simple-squiggle

decimal.mjs (127051B)

---

 /*
  *  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) {
   return toStringBinary(this, 2, sd, rm);
 };
 
 
 /*
  * Return a new Decimal whose value is the value of this Decimal rounded to a maximum of `dp`
  * decimal places using rounding mode `rm` or `rounding` if `rm` is omitted.
  *
  * If `dp` is omitted, return a new Decimal whose value is the value of this Decimal.
  *
  * [dp] {number} Decimal places. Integer, 0 to MAX_DIGITS inclusive.
  * [rm] {number} Rounding mode. Integer, 0 to 8 inclusive.
  *
  */
 P.toDecimalPlaces = P.toDP = function (dp, rm) {
   var x = this,
     Ctor = x.constructor;
 
   x = new Ctor(x);
   if (dp === void 0) return x;
 
   checkInt32(dp, 0, MAX_DIGITS);
 
   if (rm === void 0) rm = Ctor.rounding;
   else checkInt32(rm, 0, 8);
 
   return finalise(x, dp + x.e + 1, rm);
 };
 
 
 /*
  * Return a string representing the value of this Decimal in exponential notation rounded to
  * `dp` fixed decimal places using rounding mode `rounding`.
  *
  * [dp] {number} Decimal places. Integer, 0 to MAX_DIGITS inclusive.
  * [rm] {number} Rounding mode. Integer, 0 to 8 inclusive.
  *
  */
 P.toExponential = function (dp, rm) {
   var str,
     x = this,
     Ctor = x.constructor;
 
   if (dp === void 0) {
     str = finiteToString(x, true);
   } else {
     checkInt32(dp, 0, MAX_DIGITS);
 
     if (rm === void 0) rm = Ctor.rounding;
     else checkInt32(rm, 0, 8);
 
     x = finalise(new Ctor(x), dp + 1, rm);
     str = finiteToString(x, true, dp + 1);
   }
 
   return x.isNeg() && !x.isZero() ? '-' + str : str;
 };
 
 
 /*
  * Return a string representing the value of this Decimal in normal (fixed-point) notation to
  * `dp` fixed decimal places and rounded using rounding mode `rm` or `rounding` if `rm` is
  * omitted.
  *
  * As with JavaScript numbers, (-0).toFixed(0) is '0', but e.g. (-0.00001).toFixed(0) is '-0'.
  *
  * [dp] {number} Decimal places. Integer, 0 to MAX_DIGITS inclusive.
  * [rm] {number} Rounding mode. Integer, 0 to 8 inclusive.
  *
  * (-0).toFixed(0) is '0', but (-0.1).toFixed(0) is '-0'.
  * (-0).toFixed(1) is '0.0', but (-0.01).toFixed(1) is '-0.0'.
  * (-0).toFixed(3) is '0.000'.
  * (-0.5).toFixed(0) is '-0'.
  *
  */
 P.toFixed = function (dp, rm) {
   var str, y,
     x = this,
     Ctor = x.constructor;
 
   if (dp === void 0) {
     str = finiteToString(x);
   } else {
     checkInt32(dp, 0, MAX_DIGITS);
 
     if (rm === void 0) rm = Ctor.rounding;
     else checkInt32(rm, 0, 8);
 
     y = finalise(new Ctor(x), dp + x.e + 1, rm);
     str = finiteToString(y, false, dp + y.e + 1);
   }
 
   // To determine whether to add the minus sign look at the value before it was rounded,
   // i.e. look at `x` rather than `y`.
   return x.isNeg() && !x.isZero() ? '-' + str : str;
 };
 
 
 /*
  * Return an array representing the value of this Decimal as a simple fraction with an integer
  * numerator and an integer denominator.
  *
  * The denominator will be a positive non-zero value less than or equal to the specified maximum
  * denominator. If a maximum denominator is not specified, the denominator will be the lowest
  * value necessary to represent the number exactly.
  *
  * [maxD] {number|string|Decimal} Maximum denominator. Integer >= 1 and < Infinity.
  *
  */
 P.toFraction = function (maxD) {
   var d, d0, d1, d2, e, k, n, n0, n1, pr, q, r,
     x = this,
     xd = x.d,
     Ctor = x.constructor;
 
   if (!xd) return new Ctor(x);
 
   n1 = d0 = new Ctor(1);
   d1 = n0 = new Ctor(0);
 
   d = new Ctor(d1);
   e = d.e = getPrecision(xd) - x.e - 1;
   k = e % LOG_BASE;
   d.d[0] = mathpow(10, k < 0 ? LOG_BASE + k : k);
 
   if (maxD == null) {
 
     // d is 10**e, the minimum max-denominator needed.
     maxD = e > 0 ? d : n1;
   } else {
     n = new Ctor(maxD);
     if (!n.isInt() || n.lt(n1)) throw Error(invalidArgument + n);
     maxD = n.gt(d) ? (e > 0 ? d : n1) : n;
   }
 
   external = false;
   n = new Ctor(digitsToString(xd));
   pr = Ctor.precision;
   Ctor.precision = e = xd.length * LOG_BASE * 2;
 
   for (;;)  {
     q = divide(n, d, 0, 1, 1);
     d2 = d0.plus(q.times(d1));
     if (d2.cmp(maxD) == 1) break;
     d0 = d1;
     d1 = d2;
     d2 = n1;
     n1 = n0.plus(q.times(d2));
     n0 = d2;
     d2 = d;
     d = n.minus(q.times(d2));
     n = d2;
   }
 
   d2 = divide(maxD.minus(d0), d1, 0, 1, 1);
   n0 = n0.plus(d2.times(n1));
   d0 = d0.plus(d2.times(d1));
   n0.s = n1.s = x.s;
 
   // Determine which fraction is closer to x, n0/d0 or n1/d1?
   r = divide(n1, d1, e, 1).minus(x).abs().cmp(divide(n0, d0, e, 1).minus(x).abs()) < 1
       ? [n1, d1] : [n0, d0];
 
   Ctor.precision = pr;
   external = true;
 
   return r;
 };
 
 
 /*
  * Return a string representing the value of this Decimal in base 16, round to `sd` significant
  * digits using rounding mode `rm`.
  *
  * If the optional `sd` argument is present then return binary exponential notation.
  *
  * [sd] {number} Significant digits. Integer, 1 to MAX_DIGITS inclusive.
  * [rm] {number} Rounding mode. Integer, 0 to 8 inclusive.
  *
  */
 P.toHexadecimal = P.toHex = function (sd, rm) {
   return toStringBinary(this, 16, sd, rm);
 };
 
 
 /*
  * Returns a new Decimal whose value is the nearest multiple of `y` in the direction of rounding
  * mode `rm`, or `Decimal.rounding` if `rm` is omitted, to the value of this Decimal.
  *
  * The return value will always have the same sign as this Decimal, unless either this Decimal
  * or `y` is NaN, in which case the return value will be also be NaN.
  *
  * The return value is not affected by the value of `precision`.
  *
  * y {number|string|Decimal} The magnitude to round to a multiple of.
  * [rm] {number} Rounding mode. Integer, 0 to 8 inclusive.
  *
  * 'toNearest() rounding mode not an integer: {rm}'
  * 'toNearest() rounding mode out of range: {rm}'
  *
  */
 P.toNearest = function (y, rm) {
   var x = this,
     Ctor = x.constructor;
 
   x = new Ctor(x);
 
   if (y == null) {
 
     // If x is not finite, return x.
     if (!x.d) return x;
 
     y = new Ctor(1);
     rm = Ctor.rounding;
   } else {
     y = new Ctor(y);
     if (rm === void 0) {
       rm = Ctor.rounding;
     } else {
       checkInt32(rm, 0, 8);
     }
 
     // If x is not finite, return x if y is not NaN, else NaN.
     if (!x.d) return y.s ? x : y;
 
     // If y is not finite, return Infinity with the sign of x if y is Infinity, else NaN.
     if (!y.d) {
       if (y.s) y.s = x.s;
       return y;
     }
   }
 
   // If y is not zero, calculate the nearest multiple of y to x.
   if (y.d[0]) {
     external = false;
     x = divide(x, y, 0, rm, 1).times(y);
     external = true;
     finalise(x);
 
   // If y is zero, return zero with the sign of x.
   } else {
     y.s = x.s;
     x = y;
   }
 
   return x;
 };
 
 
 /*
  * Return the value of this Decimal converted to a number primitive.
  * Zero keeps its sign.
  *
  */
 P.toNumber = function () {
   return +this;
 };
 
 
 /*
  * Return a string representing the value of this Decimal in base 8, round to `sd` significant
  * digits using rounding mode `rm`.
  *
  * If the optional `sd` argument is present then return binary exponential notation.
  *
  * [sd] {number} Significant digits. Integer, 1 to MAX_DIGITS inclusive.
  * [rm] {number} Rounding mode. Integer, 0 to 8 inclusive.
  *
  */
 P.toOctal = function (sd, rm) {
   return toStringBinary(this, 8, sd, rm);
 };
 
 
 /*
  * Return a new Decimal whose value is the value of this Decimal raised to the power `y`, rounded
  * to `precision` significant digits using rounding mode `rounding`.
  *
  * ECMAScript compliant.
  *
  *   pow(x, NaN)                           = NaN
  *   pow(x, ±0)                            = 1
 
  *   pow(NaN, non-zero)                    = NaN
  *   pow(abs(x) > 1, +Infinity)            = +Infinity
  *   pow(abs(x) > 1, -Infinity)            = +0
  *   pow(abs(x) == 1, ±Infinity)           = NaN
  *   pow(abs(x) < 1, +Infinity)            = +0
  *   pow(abs(x) < 1, -Infinity)            = +Infinity
  *   pow(+Infinity, y > 0)                 = +Infinity
  *   pow(+Infinity, y < 0)                 = +0
  *   pow(-Infinity, odd integer > 0)       = -Infinity
  *   pow(-Infinity, even integer > 0)      = +Infinity
  *   pow(-Infinity, odd integer < 0)       = -0
  *   pow(-Infinity, even integer < 0)      = +0
  *   pow(+0, y > 0)                        = +0
  *   pow(+0, y < 0)                        = +Infinity
  *   pow(-0, odd integer > 0)              = -0
  *   pow(-0, even integer > 0)             = +0
  *   pow(-0, odd integer < 0)              = -Infinity
  *   pow(-0, even integer < 0)             = +Infinity
  *   pow(finite x < 0, finite non-integer) = NaN
  *
  * For non-integer or very large exponents pow(x, y) is calculated using
  *
  *   x^y = exp(y*ln(x))
  *
  * Assuming the first 15 rounding digits are each equally likely to be any digit 0-9, the
  * probability of an incorrectly rounded result
  * P([49]9{14} | [50]0{14}) = 2 * 0.2 * 10^-14 = 4e-15 = 1/2.5e+14
  * i.e. 1 in 250,000,000,000,000
  *
  * If a result is incorrectly rounded the maximum error will be 1 ulp (unit in last place).
  *
  * y {number|string|Decimal} The power to which to raise this Decimal.
  *
  */
 P.toPower = P.pow = function (y) {
   var e, k, pr, r, rm, s,
     x = this,
     Ctor = x.constructor,
     yn = +(y = new Ctor(y));
 
