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/*
* 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 ------------------------------- //
inexact, quadrant,
external = true,
decimalError = '[DecimalError] ',
invalidArgument = decimalError + 'Invalid argument: ',
precisionLimitExceeded = decimalError + 'Precision limit exceeded',
cryptoUnavailable = decimalError + 'crypto unavailable',
tag = '[object Decimal]',
mathfloor = Math.floor,
mathpow = Math.pow,
isBinary = /^0b([01]+(\.[01]*)?|\.[01]+)(p[+-]?\d+)?$/i,
isHex = /^0x([0-9a-f]+(\.[0-9a-f]*)?|\.[0-9a-f]+)(p[+-]?\d+)?$/i,
isOctal = /^0o([0-7]+(\.[0-7]*)?|\.[0-7]+)(p[+-]?\d+)?$/i,
isDecimal = /^(\d+(\.\d*)?|\.\d+)(e[+-]?\d+)?$/i,
BASE = 1e7,
LOG_BASE = 7,
MAX_SAFE_INTEGER = 9007199254740991,
LN10_PRECISION = LN10.length - 1,
PI_PRECISION = PI.length - 1,
// Decimal.prototype object
P = { toStringTag: tag };
// Decimal prototype methods
/*
* absoluteValue abs
* ceil
* clampedTo clamp
* comparedTo cmp
* cosine cos
* cubeRoot cbrt
* decimalPlaces dp
* dividedBy div
* dividedToIntegerBy divToInt
* equals eq
* floor
* greaterThan gt
* greaterThanOrEqualTo gte
* hyperbolicCosine cosh
* hyperbolicSine sinh
* hyperbolicTangent tanh
* inverseCosine acos
* inverseHyperbolicCosine acosh
* inverseHyperbolicSine asinh
* inverseHyperbolicTangent atanh
* inverseSine asin
* inverseTangent atan
* isFinite
* isInteger isInt
* isNaN
* isNegative isNeg
* isPositive isPos
* isZero
* lessThan lt
* lessThanOrEqualTo lte
* logarithm log
* [maximum] [max]
* [minimum] [min]
* minus sub
* modulo mod
* naturalExponential exp
* naturalLogarithm ln
* negated neg
* plus add
* precision sd
* round
* sine sin
* squareRoot sqrt
* tangent tan
* times mul
* toBinary
* toDecimalPlaces toDP
* toExponential
* toFixed
* toFraction
* toHexadecimal toHex
* toNearest
* toNumber
* toOctal
* toPower pow
* toPrecision
* toSignificantDigits toSD
* toString
* truncated trunc
* valueOf toJSON
*/
/*
* Return a new Decimal whose value is the absolute value of this Decimal.
*
*/
P.absoluteValue = P.abs = function () {
var x = new this.constructor(this);
if (x.s < 0) x.s = 1;
return finalise(x);
};
/*
* Return a new Decimal whose value is the value of this Decimal rounded to a whole number in the
* direction of positive Infinity.
*
*/
P.ceil = function () {
return finalise(new this.constructor(this), this.e + 1, 2);
};
/*
* Return a new Decimal whose value is the value of this Decimal clamped to the range
* delineated by `min` and `max`.
*
* min {number|string|Decimal}
* max {number|string|Decimal}
*
*/
P.clampedTo = P.clamp = function (min, max) {
var k,
x = this,
Ctor = x.constructor;
min = new Ctor(min);
max = new Ctor(max);
if (!min.s || !max.s) return new Ctor(NaN);
if (min.gt(max)) throw Error(invalidArgument + max);
k = x.cmp(min);
return k < 0 ? min : x.cmp(max) > 0 ? max : new Ctor(x);
};
/*
* Return
* 1 if the value of this Decimal is greater than the value of `y`,
* -1 if the value of this Decimal is less than the value of `y`,
* 0 if they have the same value,
* NaN if the value of either Decimal is NaN.
*
*/
P.comparedTo = P.cmp = function (y) {
var i, j, xdL, ydL,
x = this,
xd = x.d,
yd = (y = new x.constructor(y)).d,
xs = x.s,
ys = y.s;
// Either NaN or ±Infinity?
if (!xd || !yd) {
return !xs || !ys ? NaN : xs !== ys ? xs : xd === yd ? 0 : !xd ^ xs < 0 ? 1 : -1;
}
// Either zero?
if (!xd[0] || !yd[0]) return xd[0] ? xs : yd[0] ? -ys : 0;
// Signs differ?
if (xs !== ys) return xs;
// Compare exponents.
if (x.e !== y.e) return x.e > y.e ^ xs < 0 ? 1 : -1;
xdL = xd.length;
ydL = yd.length;
// Compare digit by digit.
for (i = 0, j = xdL < ydL ? xdL : ydL; i < j; ++i) {
if (xd[i] !== yd[i]) return xd[i] > yd[i] ^ xs < 0 ? 1 : -1;
}
// Compare lengths.
return xdL === ydL ? 0 : xdL > ydL ^ xs < 0 ? 1 : -1;
};
/*
* Return a new Decimal whose value is the cosine of the value in radians of this Decimal.
*
* Domain: [-Infinity, Infinity]
* Range: [-1, 1]
*
* cos(0) = 1
* cos(-0) = 1
* cos(Infinity) = NaN
* cos(-Infinity) = NaN
* cos(NaN) = NaN
*
*/
P.cosine = P.cos = function () {
var pr, rm,
x = this,
Ctor = x.constructor;
if (!x.d) return new Ctor(NaN);
// cos(0) = cos(-0) = 1
if (!x.d[0]) return new Ctor(1);
pr = Ctor.precision;
rm = Ctor.rounding;
Ctor.precision = pr + Math.max(x.e, x.sd()) + LOG_BASE;
Ctor.rounding = 1;
x = cosine(Ctor, toLessThanHalfPi(Ctor, x));
Ctor.precision = pr;
Ctor.rounding = rm;
return finalise(quadrant == 2 || quadrant == 3 ? x.neg() : x, pr, rm, true);
};
/*
*
* Return a new Decimal whose value is the cube root of the value of this Decimal, rounded to
* `precision` significant digits using rounding mode `rounding`.
