get the index of rd within w, adjusted for leading zeros. // The number of leading zeros of w is given by LOG_BASE - digits. j = i - LOG_BASE + digits; // Get the rounding digit at index j of w. rd = j < 0 ? 0 : w / mathpow(10, digits - j - 1) % 10 | 0; } } // Are there any non-zero digits after the rounding digit? isTruncated = isTruncated || sd < 0 || xd[xdi + 1] !== void 0 || (j < 0 ? w : w % mathpow(10, digits - j - 1)); // The expression `w % mathpow(10, digits - j - 1)` returns all the digits of w to the right // of the digit at (left-to-right) index j, e.g. if w is 908714 and j is 2, the expression // will give 714. roundUp = rm < 4 ? (rd || isTruncated) && (rm == 0 || rm == (x.s < 0 ? 3 : 2)) : rd > 5 || rd == 5 && (rm == 4 || isTruncated || rm == 6 && // Check whether the digit to the left of the rounding digit is odd. ((i > 0 ? j > 0 ? w / mathpow(10, digits - j) : 0 : xd[xdi - 1]) % 10) & 1 || rm == (x.s < 0 ? 8 : 7)); if (sd < 1 || !xd[0]) { xd.length = 0; if (roundUp) { // Convert sd to decimal places. sd -= x.e + 1; // 1, 0.1, 0.01, 0.001, 0.0001 etc. xd[0] = mathpow(10, (LOG_BASE - sd % LOG_BASE) % LOG_BASE); x.e = -sd || 0; } else { // Zero. xd[0] = x.e = 0; } return x; } // Remove excess digits. if (i == 0) { xd.length = xdi; k = 1; xdi--; } else { xd.length = xdi + 1; k = mathpow(10, LOG_BASE - i); // E.g. 56700 becomes 56000 if 7 is the rounding digit. // j > 0 means i > number of leading zeros of w. xd[xdi] = j > 0 ? (w / mathpow(10, digits - j) % mathpow(10, j) | 0) * k : 0; } if (roundUp) { for (;;) { // Is the digit to be rounded up in the first word of xd? if (xdi == 0) { // i will be the length of xd[0] before k is added. for (i = 1, j = xd[0]; j >= 10; j /= 10) i++; j = xd[0] += k; for (k = 1; j >= 10; j /= 10) k++; // if i != k the length has increased. if (i != k) { x.e++; if (xd[0] == BASE) xd[0] = 1; } break; } else { xd[xdi] += k; if (xd[xdi] != BASE) break; xd[xdi--] = 0; k = 1; } } } // Remove trailing zeros. for (i = xd.length; xd[--i] === 0;) xd.pop(); } if (external) { // Overflow? if (x.e > Ctor.maxE) { // Infinity. x.d = null; x.e = NaN; // Underflow? } else if (x.e < Ctor.minE) { // Zero. x.e = 0; x.d = [0]; // Ctor.underflow = true; } // else Ctor.underflow = false; } return x; } function finiteToString(x, isExp, sd) { if (!x.isFinite()) return nonFiniteToString(x); var k, e = x.e, str = digitsToString(x.d), len = str.length; if (isExp) { if (sd && (k = sd - len) > 0) { str = str.charAt(0) + '.' + str.slice(1) + getZeroString(k); } else if (len > 1) { str = str.charAt(0) + '.' + str.slice(1); } str = str + (x.e < 0 ? 'e' : 'e+') + x.e; } else if (e < 0) { str = '0.' + getZeroString(-e - 1) + str; if (sd && (k = sd - len) > 0) str += getZeroString(k); } else if (e >= len) { str += getZeroString(e + 1 - len); if (sd && (k = sd - e - 1) > 0) str = str + '.' + getZeroString(k); } else { if ((k = e + 1) < len) str = str.slice(0, k) + '.' + str.slice(k); if (sd && (k = sd - len) > 0) { if (e + 1 === len) str += '.'; str += getZeroString(k); } } return str; } // Calculate the base 10 exponent from the base 1e7 exponent. function getBase10Exponent(digits, e) { var w = digits[0]; // Add the number of digits of the first word of the digits array. for ( e *= LOG_BASE; w >= 10; w /= 10) e++; return e; } function getLn10(Ctor, sd, pr) { if (sd > LN10_PRECISION) { // Reset global state in case the exception is caught. external = true; if (pr) Ctor.precision = pr; throw Error(precisionLimitExceeded); } return finalise(new Ctor(LN10), sd, 1, true); } function getPi(Ctor, sd, rm) { if (sd > PI_PRECISION) throw Error(precisionLimitExceeded); return finalise(new Ctor(PI), sd, rm, true); } function getPrecision(digits) { var w = digits.length - 1, len = w * LOG_BASE + 1; w = digits[w]; // If non-zero... if (w) { // Subtract the number of trailing zeros of the last word. for (; w % 10 == 0; w /= 10) len--; // Add the number of digits of the first word. for (w = digits[0]; w >= 10; w /= 10) len++; } return len; } function getZeroString(k) { var zs = ''; for (; k--;) zs += '0'; return zs; } /* * Return a new Decimal whose value is the value of Decimal `x` to the power `n`, where `n` is an * integer of type number. * * Implements 'exponentiation by squaring'. Called by `pow` and `parseOther`. * */ function intPow(Ctor, x, n, pr) { var isTruncated, r = new Ctor(1), // Max n of 9007199254740991 takes 53 loop iterations. // Maximum digits array length; leaves [28, 34] guard digits. k = Math.ceil(pr / LOG_BASE + 4); external = false; for (;;) { if (n % 2) { r = r.times(x); if (truncate(r.d, k)) isTruncated = true; } n = mathfloor(n / 2); if (n === 0) { // To ensure correct rounding when r.d is truncated, increment the last word if it is zero. n = r.d.length - 1; if (isTruncated && r.d[n] === 0) ++r.d[n]; break; } x = x.times(x); truncate(x.d, k); } external = true; return r; } function isOdd(n) { return n.d[n.d.length - 1] & 1; } /* * Handle `max` and `min`. `ltgt` is 'lt' or 'gt'. */ function maxOrMin(Ctor, args, ltgt) { var y, x = new Ctor(args[0]), i = 0; for (; ++i < args.length;) { y = new Ctor(args[i]); if (!y.s) { x = y; break; } else if (x[ltgt](y)) { x = y; } } return x; } /* * Return a new Decimal whose value is the natural exponential of `x` rounded to `sd` significant * digits. * * Taylor/Maclaurin series. * * exp(x) = x^0/0! + x^1/1! + x^2/2! + x^3/3! + ... * * Argument reduction: * Repeat x = x / 32, k += 5, until |x| < 0.1 * exp(x) = exp(x / 2^k)^(2^k) * * Previously, the argument was initially reduced by * exp(x) = exp(r) * 10^k where r = x - k * ln10, k = floor(x / ln10) * to first put r in the range [0, ln10], before dividing by 32 until |x| < 0.1, but this was * found to be slower than just dividing repeatedly by 32 as above. * * Max integer argument: exp('20723265836946413') = 6.3e+9000000000000000 * Min integer argument: exp('-20723265836946411') = 1.2e-9000000000000000 * (Math object integer min/max: Math.exp(709) = 8.2e+307, Math.exp(-745) = 5e-324) * * exp(Infinity) = Infinity * exp(-Infinity) = 0 * exp(NaN) = NaN * exp(±0) = 1 * * exp(x) is non-terminating for any finite, non-zero x. * * The result will always be correctly rounded. * */ function naturalExponential(x, sd) { var denominator, guard, j, pow, sum, t, wpr, rep = 0, i = 0, k = 0, Ctor = x.constructor, rm = Ctor.rounding, pr = Ctor.precision; // 0/NaN/Infinity? if (!x.d || !x.d[0] || x.e > 17) { return new Ctor(x.d ? !x.d[0] ? 1 : x.s < 0 ? 0 : 1 / 0 : x.s ? x.s < 0 ? 0 : x : 0 / 0); } if (sd == null) { external = false; wpr = pr; } else { wpr = sd; } t = new Ctor(0.03125); // while abs(x) >= 0.1 while (x.e > -2) { // x = x / 2^5 x = x.times(t); k += 5; } // Use 2 * log10(2^k) + 5 (empirically derived) to estimate the increase in precision // necessary to ensure the first 4 rounding digits are correct. guard = Math.log(mathpow(2, k)) / Math.LN10 * 2 + 5 | 0; wpr += guard; denominator = pow = sum = new Ctor(1); Ctor.precision = wpr; for (;;) { pow = finalise(pow.times(x), wpr, 1); denominator = denominator.times(++i); t = sum.plus(divide(pow, denominator, wpr, 1)); if (digitsToString(t.d).slice(0, wpr) === digitsToString(sum.d).slice(0, wpr)) { j = k; while (j--) sum = finalise(sum.times(sum), wpr, 1); // Check to see if the first 4 rounding digits are [49]999. // If so, repeat the summation with a higher precision, otherwise // e.g. with precision: 18, rounding: 1 // exp(18.404272462595034083567793919843761) = 98372560.1229999999 (should be 98372560.123) // `wpr - guard` is the index of first rounding digit. if (sd == null) { if (rep < 3 && checkRoundingDigits(sum.d, wpr - guard, rm, rep)) { Ctor.precision = wpr += 10; denominator = pow = t = new Ctor(1); i = 0; rep++; } else { return finalise(sum, Ctor.precision = pr, rm, external = true); } } else { Ctor.precision = pr; return sum; } } sum = t; } } /* * Return a new Decimal whose value is the natural logarithm of `x` rounded to `sd` significant * digits. * * ln(-n) = NaN * ln(0) = -Infinity * ln(-0) = -Infinity * ln(1) = 0 * ln(Infinity) = Infinity * ln(-Infinity) = NaN * ln(NaN) = NaN * * ln(n) (n != 1) is non-terminating. * */ function naturalLogarithm(y, sd) { var c, c0, denominator, e, numerator, rep, sum, t, wpr, x1, x2, n = 1, guard = 10, x = y, xd = x.d, Ctor = x.constructor, rm = Ctor.rounding, pr = Ctor.precision; // Is x negative or Infinity, NaN, 0 or 1? if (x.s < 0 || !xd || !xd[0] || !x.e && xd[0] == 1 && xd.length == 1) { return new Ctor(xd && !xd[0] ? -1 / 0 : x.s != 1 ? NaN : xd ? 