   // Either ±Infinity, NaN or ±0?
   if (!x.d || !y.d || !x.d[0] || !y.d[0]) return new Ctor(mathpow(+x, yn));
 
   x = new Ctor(x);
 
   if (x.eq(1)) return x;
 
   pr = Ctor.precision;
   rm = Ctor.rounding;
 
   if (y.eq(1)) return finalise(x, pr, rm);
 
   // y exponent
   e = mathfloor(y.e / LOG_BASE);
 
   // If y is a small integer use the 'exponentiation by squaring' algorithm.
   if (e >= y.d.length - 1 && (k = yn < 0 ? -yn : yn) <= MAX_SAFE_INTEGER) {
     r = intPow(Ctor, x, k, pr);
     return y.s < 0 ? new Ctor(1).div(r) : finalise(r, pr, rm);
   }
 
   s = x.s;
 
   // if x is negative
   if (s < 0) {
 
     // if y is not an integer
     if (e < y.d.length - 1) return new Ctor(NaN);
 
     // Result is positive if x is negative and the last digit of integer y is even.
     if ((y.d[e] & 1) == 0) s = 1;
 
     // if x.eq(-1)
     if (x.e == 0 && x.d[0] == 1 && x.d.length == 1) {
       x.s = s;
       return x;
     }
   }
 
   // Estimate result exponent.
   // x^y = 10^e,  where e = y * log10(x)
   // log10(x) = log10(x_significand) + x_exponent
   // log10(x_significand) = ln(x_significand) / ln(10)
   k = mathpow(+x, yn);
   e = k == 0 || !isFinite(k)
     ? mathfloor(yn * (Math.log('0.' + digitsToString(x.d)) / Math.LN10 + x.e + 1))
     : new Ctor(k + '').e;
 
   // Exponent estimate may be incorrect e.g. x: 0.999999999999999999, y: 2.29, e: 0, r.e: -1.
 
   // Overflow/underflow?
   if (e > Ctor.maxE + 1 || e < Ctor.minE - 1) return new Ctor(e > 0 ? s / 0 : 0);
 
   external = false;
   Ctor.rounding = x.s = 1;
 
   // Estimate the extra guard digits needed to ensure five correct rounding digits from
   // naturalLogarithm(x). Example of failure without these extra digits (precision: 10):
   // new Decimal(2.32456).pow('2087987436534566.46411')
   // should be 1.162377823e+764914905173815, but is 1.162355823e+764914905173815
   k = Math.min(12, (e + '').length);
 
   // r = x^y = exp(y*ln(x))
   r = naturalExponential(y.times(naturalLogarithm(x, pr + k)), pr);
 
   // r may be Infinity, e.g. (0.9999999999999999).pow(-1e+40)
   if (r.d) {
 
     // Truncate to the required precision plus five rounding digits.
     r = finalise(r, pr + 5, 1);
 
     // If the rounding digits are [49]9999 or [50]0000 increase the precision by 10 and recalculate
     // the result.
     if (checkRoundingDigits(r.d, pr, rm)) {
       e = pr + 10;
 
       // Truncate to the increased precision plus five rounding digits.
       r = finalise(naturalExponential(y.times(naturalLogarithm(x, e + k)), e), e + 5, 1);
 
       // Check for 14 nines from the 2nd rounding digit (the first rounding digit may be 4 or 9).
       if (+digitsToString(r.d).slice(pr + 1, pr + 15) + 1 == 1e14) {
         r = finalise(r, pr + 1, 0);
       }
     }
   }
 
   r.s = s;
   external = true;
   Ctor.rounding = rm;
 
   return finalise(r, pr, rm);
 };
 
 
 /*
  * Return a string representing the value of this Decimal rounded to `sd` significant digits
  * using rounding mode `rounding`.
  *
  * Return exponential notation if `sd` is less than the number of digits necessary to represent
  * the integer part of the value in normal notation.
  *
  * [sd] {number} Significant digits. Integer, 1 to MAX_DIGITS inclusive.
  * [rm] {number} Rounding mode. Integer, 0 to 8 inclusive.
  *
  */
 P.toPrecision = function (sd, rm) {
   var str,
     x = this,
     Ctor = x.constructor;
 
   if (sd === void 0) {
     str = finiteToString(x, x.e <= Ctor.toExpNeg || x.e >= Ctor.toExpPos);
   } else {
     checkInt32(sd, 1, MAX_DIGITS);
 
     if (rm === void 0) rm = Ctor.rounding;
     else checkInt32(rm, 0, 8);
 
     x = finalise(new Ctor(x), sd, rm);
     str = finiteToString(x, sd <= x.e || x.e <= Ctor.toExpNeg, sd);
   }
 
   return x.isNeg() && !x.isZero() ? '-' + str : str;
 };
 
 
 /*
  * Return a new Decimal whose value is the value of this Decimal rounded to a maximum of `sd`
  * significant digits using rounding mode `rm`, or to `precision` and `rounding` respectively if
  * omitted.
  *
  * [sd] {number} Significant digits. Integer, 1 to MAX_DIGITS inclusive.
  * [rm] {number} Rounding mode. Integer, 0 to 8 inclusive.
  *
  * 'toSD() digits out of range: {sd}'
  * 'toSD() digits not an integer: {sd}'
  * 'toSD() rounding mode not an integer: {rm}'
  * 'toSD() rounding mode out of range: {rm}'
  *
  */
 P.toSignificantDigits = P.toSD = function (sd, rm) {
   var x = this,
     Ctor = x.constructor;
 
   if (sd === void 0) {
     sd = Ctor.precision;
     rm = Ctor.rounding;
   } else {
     checkInt32(sd, 1, MAX_DIGITS);
 
     if (rm === void 0) rm = Ctor.rounding;
     else checkInt32(rm, 0, 8);
   }
 
   return finalise(new Ctor(x), sd, rm);
 };
 
 
 /*
  * Return a string representing the value of this Decimal.
  *
  * Return exponential notation if this Decimal has a positive exponent equal to or greater than
  * `toExpPos`, or a negative exponent equal to or less than `toExpNeg`.
  *
  */
 P.toString = function () {
   var x = this,
     Ctor = x.constructor,
     str = finiteToString(x, x.e <= Ctor.toExpNeg || x.e >= Ctor.toExpPos);
 
   return x.isNeg() && !x.isZero() ? '-' + str : str;
 };
 
 
 /*
  * Return a new Decimal whose value is the value of this Decimal truncated to a whole number.
  *
  */
 P.truncated = P.trunc = function () {
   return finalise(new this.constructor(this), this.e + 1, 1);
 };
 
 
 /*
  * Return a string representing the value of this Decimal.
  * Unlike `toString`, negative zero will include the minus sign.
  *
  */
 P.valueOf = P.toJSON = function () {
   var x = this,
     Ctor = x.constructor,
     str = finiteToString(x, x.e <= Ctor.toExpNeg || x.e >= Ctor.toExpPos);
 
   return x.isNeg() ? '-' + str : str;
 };
 
 
 // Helper functions for Decimal.prototype (P) and/or Decimal methods, and their callers.
 
 
 /*
  *  digitsToString           P.cubeRoot, P.logarithm, P.squareRoot, P.toFraction, P.toPower,
  *                           finiteToString, naturalExponential, naturalLogarithm
  *  checkInt32               P.toDecimalPlaces, P.toExponential, P.toFixed, P.toNearest,
  *                           P.toPrecision, P.toSignificantDigits, toStringBinary, random
  *  checkRoundingDigits      P.logarithm, P.toPower, naturalExponential, naturalLogarithm
  *  convertBase              toStringBinary, parseOther
  *  cos                      P.cos
  *  divide                   P.atanh, P.cubeRoot, P.dividedBy, P.dividedToIntegerBy,
  *                           P.logarithm, P.modulo, P.squareRoot, P.tan, P.tanh, P.toFraction,
  *                           P.toNearest, toStringBinary, naturalExponential, naturalLogarithm,
  *                           taylorSeries, atan2, parseOther
  *  finalise                 P.absoluteValue, P.atan, P.atanh, P.ceil, P.cos, P.cosh,
  *                           P.cubeRoot, P.dividedToIntegerBy, P.floor, P.logarithm, P.minus,
  *                           P.modulo, P.negated, P.plus, P.round, P.sin, P.sinh, P.squareRoot,
  *                           P.tan, P.times, P.toDecimalPlaces, P.toExponential, P.toFixed,
  *                           P.toNearest, P.toPower, P.toPrecision, P.toSignificantDigits,
  *                           P.truncated, divide, getLn10, getPi, naturalExponential,
  *                           naturalLogarithm, ceil, floor, round, trunc
  *  finiteToString           P.toExponential, P.toFixed, P.toPrecision, P.toString, P.valueOf,
  *                           toStringBinary
  *  getBase10Exponent        P.minus, P.plus, P.times, parseOther
  *  getLn10                  P.logarithm, naturalLogarithm
  *  getPi                    P.acos, P.asin, P.atan, toLessThanHalfPi, atan2
  *  getPrecision             P.precision, P.toFraction
  *  getZeroString            digitsToString, finiteToString
  *  intPow                   P.toPower, parseOther
  *  isOdd                    toLessThanHalfPi
  *  maxOrMin                 max, min
  *  naturalExponential       P.naturalExponential, P.toPower
  *  naturalLogarithm         P.acosh, P.asinh, P.atanh, P.logarithm, P.naturalLogarithm,
  *                           P.toPower, naturalExponential
  *  nonFiniteToString        finiteToString, toStringBinary
  *  parseDecimal             Decimal
  *  parseOther               Decimal
  *  sin                      P.sin
  *  taylorSeries             P.cosh, P.sinh, cos, sin
  *  toLessThanHalfPi         P.cos, P.sin
  *  toStringBinary           P.toBinary, P.toHexadecimal, P.toOctal
  *  truncate                 intPow
  *
  *  Throws:                  P.logarithm, P.precision, P.toFraction, checkInt32, getLn10, getPi,
  *                           naturalLogarithm, config, parseOther, random, Decimal
  */
 
 
 function digitsToString(d) {
   var i, k, ws,
     indexOfLastWord = d.length - 1,
     str = '',
     w = d[0];
 
   if (indexOfLastWord > 0) {
     str += w;
     for (i = 1; i < indexOfLastWord; i++) {
       ws = d[i] + '';
       k = LOG_BASE - ws.length;
       if (k) str += getZeroString(k);
       str += ws;
     }
 
     w = d[i];
     ws = w + '';
     k = LOG_BASE - ws.length;
     if (k) str += getZeroString(k);
   } else if (w === 0) {
     return '0';
   }
 