*
* cbrt(0) = 0
* cbrt(-0) = -0
* cbrt(1) = 1
* cbrt(-1) = -1
* cbrt(N) = N
* cbrt(-I) = -I
* cbrt(I) = I
*
* Math.cbrt(x) = (x < 0 ? -Math.pow(-x, 1/3) : Math.pow(x, 1/3))
*
*/
P.cubeRoot = P.cbrt = function () {
var e, m, n, r, rep, s, sd, t, t3, t3plusx,
x = this,
Ctor = x.constructor;
if (!x.isFinite() || x.isZero()) return new Ctor(x);
external = false;
// Initial estimate.
s = x.s * mathpow(x.s * x, 1 / 3);
// Math.cbrt underflow/overflow?
// Pass x to Math.pow as integer, then adjust the exponent of the result.
if (!s || Math.abs(s) == 1 / 0) {
n = digitsToString(x.d);
e = x.e;
// Adjust n exponent so it is a multiple of 3 away from x exponent.
if (s = (e - n.length + 1) % 3) n += (s == 1 || s == -2 ? '0' : '00');
s = mathpow(n, 1 / 3);
// Rarely, e may be one less than the result exponent value.
e = mathfloor((e + 1) / 3) - (e % 3 == (e < 0 ? -1 : 2));
if (s == 1 / 0) {
n = '5e' + e;
} else {
n = s.toExponential();
n = n.slice(0, n.indexOf('e') + 1) + e;
}
r = new Ctor(n);
r.s = x.s;
} else {
r = new Ctor(s.toString());
}
sd = (e = Ctor.precision) + 3;
// Halley's method.
// TODO? Compare Newton's method.
for (;;) {
t = r;
t3 = t.times(t).times(t);
t3plusx = t3.plus(x);
r = divide(t3plusx.plus(x).times(t), t3plusx.plus(t3), sd + 2, 1);
// TODO? Replace with for-loop and checkRoundingDigits.
if (digitsToString(t.d).slice(0, sd) === (n = digitsToString(r.d)).slice(0, sd)) {
n = n.slice(sd - 3, sd + 1);
// The 4th rounding digit may be in error by -1 so if the 4 rounding digits are 9999 or 4999
// , i.e. approaching a rounding boundary, continue the iteration.
if (n == '9999' || !rep && n == '4999') {
// On the first iteration only, check to see if rounding up gives the exact result as the
// nines may infinitely repeat.
if (!rep) {
finalise(t, e + 1, 0);
if (t.times(t).times(t).eq(x)) {
r = t;
break;
}
}
sd += 4;
rep = 1;
} else {
// If the rounding digits are null, 0{0,4} or 50{0,3}, check for an exact result.
// If not, then there are further digits and m will be truthy.
if (!+n || !+n.slice(1) && n.charAt(0) == '5') {
// Truncate to the first rounding digit.
finalise(r, e + 1, 1);
m = !r.times(r).times(r).eq(x);
}
break;
}
}
}
external = true;
return finalise(r, e, Ctor.rounding, m);
};
/*
* Return the number of decimal places of the value of this Decimal.
*
*/
P.decimalPlaces = P.dp = function () {
var w,
d = this.d,
n = NaN;
if (d) {
w = d.length - 1;
n = (w - mathfloor(this.e / LOG_BASE)) * LOG_BASE;
// Subtract the number of trailing zeros of the last word.
w = d[w];
if (w) for (; w % 10 == 0; w /= 10) n--;
if (n < 0) n = 0;
}
return n;
};
/*
* n / 0 = I
* n / N = N
* n / I = 0
* 0 / n = 0
* 0 / 0 = N
* 0 / N = N
* 0 / I = 0
* N / n = N
* N / 0 = N
* N / N = N
* N / I = N
* I / n = I
* I / 0 = I
* I / N = N
* I / I = N
*
* Return a new Decimal whose value is the value of this Decimal divided by `y`, rounded to
* `precision` significant digits using rounding mode `rounding`.
*
*/
P.dividedBy = P.div = function (y) {
return divide(this, new this.constructor(y));
};
/*
* Return a new Decimal whose value is the integer part of dividing the value of this Decimal
* by the value of `y`, rounded to `precision` significant digits using rounding mode `rounding`.
*
*/
P.dividedToIntegerBy = P.divToInt = function (y) {
var x = this,
Ctor = x.constructor;
return finalise(divide(x, new Ctor(y), 0, 1, 1), Ctor.precision, Ctor.rounding);
};
/*
* Return true if the value of this Decimal is equal to the value of `y`, otherwise return false.
*
*/
P.equals = P.eq = function (y) {
return this.cmp(y) === 0;
};
/*
* Return a new Decimal whose value is the value of this Decimal rounded to a whole number in the
* direction of negative Infinity.
*
*/
P.floor = function () {
return finalise(new this.constructor(this), this.e + 1, 3);
};
/*
* Return true if the value of this Decimal is greater than the value of `y`, otherwise return
* false.
*
*/
P.greaterThan = P.gt = function (y) {
return this.cmp(y) > 0;
};
/*
* Return true if the value of this Decimal is greater than or equal to the value of `y`,
* otherwise return false.
*
*/
P.greaterThanOrEqualTo = P.gte = function (y) {
var k = this.cmp(y);
return k == 1 || k === 0;
};
/*
* Return a new Decimal whose value is the hyperbolic cosine of the value in radians of this
* Decimal.
*
* Domain: [-Infinity, Infinity]
* Range: [1, Infinity]
*
* cosh(x) = 1 + x^2/2! + x^4/4! + x^6/6! + ...
*
* cosh(0) = 1
* cosh(-0) = 1
* cosh(Infinity) = Infinity
* cosh(-Infinity) = Infinity
* cosh(NaN) = NaN
*
* x time taken (ms) result
* 1000 9 9.8503555700852349694e+433
* 10000 25 4.4034091128314607936e+4342
* 100000 171 1.4033316802130615897e+43429
* 1000000 3817 1.5166076984010437725e+434294
* 10000000 abandoned after 2 minute wait
*
* TODO? Compare performance of cosh(x) = 0.5 * (exp(x) + exp(-x))
*
*/
P.hyperbolicCosine = P.cosh = function () {
var k, n, pr, rm, len,
x = this,
Ctor = x.constructor,
one = new Ctor(1);
if (!x.isFinite()) return new Ctor(x.s ? 1 / 0 : NaN);
if (x.isZero()) return one;
pr = Ctor.precision;
rm = Ctor.rounding;
Ctor.precision = pr + Math.max(x.e, x.sd()) + 4;
Ctor.rounding = 1;
len = x.d.length;
// Argument reduction: cos(4x) = 1 - 8cos^2(x) + 8cos^4(x) + 1
// i.e. cos(x) = 1 - cos^2(x/4)(8 - 8cos^2(x/4))
// Estimate the optimum number of times to use the argument reduction.