0 : x); } if (sd == null) { external = false; wpr = pr; } else { wpr = sd; } Ctor.precision = wpr += guard; c = digitsToString(xd); c0 = c.charAt(0); if (Math.abs(e = x.e) < 1.5e15) { // Argument reduction. // The series converges faster the closer the argument is to 1, so using // ln(a^b) = b * ln(a), ln(a) = ln(a^b) / b // multiply the argument by itself until the leading digits of the significand are 7, 8, 9, // 10, 11, 12 or 13, recording the number of multiplications so the sum of the series can // later be divided by this number, then separate out the power of 10 using // ln(a*10^b) = ln(a) + b*ln(10). // max n is 21 (gives 0.9, 1.0 or 1.1) (9e15 / 21 = 4.2e14). //while (c0 < 9 && c0 != 1 || c0 == 1 && c.charAt(1) > 1) { // max n is 6 (gives 0.7 - 1.3) while (c0 < 7 && c0 != 1 || c0 == 1 && c.charAt(1) > 3) { x = x.times(y); c = digitsToString(x.d); c0 = c.charAt(0); n++; } e = x.e; if (c0 > 1) { x = new Ctor('0.' + c); e++; } else { x = new Ctor(c0 + '.' + c.slice(1)); } } else { // The argument reduction method above may result in overflow if the argument y is a massive // number with exponent >= 1500000000000000 (9e15 / 6 = 1.5e15), so instead recall this // function using ln(x*10^e) = ln(x) + e*ln(10). t = getLn10(Ctor, wpr + 2, pr).times(e + ''); x = naturalLogarithm(new Ctor(c0 + '.' + c.slice(1)), wpr - guard).plus(t); Ctor.precision = pr; return sd == null ? finalise(x, pr, rm, external = true) : x; } // x1 is x reduced to a value near 1. x1 = x; // Taylor series. // ln(y) = ln((1 + x)/(1 - x)) = 2(x + x^3/3 + x^5/5 + x^7/7 + ...) // where x = (y - 1)/(y + 1) (|x| < 1) sum = numerator = x = divide(x.minus(1), x.plus(1), wpr, 1); x2 = finalise(x.times(x), wpr, 1); denominator = 3; for (;;) { numerator = finalise(numerator.times(x2), wpr, 1); t = sum.plus(divide(numerator, new Ctor(denominator), wpr, 1)); if (digitsToString(t.d).slice(0, wpr) === digitsToString(sum.d).slice(0, wpr)) { sum = sum.times(2); // Reverse the argument reduction. Check that e is not 0 because, besides preventing an // unnecessary calculation, -0 + 0 = +0 and to ensure correct rounding -0 needs to stay -0. if (e !== 0) sum = sum.plus(getLn10(Ctor, wpr + 2, pr).times(e + '')); sum = divide(sum, new Ctor(n), wpr, 1); // Is rm > 3 and the first 4 rounding digits 4999, or rm < 4 (or the summation has // been repeated previously) and the first 4 rounding digits 9999? // If so, restart the summation with a higher precision, otherwise // e.g. with precision: 12, rounding: 1 // ln(135520028.6126091714265381533) = 18.7246299999 when it should be 18.72463. // `wpr - guard` is the index of first rounding digit. if (sd == null) { if (checkRoundingDigits(sum.d, wpr - guard, rm, rep)) { Ctor.precision = wpr += guard; t = numerator = x = divide(x1.minus(1), x1.plus(1), wpr, 1); x2 = finalise(x.times(x), wpr, 1); denominator = rep = 1; } else { return finalise(sum, Ctor.precision = pr, rm, external = true); } } else { Ctor.precision = pr; return sum; } } sum = t; denominator += 2; } } // ±Infinity, NaN. function nonFiniteToString(x) { // Unsigned. return String(x.s * x.s / 0); } /* * Parse the value of a new Decimal `x` from string `str`. */ function parseDecimal(x, str) { var e, i, len; // Decimal point? if ((e = str.indexOf('.')) > -1) str = str.replace('.', ''); // Exponential form? if ((i = str.search(/e/i)) > 0) { // Determine exponent. if (e < 0) e = i; e += +str.slice(i + 1); str = str.substring(0, i); } else if (e < 0) { // Integer. e = str.length; } // Determine leading zeros. for (i = 0; str.charCodeAt(i) === 48; i++); // Determine trailing zeros. for (len = str.length; str.charCodeAt(len - 1) === 48; --len); str = str.slice(i, len); if (str) { len -= i; x.e = e = e - i - 1; x.d = []; // Transform base // e is the base 10 exponent. // i is where to slice str to get the first word of the digits array. i = (e + 1) % LOG_BASE; if (e < 0) i += LOG_BASE; if (i < len) { if (i) x.d.push(+str.slice(0, i)); for (len -= LOG_BASE; i < len;) x.d.push(+str.slice(i, i += LOG_BASE)); str = str.slice(i); i = LOG_BASE - str.length; } else { i -= len; } for (; i--;) str += '0'; x.d.push(+str); if (external) { // Overflow? if (x.e > x.constructor.maxE) { // Infinity. x.d = null; x.e = NaN; // Underflow? } else if (x.e < x.constructor.minE) { // Zero. x.e = 0; x.d = [0]; // x.constructor.underflow = true; } // else x.constructor.underflow = false; } } else { // Zero. x.e = 0; x.d = [0]; } return x; } /* * Parse the value of a new Decimal `x` from a string `str`, which is not a decimal value. */ function parseOther(x, str) { var base, Ctor, divisor, i, isFloat, len, p, xd, xe; if (str === 'Infinity' || str === 'NaN') { if (!