   // Remove trailing zeros of last w.
   for (; w % 10 === 0;) w /= 10;
 
   return str + w;
 }
 
 
 function checkInt32(i, min, max) {
   if (i !== ~~i || i < min || i > max) {
     throw Error(invalidArgument + i);
   }
 }
 
 
 /*
  * Check 5 rounding digits if `repeating` is null, 4 otherwise.
  * `repeating == null` if caller is `log` or `pow`,
  * `repeating != null` if caller is `naturalLogarithm` or `naturalExponential`.
  */
 function checkRoundingDigits(d, i, rm, repeating) {
   var di, k, r, rd;
 
   // Get the length of the first word of the array d.
   for (k = d[0]; k >= 10; k /= 10) --i;
 
   // Is the rounding digit in the first word of d?
   if (--i < 0) {
     i += LOG_BASE;
     di = 0;
   } else {
     di = Math.ceil((i + 1) / LOG_BASE);
     i %= LOG_BASE;
   }
 
   // i is the index (0 - 6) of the rounding digit.
   // E.g. if within the word 3487563 the first rounding digit is 5,
   // then i = 4, k = 1000, rd = 3487563 % 1000 = 563
   k = mathpow(10, LOG_BASE - i);
   rd = d[di] % k | 0;
 
   if (repeating == null) {
     if (i < 3) {
       if (i == 0) rd = rd / 100 | 0;
       else if (i == 1) rd = rd / 10 | 0;
       r = rm < 4 && rd == 99999 || rm > 3 && rd == 49999 || rd == 50000 || rd == 0;
     } else {
       r = (rm < 4 && rd + 1 == k || rm > 3 && rd + 1 == k / 2) &&
         (d[di + 1] / k / 100 | 0) == mathpow(10, i - 2) - 1 ||
           (rd == k / 2 || rd == 0) && (d[di + 1] / k / 100 | 0) == 0;
     }
   } else {
     if (i < 4) {
       if (i == 0) rd = rd / 1000 | 0;
       else if (i == 1) rd = rd / 100 | 0;
       else if (i == 2) rd = rd / 10 | 0;
       r = (repeating || rm < 4) && rd == 9999 || !repeating && rm > 3 && rd == 4999;
     } else {
       r = ((repeating || rm < 4) && rd + 1 == k ||
       (!repeating && rm > 3) && rd + 1 == k / 2) &&
         (d[di + 1] / k / 1000 | 0) == mathpow(10, i - 3) - 1;
     }
   }
 
   return r;
 }
 
 
 // Convert string of `baseIn` to an array of numbers of `baseOut`.
 // Eg. convertBase('255', 10, 16) returns [15, 15].
 // Eg. convertBase('ff', 16, 10) returns [2, 5, 5].
 function convertBase(str, baseIn, baseOut) {
   var j,
     arr = [0],
     arrL,
     i = 0,
     strL = str.length;
 
   for (; i < strL;) {
     for (arrL = arr.length; arrL--;) arr[arrL] *= baseIn;
     arr[0] += NUMERALS.indexOf(str.charAt(i++));
     for (j = 0; j < arr.length; j++) {
       if (arr[j] > baseOut - 1) {
         if (arr[j + 1] === void 0) arr[j + 1] = 0;
         arr[j + 1] += arr[j] / baseOut | 0;
         arr[j] %= baseOut;
       }
     }
   }
 
   return arr.reverse();
 }
 
 
 /*
  * cos(x) = 1 - x^2/2! + x^4/4! - ...
  * |x| < pi/2
  *
  */
 function cosine(Ctor, x) {
   var k, len, y;
 
   if (x.isZero()) return x;
 
   // Argument reduction: cos(4x) = 8*(cos^4(x) - cos^2(x)) + 1
   // i.e. cos(x) = 8*(cos^4(x/4) - cos^2(x/4)) + 1
 
   // Estimate the optimum number of times to use the argument reduction.
   len = x.d.length;
   if (len < 32) {
     k = Math.ceil(len / 3);
     y = (1 / tinyPow(4, k)).toString();
   } else {
     k = 16;
     y = '2.3283064365386962890625e-10';
   }
 
   Ctor.precision += k;
 
   x = taylorSeries(Ctor, 1, x.times(y), new Ctor(1));
 
   // Reverse argument reduction
   for (var i = k; i--;) {
     var cos2x = x.times(x);
     x = cos2x.times(cos2x).minus(cos2x).times(8).plus(1);
   }
 
   Ctor.precision -= k;
 
   return x;
 }
 
 
 /*
  * Perform division in the specified base.
  */
 var divide = (function () {
 
   // Assumes non-zero x and k, and hence non-zero result.
   function multiplyInteger(x, k, base) {
     var temp,
       carry = 0,
       i = x.length;
 
     for (x = x.slice(); i--;) {
       temp = x[i] * k + carry;
       x[i] = temp % base | 0;
       carry = temp / base | 0;
     }
 
     if (carry) x.unshift(carry);
 
     return x;
   }
 
   function compare(a, b, aL, bL) {
     var i, r;
 
     if (aL != bL) {
       r = aL > bL ? 1 : -1;
     } else {
       for (i = r = 0; i < aL; i++) {
         if (a[i] != b[i]) {
           r = a[i] > b[i] ? 1 : -1;
           break;
         }
       }
     }
 
     return r;
   }
 
   function subtract(a, b, aL, base) {
     var i = 0;
 
     // Subtract b from a.
     for (; aL--;) {
       a[aL] -= i;
       i = a[aL] < b[aL] ? 1 : 0;
       a[aL] = i * base + a[aL] - b[aL];
     }
 
     // Remove leading zeros.
     for (; !a[0] && a.length > 1;) a.shift();
   }
 
   return function (x, y, pr, rm, dp, base) {
     var cmp, e, i, k, logBase, more, prod, prodL, q, qd, rem, remL, rem0, sd, t, xi, xL, yd0,
       yL, yz,
       Ctor = x.constructor,
       sign = x.s == y.s ? 1 : -1,
       xd = x.d,
       yd = y.d;
 
     // Either NaN, Infinity or 0?
     if (!xd || !xd[0] || !yd || !yd[0]) {
 
       return new Ctor(// Return NaN if either NaN, or both Infinity or 0.
         !x.s || !y.s || (xd ? yd && xd[0] == yd[0] : !yd) ? NaN :
 
         // Return ±0 if x is 0 or y is ±Infinity, or return ±Infinity as y is 0.
         xd && xd[0] == 0 || !yd ? sign * 0 : sign / 0);
     }
 
     if (base) {
       logBase = 1;
       e = x.e - y.e;
     } else {
       base = BASE;
       logBase = LOG_BASE;
       e = mathfloor(x.e / logBase) - mathfloor(y.e / logBase);
     }
 
     yL = yd.length;
     xL = xd.length;
     q = new Ctor(sign);
     qd = q.d = [];
 
     // Result exponent may be one less than e.
     // The digit array of a Decimal from toStringBinary may have trailing zeros.
     for (i = 0; yd[i] == (xd[i] || 0); i++);
 
     if (yd[i] > (xd[i] || 0)) e--;
 
     if (pr == null) {
       sd = pr = Ctor.precision;
       rm = Ctor.rounding;
     } else if (dp) {
       sd = pr + (x.e - y.e) + 1;
     } else {
       sd = pr;
     }
 
     if (sd < 0) {
       qd.push(1);
       more = true;
     } else {
 
       // Convert precision in number of base 10 digits to base 1e7 digits.
       sd = sd / logBase + 2 | 0;
       i = 0;
 
       // divisor < 1e7
       if (yL == 1) {
         k = 0;
         yd = yd[0];
         sd++;
 
         // k is the carry.
         for (; (i < xL || k) && sd--; i++) {
           t = k * base + (xd[i] || 0);
           qd[i] = t / yd | 0;
           k = t % yd | 0;
         }
 
         more = k || i < xL;
 
       // divisor >= 1e7
       } else {
 
         // Normalise xd and yd so highest order digit of yd is >= base/2
         k = base / (yd[0] + 1) | 0;
 
         if (k > 1) {
           yd = multiplyInteger(yd, k, base);
           xd = multiplyInteger(xd, k, base);
           yL = yd.length;
           xL = xd.length;
         }
 
         xi = yL;
         rem = xd.slice(0, yL);
         remL = rem.length;
 
         // Add zeros to make remainder as long as divisor.
         for (; remL < yL;) rem[remL++] = 0;
 
         yz = yd.slice();
         yz.unshift(0);
         yd0 = yd[0];
 
         if (yd[1] >= base / 2) ++yd0;
 
         do {
           k = 0;
 
           // Compare divisor and remainder.
           cmp = compare(yd, rem, yL, remL);
 
           // If divisor < remainder.
           if (cmp < 0) {
 
             // Calculate trial digit, k.
             rem0 = rem[0];
             if (yL != remL) rem0 = rem0 * base + (rem[1] || 0);
 
             // k will be how many times the divisor goes into the current remainder.
             k = rem0 / yd0 | 0;
 
             //  Algorithm:
             //  1. product = divisor * trial digit (k)
             //  2. if product > remainder: product -= divisor, k--
             //  3. remainder -= product
             //  4. if product was < remainder at 2:
             //    5. compare new remainder and divisor
             //    6. If remainder > divisor: remainder -= divisor, k++
 
             if (k > 1) {
               if (k >= base) k = base - 1;
 
               // product = divisor * trial digit.
               prod = multiplyInteger(yd, k, base);
               prodL = prod.length;
               remL = rem.length;
 
               // Compare product and remainder.
               cmp = compare(prod, rem, prodL, remL);
 
               // product > remainder.
               if (cmp == 1) {
                 k--;
 
                 // Subtract divisor from product.
                 subtract(prod, yL < prodL ? yz : yd, prodL, base);
               }
             } else {
 