// TODO? Estimation reused from cosine() and may not be optimal here.
if (len < 32) {
k = Math.ceil(len / 3);
n = (1 / tinyPow(4, k)).toString();
} else {
k = 16;
n = '2.3283064365386962890625e-10';
}
x = taylorSeries(Ctor, 1, x.times(n), new Ctor(1), true);
// Reverse argument reduction
var cosh2_x,
i = k,
d8 = new Ctor(8);
for (; i--;) {
cosh2_x = x.times(x);
x = one.minus(cosh2_x.times(d8.minus(cosh2_x.times(d8))));
}
return finalise(x, Ctor.precision = pr, Ctor.rounding = rm, true);
};
/*
* Return a new Decimal whose value is the hyperbolic sine of the value in radians of this
* Decimal.
*
* Domain: [-Infinity, Infinity]
* Range: [-Infinity, Infinity]
*
* sinh(x) = x + x^3/3! + x^5/5! + x^7/7! + ...
*
* sinh(0) = 0
* sinh(-0) = -0
* sinh(Infinity) = Infinity
* sinh(-Infinity) = -Infinity
* sinh(NaN) = NaN
*
* x time taken (ms)
* 10 2 ms
* 100 5 ms
* 1000 14 ms
* 10000 82 ms
* 100000 886 ms 1.4033316802130615897e+43429
* 200000 2613 ms
* 300000 5407 ms
* 400000 8824 ms
* 500000 13026 ms 8.7080643612718084129e+217146
* 1000000 48543 ms
*
* TODO? Compare performance of sinh(x) = 0.5 * (exp(x) - exp(-x))
*
*/
P.hyperbolicSine = P.sinh = function () {
var k, pr, rm, len,
x = this,
Ctor = x.constructor;
if (!x.isFinite() || x.isZero()) return new Ctor(x);
pr = Ctor.precision;
rm = Ctor.rounding;
Ctor.precision = pr + Math.max(x.e, x.sd()) + 4;
Ctor.rounding = 1;
len = x.d.length;
if (len < 3) {
x = taylorSeries(Ctor, 2, x, x, true);
} else {
// Alternative argument reduction: sinh(3x) = sinh(x)(3 + 4sinh^2(x))
// i.e. sinh(x) = sinh(x/3)(3 + 4sinh^2(x/3))
// 3 multiplications and 1 addition
// Argument reduction: sinh(5x) = sinh(x)(5 + sinh^2(x)(20 + 16sinh^2(x)))
// i.e. sinh(x) = sinh(x/5)(5 + sinh^2(x/5)(20 + 16sinh^2(x/5)))
// 4 multiplications and 2 additions
// Estimate the optimum number of times to use the argument reduction.
k = 1.4 * Math.sqrt(len);
k = k > 16 ? 16 : k | 0;
x = x.times(1 / tinyPow(5, k));
x = taylorSeries(Ctor, 2, x, x, true);
// Reverse argument reduction
var sinh2_x,
d5 = new Ctor(5),
d16 = new Ctor(16),
d20 = new Ctor(20);
for (; k--;) {
sinh2_x = x.times(x);
x = x.times(d5.plus(sinh2_x.times(d16.times(sinh2_x).plus(d20))));
}
}
Ctor.precision = pr;
Ctor.rounding = rm;
return finalise(x, pr, rm, true);
};
/*
* Return a new Decimal whose value is the hyperbolic tangent of the value in radians of this
* Decimal.
*
* Domain: [-Infinity, Infinity]
* Range: [-1, 1]
*
* tanh(x) = sinh(x) / cosh(x)
*
* tanh(0) = 0
* tanh(-0) = -0
* tanh(Infinity) = 1
* tanh(-Infinity) = -1
* tanh(NaN) = NaN
*
*/
P.hyperbolicTangent = P.tanh = function () {
var pr, rm,
x = this,
Ctor = x.constructor;
if (!x.isFinite()) return new Ctor(x.s);
if (x.isZero()) return new Ctor(x);
pr = Ctor.precision;
rm = Ctor.rounding;
Ctor.precision = pr + 7;
Ctor.rounding = 1;
return divide(x.sinh(), x.cosh(), Ctor.precision = pr, Ctor.rounding = rm);
};
/*
* Return a new Decimal whose value is the arccosine (inverse cosine) in radians of the value of
* this Decimal.
*
* Domain: [-1, 1]
* Range: [0, pi]
*
* acos(x) = pi/2 - asin(x)
*
* acos(0) = pi/2
* acos(-0) = pi/2
* acos(1) = 0
* acos(-1) = pi
* acos(1/2) = pi/3
* acos(-1/2) = 2*pi/3
* acos(|x| > 1) = NaN
* acos(NaN) = NaN
*
*/
P.inverseCosine = P.acos = function () {
var halfPi,
x = this,
Ctor = x.constructor,
k = x.abs().cmp(1),
pr = Ctor.precision,
rm = Ctor.rounding;
if (k !== -1) {
return k === 0
// |x| is 1
? x.isNeg() ? getPi(Ctor, pr, rm) : new Ctor(0)
// |x| > 1 or x is NaN
: new Ctor(NaN);
}
if (x.isZero()) return getPi(Ctor, pr + 4, rm).times(0.5);
// TODO? Special case acos(0.5) = pi/3 and acos(-0.5) = 2*pi/3
Ctor.precision = pr + 6;
Ctor.rounding = 1;
x = x.asin();
halfPi = getPi(Ctor, pr + 4, rm).times(0.5);
Ctor.precision = pr;
Ctor.rounding = rm;
return halfPi.minus(x);
};
/*
* Return a new Decimal whose value is the inverse of the hyperbolic cosine in radians of the
* value of this Decimal.