+str) x.s = NaN; x.e = NaN; x.d = null; return x; } if (isHex.test(str)) { base = 16; str = str.toLowerCase(); } else if (isBinary.test(str)) { base = 2; } else if (isOctal.test(str)) { base = 8; } else { throw Error(invalidArgument + str); } // Is there a binary exponent part? i = str.search(/p/i); if (i > 0) { p = +str.slice(i + 1); str = str.substring(2, i); } else { str = str.slice(2); } // Convert `str` as an integer then divide the result by `base` raised to a power such that the // fraction part will be restored. i = str.indexOf('.'); isFloat = i >= 0; Ctor = x.constructor; if (isFloat) { str = str.replace('.', ''); len = str.length; i = len - i; // log[10](16) = 1.2041... , log[10](88) = 1.9444.... divisor = intPow(Ctor, new Ctor(base), i, i * 2); } xd = convertBase(str, base, BASE); xe = xd.length - 1; // Remove trailing zeros. for (i = xe; xd[i] === 0; --i) xd.pop(); if (i < 0) return new Ctor(x.s * 0); x.e = getBase10Exponent(xd, xe); x.d = xd; external = false; // At what precision to perform the division to ensure exact conversion? // maxDecimalIntegerPartDigitCount = ceil(log[10](b) * otherBaseIntegerPartDigitCount) // log[10](2) = 0.30103, log[10](8) = 0.90309, log[10](16) = 1.20412 // E.g. ceil(1.2 * 3) = 4, so up to 4 decimal digits are needed to represent 3 hex int digits. // maxDecimalFractionPartDigitCount = {Hex:4|Oct:3|Bin:1} * otherBaseFractionPartDigitCount // Therefore using 4 * the number of digits of str will always be enough. if (isFloat) x = divide(x, divisor, len * 4); // Multiply by the binary exponent part if present. if (p) x = x.times(Math.abs(p) < 54 ? Math.pow(2, p) : Decimal.pow(2, p)); external = true; return x; } /* * sin(x) = x - x^3/3! + x^5/5! - ... * |x| < pi/2 * */ function sine(Ctor, x) { var k, len = x.d.length; if (len < 3) return taylorSeries(Ctor, 2, x, x); // Argument reduction: sin(5x) = 16*sin^5(x) - 20*sin^3(x) + 5*sin(x) // i.e. sin(x) = 16*sin^5(x/5) - 20*sin^3(x/5) + 5*sin(x/5) // and sin(x) = sin(x/5)(5 + sin^2(x/5)(16sin^2(x/5) - 20)) // Estimate the optimum number of times to use the argument reduction. k = 1.4 * Math.sqrt(len); k = k > 16 ? 16 : k | 0; // Max k before Math.pow precision loss is 22 x = x.times(Math.pow(5, -k)); x = taylorSeries(Ctor, 2, x, x); // Reverse argument reduction var sin2_x, d5 = new Ctor(5), d16 = new Ctor(16), d20 = new Ctor(20); for (; k--;) { sin2_x = x.times(x); x = x.times(d5.plus(sin2_x.times(d16.times(sin2_x).minus(d20)))); } return x; } // Calculate Taylor series for `cos`, `cosh`, `sin` and `sinh`. function taylorSeries(Ctor, n, x, y, isHyperbolic) { var j, t, u, x2, i = 1, pr = Ctor.precision, k = Math.ceil(pr / LOG_BASE); external = false; x2 = x.times(x); u = new Ctor(y); for (;;) { t = divide(u.times(x2), new Ctor(n++ * n++), pr, 1); u = isHyperbolic ? y.plus(t) : y.minus(t); y = divide(t.times(x2), new Ctor(n++ * n++), pr, 1); t = u.plus(y); if (t.d[k] !== void 0) { for (j = k; t.d[j] === u.d[j] && j--;); if (j == -1) break; } j = u; u = y; y = t; t = j; i++; } external = true; t.d.length = k + 1; return t; } // Return the absolute value of `x` reduced to less than or equal to half pi. function toLessThanHalfPi(Ctor, x) { var t, isNeg = x.s < 0, pi = getPi(Ctor, Ctor.precision, 1), halfPi = pi.times(0.5); x = x.abs(); if (x.lte(halfPi)) { quadrant = isNeg ? 4 : 1; return x; } t = x.divToInt(pi); if (t.isZero()) { quadrant = isNeg ? 3 : 2; } else { x = x.minus(t.times(pi)); // 0 <= x < pi if (x.lte(halfPi)) { quadrant = isOdd(t) ? (isNeg ? 2 : 3) : (isNeg ? 4 : 1); return x; } quadrant = isOdd(t) ? (isNeg ? 1 : 4) : (isNeg ? 3 : 2); } return x.minus(pi).abs(); } /* * Return the value of Decimal `x` as a string in base `baseOut`. * * If the optional `sd` argument is present include a binary exponent suffix. */ function toStringBinary(x, baseOut, sd, rm) { var base, e, i, k, len, roundUp, str, xd, y, Ctor = x.constructor, isExp = sd !== void 0; if (isExp) { checkInt32(sd, 1, MAX_DIGITS); if (rm === void 0) rm = Ctor.rounding; else checkInt32(rm, 0, 8); } else { sd = Ctor.precision; rm = Ctor.rounding; } if (!x.isFinite()) { str = nonFiniteToString(x); } else { str = finiteToString(x); i = str.indexOf('.'); // Use exponential notation according to `toExpPos` and `toExpNeg`? No, but if required: // maxBinaryExponent = floor((decimalExponent + 1) * log[2](10)) // minBinaryExponent = floor(decimalExponent * log[2](10)) // log[2](10) = 3.321928094887362347870319429489390175864 if (isExp) { base = 2; if (baseOut == 16) { sd = sd * 4 - 3; } else if (baseOut == 8) { sd = sd * 3 - 2; } } else { base = baseOut; } // Convert the number as an integer then divide the result by its base raised to a power such // that the fraction part will be restored. // Non-integer. if (i >= 0) { str = str.replace('.', ''); y = new Ctor(1); y.e = str.length - i; y.d = convertBase(finiteToString(y), 10, base); y.e = y.d.length; } xd = convertBase(str, 10, base); e = len = xd.length; // Remove trailing zeros. for (; xd[--len] == 0;) xd.pop(); if (!xd[0]) { str = isExp ? '0p+0' : '0'; } else { if (i < 0) { e--; } else { x = new Ctor(x); x.d = xd; x.e = e; x = divide(x, y, sd, rm, 0, base); xd = x.d; e = x.e; roundUp = inexact; } // The rounding digit, i.e. the digit after the digit that may be rounded up. i = xd[sd]; k = base / 2; roundUp = roundUp || xd[sd + 1] !