               // cmp is -1.
               // If k is 0, there is no need to compare yd and rem again below, so change cmp to 1
               // to avoid it. If k is 1 there is a need to compare yd and rem again below.
               if (k == 0) cmp = k = 1;
               prod = yd.slice();
             }
 
             prodL = prod.length;
             if (prodL < remL) prod.unshift(0);
 
             // Subtract product from remainder.
             subtract(rem, prod, remL, base);
 
             // If product was < previous remainder.
             if (cmp == -1) {
               remL = rem.length;
 
               // Compare divisor and new remainder.
               cmp = compare(yd, rem, yL, remL);
 
               // If divisor < new remainder, subtract divisor from remainder.
               if (cmp < 1) {
                 k++;
 
                 // Subtract divisor from remainder.
                 subtract(rem, yL < remL ? yz : yd, remL, base);
               }
             }
 
             remL = rem.length;
           } else if (cmp === 0) {
             k++;
             rem = [0];
           }    // if cmp === 1, k will be 0
 
           // Add the next digit, k, to the result array.
           qd[i++] = k;
 
           // Update the remainder.
           if (cmp && rem[0]) {
             rem[remL++] = xd[xi] || 0;
           } else {
             rem = [xd[xi]];
             remL = 1;
           }
 
         } while ((xi++ < xL || rem[0] !== void 0) && sd--);
 
         more = rem[0] !== void 0;
       }
 
       // Leading zero?
       if (!qd[0]) qd.shift();
     }
 
     // logBase is 1 when divide is being used for base conversion.
     if (logBase == 1) {
       q.e = e;
       inexact = more;
     } else {
 
       // To calculate q.e, first get the number of digits of qd[0].
       for (i = 1, k = qd[0]; k >= 10; k /= 10) i++;
       q.e = i + e * logBase - 1;
 
       finalise(q, dp ? pr + q.e + 1 : pr, rm, more);
     }
 
     return q;
   };
 })();
 
 
 /*
  * Round `x` to `sd` significant digits using rounding mode `rm`.
  * Check for over/under-flow.
  */
  function finalise(x, sd, rm, isTruncated) {
   var digits, i, j, k, rd, roundUp, w, xd, xdi,
     Ctor = x.constructor;
 
   // Don't round if sd is null or undefined.
   out: if (sd != null) {
     xd = x.d;
 
     // Infinity/NaN.
     if (!xd) return x;
 
     // rd: the rounding digit, i.e. the digit after the digit that may be rounded up.
     // w: the word of xd containing rd, a base 1e7 number.
     // xdi: the index of w within xd.
     // digits: the number of digits of w.
     // i: what would be the index of rd within w if all the numbers were 7 digits long (i.e. if
     // they had leading zeros)
     // j: if > 0, the actual index of rd within w (if < 0, rd is a leading zero).
 
     // Get the length of the first word of the digits array xd.
     for (digits = 1, k = xd[0]; k >= 10; k /= 10) digits++;
     i = sd - digits;
 
     // Is the rounding digit in the first word of xd?
     if (i < 0) {
       i += LOG_BASE;
       j = sd;
       w = xd[xdi = 0];
 
       // Get the rounding digit at index j of w.
       rd = w / mathpow(10, digits - j - 1) % 10 | 0;
     } else {
       xdi = Math.ceil((i + 1) / LOG_BASE);
       k = xd.length;
       if (xdi >= k) {
         if (isTruncated) {
 
           // Needed by `naturalExponential`, `naturalLogarithm` and `squareRoot`.
           for (; k++ <= xdi;) xd.push(0);
           w = rd = 0;
           digits = 1;
           i %= LOG_BASE;
           j = i - LOG_BASE + 1;
         } else {
           break out;
         }
       } else {
         w = k = xd[xdi];
 
         // Get the number of digits of w.
         for (digits = 1; k >= 10; k /= 10) digits++;
 
         // Get the index of rd within w.
         i %= LOG_BASE;
 
         // Get the index of rd within w, adjusted for leading zeros.
         // The number of leading zeros of w is given by LOG_BASE - digits.
         j = i - LOG_BASE + digits;
 
         // Get the rounding digit at index j of w.
         rd = j < 0 ? 0 : w / mathpow(10, digits - j - 1) % 10 | 0;
       }
     }
 
     // Are there any non-zero digits after the rounding digit?
     isTruncated = isTruncated || sd < 0 ||
       xd[xdi + 1] !== void 0 || (j < 0 ? w : w % mathpow(10, digits - j - 1));
 
     // The expression `w % mathpow(10, digits - j - 1)` returns all the digits of w to the right
     // of the digit at (left-to-right) index j, e.g. if w is 908714 and j is 2, the expression
     // will give 714.
 
     roundUp = rm < 4
       ? (rd || isTruncated) && (rm == 0 || rm == (x.s < 0 ? 3 : 2))
       : rd > 5 || rd == 5 && (rm == 4 || isTruncated || rm == 6 &&
 
         // Check whether the digit to the left of the rounding digit is odd.
         ((i > 0 ? j > 0 ? w / mathpow(10, digits - j) : 0 : xd[xdi - 1]) % 10) & 1 ||
           rm == (x.s < 0 ? 8 : 7));
 
     if (sd < 1 || !xd[0]) {
       xd.length = 0;
       if (roundUp) {
 
         // Convert sd to decimal places.
         sd -= x.e + 1;
 
         // 1, 0.1, 0.01, 0.001, 0.0001 etc.
         xd[0] = mathpow(10, (LOG_BASE - sd % LOG_BASE) % LOG_BASE);
         x.e = -sd || 0;
       } else {
 
         // Zero.
         xd[0] = x.e = 0;
       }
 
       return x;
     }
 
     // Remove excess digits.
     if (i == 0) {
       xd.length = xdi;
       k = 1;
       xdi--;
     } else {
       xd.length = xdi + 1;
       k = mathpow(10, LOG_BASE - i);
 
       // E.g. 56700 becomes 56000 if 7 is the rounding digit.
       // j > 0 means i > number of leading zeros of w.
       xd[xdi] = j > 0 ? (w / mathpow(10, digits - j) % mathpow(10, j) | 0) * k : 0;
     }
 
     if (roundUp) {
       for (;;) {
 
         // Is the digit to be rounded up in the first word of xd?
         if (xdi == 0) {
 
           // i will be the length of xd[0] before k is added.
           for (i = 1, j = xd[0]; j >= 10; j /= 10) i++;
           j = xd[0] += k;
           for (k = 1; j >= 10; j /= 10) k++;
 
           // if i != k the length has increased.
           if (i != k) {
             x.e++;
             if (xd[0] == BASE) xd[0] = 1;
           }
 
           break;
         } else {
           xd[xdi] += k;
           if (xd[xdi] != BASE) break;
           xd[xdi--] = 0;
           k = 1;
         }
       }
     }
 
     // Remove trailing zeros.
     for (i = xd.length; xd[--i] === 0;) xd.pop();
   }
 
   if (external) {
 
     // Overflow?
     if (x.e > Ctor.maxE) {
 
       // Infinity.
       x.d = null;
       x.e = NaN;
 
     // Underflow?
     } else if (x.e < Ctor.minE) {
 
       // Zero.
       x.e = 0;
       x.d = [0];
       // Ctor.underflow = true;
     } // else Ctor.underflow = false;
   }
 
   return x;
 }
 
 
 function finiteToString(x, isExp, sd) {
   if (!x.isFinite()) return nonFiniteToString(x);
   var k,
     e = x.e,
     str = digitsToString(x.d),
     len = str.length;
 
   if (isExp) {
     if (sd && (k = sd - len) > 0) {
       str = str.charAt(0) + '.' + str.slice(1) + getZeroString(k);
     } else if (len > 1) {
       str = str.charAt(0) + '.' + str.slice(1);
     }
 
     str = str + (x.e < 0 ? 'e' : 'e+') + x.e;
   } else if (e < 0) {
     str = '0.' + getZeroString(-e - 1) + str;
     if (sd && (k = sd - len) > 0) str += getZeroString(k);
   } else if (e >= len) {
     str += getZeroString(e + 1 - len);
     if (sd && (k = sd - e - 1) > 0) str = str + '.' + getZeroString(k);
   } else {
     if ((k = e + 1) < len) str = str.slice(0, k) + '.' + str.slice(k);
     if (sd && (k = sd - len) > 0) {
       if (e + 1 === len) str += '.';
       str += getZeroString(k);
     }
   }
 
   return str;
 }
 
 
 // Calculate the base 10 exponent from the base 1e7 exponent.
 function getBase10Exponent(digits, e) {
   var w = digits[0];
 
   // Add the number of digits of the first word of the digits array.
   for ( e *= LOG_BASE; w >= 10; w /= 10) e++;
   return e;
 }
 
 
 function getLn10(Ctor, sd, pr) {
   if (sd > LN10_PRECISION) {
 
     // Reset global state in case the exception is caught.
     external = true;
     if (pr) Ctor.precision = pr;
     throw Error(precisionLimitExceeded);
   }
   return finalise(new Ctor(LN10), sd, 1, true);
 }
 
 
 function getPi(Ctor, sd, rm) {
   if (sd > PI_PRECISION) throw Error(precisionLimitExceeded);
   return finalise(new Ctor(PI), sd, rm, true);
 }
 
 
 function getPrecision(digits) {
   var w = digits.length - 1,
     len = w * LOG_BASE + 1;
 
   w = digits[w];
 
   // If non-zero...
   if (w) {
 
     // Subtract the number of trailing zeros of the last word.
     for (; w % 10 == 0; w /= 10) len--;
 
     // Add the number of digits of the first word.
     for (w = digits[0]; w >= 10; w /= 10) len++;
   }
 
   return len;
 }
 
 
 function getZeroString(k) {
   var zs = '';
   for (; k--;) zs += '0';
   return zs;
 }
 
 
 /*
  * Return a new Decimal whose value is the value of Decimal `x` to the power `n`, where `n` is an
  * integer of type number.
  *
  * Implements 'exponentiation by squaring'. Called by `pow` and `parseOther`.
  *
  */
 function intPow(Ctor, x, n, pr) {
   var isTruncated,
     r = new Ctor(1),
 