*
* Domain: [1, Infinity]
* Range: [0, Infinity]
*
* acosh(x) = ln(x + sqrt(x^2 - 1))
*
* acosh(x < 1) = NaN
* acosh(NaN) = NaN
* acosh(Infinity) = Infinity
* acosh(-Infinity) = NaN
* acosh(0) = NaN
* acosh(-0) = NaN
* acosh(1) = 0
* acosh(-1) = NaN
*
*/
P.inverseHyperbolicCosine = P.acosh = function () {
var pr, rm,
x = this,
Ctor = x.constructor;
if (x.lte(1)) return new Ctor(x.eq(1) ? 0 : NaN);
if (!x.isFinite()) return new Ctor(x);
pr = Ctor.precision;
rm = Ctor.rounding;
Ctor.precision = pr + Math.max(Math.abs(x.e), x.sd()) + 4;
Ctor.rounding = 1;
external = false;
x = x.times(x).minus(1).sqrt().plus(x);
external = true;
Ctor.precision = pr;
Ctor.rounding = rm;
return x.ln();
};
/*
* Return a new Decimal whose value is the inverse of the hyperbolic sine in radians of the value
* of this Decimal.
*
* Domain: [-Infinity, Infinity]
* Range: [-Infinity, Infinity]
*
* asinh(x) = ln(x + sqrt(x^2 + 1))
*
* asinh(NaN) = NaN
* asinh(Infinity) = Infinity
* asinh(-Infinity) = -Infinity
* asinh(0) = 0
* asinh(-0) = -0
*
*/
P.inverseHyperbolicSine = P.asinh = function () {
var pr, rm,
x = this,
Ctor = x.constructor;
if (!x.isFinite() || x.isZero()) return new Ctor(x);
pr = Ctor.precision;
rm = Ctor.rounding;
Ctor.precision = pr + 2 * Math.max(Math.abs(x.e), x.sd()) + 6;
Ctor.rounding = 1;
external = false;
x = x.times(x).plus(1).sqrt().plus(x);
external = true;
Ctor.precision = pr;
Ctor.rounding = rm;
return x.ln();
};
/*
* Return a new Decimal whose value is the inverse of the hyperbolic tangent in radians of the
* value of this Decimal.
*
* Domain: [-1, 1]
* Range: [-Infinity, Infinity]
*
* atanh(x) = 0.5 * ln((1 + x) / (1 - x))
*
* atanh(|x| > 1) = NaN
* atanh(NaN) = NaN
* atanh(Infinity) = NaN
* atanh(-Infinity) = NaN
* atanh(0) = 0
* atanh(-0) = -0
* atanh(1) = Infinity
* atanh(-1) = -Infinity
*
*/
P.inverseHyperbolicTangent = P.atanh = function () {
var pr, rm, wpr, xsd,
x = this,
Ctor = x.constructor;
if (!x.isFinite()) return new Ctor(NaN);
if (x.e >= 0) return new Ctor(x.abs().eq(1) ? x.s / 0 : x.isZero() ? x : NaN);
pr = Ctor.precision;
rm = Ctor.rounding;
xsd = x.sd();
if (Math.max(xsd, pr) < 2 * -x.e - 1) return finalise(new Ctor(x), pr, rm, true);
Ctor.precision = wpr = xsd - x.e;
x = divide(x.plus(1), new Ctor(1).minus(x), wpr + pr, 1);
Ctor.precision = pr + 4;
Ctor.rounding = 1;
x = x.ln();
Ctor.precision = pr;
Ctor.rounding = rm;
return x.times(0.5);
};
/*
* Return a new Decimal whose value is the arcsine (inverse sine) in radians of the value of this
* Decimal.
*
* Domain: [-Infinity, Infinity]
* Range: [-pi/2, pi/2]
*
* asin(x) = 2*atan(x/(1 + sqrt(1 - x^2)))
*
* asin(0) = 0
* asin(-0) = -0
* asin(1/2) = pi/6
* asin(-1/2) = -pi/6
* asin(1) = pi/2
* asin(-1) = -pi/2
* asin(|x| > 1) = NaN
* asin(NaN) = NaN
*
* TODO? Compare performance of Taylor series.
*
*/
P.inverseSine = P.asin = function () {
var halfPi, k,
pr, rm,
x = this,
Ctor = x.constructor;
if (x.isZero()) return new Ctor(x);
k = x.abs().cmp(1);
pr = Ctor.precision;
rm = Ctor.rounding;
if (k !== -1) {
// |x| is 1
if (k === 0) {
halfPi = getPi(Ctor, pr + 4, rm).times(0.5);
halfPi.s = x.s;
return halfPi;
}
// |x| > 1 or x is NaN
return new Ctor(NaN);
}
// TODO? Special case asin(1/2) = pi/6 and asin(-1/2) = -pi/6
Ctor.precision = pr + 6;
Ctor.rounding = 1;
x = x.div(new Ctor(1).minus(x.times(x)).sqrt().plus(1)).atan();
Ctor.precision = pr;
Ctor.rounding = rm;
return x.times(2);
};
/*
* Return a new Decimal whose value is the arctangent (inverse tangent) in radians of the value
* of this Decimal.
*
* Domain: [-Infinity, Infinity]
* Range: [-pi/2, pi/2]
*
* atan(x) = x - x^3/3 + x^5/5 - x^7/7 + ...