== void 0; roundUp = rm < 4 ? (i !== void 0 || roundUp) && (rm === 0 || rm === (x.s < 0 ? 3 : 2)) : i > k || i === k && (rm === 4 || roundUp || rm === 6 && xd[sd - 1] & 1 || rm === (x.s < 0 ? 8 : 7)); xd.length = sd; if (roundUp) { // Rounding up may mean the previous digit has to be rounded up and so on. for (; ++xd[--sd] > base - 1;) { xd[sd] = 0; if (!sd) { ++e; xd.unshift(1); } } } // Determine trailing zeros. for (len = xd.length; !xd[len - 1]; --len); // E.g. [4, 11, 15] becomes 4bf. for (i = 0, str = ''; i < len; i++) str += NUMERALS.charAt(xd[i]); // Add binary exponent suffix? if (isExp) { if (len > 1) { if (baseOut == 16 || baseOut == 8) { i = baseOut == 16 ? 4 : 3; for (--len; len % i; len++) str += '0'; xd = convertBase(str, base, baseOut); for (len = xd.length; !xd[len - 1]; --len); // xd[0] will always be be 1 for (i = 1, str = '1.'; i < len; i++) str += NUMERALS.charAt(xd[i]); } else { str = str.charAt(0) + '.' + str.slice(1); } } str = str + (e < 0 ? 'p' : 'p+') + e; } else if (e < 0) { for (; ++e;) str = '0' + str; str = '0.' + str; } else { if (++e > len) for (e -= len; e-- ;) str += '0'; else if (e < len) str = str.slice(0, e) + '.' + str.slice(e); } } str = (baseOut == 16 ? '0x' : baseOut == 2 ? '0b' : baseOut == 8 ? '0o' : '') + str; } return x.s < 0 ? '-' + str : str; } // Does not strip trailing zeros. function truncate(arr, len) { if (arr.length > len) { arr.length = len; return true; } } // Decimal methods /* * abs * acos * acosh * add * asin * asinh * atan * atanh * atan2 * cbrt * ceil * clone * config * cos * cosh * div * exp * floor * hypot * ln * log * log2 * log10 * max * min * mod * mul * pow * random * round * set * sign * sin * sinh * sqrt * sub * tan * tanh * trunc */ /* * Return a new Decimal whose value is the absolute value of `x`. * * x {number|string|Decimal} * */ function abs(x) { return new this(x).abs(); } /* * Return a new Decimal whose value is the arccosine in radians of `x`. * * x {number|string|Decimal} * */ function acos(x) { return new this(x).acos(); } /* * Return a new Decimal whose value is the inverse of the hyperbolic cosine of `x`, rounded to * `precision` significant digits using rounding mode `rounding`. * * x {number|string|Decimal} A value in radians. * */ function acosh(x) { return new this(x).acosh(); } /* * Return a new Decimal whose value is the sum of `x` and `y`, rounded to `precision` significant * digits using rounding mode `rounding`. * * x {number|string|Decimal} * y {number|string|Decimal} * */ function add(x, y) { return new this(x).plus(y); } /* * Return a new Decimal whose value is the arcsine in radians of `x`, rounded to `precision` * significant digits using rounding mode `rounding`. * * x {number|string|Decimal} * */ function asin(x) { return new this(x).asin(); } /* * Return a new Decimal whose value is the inverse of the hyperbolic sine of `x`, rounded to * `precision` significant digits using rounding mode `rounding`. * * x {number|string|Decimal} A value in radians. * */ function asinh(x) { return new this(x).asinh(); } /* * Return a new Decimal whose value is the arctangent in radians of `x`, rounded to `precision` * significant digits using rounding mode `rounding`. * * x {number|string|Decimal} * */ function atan(x) { return new this(x).atan(); } /* * Return a new Decimal whose value is the inverse of the hyperbolic tangent of `x`, rounded to * `precision` significant digits using rounding mode `rounding`. * * x {number|string|Decimal} A value in radians. * */ function atanh(x) { return new this(x).atanh(); } /* * Return a new Decimal whose value is the arctangent in radians of `y/x` in the range -pi to pi * (inclusive), rounded to `precision` significant digits using rounding mode `rounding`. * * Domain: [-Infinity, Infinity] * Range: [-pi, pi] * * y {number|string|Decimal} The y-coordinate. * x {number|string|Decimal} The x-coordinate. * * atan2(±0, -0) = ±pi * atan2(±0, +0) = ±0 * atan2(±0, -x) = ±pi for x > 0 * atan2(±0, x) = ±0 for x > 0 * atan2(-y, ±0) = -pi/2 for y > 0 * atan2(y, ±0) = pi/2 for y > 0 * atan2(±y, -Infinity) = ±pi for finite y > 0 * atan2(±y, +Infinity) = ±0 for finite y > 0 * atan2(±Infinity, x) = ±pi/2 for finite x * atan2(±Infinity, -Infinity) = ±3*pi/4 * atan2(±Infinity, +Infinity) = ±pi/4 * atan2(NaN, x) = NaN * atan2(y, NaN) = NaN * */ function atan2(y, x) { y = new this(y); x = new this(x); var r, pr = this.precision, rm = this.rounding, wpr = pr + 4; // Either NaN if (!y.s || !x.s) { r = new this(NaN); // Both ±Infinity } else if (!y.d && !x.d) { r = getPi(this, wpr, 1).times(x.s > 0 ? 