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;(function (globalScope) {
'use strict';
/*
* decimal.js v10.3.1
* An arbitrary-precision Decimal type for JavaScript.
* https://github.com/MikeMcl/decimal.js
* Copyright (c) 2021 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.
// AMD.
if (typeof define == 'function' && define.amd) {
define(function () {
return Decimal;
});
// Node and other environments that support module.exports.
} else if (typeof module != 'undefined' && module.exports) {
if (typeof Symbol == 'function' && typeof Symbol.iterator == 'symbol') {
P[Symbol['for']('nodejs.util.inspect.custom')] = P.toString;
P[Symbol.toStringTag] = 'Decimal';
}
module.exports = Decimal;
// Browser.
} else {
if (!globalScope) {
globalScope = typeof self != 'undefined' && self && self.self == self ? self : window;
}
noConflict = globalScope.Decimal;
Decimal.noConflict = function () {
globalScope.Decimal = noConflict;
return Decimal;
};
globalScope.Decimal = Decimal;
}
})(this);