*
* atan(0) = 0
* atan(-0) = -0
* atan(1) = pi/4
* atan(-1) = -pi/4
* atan(Infinity) = pi/2
* atan(-Infinity) = -pi/2
* atan(NaN) = NaN
*
*/
P.inverseTangent = P.atan = function () {
var i, j, k, n, px, t, r, wpr, x2,
x = this,
Ctor = x.constructor,
pr = Ctor.precision,
rm = Ctor.rounding;
if (!x.isFinite()) {
if (!x.s) return new Ctor(NaN);
if (pr + 4 <= PI_PRECISION) {
r = getPi(Ctor, pr + 4, rm).times(0.5);
r.s = x.s;
return r;
}
} else if (x.isZero()) {
return new Ctor(x);
} else if (x.abs().eq(1) && pr + 4 <= PI_PRECISION) {
r = getPi(Ctor, pr + 4, rm).times(0.25);
r.s = x.s;
return r;
}
Ctor.precision = wpr = pr + 10;
Ctor.rounding = 1;
// TODO? if (x >= 1 && pr <= PI_PRECISION) atan(x) = halfPi * x.s - atan(1 / x);
// Argument reduction
// Ensure |x| < 0.42
// atan(x) = 2 * atan(x / (1 + sqrt(1 + x^2)))
k = Math.min(28, wpr / LOG_BASE + 2 | 0);
for (i = k; i; --i) x = x.div(x.times(x).plus(1).sqrt().plus(1));
external = false;
j = Math.ceil(wpr / LOG_BASE);
n = 1;
x2 = x.times(x);
r = new Ctor(x);
px = x;
// atan(x) = x - x^3/3 + x^5/5 - x^7/7 + ...
for (; i !== -1;) {
px = px.times(x2);
t = r.minus(px.div(n += 2));
px = px.times(x2);
r = t.plus(px.div(n += 2));
if (r.d[j] !== void 0) for (i = j; r.d[i] === t.d[i] && i--;);
}
if (k) r = r.times(2 << (k - 1));
external = true;
return finalise(r, Ctor.precision = pr, Ctor.rounding = rm, true);
};
/*
* Return true if the value of this Decimal is a finite number, otherwise return false.
*
*/
P.isFinite = function () {
return !!this.d;
};
/*
* Return true if the value of this Decimal is an integer, otherwise return false.
*
*/
P.isInteger = P.isInt = function () {
return !!this.d && mathfloor(this.e / LOG_BASE) > this.d.length - 2;
};
/*
* Return true if the value of this Decimal is NaN, otherwise return false.
*
*/
P.isNaN = function () {
return !this.s;
};
/*
* Return true if the value of this Decimal is negative, otherwise return false.
*
*/
P.isNegative = P.isNeg = function () {
return this.s < 0;
};
/*
* Return true if the value of this Decimal is positive, otherwise return false.
*
*/
P.isPositive = P.isPos = function () {
return this.s > 0;
};
/*
* Return true if the value of this Decimal is 0 or -0, otherwise return false.
*
*/
P.isZero = function () {
return !!this.d && this.d[0] === 0;
};
/*
* Return true if the value of this Decimal is less than `y`, otherwise return false.
*
*/
P.lessThan = P.lt = function (y) {
return this.cmp(y) < 0;
};
/*
* Return true if the value of this Decimal is less than or equal to `y`, otherwise return false.
*
*/
P.lessThanOrEqualTo = P.lte = function (y) {
return this.cmp(y) < 1;
};
/*
* Return the logarithm of the value of this Decimal to the specified base, rounded to `precision`
* significant digits using rounding mode `rounding`.
*
* If no base is specified, return log[10](arg).
*
* log[base](arg) = ln(arg) / ln(base)
*
* The result will always be correctly rounded if the base of the log is 10, and 'almost always'
* otherwise:
*
* Depending on the rounding mode, the result may be incorrectly rounded if the first fifteen
* rounding digits are [49]99999999999999 or [50]00000000000000. In that case, the maximum error
* between the result and the correctly rounded result will be one ulp (unit in the last place).
*
* log[-b](a) = NaN
* log[0](a) = NaN
* log[1](a) = NaN
* log[NaN](a) = NaN
* log[Infinity](a) = NaN
* log[b](0) = -Infinity
* log[b](-0) = -Infinity
* log[b](-a) = NaN
* log[b](1) = 0
* log[b](Infinity) = Infinity
* log[b](NaN) = NaN
*
* [base] {number|string|Decimal} The base of the logarithm.
*
*/
P.logarithm = P.log = function (base) {
var isBase10, d, denominator, k, inf, num, sd, r,
arg = this,
Ctor = arg.constructor,
pr = Ctor.precision,
rm = Ctor.rounding,
guard = 5;
// Default base is 10.
if (base == null) {
base = new Ctor(10);
isBase10 = true;
} else {
base = new Ctor(base);
d = base.d;
// Return NaN if base is negative, or non-finite, or is 0 or 1.
if (base.s < 0 || !d || !d[0] || base.eq(1)) return new Ctor(NaN);
isBase10 = base.eq(10);
}
d = arg.d;
// Is arg negative, non-finite, 0 or 1?
if (arg.s < 0 || !d || !d[0] || arg.eq(1)) {
return new Ctor(d && !d[0] ? -1 / 0 : arg.s != 1 ? NaN : d ? 0 : 1 / 0);
}
// The result will have a non-terminating decimal expansion if base is 10 and arg is not an
// integer power of 10.
if (isBase10) {
if (d.length > 1) {
inf = true;
} else {
for (k = d[0]; k % 10 === 0;) k /= 10;
inf = k !== 1;
}
}
external = false;
sd = pr + guard;
num = naturalLogarithm(arg, sd);
denominator = isBase10 ? getLn10(Ctor, sd + 10) : naturalLogarithm(base, sd);
// The result will have 5 rounding digits.
r = divide(num, denominator, sd, 1);
// If at a rounding boundary, i.e. the result's rounding digits are [49]9999 or [50]0000,
// calculate 10 further digits.
//
// If the result is known to have an infinite decimal expansion, repeat this until it is clear
// that the result is above or below the boundary. Otherwise, if after calculating the 10
// further digits, the last 14 are nines, round up and assume the result is exact.
// Also assume the result is exact if the last 14 are zero.
//
// Example of a result that will be incorrectly rounded:
// log[1048576](4503599627370502) = 2.60000000000000009610279511444746...
// The above result correctly rounded using ROUND_CEIL to 1 decimal place should be 2.7, but it
// will be given as 2.6 as there are 15 zeros immediately after the requested decimal place, so
// the exact result would be assumed to be 2.6, which rounded using ROUND_CEIL to 1 decimal
// place is still 2.6.
if (checkRoundingDigits(r.d, k = pr, rm)) {
do {
sd += 10;
num = naturalLogarithm(arg, sd);
denominator = isBase10 ? getLn10(Ctor, sd + 10) : naturalLogarithm(base, sd);
r = divide(num, denominator, sd, 1);
if (!inf) {
// Check for 14 nines from the 2nd rounding digit, as the first may be 4.
if (+digitsToString(r.d).slice(k + 1, k + 15) + 1 == 1e14) {
r = finalise(r, pr + 1, 0);
}
break;
}
} while (checkRoundingDigits(r.d, k += 10, rm));
}
external = true;
return finalise(r, pr, rm);
};
/*
* Return a new Decimal whose value is the maximum of the arguments and the value of this Decimal.