0.25 : 0.75); r.s = y.s; // x is ±Infinity or y is ±0 } else if (!x.d || y.isZero()) { r = x.s < 0 ? getPi(this, pr, rm) : new this(0); r.s = y.s; // y is ±Infinity or x is ±0 } else if (!y.d || x.isZero()) { r = getPi(this, wpr, 1).times(0.5); r.s = y.s; // Both non-zero and finite } else if (x.s < 0) { this.precision = wpr; this.rounding = 1; r = this.atan(divide(y, x, wpr, 1)); x = getPi(this, wpr, 1); this.precision = pr; this.rounding = rm; r = y.s < 0 ? r.minus(x) : r.plus(x); } else { r = this.atan(divide(y, x, wpr, 1)); } return r; } /* * Return a new Decimal whose value is the cube root of `x`, rounded to `precision` significant * digits using rounding mode `rounding`. * * x {number|string|Decimal} * */ function cbrt(x) { return new this(x).cbrt(); } /* * Return a new Decimal whose value is `x` rounded to an integer using `ROUND_CEIL`. * * x {number|string|Decimal} * */ function ceil(x) { return finalise(x = new this(x), x.e + 1, 2); } /* * Configure global settings for a Decimal constructor. * * `obj` is an object with one or more of the following properties, * * precision {number} * rounding {number} * toExpNeg {number} * toExpPos {number} * maxE {number} * minE {number} * modulo {number} * crypto {boolean|number} * defaults {true} * * E.g. Decimal.config({ precision: 20, rounding: 4 }) * */ function config(obj) { if (!obj || typeof obj !== 'object') throw Error(decimalError + 'Object expected'); var i, p, v, useDefaults = obj.defaults === true, ps = [ 'precision', 1, MAX_DIGITS, 'rounding', 0, 8, 'toExpNeg', -EXP_LIMIT, 0, 'toExpPos', 0, EXP_LIMIT, 'maxE', 0, EXP_LIMIT, 'minE', -EXP_LIMIT, 0, 'modulo', 0, 9 ]; for (i = 0; i < ps.length; i += 3) { if (p = ps[i], useDefaults) this[p] = DEFAULTS[p]; if ((v = obj[p]) !== void 0) { if (mathfloor(v) === v && v >= ps[i + 1] && v <= ps[i + 2]) this[p] = v; else throw Error(invalidArgument + p + ': ' + v); } } if (p = 'crypto', useDefaults) this[p] = DEFAULTS[p]; if ((v = obj[p]) !== void 0) { if (v === true || v === false || v === 0 || v === 1) { if (v) { if (typeof crypto != 'undefined' && crypto && (crypto.getRandomValues || crypto.randomBytes)) { this[p] = true; } else { throw Error(cryptoUnavailable); } } else { this[p] = false; } } else { throw Error(invalidArgument + p + ': ' + v); } } return this; } /* * Return a new Decimal whose value is the cosine of `x`, rounded to `precision` significant * digits using rounding mode `rounding`. * * x {number|string|Decimal} A value in radians. * */ function cos(x) { return new this(x).cos(); } /* * Return a new Decimal whose value is the hyperbolic cosine of `x`, rounded to precision * significant digits using rounding mode `rounding`. * * x {number|string|Decimal} A value in radians. * */ function cosh(x) { return new this(x).cosh(); } /* * Create and return a Decimal constructor with the same configuration properties as this Decimal * constructor. * */ function clone(obj) { var i, p, ps; /* * The Decimal constructor and exported function. * Return a new Decimal instance. * * v {number|string|Decimal} A numeric value. * */ function Decimal(v) { var e, i, t, x = this; // Decimal called without new. if (!(x instanceof Decimal)) return new Decimal(v); // Retain a reference to this Decimal constructor, and shadow Decimal.prototype.constructor // which points to Object. x.constructor = Decimal; // Duplicate. if (v instanceof Decimal) { x.s = v.s; x.e = v.e; x.d = (v = v.d) ? v.slice() : v; return; } t = typeof v; if (t === 'number') { if (v === 0) { x.s = 1 / v < 0 ? -1 : 1; x.e = 0; x.d = [0]; return; } if (v < 0) { v = -v; x.s = -1; } else { x.s = 1; } // Fast path for small integers. if (v === ~~v && v < 1e7) { for (e = 0, i = v; i >= 10; i /= 10) e++; x.e = e; x.d = [v]; return; // Infinity, NaN. } else if (v * 0 !== 0) { if (!v) x.s = NaN; x.e = NaN; x.d = null; return; } return parseDecimal(x, v.toString()); } else if (t !