     // Max n of 9007199254740991 takes 53 loop iterations.
     // Maximum digits array length; leaves [28, 34] guard digits.
     k = Math.ceil(pr / LOG_BASE + 4);
 
   external = false;
 
   for (;;) {
     if (n % 2) {
       r = r.times(x);
       if (truncate(r.d, k)) isTruncated = true;
     }
 
     n = mathfloor(n / 2);
     if (n === 0) {
 
       // To ensure correct rounding when r.d is truncated, increment the last word if it is zero.
       n = r.d.length - 1;
       if (isTruncated && r.d[n] === 0) ++r.d[n];
       break;
     }
 
     x = x.times(x);
     truncate(x.d, k);
   }
 
   external = true;
 
   return r;
 }
 
 
 function isOdd(n) {
   return n.d[n.d.length - 1] & 1;
 }
 
 
 /*
  * Handle `max` and `min`. `ltgt` is 'lt' or 'gt'.
  */
 function maxOrMin(Ctor, args, ltgt) {
   var y,
     x = new Ctor(args[0]),
     i = 0;
 
   for (; ++i < args.length;) {
     y = new Ctor(args[i]);
     if (!y.s) {
       x = y;
       break;
     } else if (x[ltgt](y)) {
       x = y;
     }
   }
 
   return x;
 }
 
 
 /*
  * Return a new Decimal whose value is the natural exponential of `x` rounded to `sd` significant
  * digits.
  *
  * Taylor/Maclaurin series.
  *
  * exp(x) = x^0/0! + x^1/1! + x^2/2! + x^3/3! + ...
  *
  * Argument reduction:
  *   Repeat x = x / 32, k += 5, until |x| < 0.1
  *   exp(x) = exp(x / 2^k)^(2^k)
  *
  * Previously, the argument was initially reduced by
  * exp(x) = exp(r) * 10^k  where r = x - k * ln10, k = floor(x / ln10)
  * to first put r in the range [0, ln10], before dividing by 32 until |x| < 0.1, but this was
  * found to be slower than just dividing repeatedly by 32 as above.
  *
  * Max integer argument: exp('20723265836946413') = 6.3e+9000000000000000
  * Min integer argument: exp('-20723265836946411') = 1.2e-9000000000000000
  * (Math object integer min/max: Math.exp(709) = 8.2e+307, Math.exp(-745) = 5e-324)
  *
  *  exp(Infinity)  = Infinity
  *  exp(-Infinity) = 0
  *  exp(NaN)       = NaN
  *  exp(±0)        = 1
  *
  *  exp(x) is non-terminating for any finite, non-zero x.
  *
  *  The result will always be correctly rounded.
  *
  */
 function naturalExponential(x, sd) {
   var denominator, guard, j, pow, sum, t, wpr,
     rep = 0,
     i = 0,
     k = 0,
     Ctor = x.constructor,
     rm = Ctor.rounding,
     pr = Ctor.precision;
 
   // 0/NaN/Infinity?
   if (!x.d || !x.d[0] || x.e > 17) {
 
     return new Ctor(x.d
       ? !x.d[0] ? 1 : x.s < 0 ? 0 : 1 / 0
       : x.s ? x.s < 0 ? 0 : x : 0 / 0);
   }
 
   if (sd == null) {
     external = false;
     wpr = pr;
   } else {
     wpr = sd;
   }
 
   t = new Ctor(0.03125);
 
   // while abs(x) >= 0.1
   while (x.e > -2) {
 
     // x = x / 2^5
     x = x.times(t);
     k += 5;
   }
 
   // Use 2 * log10(2^k) + 5 (empirically derived) to estimate the increase in precision
   // necessary to ensure the first 4 rounding digits are correct.
   guard = Math.log(mathpow(2, k)) / Math.LN10 * 2 + 5 | 0;
   wpr += guard;
   denominator = pow = sum = new Ctor(1);
   Ctor.precision = wpr;
 
   for (;;) {
     pow = finalise(pow.times(x), wpr, 1);
     denominator = denominator.times(++i);
     t = sum.plus(divide(pow, denominator, wpr, 1));
 
     if (digitsToString(t.d).slice(0, wpr) === digitsToString(sum.d).slice(0, wpr)) {
       j = k;
       while (j--) sum = finalise(sum.times(sum), wpr, 1);
 
       // Check to see if the first 4 rounding digits are [49]999.
       // If so, repeat the summation with a higher precision, otherwise
       // e.g. with precision: 18, rounding: 1
       // exp(18.404272462595034083567793919843761) = 98372560.1229999999 (should be 98372560.123)
       // `wpr - guard` is the index of first rounding digit.
       if (sd == null) {
 
         if (rep < 3 && checkRoundingDigits(sum.d, wpr - guard, rm, rep)) {
           Ctor.precision = wpr += 10;
           denominator = pow = t = new Ctor(1);
           i = 0;
           rep++;
         } else {
           return finalise(sum, Ctor.precision = pr, rm, external = true);
         }
       } else {
         Ctor.precision = pr;
         return sum;
       }
     }
 
     sum = t;
   }
 }
 
 
 /*
  * Return a new Decimal whose value is the natural logarithm of `x` rounded to `sd` significant
  * digits.
  *
  *  ln(-n)        = NaN
  *  ln(0)         = -Infinity
  *  ln(-0)        = -Infinity
  *  ln(1)         = 0
  *  ln(Infinity)  = Infinity
  *  ln(-Infinity) = NaN
  *  ln(NaN)       = NaN
  *
  *  ln(n) (n != 1) is non-terminating.
  *
  */
 function naturalLogarithm(y, sd) {
   var c, c0, denominator, e, numerator, rep, sum, t, wpr, x1, x2,
     n = 1,
     guard = 10,
     x = y,
     xd = x.d,
     Ctor = x.constructor,
     rm = Ctor.rounding,
     pr = Ctor.precision;
 
   // Is x negative or Infinity, NaN, 0 or 1?
   if (x.s < 0 || !xd || !xd[0] || !x.e && xd[0] == 1 && xd.length == 1) {
     return new Ctor(xd && !xd[0] ? -1 / 0 : x.s != 1 ? NaN : xd ? 0 : x);
   }
 
   if (sd == null) {
     external = false;
     wpr = pr;
   } else {
     wpr = sd;
   }
 
   Ctor.precision = wpr += guard;
   c = digitsToString(xd);
   c0 = c.charAt(0);
 
   if (Math.abs(e = x.e) < 1.5e15) {
 
     // Argument reduction.
     // The series converges faster the closer the argument is to 1, so using
     // ln(a^b) = b * ln(a),   ln(a) = ln(a^b) / b
     // multiply the argument by itself until the leading digits of the significand are 7, 8, 9,
     // 10, 11, 12 or 13, recording the number of multiplications so the sum of the series can
     // later be divided by this number, then separate out the power of 10 using
     // ln(a*10^b) = ln(a) + b*ln(10).
 
     // max n is 21 (gives 0.9, 1.0 or 1.1) (9e15 / 21 = 4.2e14).
     //while (c0 < 9 && c0 != 1 || c0 == 1 && c.charAt(1) > 1) {
     // max n is 6 (gives 0.7 - 1.3)
     while (c0 < 7 && c0 != 1 || c0 == 1 && c.charAt(1) > 3) {
       x = x.times(y);
       c = digitsToString(x.d);
       c0 = c.charAt(0);
       n++;
     }
 
     e = x.e;
 
     if (c0 > 1) {
       x = new Ctor('0.' + c);
       e++;
     } else {
       x = new Ctor(c0 + '.' + c.slice(1));
     }
   } else {
 
     // The argument reduction method above may result in overflow if the argument y is a massive
     // number with exponent >= 1500000000000000 (9e15 / 6 = 1.5e15), so instead recall this
     // function using ln(x*10^e) = ln(x) + e*ln(10).
     t = getLn10(Ctor, wpr + 2, pr).times(e + '');
     x = naturalLogarithm(new Ctor(c0 + '.' + c.slice(1)), wpr - guard).plus(t);
     Ctor.precision = pr;
 
     return sd == null ? finalise(x, pr, rm, external = true) : x;
   }
 
   // x1 is x reduced to a value near 1.
   x1 = x;
 
   // Taylor series.
   // ln(y) = ln((1 + x)/(1 - x)) = 2(x + x^3/3 + x^5/5 + x^7/7 + ...)
   // where x = (y - 1)/(y + 1)    (|x| < 1)
   sum = numerator = x = divide(x.minus(1), x.plus(1), wpr, 1);
   x2 = finalise(x.times(x), wpr, 1);
   denominator = 3;
 
   for (;;) {
     numerator = finalise(numerator.times(x2), wpr, 1);
     t = sum.plus(divide(numerator, new Ctor(denominator), wpr, 1));
 
     if (digitsToString(t.d).slice(0, wpr) === digitsToString(sum.d).slice(0, wpr)) {
       sum = sum.times(2);
 
       // Reverse the argument reduction. Check that e is not 0 because, besides preventing an
       // unnecessary calculation, -0 + 0 = +0 and to ensure correct rounding -0 needs to stay -0.
       if (e !== 0) sum = sum.plus(getLn10(Ctor, wpr + 2, pr).times(e + ''));
       sum = divide(sum, new Ctor(n), wpr, 1);
 
       // Is rm > 3 and the first 4 rounding digits 4999, or rm < 4 (or the summation has
       // been repeated previously) and the first 4 rounding digits 9999?
       // If so, restart the summation with a higher precision, otherwise
       // e.g. with precision: 12, rounding: 1
       // ln(135520028.6126091714265381533) = 18.7246299999 when it should be 18.72463.
       // `wpr - guard` is the index of first rounding digit.
       if (sd == null) {
         if (checkRoundingDigits(sum.d, wpr - guard, rm, rep)) {
           Ctor.precision = wpr += guard;
           t = numerator = x = divide(x1.minus(1), x1.plus(1), wpr, 1);
           x2 = finalise(x.times(x), wpr, 1);
           denominator = rep = 1;
         } else {
           return finalise(sum, Ctor.precision = pr, rm, external = true);
         }
       } else {
         Ctor.precision = pr;
         return sum;
       }
     }
 
     sum = t;
     denominator += 2;
   }
 }
 
 
 // ±Infinity, NaN.
 function nonFiniteToString(x) {
   // Unsigned.
   return String(x.s * x.s / 0);
 }
 
 
 /*
  * Parse the value of a new Decimal `x` from string `str`.
  */
 function parseDecimal(x, str) {
   var e, i, len;
 
   // Decimal point?
   if ((e = str.indexOf('.')) > -1) str = str.replace('.', '');
 
   // Exponential form?
   if ((i = str.search(/e/i)) > 0) {
 
     // Determine exponent.
     if (e < 0) e = i;
     e += +str.slice(i + 1);
     str = str.substring(0, i);
   } else if (e < 0) {
 