*
* arguments {number|string|Decimal}
*
P.max = function () {
Array.prototype.push.call(arguments, this);
return maxOrMin(this.constructor, arguments, 'lt');
};
*/
/*
* Return a new Decimal whose value is the minimum of the arguments and the value of this Decimal.
*
* arguments {number|string|Decimal}
*
P.min = function () {
Array.prototype.push.call(arguments, this);
return maxOrMin(this.constructor, arguments, 'gt');
};
*/
/*
* n - 0 = n
* n - N = N
* n - I = -I
* 0 - n = -n
* 0 - 0 = 0
* 0 - N = N
* 0 - I = -I
* N - n = N
* N - 0 = N
* N - N = N
* N - I = N
* I - n = I
* I - 0 = I
* I - N = N
* I - I = N
*
* Return a new Decimal whose value is the value of this Decimal minus `y`, rounded to `precision`
* significant digits using rounding mode `rounding`.
*
*/
P.minus = P.sub = function (y) {
var d, e, i, j, k, len, pr, rm, xd, xe, xLTy, yd,
x = this,
Ctor = x.constructor;
y = new Ctor(y);
// If either is not finite...
if (!x.d || !y.d) {
// Return NaN if either is NaN.
if (!x.s || !y.s) y = new Ctor(NaN);
// Return y negated if x is finite and y is ±Infinity.
else if (x.d) y.s = -y.s;
// Return x if y is finite and x is ±Infinity.
// Return x if both are ±Infinity with different signs.
// Return NaN if both are ±Infinity with the same sign.
else y = new Ctor(y.d || x.s !== y.s ? x : NaN);
return y;
}
// If signs differ...
if (x.s != y.s) {
y.s = -y.s;
return x.plus(y);
}
xd = x.d;
yd = y.d;
pr = Ctor.precision;
rm = Ctor.rounding;
// If either is zero...
if (!xd[0] || !yd[0]) {
// Return y negated if x is zero and y is non-zero.
if (yd[0]) y.s = -y.s;
// Return x if y is zero and x is non-zero.
else if (xd[0]) y = new Ctor(x);
// Return zero if both are zero.
// From IEEE 754 (2008) 6.3: 0 - 0 = -0 - -0 = -0 when rounding to -Infinity.
else return new Ctor(rm === 3 ? -0 : 0);
return external ? finalise(y, pr, rm) : y;
}
// x and y are finite, non-zero numbers with the same sign.
// Calculate base 1e7 exponents.
e = mathfloor(y.e / LOG_BASE);
xe = mathfloor(x.e / LOG_BASE);
xd = xd.slice();
k = xe - e;
// If base 1e7 exponents differ...
if (k) {
xLTy = k < 0;
if (xLTy) {
d = xd;
k = -k;
len = yd.length;
} else {
d = yd;
e = xe;
len = xd.length;
}
// Numbers with massively different exponents would result in a very high number of
// zeros needing to be prepended, but this can be avoided while still ensuring correct
// rounding by limiting the number of zeros to `Math.ceil(pr / LOG_BASE) + 2`.
i = Math.max(Math.ceil(pr / LOG_BASE), len) + 2;
if (k > i) {
k = i;
d.length = 1;
}
// Prepend zeros to equalise exponents.
d.reverse();
for (i = k; i--;) d.push(0);
d.reverse();
// Base 1e7 exponents equal.
} else {
// Check digits to determine which is the bigger number.
i = xd.length;
len = yd.length;
xLTy = i < len;
if (xLTy) len = i;
for (i = 0; i < len; i++) {
if (xd[i] != yd[i]) {
xLTy = xd[i] < yd[i];
break;
}
}
k = 0;
}
if (xLTy) {
d = xd;
xd = yd;
yd = d;
y.s = -y.s;
}
len = xd.length;
// Append zeros to `xd` if shorter.
// Don't add zeros to `yd` if shorter as subtraction only needs to start at `yd` length.
for (i = yd.length - len; i > 0; --i) xd[len++] = 0;
// Subtract yd from xd.
for (i = yd.length; i > k;) {
if (xd[--i] < yd[i]) {
for (j = i; j && xd[--j] === 0;) xd[j] = BASE - 1;
--xd[j];
xd[i] += BASE;
}
xd[i] -= yd[i];
}
// Remove trailing zeros.
for (; xd[--len] === 0;) xd.pop();
// Remove leading zeros and adjust exponent accordingly.
for (; xd[0] === 0; xd.shift()) --e;
// Zero?
if (!xd[0]) return new Ctor(rm === 3 ? -0 : 0);
y.d = xd;
y.e = getBase10Exponent(xd, e);
return external ? finalise(y, pr, rm) : y;
};
/*
* n % 0 = N
* n % N = N
* n % I = n
* 0 % n = 0
* -0 % n = -0
* 0 % 0 = N
* 0 % N = N
* 0 % I = 0
* N % n = N
* N % 0 = N
* N % N = N
* N % I = N
* I % n = N
* I % 0 = N
* I % N = N
* I % I = N
*
* Return a new Decimal whose value is the value of this Decimal modulo `y`, rounded to
* `precision` significant digits using rounding mode `rounding`.
*
* The result depends on the modulo mode.
*
*/
P.modulo = P.mod = function (y) {
var q,
x = this,
Ctor = x.constructor;
y = new Ctor(y);
// Return NaN if x is ±Infinity or NaN, or y is NaN or ±0.
if (!x.d || !y.s || y.d && !y.d[0]) return new Ctor(NaN);
// Return x if y is ±Infinity or x is ±0.
if (!y.d || x.d && !x.d[0]) {
return finalise(new Ctor(x), Ctor.precision, Ctor.rounding);
}
// Prevent rounding of intermediate calculations.
external = false;
if (Ctor.modulo == 9) {
// Euclidian division: q = sign(y) * floor(x / abs(y))
// result = x - q * y where 0 <= result < abs(y)
q = divide(x, y.abs(), 0, 3, 1);
q.s *= y.s;
} else {
q = divide(x, y, 0, Ctor.modulo, 1);
}
q = q.times(y);
external = true;
return x.minus(q);
};
/*
* Return a new Decimal whose value is the natural exponential of the value of this Decimal,
* i.e. the base e raised to the power the value of this Decimal, rounded to `precision`
* significant digits using rounding mode `rounding`.