== 'string') { throw Error(invalidArgument + v); } // Minus sign? if (v.charCodeAt(0) === 45) { v = v.slice(1); x.s = -1; } else { x.s = 1; } return isDecimal.test(v) ? parseDecimal(x, v) : parseOther(x, v); } Decimal.prototype = P; Decimal.ROUND_UP = 0; Decimal.ROUND_DOWN = 1; Decimal.ROUND_CEIL = 2; Decimal.ROUND_FLOOR = 3; Decimal.ROUND_HALF_UP = 4; Decimal.ROUND_HALF_DOWN = 5; Decimal.ROUND_HALF_EVEN = 6; Decimal.ROUND_HALF_CEIL = 7; Decimal.ROUND_HALF_FLOOR = 8; Decimal.EUCLID = 9; Decimal.config = Decimal.set = config; Decimal.clone = clone; Decimal.isDecimal = isDecimalInstance; Decimal.abs = abs; Decimal.acos = acos; Decimal.acosh = acosh; // ES6 Decimal.add = add; Decimal.asin = asin; Decimal.asinh = asinh; // ES6 Decimal.atan = atan; Decimal.atanh = atanh; // ES6 Decimal.atan2 = atan2; Decimal.cbrt = cbrt; // ES6 Decimal.ceil = ceil; Decimal.cos = cos; Decimal.cosh = cosh; // ES6 Decimal.div = div; Decimal.exp = exp; Decimal.floor = floor; Decimal.hypot = hypot; // ES6 Decimal.ln = ln; Decimal.log = log; Decimal.log10 = log10; // ES6 Decimal.log2 = log2; // ES6 Decimal.max = max; Decimal.min = min; Decimal.mod = mod; Decimal.mul = mul; Decimal.pow = pow; Decimal.random = random; Decimal.round = round; Decimal.sign = sign; // ES6 Decimal.sin = sin; Decimal.sinh = sinh; // ES6 Decimal.sqrt = sqrt; Decimal.sub = sub; Decimal.tan = tan; Decimal.tanh = tanh; // ES6 Decimal.trunc = trunc; // ES6 if (obj === void 0) obj = {}; if (obj) { if (obj.defaults !== true) { ps = ['precision', 'rounding', 'toExpNeg', 'toExpPos', 'maxE', 'minE', 'modulo', 'crypto']; for (i = 0; i < ps.length;) if (!obj.hasOwnProperty(p = ps[i++])) obj[p] = this[p]; } } Decimal.config(obj); return Decimal; } /* * Return a new Decimal whose value is `x` divided by `y`, rounded to `precision` significant * digits using rounding mode `rounding`. * * x {number|string|Decimal} * y {number|string|Decimal} * */ function div(x, y) { return new this(x).div(y); } /* * Return a new Decimal whose value is the natural exponential of `x`, rounded to `precision` * significant digits using rounding mode `rounding`. * * x {number|string|Decimal} The power to which to raise the base of the natural log. * */ function exp(x) { return new this(x).exp(); } /* * Return a new Decimal whose value is `x` round to an integer using `ROUND_FLOOR`. * * x {number|string|Decimal} * */ function floor(x) { return finalise(x = new this(x), x.e + 1, 3); } /* * Return a new Decimal whose value is the square root of the sum of the squares of the arguments, * rounded to `precision` significant digits using rounding mode `rounding`. * * hypot(a, b, ...) = sqrt(a^2 + b^2 + ...) * */ function hypot() { var i, n, t = new this(0); external = false; for (i = 0; i < arguments.length;) { n = new this(arguments[i++]); if (!n.d) { if (n.s) { external = true; return new this(1 / 0); } t = n; } else if (t.d) { t = t.plus(n.times(n)); } } external = true; return t.sqrt(); } /* * Return true if object is a Decimal instance (where Decimal is any Decimal constructor), * otherwise return false. * */ function isDecimalInstance(obj) { return obj instanceof Decimal || obj && obj.name === '[object Decimal]' || false; } /* * Return a new Decimal whose value is the natural logarithm of `x`, rounded to `precision` * significant digits using rounding mode `rounding`. * * x {number|string|Decimal} * */ function ln(x) { return new this(x).ln(); } /* * Return a new Decimal whose value is the log of `x` to the base `y`, or to base 10 if no base * is specified, rounded to `precision` significant digits using rounding mode `rounding`. * * log[y](x) * * x {number|string|Decimal} The argument of the logarithm. * y {number|string|Decimal} The base of the logarithm. * */ function log(x, y) { return new this(x).log(y); } /* * Return a new Decimal whose value is the base 2 logarithm of `x`, rounded to `precision` * significant digits using rounding mode `rounding`. * * x {number|string|Decimal} * */ function log2(x) { return new this(x).log(2); } /* * Return a new Decimal whose value is the base 10 logarithm of `x`, rounded to `precision` * significant digits using rounding mode `rounding`. * * x {number|string|Decimal} * */ function log10(x) { return new this(x).log(10); } /* * Return a new Decimal whose value is the maximum of the arguments. * * arguments {number|string|Decimal} * */ function max() { return maxOrMin(this, arguments, 'lt'); } /* * Return a new Decimal whose value is the minimum of the arguments. * * arguments {number|string|Decimal} * */ function min() { return maxOrMin(this, arguments, 'gt'); } /* * Return a new Decimal whose value is `x` modulo `y`, rounded to `precision` significant digits * using rounding mode `rounding`. * * x {number|string|Decimal} * y {number|string|Decimal} * */ function mod(x, y) { return new this(x).mod(y); } /* * Return a new Decimal whose value is `x` multiplied by `y`, rounded to `precision` significant * digits using rounding mode `rounding`. * * x {number|string|Decimal} * y {number|string|Decimal} * */ function mul(x, y) { return new this(x).mul(y); } /* * Return a new Decimal whose value is `x` raised to the power `y`, rounded to precision * significant