     // Integer.
     e = str.length;
   }
 
   // Determine leading zeros.
   for (i = 0; str.charCodeAt(i) === 48; i++);
 
   // Determine trailing zeros.
   for (len = str.length; str.charCodeAt(len - 1) === 48; --len);
   str = str.slice(i, len);
 
   if (str) {
     len -= i;
     x.e = e = e - i - 1;
     x.d = [];
 
     // Transform base
 
     // e is the base 10 exponent.
     // i is where to slice str to get the first word of the digits array.
     i = (e + 1) % LOG_BASE;
     if (e < 0) i += LOG_BASE;
 
     if (i < len) {
       if (i) x.d.push(+str.slice(0, i));
       for (len -= LOG_BASE; i < len;) x.d.push(+str.slice(i, i += LOG_BASE));
       str = str.slice(i);
       i = LOG_BASE - str.length;
     } else {
       i -= len;
     }
 
     for (; i--;) str += '0';
     x.d.push(+str);
 
     if (external) {
 
       // Overflow?
       if (x.e > x.constructor.maxE) {
 
         // Infinity.
         x.d = null;
         x.e = NaN;
 
       // Underflow?
       } else if (x.e < x.constructor.minE) {
 
         // Zero.
         x.e = 0;
         x.d = [0];
         // x.constructor.underflow = true;
       } // else x.constructor.underflow = false;
     }
   } else {
 
     // Zero.
     x.e = 0;
     x.d = [0];
   }
 
   return x;
 }
 
 
 /*
  * Parse the value of a new Decimal `x` from a string `str`, which is not a decimal value.
  */
 function parseOther(x, str) {
   var base, Ctor, divisor, i, isFloat, len, p, xd, xe;
 
   if (str.indexOf('_') > -1) {
     str = str.replace(/(\d)_(?=\d)/g, '$1');
     if (isDecimal.test(str)) return parseDecimal(x, str);
   } else if (str === 'Infinity' || str === 'NaN') {
     if (!+str) x.s = NaN;
     x.e = NaN;
     x.d = null;
     return x;
   }
 
   if (isHex.test(str))  {
     base = 16;
     str = str.toLowerCase();
   } else if (isBinary.test(str))  {
     base = 2;
   } else if (isOctal.test(str))  {
     base = 8;
   } else {
     throw Error(invalidArgument + str);
   }
 
   // Is there a binary exponent part?
   i = str.search(/p/i);
 
   if (i > 0) {
     p = +str.slice(i + 1);
     str = str.substring(2, i);
   } else {
     str = str.slice(2);
   }
 
   // Convert `str` as an integer then divide the result by `base` raised to a power such that the
   // fraction part will be restored.
   i = str.indexOf('.');
   isFloat = i >= 0;
   Ctor = x.constructor;
 
   if (isFloat) {
     str = str.replace('.', '');
     len = str.length;
     i = len - i;
 
     // log[10](16) = 1.2041... , log[10](88) = 1.9444....
     divisor = intPow(Ctor, new Ctor(base), i, i * 2);
   }
 
   xd = convertBase(str, base, BASE);
   xe = xd.length - 1;
 
   // Remove trailing zeros.
   for (i = xe; xd[i] === 0; --i) xd.pop();
   if (i < 0) return new Ctor(x.s * 0);
   x.e = getBase10Exponent(xd, xe);
   x.d = xd;
   external = false;
 
   // At what precision to perform the division to ensure exact conversion?
   // maxDecimalIntegerPartDigitCount = ceil(log[10](b) * otherBaseIntegerPartDigitCount)
   // log[10](2) = 0.30103, log[10](8) = 0.90309, log[10](16) = 1.20412
   // E.g. ceil(1.2 * 3) = 4, so up to 4 decimal digits are needed to represent 3 hex int digits.
   // maxDecimalFractionPartDigitCount = {Hex:4|Oct:3|Bin:1} * otherBaseFractionPartDigitCount
   // Therefore using 4 * the number of digits of str will always be enough.
   if (isFloat) x = divide(x, divisor, len * 4);
 
   // Multiply by the binary exponent part if present.
   if (p) x = x.times(Math.abs(p) < 54 ? mathpow(2, p) : Decimal.pow(2, p));
   external = true;
 
   return x;
 }
 
 
 /*
  * sin(x) = x - x^3/3! + x^5/5! - ...
  * |x| < pi/2
  *
  */
 function sine(Ctor, x) {
   var k,
     len = x.d.length;
 
   if (len < 3) {
     return x.isZero() ? x : taylorSeries(Ctor, 2, x, x);
   }
 
   // Argument reduction: sin(5x) = 16*sin^5(x) - 20*sin^3(x) + 5*sin(x)
   // i.e. sin(x) = 16*sin^5(x/5) - 20*sin^3(x/5) + 5*sin(x/5)
   // and  sin(x) = sin(x/5)(5 + sin^2(x/5)(16sin^2(x/5) - 20))
 
   // Estimate the optimum number of times to use the argument reduction.
   k = 1.4 * Math.sqrt(len);
   k = k > 16 ? 16 : k | 0;
 
   x = x.times(1 / tinyPow(5, k));
   x = taylorSeries(Ctor, 2, x, x);
 
   // Reverse argument reduction
   var sin2_x,
     d5 = new Ctor(5),
     d16 = new Ctor(16),
     d20 = new Ctor(20);
   for (; k--;) {
     sin2_x = x.times(x);
     x = x.times(d5.plus(sin2_x.times(d16.times(sin2_x).minus(d20))));
   }
 
   return x;
 }
 
 
 // Calculate Taylor series for `cos`, `cosh`, `sin` and `sinh`.
 function taylorSeries(Ctor, n, x, y, isHyperbolic) {
   var j, t, u, x2,
     i = 1,
     pr = Ctor.precision,
     k = Math.ceil(pr / LOG_BASE);
 
   external = false;
   x2 = x.times(x);
   u = new Ctor(y);
 
   for (;;) {
     t = divide(u.times(x2), new Ctor(n++ * n++), pr, 1);
     u = isHyperbolic ? y.plus(t) : y.minus(t);
     y = divide(t.times(x2), new Ctor(n++ * n++), pr, 1);
     t = u.plus(y);
 
     if (t.d[k] !== void 0) {
       for (j = k; t.d[j] === u.d[j] && j--;);
       if (j == -1) break;
     }
 
     j = u;
     u = y;
     y = t;
     t = j;
     i++;
   }
 
   external = true;
   t.d.length = k + 1;
 
   return t;
 }
 
 
 // Exponent e must be positive and non-zero.
 function tinyPow(b, e) {
   var n = b;
   while (--e) n *= b;
   return n;
 }
 
 
 // Return the absolute value of `x` reduced to less than or equal to half pi.
 function toLessThanHalfPi(Ctor, x) {
   var t,
     isNeg = x.s < 0,
     pi = getPi(Ctor, Ctor.precision, 1),
     halfPi = pi.times(0.5);
 
   x = x.abs();
 
   if (x.lte(halfPi)) {
     quadrant = isNeg ? 4 : 1;
     return x;
   }
 
   t = x.divToInt(pi);
 
   if (t.isZero()) {
     quadrant = isNeg ? 3 : 2;
   } else {
     x = x.minus(t.times(pi));
 
     // 0 <= x < pi
     if (x.lte(halfPi)) {
       quadrant = isOdd(t) ? (isNeg ? 2 : 3) : (isNeg ? 4 : 1);
       return x;
     }
 
     quadrant = isOdd(t) ? (isNeg ? 1 : 4) : (isNeg ? 3 : 2);
   }
 
   return x.minus(pi).abs();
 }
 
 
 /*
  * Return the value of Decimal `x` as a string in base `baseOut`.
  *
  * If the optional `sd` argument is present include a binary exponent suffix.
  */
 function toStringBinary(x, baseOut, sd, rm) {
   var base, e, i, k, len, roundUp, str, xd, y,
     Ctor = x.constructor,
     isExp = sd !== void 0;
 
   if (isExp) {
     checkInt32(sd, 1, MAX_DIGITS);
     if (rm === void 0) rm = Ctor.rounding;
     else checkInt32(rm, 0, 8);
   } else {
     sd = Ctor.precision;
     rm = Ctor.rounding;
   }
 
   if (!x.isFinite()) {
     str = nonFiniteToString(x);
   } else {
     str = finiteToString(x);
     i = str.indexOf('.');
 
     // Use exponential notation according to `toExpPos` and `toExpNeg`? No, but if required:
     // maxBinaryExponent = floor((decimalExponent + 1) * log[2](10))
     // minBinaryExponent = floor(decimalExponent * log[2](10))
     // log[2](10) = 3.321928094887362347870319429489390175864
 
     if (isExp) {
       base = 2;
       if (baseOut == 16) {
         sd = sd * 4 - 3;
       } else if (baseOut == 8) {
         sd = sd * 3 - 2;
       }
     } else {
       base = baseOut;
     }
 
     // Convert the number as an integer then divide the result by its base raised to a power such
     // that the fraction part will be restored.
 