*
*/
P.naturalExponential = P.exp = function () {
return naturalExponential(this);
};
/*
* Return a new Decimal whose value is the natural logarithm of the value of this Decimal,
* rounded to `precision` significant digits using rounding mode `rounding`.
*
*/
P.naturalLogarithm = P.ln = function () {
return naturalLogarithm(this);
};
/*
* Return a new Decimal whose value is the value of this Decimal negated, i.e. as if multiplied by
* -1.
*
*/
P.negated = P.neg = function () {
var x = new this.constructor(this);
x.s = -x.s;
return finalise(x);
};
/*
* n + 0 = n
* n + N = N
* n + I = I
* 0 + n = n
* 0 + 0 = 0
* 0 + N = N
* 0 + I = I
* N + n = N
* N + 0 = N
* N + N = N
* N + I = N
* I + n = I
* I + 0 = I
* I + N = N
* I + I = I
*
* Return a new Decimal whose value is the value of this Decimal plus `y`, rounded to `precision`
* significant digits using rounding mode `rounding`.
*
*/
P.plus = P.add = function (y) {
var carry, d, e, i, k, len, pr, rm, xd, yd,
x = this,
Ctor = x.constructor;
y = new Ctor(y);
// If either is not finite...
if (!x.d || !y.d) {
// Return NaN if either is NaN.
if (!x.s || !y.s) y = new Ctor(NaN);
// Return x if y is finite and x is ±Infinity.
// Return x if both are ±Infinity with the same sign.
// Return NaN if both are ±Infinity with different signs.
// Return y if x is finite and y is ±Infinity.
else if (!x.d) y = new Ctor(y.d || x.s === y.s ? x : NaN);
return y;
}
// If signs differ...
if (x.s != y.s) {
y.s = -y.s;
return x.minus(y);
}
xd = x.d;
yd = y.d;
pr = Ctor.precision;
rm = Ctor.rounding;
// If either is zero...
if (!xd[0] || !yd[0]) {
// Return x if y is zero.
// Return y if y is non-zero.
if (!yd[0]) y = new Ctor(x);
return external ? finalise(y, pr, rm) : y;
}
// x and y are finite, non-zero numbers with the same sign.
// Calculate base 1e7 exponents.
k = mathfloor(x.e / LOG_BASE);
e = mathfloor(y.e / LOG_BASE);
xd = xd.slice();
i = k - e;
// If base 1e7 exponents differ...
if (i) {
if (i < 0) {
d = xd;
i = -i;
len = yd.length;
} else {
d = yd;
e = k;
len = xd.length;
}
// Limit number of zeros prepended to max(ceil(pr / LOG_BASE), len) + 1.
k = Math.ceil(pr / LOG_BASE);
len = k > len ? k + 1 : len + 1;
if (i > len) {
i = len;
d.length = 1;
}
// Prepend zeros to equalise exponents. Note: Faster to use reverse then do unshifts.
d.reverse();
for (; i--;) d.push(0);
d.reverse();
}
len = xd.length;
i = yd.length;
// If yd is longer than xd, swap xd and yd so xd points to the longer array.
if (len - i < 0) {
i = len;
d = yd;
yd = xd;
xd = d;
}
// Only start adding at yd.length - 1 as the further digits of xd can be left as they are.
for (carry = 0; i;) {
carry = (xd[--i] = xd[i] + yd[i] + carry) / BASE | 0;
xd[i] %= BASE;
}
if (carry) {
xd.unshift(carry);
++e;
}
// Remove trailing zeros.
// No need to check for zero, as +x + +y != 0 && -x + -y != 0
for (len = xd.length; xd[--len] == 0;) xd.pop();
y.d = xd;
y.e = getBase10Exponent(xd, e);
return external ? finalise(y, pr, rm) : y;
};
/*
* Return the number of significant digits of the value of this Decimal.
*
* [z] {boolean|number} Whether to count integer-part trailing zeros: true, false, 1 or 0.
*
*/
P.precision = P.sd = function (z) {
var k,
x = this;
if (z !== void 0 && z !== !!z && z !== 1 && z !== 0) throw Error(invalidArgument + z);
if (x.d) {
k = getPrecision(x.d);
if (z && x.e + 1 > k) k = x.e + 1;
} else {
k = NaN;
}
return k;
};
/*
* Return a new Decimal whose value is the value of this Decimal rounded to a whole number using
* rounding mode `rounding`.
*
*/
P.round = function () {
var x = this,
Ctor = x.constructor;
return finalise(new Ctor(x), x.e + 1, Ctor.rounding);
};
/*
* Return a new Decimal whose value is the sine of the value in radians of this Decimal.
*
* Domain: [-Infinity, Infinity]
* Range: [-1, 1]
*
* sin(x) = x - x^3/3! + x^5/5! - ...
*
* sin(0) = 0
* sin(-0) = -0
* sin(Infinity) = NaN
* sin(-Infinity) = NaN
* sin(NaN) = NaN
*
*/
P.sine = P.sin = function () {
var pr, rm,
x = this,
Ctor = x.constructor;
if (!x.isFinite()) return new Ctor(NaN);
if (x.isZero()) return new Ctor(x);
pr = Ctor.precision;
rm = Ctor.rounding;
Ctor.precision = pr + Math.max(x.e, x.sd()) + LOG_BASE;
Ctor.rounding = 1;
x = sine(Ctor, toLessThanHalfPi(Ctor, x));
Ctor.precision = pr;
Ctor.rounding = rm;
return finalise(quadrant > 2 ? x.neg() : x, pr, rm, true);
};
/*
* Return a new Decimal whose value is the square root of this Decimal, rounded to `precision`
* significant digits using rounding mode `rounding`.
