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| author | Minteck <contact@minteck.org> | 2023-02-23 19:34:56 +0100 |
|---|---|---|
| committer | Minteck <contact@minteck.org> | 2023-02-23 19:34:56 +0100 |
| commit | 3d1cd02f27518f1a04374c7c8320cd5d82ede6e9 (patch) | |
| tree | 75be5fba4368472fb11c8015aee026b2b9a71888 /includes/external/school/node_modules/decimal.js/decimal.mjs | |
| parent | 8cc1f13c17fa2fb5a4410542d39e650e02945634 (diff) | |
| download | pluralconnect-3d1cd02f27518f1a04374c7c8320cd5d82ede6e9.tar.gz pluralconnect-3d1cd02f27518f1a04374c7c8320cd5d82ede6e9.tar.bz2 pluralconnect-3d1cd02f27518f1a04374c7c8320cd5d82ede6e9.zip | |
Updated 40 files, added 37 files, deleted 1103 files and renamed 3905 files (automated)
Diffstat (limited to 'includes/external/school/node_modules/decimal.js/decimal.mjs')
| -rw-r--r-- | includes/external/school/node_modules/decimal.js/decimal.mjs | 4898 |
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diff --git a/includes/external/school/node_modules/decimal.js/decimal.mjs b/includes/external/school/node_modules/decimal.js/decimal.mjs new file mode 100644 index 0000000..5d9101b --- /dev/null +++ b/includes/external/school/node_modules/decimal.js/decimal.mjs @@ -0,0 +1,4898 @@ +/*!
+ * decimal.js v10.4.1
+ * An arbitrary-precision Decimal type for JavaScript.
+ * https://github.com/MikeMcl/decimal.js
+ * Copyright (c) 2022 Michael Mclaughlin <M8ch88l@gmail.com>
+ * MIT 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',
+ tag = '[object Decimal]',
+
+ 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 = { toStringTag: tag };
+
+
+// Decimal prototype methods
+
+
+/*
+ * absoluteValue abs
+ * ceil
+ * clampedTo clamp
+ * 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 a new Decimal whose value is the value of this Decimal clamped to the range
+ * delineated by `min` and `max`.
+ *
+ * min {number|string|Decimal}
+ * max {number|string|Decimal}
+ *
+ */
+P.clampedTo = P.clamp = function (min, max) {
+ var k,
+ x = this,
+ Ctor = x.constructor;
+ min = new Ctor(min);
+ max = new Ctor(max);
+ if (!min.s || !max.s) return new Ctor(NaN);
+ if (min.gt(max)) throw Error(invalidArgument + max);
+ k = x.cmp(min);
+ return k < 0 ? min : x.cmp(max) > 0 ? max : new Ctor(x);
+};
+
+
+/*
+ * 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 * mathpow(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 = mathpow(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 = (1 / tinyPow(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(1 / tinyPow(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 = '5e' + 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;
+};
+
+
+// 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, len, y;
+
+ if (x.isZero()) return x;
+
+ // 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.
+ len = x.d.length;
+ if (len < 32) {
+ k = Math.ceil(len / 3);
+ y = (1 / tinyPow(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.indexOf('_') > -1) {
+ str = str.replace(/(\d)_(?=\d)/g, '$1');
+ if (isDecimal.test(str)) return parseDecimal(x, str);
+ } else 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 ? mathpow(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 x.isZero() ? x : 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;
+
+ x = x.times(1 / tinyPow(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;
+}
+
+
+// Exponent e must be positive and non-zero.
+function tinyPow(b, e) {
+ var n = b;
+ while (--e) n *= b;
+ return n;
+}
+
+
+// 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
+ * clamp
+ * 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
+ * sum
+ * 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);
+}
+
+
+/*
+ * Return a new Decimal whose value is `x` clamped to the range delineated by `min` and `max`.
+ *
+ * x {number|string|Decimal}
+ * min {number|string|Decimal}
+ * max {number|string|Decimal}
+ *
+ */
+function clamp(x, min, max) {
+ return new this(x).clamp(min, max);
+}
+
+
+/*
+ * 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 (isDecimalInstance(v)) {
+ x.s = v.s;
+
+ if (external) {
+ if (!v.d || v.e > Decimal.maxE) {
+
+ // Infinity.
+ x.e = NaN;
+ x.d = null;
+ } else if (v.e < Decimal.minE) {
+
+ // Zero.
+ x.e = 0;
+ x.d = [0];
+ } else {
+ x.e = v.e;
+ x.d = v.d.slice();
+ }
+ } else {
+ x.e = v.e;
+ x.d = v.d ? v.d.slice() : v.d;
+ }
+
+ 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++;
+
+ if (external) {
+ if (e > Decimal.maxE) {
+ x.e = NaN;
+ x.d = null;
+ } else if (e < Decimal.minE) {
+ x.e = 0;
+ x.d = [0];
+ } else {
+ x.e = e;
+ x.d = [v];
+ }
+ } else {
+ 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 ((i = v.charCodeAt(0)) === 45) {
+ v = v.slice(1);
+ x.s = -1;
+ } else {
+ // Plus sign?
+ if (i === 43) v = v.slice(1);
+ 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.clamp = clamp;
+ 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.sum = sum;
+ 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 + ...)
+ *
+ * arguments {number|string|Decimal}
+ *
+ */
+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.toStringTag === tag || 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
+ *
+ * x {number|string|Decimal}
+ *
+ */
+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 sum of the arguments, rounded to `precision`
+ * significant digits using rounding mode `rounding`.
+ *
+ * Only the result is rounded, not the intermediate calculations.
+ *
+ * arguments {number|string|Decimal}
+ *
+ */
+function sum() {
+ var i = 0,
+ args = arguments,
+ x = new this(args[i]);
+
+ external = false;
+ for (; x.s && ++i < args.length;) x = x.plus(args[i]);
+ external = true;
+
+ return finalise(x, this.precision, this.rounding);
+}
+
+
+/*
+ * 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);
+}
+
+
+P[Symbol.for('nodejs.util.inspect.custom')] = P.toString;
+P[Symbol.toStringTag] = 'Decimal';
+
+// Create and configure initial Decimal constructor.
+export var Decimal = P.constructor = clone(DEFAULTS);
+
+// Create the internal constants from their string values.
+LN10 = new Decimal(LN10);
+PI = new Decimal(PI);
+
+export default Decimal;
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