digits using rounding mode `rounding`. * * x {number|string|Decimal} The base. * y {number|string|Decimal} The exponent. * */ function pow(x, y) { return new this(x).pow(y); } /* * Returns a new Decimal with a random value equal to or greater than 0 and less than 1, and with * `sd`, or `Decimal.precision` if `sd` is omitted, significant digits (or less if trailing zeros * are produced). * * [sd] {number} Significant digits. Integer, 0 to MAX_DIGITS inclusive. * */ function random(sd) { var d, e, k, n, i = 0, r = new this(1), rd = []; if (sd === void 0) sd = this.precision; else checkInt32(sd, 1, MAX_DIGITS); k = Math.ceil(sd / LOG_BASE); if (!this.crypto) { for (; i < k;) rd[i++] = Math.random() * 1e7 | 0; // Browsers supporting crypto.getRandomValues. } else if (crypto.getRandomValues) { d = crypto.getRandomValues(new Uint32Array(k)); for (; i < k;) { n = d[i]; // 0 <= n < 4294967296 // Probability n >= 4.29e9, is 4967296 / 4294967296 = 0.00116 (1 in 865). if (n >= 4.29e9) { d[i] = crypto.getRandomValues(new Uint32Array(1))[0]; } else { // 0 <= n <= 4289999999 // 0 <= (n % 1e7) <= 9999999 rd[i++] = n % 1e7; } } // Node.js supporting crypto.randomBytes. } else if (crypto.randomBytes) { // buffer d = crypto.randomBytes(k *= 4); for (; i < k;) { // 0 <= n < 2147483648 n = d[i] + (d[i + 1] << 8) + (d[i + 2] << 16) + ((d[i + 3] & 0x7f) << 24); // Probability n >= 2.14e9, is 7483648 / 2147483648 = 0.0035 (1 in 286). if (n >= 2.14e9) { crypto.randomBytes(4).copy(d, i); } else { // 0 <= n <= 2139999999 // 0 <= (n % 1e7) <= 9999999 rd.push(n % 1e7); i += 4; } } i = k / 4; } else { throw Error(cryptoUnavailable); } k = rd[--i]; sd %= LOG_BASE; // Convert trailing digits to zeros according to sd. if (k && sd) { n = mathpow(10, LOG_BASE - sd); rd[i] = (k / n | 0) * n; } // Remove trailing words which are zero. for (; rd[i] === 0; i--) rd.pop(); // Zero? if (i < 0) { e = 0; rd = [0]; } else { e = -1; // Remove leading words which are zero and adjust exponent accordingly. for (; rd[0] === 0; e -= LOG_BASE) rd.shift(); // Count the digits of the first word of rd to determine leading zeros. for (k = 1, n = rd[0]; n >= 10; n /= 10) k++; // Adjust the exponent for leading zeros of the first word of rd. if (k < LOG_BASE) e -= LOG_BASE - k; } r.e = e; r.d = rd; return r; } /* * Return a new Decimal whose value is `x` rounded to an integer using rounding mode `rounding`. * * To emulate `Math.round`, set rounding to 7 (ROUND_HALF_CEIL). * * x {number|string|Decimal} * */ function round(x) { return finalise(x = new this(x), x.e + 1, this.rounding); } /* * Return * 1 if x > 0, * -1 if x < 0, * 0 if x is 0, * -0 if x is -0, * NaN otherwise * */ function sign(x) { x = new this(x); return x.d ? (x.d[0] ? x.s : 0 * x.s) : x.s || NaN; } /* * Return a new Decimal whose value is the sine of `x`, rounded to `precision` significant digits * using rounding mode `rounding`. * * x {number|string|Decimal} A value in radians. * */ function sin(x) { return new this(x).sin(); } /* * Return a new Decimal whose value is the hyperbolic sine of `x`, rounded to `precision` * significant digits using rounding mode `rounding`. * * x {number|string|Decimal} A value in radians. * */ function sinh(x) { return new this(x).sinh(); } /* * Return a new Decimal whose value is the square root of `x`, rounded to `precision` significant * digits using rounding mode `rounding`. * * x {number|string|Decimal} * */ function sqrt(x) { return new this(x).sqrt(); } /* * Return a new Decimal whose value is `x` minus `y`, rounded to `precision` significant digits * using rounding mode `rounding`. * * x {number|string|Decimal} * y {number|string|Decimal} * */ function sub(x, y) { return new this(x).sub(y); } /* * Return a new Decimal whose value is the tangent of `x`, rounded to `precision` significant * digits using rounding mode `rounding`. * * x {number|string|Decimal} A value in radians. * */ function tan(x) { return new this(x).tan(); } /* * Return a new Decimal whose value is the hyperbolic tangent of `x`, rounded to `precision` * significant digits using rounding mode `rounding`. * * x {number|string|Decimal} A value in radians. * */ function tanh(x) { return new this(x).tanh(); } /* * Return a new Decimal whose value is `x` truncated to an integer. * * x {number|string|Decimal} * */ function trunc(x) { return finalise(x = new this(x), x.e + 1, 1); } // Create and configure initial Decimal constructor. export var Decimal = clone(DEFAULTS); // Create the internal constants from their string values. LN10 = new Decimal(LN10); PI = new Decimal(PI); export default Decimal;