*
* sqrt(-n) = N
* sqrt(N) = N
* sqrt(-I) = N
* sqrt(I) = I
* sqrt(0) = 0
* sqrt(-0) = -0
*
*/
P.squareRoot = P.sqrt = function () {
var m, n, sd, r, rep, t,
x = this,
d = x.d,
e = x.e,
s = x.s,
Ctor = x.constructor;
// Negative/NaN/Infinity/zero?
if (s !== 1 || !d || !d[0]) {
return new Ctor(!s || s < 0 && (!d || d[0]) ? NaN : d ? x : 1 / 0);
}
external = false;
// Initial estimate.
s = Math.sqrt(+x);
// Math.sqrt underflow/overflow?
// Pass x to Math.sqrt as integer, then adjust the exponent of the result.
if (s == 0 || s == 1 / 0) {
n = digitsToString(d);
if ((n.length + e) % 2 == 0) n += '0';
s = Math.sqrt(n);
e = mathfloor((e + 1) / 2) - (e < 0 || e % 2);
if (s == 1 / 0) {
n = '5e' + e;
} else {
n = s.toExponential();
n = n.slice(0, n.indexOf('e') + 1) + e;
}
r = new Ctor(n);
} else {
r = new Ctor(s.toString());
}
sd = (e = Ctor.precision) + 3;
// Newton-Raphson iteration.
for (;;) {
t = r;
r = t.plus(divide(x, t, sd + 2, 1)).times(0.5);
// TODO? Replace with for-loop and checkRoundingDigits.
if (digitsToString(t.d).slice(0, sd) === (n = digitsToString(r.d)).slice(0, sd)) {
n = n.slice(sd - 3, sd + 1);
// The 4th rounding digit may be in error by -1 so if the 4 rounding digits are 9999 or
// 4999, i.e. approaching a rounding boundary, continue the iteration.
if (n == '9999' || !rep && n == '4999') {
// On the first iteration only, check to see if rounding up gives the exact result as the
// nines may infinitely repeat.
if (!rep) {
finalise(t, e + 1, 0);
if (t.times(t).eq(x)) {
r = t;
break;
}
}
sd += 4;
rep = 1;
} else {
// If the rounding digits are null, 0{0,4} or 50{0,3}, check for an exact result.
// If not, then there are further digits and m will be truthy.
if (!+n || !+n.slice(1) && n.charAt(0) == '5') {
// Truncate to the first rounding digit.
finalise(r, e + 1, 1);
m = !r.times(r).eq(x);
}
break;
}
}
}
external = true;
return finalise(r, e, Ctor.rounding, m);
};
/*
* Return a new Decimal whose value is the tangent of the value in radians of this Decimal.
*
* Domain: [-Infinity, Infinity]
* Range: [-Infinity, Infinity]
*
* tan(0) = 0
* tan(-0) = -0
* tan(Infinity) = NaN
* tan(-Infinity) = NaN
* tan(NaN) = NaN
*
*/
P.tangent = P.tan = function () {
var pr, rm,
x = this,
Ctor = x.constructor;
if (!x.isFinite()) return new Ctor(NaN);
if (x.isZero()) return new Ctor(x);
pr = Ctor.precision;
rm = Ctor.rounding;
Ctor.precision = pr + 10;
Ctor.rounding = 1;
x = x.sin();
x.s = 1;
x = divide(x, new Ctor(1).minus(x.times(x)).sqrt(), pr + 10, 0);
Ctor.precision = pr;
Ctor.rounding = rm;
return finalise(quadrant == 2 || quadrant == 4 ? x.neg() : x, pr, rm, true);
};
/*
* n * 0 = 0
* n * N = N
* n * I = I
* 0 * n = 0
* 0 * 0 = 0
* 0 * N = N
* 0 * I = N
* N * n = N
* N * 0 = N
* N * N = N
* N * I = N
* I * n = I
* I * 0 = N
* I * N = N
* I * I = I
*
* Return a new Decimal whose value is this Decimal times `y`, rounded to `precision` significant
* digits using rounding mode `rounding`.
*
*/
P.times = P.mul = function (y) {
var carry, e, i, k, r, rL, t, xdL, ydL,
x = this,
Ctor = x.constructor,
xd = x.d,
yd = (y = new Ctor(y)).d;
y.s *= x.s;
// If either is NaN, ±Infinity or ±0...
if (!xd || !xd[0] || !yd || !yd[0]) {
return new Ctor(!y.s || xd && !xd[0] && !yd || yd && !yd[0] && !xd
// Return NaN if either is NaN.
// Return NaN if x is ±0 and y is ±Infinity, or y is ±0 and x is ±Infinity.
? NaN
// Return ±Infinity if either is ±Infinity.
// Return ±0 if either is ±0.
: !xd || !yd ? y.s / 0 : y.s * 0);
}
e = mathfloor(x.e / LOG_BASE) + mathfloor(y.e / LOG_BASE);
xdL = xd.length;
ydL = yd.length;
// Ensure xd points to the longer array.
if (xdL < ydL) {
r = xd;
xd = yd;
yd = r;
rL = xdL;
xdL = ydL;
ydL = rL;
}
// Initialise the result array with zeros.
r = [];
rL = xdL + ydL;
for (i = rL; i--;) r.push(0);
// Multiply!
for (i = ydL; --i >= 0;) {
carry = 0;
for (k = xdL + i; k > i;) {
t = r[k] + yd[i] * xd[k - i - 1] + carry;
r[k--] = t % BASE | 0;
carry = t / BASE | 0;
}
r[k] = (r[k] + carry) % BASE | 0;
}
// Remove trailing zeros.
for (; !r[--rL];) r.pop();
if (carry) ++e;
else r.shift();
y.d = r;
y.e = getBase10Exponent(r, e);
return external ? finalise(y, Ctor.precision, Ctor.rounding) : y;
};
/*
* Return a string representing the value of this Decimal in base 2, round to `sd` significant
* digits using rounding mode `rm`.
*
* If the optional `sd` argument is present then return binary exponential notation.
*
* [sd] {number} Significant digits. Integer, 1 to MAX_DIGITS inclusive.
* [rm] {number} Rounding mode. Integer, 0 to 8 inclusive.
*
*/
P.toBinary = function (sd, rm) {