exam_mp/utils/decimal.js

/*!
 *  decimal.js v10.4.3
 *  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 ------------------------------- //


	Decimal, inexact, noConflict, 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);
}


// Create and configure initial Decimal constructor.
Decimal = clone(DEFAULTS);
Decimal.prototype.constructor = Decimal;
Decimal['default'] = Decimal.Decimal = Decimal;

// Create the internal constants from their string values.
LN10 = new Decimal(LN10);
PI = new Decimal(PI);

export default Decimal