     // Non-integer.
     if (i >= 0) {
       str = str.replace('.', '');
       y = new Ctor(1);
       y.e = str.length - i;
       y.d = convertBase(finiteToString(y), 10, base);
       y.e = y.d.length;
     }
 
     xd = convertBase(str, 10, base);
     e = len = xd.length;
 
     // Remove trailing zeros.
     for (; xd[--len] == 0;) xd.pop();
 
     if (!xd[0]) {
       str = isExp ? '0p+0' : '0';
     } else {
       if (i < 0) {
         e--;
       } else {
         x = new Ctor(x);
         x.d = xd;
         x.e = e;
         x = divide(x, y, sd, rm, 0, base);
         xd = x.d;
         e = x.e;
         roundUp = inexact;
       }
 
       // The rounding digit, i.e. the digit after the digit that may be rounded up.
       i = xd[sd];
       k = base / 2;
       roundUp = roundUp || xd[sd + 1] !== void 0;
 
       roundUp = rm < 4
         ? (i !== void 0 || roundUp) && (rm === 0 || rm === (x.s < 0 ? 3 : 2))
         : i > k || i === k && (rm === 4 || roundUp || rm === 6 && xd[sd - 1] & 1 ||
           rm === (x.s < 0 ? 8 : 7));
 
       xd.length = sd;
 
       if (roundUp) {
 
         // Rounding up may mean the previous digit has to be rounded up and so on.
         for (; ++xd[--sd] > base - 1;) {
           xd[sd] = 0;
           if (!sd) {
             ++e;
             xd.unshift(1);
           }
         }
       }
 
       // Determine trailing zeros.
       for (len = xd.length; !xd[len - 1]; --len);
 
       // E.g. [4, 11, 15] becomes 4bf.
       for (i = 0, str = ''; i < len; i++) str += NUMERALS.charAt(xd[i]);
 
       // Add binary exponent suffix?
       if (isExp) {
         if (len > 1) {
           if (baseOut == 16 || baseOut == 8) {
             i = baseOut == 16 ? 4 : 3;
             for (--len; len % i; len++) str += '0';
             xd = convertBase(str, base, baseOut);
             for (len = xd.length; !xd[len - 1]; --len);
 
             // xd[0] will always be be 1
             for (i = 1, str = '1.'; i < len; i++) str += NUMERALS.charAt(xd[i]);
           } else {
             str = str.charAt(0) + '.' + str.slice(1);
           }
         }
 
         str =  str + (e < 0 ? 'p' : 'p+') + e;
       } else if (e < 0) {
         for (; ++e;) str = '0' + str;
         str = '0.' + str;
       } else {
         if (++e > len) for (e -= len; e-- ;) str += '0';
         else if (e < len) str = str.slice(0, e) + '.' + str.slice(e);
       }
     }
 
     str = (baseOut == 16 ? '0x' : baseOut == 2 ? '0b' : baseOut == 8 ? '0o' : '') + str;
   }
 
   return x.s < 0 ? '-' + str : str;
 }
 
 
 // Does not strip trailing zeros.
 function truncate(arr, len) {
   if (arr.length > len) {
     arr.length = len;
     return true;
   }
 }
 
 
 // Decimal methods
 
 
 /*
  *  abs
  *  acos
  *  acosh
  *  add
  *  asin
  *  asinh
  *  atan
  *  atanh
  *  atan2
  *  cbrt
  *  ceil
  *  clamp
  *  clone
  *  config
  *  cos
  *  cosh
  *  div
  *  exp
  *  floor
  *  hypot
  *  ln
  *  log
  *  log2
  *  log10
  *  max
  *  min
  *  mod
  *  mul
  *  pow
  *  random
  *  round
  *  set
  *  sign
  *  sin
  *  sinh
  *  sqrt
  *  sub
  *  sum
  *  tan
  *  tanh
  *  trunc
  */
 
 
 /*
  * Return a new Decimal whose value is the absolute value of `x`.
  *
  * x {number|string|Decimal}
  *
  */
 function abs(x) {
   return new this(x).abs();
 }
 
 
 /*
  * Return a new Decimal whose value is the arccosine in radians of `x`.
  *
  * x {number|string|Decimal}
  *
  */
 function acos(x) {
   return new this(x).acos();
 }
 
 
 /*
  * Return a new Decimal whose value is the inverse of the hyperbolic cosine of `x`, rounded to
  * `precision` significant digits using rounding mode `rounding`.
  *
  * x {number|string|Decimal} A value in radians.
  *
  */
 function acosh(x) {
   return new this(x).acosh();
 }
 
 
 /*
  * Return a new Decimal whose value is the sum of `x` and `y`, rounded to `precision` significant
  * digits using rounding mode `rounding`.
  *
  * x {number|string|Decimal}
  * y {number|string|Decimal}
  *
  */
 function add(x, y) {
   return new this(x).plus(y);
 }
 
 
 /*
  * Return a new Decimal whose value is the arcsine in radians of `x`, rounded to `precision`
  * significant digits using rounding mode `rounding`.
  *
  * x {number|string|Decimal}
  *
  */
 function asin(x) {
   return new this(x).asin();
 }
 
 
 /*
  * Return a new Decimal whose value is the inverse of the hyperbolic sine of `x`, rounded to
  * `precision` significant digits using rounding mode `rounding`.
  *
  * x {number|string|Decimal} A value in radians.
  *
  */
 function asinh(x) {
   return new this(x).asinh();
 }
 
 
 /*
  * Return a new Decimal whose value is the arctangent in radians of `x`, rounded to `precision`
  * significant digits using rounding mode `rounding`.
  *
  * x {number|string|Decimal}
  *
  */
 function atan(x) {
   return new this(x).atan();
 }
 
 
 /*
  * Return a new Decimal whose value is the inverse of the hyperbolic tangent of `x`, rounded to
  * `precision` significant digits using rounding mode `rounding`.
  *
  * x {number|string|Decimal} A value in radians.
  *
  */
 function atanh(x) {
   return new this(x).atanh();
 }
 
 
 /*
  * Return a new Decimal whose value is the arctangent in radians of `y/x` in the range -pi to pi
  * (inclusive), rounded to `precision` significant digits using rounding mode `rounding`.
  *
  * Domain: [-Infinity, Infinity]
  * Range: [-pi, pi]
  *
  * y {number|string|Decimal} The y-coordinate.
  * x {number|string|Decimal} The x-coordinate.
  *
  * atan2(±0, -0)               = ±pi
  * atan2(±0, +0)               = ±0
  * atan2(±0, -x)               = ±pi for x > 0
  * atan2(±0, x)                = ±0 for x > 0
  * atan2(-y, ±0)               = -pi/2 for y > 0
  * atan2(y, ±0)                = pi/2 for y > 0
  * atan2(±y, -Infinity)        = ±pi for finite y > 0
  * atan2(±y, +Infinity)        = ±0 for finite y > 0
  * atan2(±Infinity, x)         = ±pi/2 for finite x
  * atan2(±Infinity, -Infinity) = ±3*pi/4
  * atan2(±Infinity, +Infinity) = ±pi/4
  * atan2(NaN, x) = NaN
  * atan2(y, NaN) = NaN
  *
  */
 function atan2(y, x) {
   y = new this(y);
   x = new this(x);
   var r,
     pr = this.precision,
     rm = this.rounding,
     wpr = pr + 4;
 
   // Either NaN
   if (!y.s || !x.s) {
     r = new this(NaN);
 
   // Both ±Infinity
   } else if (!y.d && !x.d) {
     r = getPi(this, wpr, 1).times(x.s > 0 ? 0.25 : 0.75);
     r.s = y.s;
 
   // x is ±Infinity or y is ±0
   } else if (!x.d || y.isZero()) {
     r = x.s < 0 ? getPi(this, pr, rm) : new this(0);
     r.s = y.s;
 
   // y is ±Infinity or x is ±0
   } else if (!y.d || x.isZero()) {
     r = getPi(this, wpr, 1).times(0.5);
     r.s = y.s;
 
   // Both non-zero and finite
   } else if (x.s < 0) {
     this.precision = wpr;
     this.rounding = 1;
     r = this.atan(divide(y, x, wpr, 1));
     x = getPi(this, wpr, 1);
     this.precision = pr;
     this.rounding = rm;
     r = y.s < 0 ? r.minus(x) : r.plus(x);
   } else {
     r = this.atan(divide(y, x, wpr, 1));
   }
 
   return r;
 }
 
 
 /*
  * Return a new Decimal whose value is the cube root of `x`, rounded to `precision` significant
  * digits using rounding mode `rounding`.
  *
  * x {number|string|Decimal}
  *
  */
 function cbrt(x) {
   return new this(x).cbrt();
 }
 
 
 /*
  * Return a new Decimal whose value is `x` rounded to an integer using `ROUND_CEIL`.
  *
  * x {number|string|Decimal}
  *
  */
 function ceil(x) {
   return finalise(x = new this(x), x.e + 1, 2);
 }
 
 
 /*
  * Return a new Decimal whose value is `x` clamped to the range delineated by `min` and `max`.
  *
  * x {number|string|Decimal}
  * min {number|string|Decimal}
  * max {number|string|Decimal}
  *
  */
 function clamp(x, min, max) {
   return new this(x).clamp(min, max);
 }
 
 
 /*
  * Configure global settings for a Decimal constructor.
  *
  * `obj` is an object with one or more of the following properties,
  *
  *   precision  {number}
  *   rounding   {number}
  *   toExpNeg   {number}
  *   toExpPos   {number}
  *   maxE       {number}
  *   minE       {number}
  *   modulo     {number}
  *   crypto     {boolean|number}
  *   defaults   {true}
  *
  * E.g. Decimal.config({ precision: 20, rounding: 4 })
  *
  */
 function config(obj) {
   if (!obj || typeof obj !== 'object') throw Error(decimalError + 'Object expected');
   var i, p, v,
     useDefaults = obj.defaults === true,
     ps = [
       'precision', 1, MAX_DIGITS,
       'rounding', 0, 8,
       'toExpNeg', -EXP_LIMIT, 0,
       'toExpPos', 0, EXP_LIMIT,
       'maxE', 0, EXP_LIMIT,
       'minE', -EXP_LIMIT, 0,
       'modulo', 0, 9
     ];
 
   for (i = 0; i < ps.length; i += 3) {
     if (p = ps[i], useDefaults) this[p] = DEFAULTS[p];
     if ((v = obj[p]) !== void 0) {
       if (mathfloor(v) === v && v >= ps[i + 1] && v <= ps[i + 2]) this[p] = v;
       else throw Error(invalidArgument + p + ': ' + v);
     }
   }
 
   if (p = 'crypto', useDefaults) this[p] = DEFAULTS[p];
   if ((v = obj[p]) !== void 0) {
     if (v === true || v === false || v === 0 || v === 1) {
       if (v) {
         if (typeof crypto != 'undefined' && crypto &&
           (crypto.getRandomValues || crypto.randomBytes)) {
           this[p] = true;
         } else {
           throw Error(cryptoUnavailable);
         }
       } else {
         this[p] = false;
       }
     } else {
       throw Error(invalidArgument + p + ': ' + v);
     }
   }
 
   return this;
 }
 
 
 /*
  * Return a new Decimal whose value is the cosine of `x`, rounded to `precision` significant
  * digits using rounding mode `rounding`.
  *
  * x {number|string|Decimal} A value in radians.
  *
  */
 function cos(x) {
   return new this(x).cos();
 }
 
 
 /*
  * Return a new Decimal whose value is the hyperbolic cosine of `x`, rounded to precision
  * significant digits using rounding mode `rounding`.
  *
  * x {number|string|Decimal} A value in radians.
  *
  */
 function cosh(x) {
   return new this(x).cosh();
 }
 
 
 /*
  * Create and return a Decimal constructor with the same configuration properties as this Decimal
  * constructor.
  *
  */
 function clone(obj) {
   var i, p, ps;
 
   /*
    * The Decimal constructor and exported function.
    * Return a new Decimal instance.
    *
    * v {number|string|Decimal} A numeric value.
    *
    */
   function Decimal(v) {
     var e, i, t,
       x = this;
 
     // Decimal called without new.
     if (!(x instanceof Decimal)) return new Decimal(v);
 
     // Retain a reference to this Decimal constructor, and shadow Decimal.prototype.constructor
     // which points to Object.
     x.constructor = Decimal;
 
     // Duplicate.
     if (isDecimalInstance(v)) {
       x.s = v.s;
 
       if (external) {
         if (!v.d || v.e > Decimal.maxE) {
 
           // Infinity.
           x.e = NaN;
           x.d = null;
         } else if (v.e < Decimal.minE) {
 
           // Zero.
           x.e = 0;
           x.d = [0];
         } else {
           x.e = v.e;
           x.d = v.d.slice();
         }
       } else {
         x.e = v.e;
         x.d = v.d ? v.d.slice() : v.d;
       }
 
       return;
     }
 
     t = typeof v;
 
     if (t === 'number') {
       if (v === 0) {
         x.s = 1 / v < 0 ? -1 : 1;
         x.e = 0;
         x.d = [0];
         return;
       }
 
       if (v < 0) {
         v = -v;
         x.s = -1;
       } else {
         x.s = 1;
       }
 
       // Fast path for small integers.
       if (v === ~~v && v < 1e7) {
         for (e = 0, i = v; i >= 10; i /= 10) e++;
 
         if (external) {
           if (e > Decimal.maxE) {
             x.e = NaN;
             x.d = null;
           } else if (e < Decimal.minE) {
             x.e = 0;
             x.d = [0];
           } else {
             x.e = e;
             x.d = [v];
           }
         } else {
           x.e = e;
           x.d = [v];
         }
 
         return;
 
       // Infinity, NaN.
       } else if (v * 0 !== 0) {
         if (!v) x.s = NaN;
         x.e = NaN;
         x.d = null;
         return;
       }
 
       return parseDecimal(x, v.toString());
 
     } else if (t !== 'string') {
       throw Error(invalidArgument + v);
     }
 
     // Minus sign?
     if ((i = v.charCodeAt(0)) === 45) {
       v = v.slice(1);
       x.s = -1;
     } else {
       // Plus sign?
       if (i === 43) v = v.slice(1);
       x.s = 1;
     }
 
     return isDecimal.test(v) ? parseDecimal(x, v) : parseOther(x, v);
   }
 
   Decimal.prototype = P;
 
   Decimal.ROUND_UP = 0;
   Decimal.ROUND_DOWN = 1;
   Decimal.ROUND_CEIL = 2;
   Decimal.ROUND_FLOOR = 3;
   Decimal.ROUND_HALF_UP = 4;
   Decimal.ROUND_HALF_DOWN = 5;
   Decimal.ROUND_HALF_EVEN = 6;
   Decimal.ROUND_HALF_CEIL = 7;
   Decimal.ROUND_HALF_FLOOR = 8;
   Decimal.EUCLID = 9;
 
   Decimal.config = Decimal.set = config;
   Decimal.clone = clone;
   Decimal.isDecimal = isDecimalInstance;
 
   Decimal.abs = abs;
   Decimal.acos = acos;
   Decimal.acosh = acosh;        // ES6
   Decimal.add = add;
   Decimal.asin = asin;
   Decimal.asinh = asinh;        // ES6
   Decimal.atan = atan;
   Decimal.atanh = atanh;        // ES6
   Decimal.atan2 = atan2;
   Decimal.cbrt = cbrt;          // ES6
   Decimal.ceil = ceil;
   Decimal.clamp = clamp;
   Decimal.cos = cos;
   Decimal.cosh = cosh;          // ES6
   Decimal.div = div;
   Decimal.exp = exp;
   Decimal.floor = floor;
   Decimal.hypot = hypot;        // ES6
   Decimal.ln = ln;
   Decimal.log = log;
   Decimal.log10 = log10;        // ES6
   Decimal.log2 = log2;          // ES6
   Decimal.max = max;
   Decimal.min = min;
   Decimal.mod = mod;
   Decimal.mul = mul;
   Decimal.pow = pow;
   Decimal.random = random;
   Decimal.round = round;
   Decimal.sign = sign;          // ES6
   Decimal.sin = sin;
   Decimal.sinh = sinh;          // ES6
   Decimal.sqrt = sqrt;
   Decimal.sub = sub;
   Decimal.sum = sum;
   Decimal.tan = tan;
   Decimal.tanh = tanh;          // ES6
   Decimal.trunc = trunc;        // ES6
 
   if (obj === void 0) obj = {};
   if (obj) {
     if (obj.defaults !== true) {
       ps = ['precision', 'rounding', 'toExpNeg', 'toExpPos', 'maxE', 'minE', 'modulo', 'crypto'];
       for (i = 0; i < ps.length;) if (!obj.hasOwnProperty(p = ps[i++])) obj[p] = this[p];
     }
   }
 
   Decimal.config(obj);
 
   return Decimal;
 }
 
 
 /*
  * Return a new Decimal whose value is `x` divided by `y`, rounded to `precision` significant
  * digits using rounding mode `rounding`.
  *
  * x {number|string|Decimal}
  * y {number|string|Decimal}
  *
  */
 function div(x, y) {
   return new this(x).div(y);
 }
 
 
 /*
  * Return a new Decimal whose value is the natural exponential of `x`, rounded to `precision`
  * significant digits using rounding mode `rounding`.
  *
  * x {number|string|Decimal} The power to which to raise the base of the natural log.
  *
  */
 function exp(x) {
   return new this(x).exp();
 }
 
 
 /*
  * Return a new Decimal whose value is `x` round to an integer using `ROUND_FLOOR`.
  *
  * x {number|string|Decimal}
  *
  */
 function floor(x) {
   return finalise(x = new this(x), x.e + 1, 3);
 }
 
 
 /*
  * Return a new Decimal whose value is the square root of the sum of the squares of the arguments,
  * rounded to `precision` significant digits using rounding mode `rounding`.
  *
  * hypot(a, b, ...) = sqrt(a^2 + b^2 + ...)
  *
  * arguments {number|string|Decimal}
  *
  */
 function hypot() {
   var i, n,
     t = new this(0);
 
   external = false;
 
   for (i = 0; i < arguments.length;) {
     n = new this(arguments[i++]);
     if (!n.d) {
       if (n.s) {
         external = true;
         return new this(1 / 0);
       }
       t = n;
     } else if (t.d) {
       t = t.plus(n.times(n));
     }
   }
 
   external = true;
 
   return t.sqrt();
 }
 
 
 /*
  * Return true if object is a Decimal instance (where Decimal is any Decimal constructor),
  * otherwise return false.
  *
  */
 function isDecimalInstance(obj) {
   return obj instanceof Decimal || obj && obj.toStringTag === tag || false;
 }
 
 
 /*
  * Return a new Decimal whose value is the natural logarithm of `x`, rounded to `precision`
  * significant digits using rounding mode `rounding`.
  *
  * x {number|string|Decimal}
  *
  */
 function ln(x) {
   return new this(x).ln();
 }
 
 
 /*
  * Return a new Decimal whose value is the log of `x` to the base `y`, or to base 10 if no base
  * is specified, rounded to `precision` significant digits using rounding mode `rounding`.
  *
  * log[y](x)
  *
  * x {number|string|Decimal} The argument of the logarithm.
  * y {number|string|Decimal} The base of the logarithm.
  *
  */
 function log(x, y) {
   return new this(x).log(y);
 }
 
 
 /*
  * Return a new Decimal whose value is the base 2 logarithm of `x`, rounded to `precision`
  * significant digits using rounding mode `rounding`.
  *
  * x {number|string|Decimal}
  *
  */
 function log2(x) {
   return new this(x).log(2);
 }
 
 
 /*
  * Return a new Decimal whose value is the base 10 logarithm of `x`, rounded to `precision`
  * significant digits using rounding mode `rounding`.
  *
  * x {number|string|Decimal}
  *
  */
 function log10(x) {
   return new this(x).log(10);
 }
 
 
 /*
  * Return a new Decimal whose value is the maximum of the arguments.
  *
  * arguments {number|string|Decimal}
  *
  */
 function max() {
   return maxOrMin(this, arguments, 'lt');
 }
 
 
 /*
  * Return a new Decimal whose value is the minimum of the arguments.
  *
  * arguments {number|string|Decimal}
  *
  */
 function min() {
   return maxOrMin(this, arguments, 'gt');
 }
 
 
 /*
  * Return a new Decimal whose value is `x` modulo `y`, rounded to `precision` significant digits
  * using rounding mode `rounding`.
  *
  * x {number|string|Decimal}
  * y {number|string|Decimal}
  *
  */
 function mod(x, y) {
   return new this(x).mod(y);
 }
 
 
 /*
  * Return a new Decimal whose value is `x` multiplied by `y`, rounded to `precision` significant
  * digits using rounding mode `rounding`.
  *
  * x {number|string|Decimal}
  * y {number|string|Decimal}
  *
  */
 function mul(x, y) {
   return new this(x).mul(y);
 }
 
 
 /*
  * Return a new Decimal whose value is `x` raised to the power `y`, rounded to precision
  * significant digits using rounding mode `rounding`.
  *
  * x {number|string|Decimal} The base.
  * y {number|string|Decimal} The exponent.
  *
  */
 function pow(x, y) {
   return new this(x).pow(y);
 }
 
 
 /*
  * Returns a new Decimal with a random value equal to or greater than 0 and less than 1, and with
  * `sd`, or `Decimal.precision` if `sd` is omitted, significant digits (or less if trailing zeros
  * are produced).
  *
  * [sd] {number} Significant digits. Integer, 0 to MAX_DIGITS inclusive.
  *
  */
 function random(sd) {
   var d, e, k, n,
     i = 0,
     r = new this(1),
     rd = [];
 
   if (sd === void 0) sd = this.precision;
   else checkInt32(sd, 1, MAX_DIGITS);
 
   k = Math.ceil(sd / LOG_BASE);
 
   if (!this.crypto) {
     for (; i < k;) rd[i++] = Math.random() * 1e7 | 0;
 
   // Browsers supporting crypto.getRandomValues.
   } else if (crypto.getRandomValues) {
     d = crypto.getRandomValues(new Uint32Array(k));
 
     for (; i < k;) {
       n = d[i];
 
       // 0 <= n < 4294967296
       // Probability n >= 4.29e9, is 4967296 / 4294967296 = 0.00116 (1 in 865).
       if (n >= 4.29e9) {
         d[i] = crypto.getRandomValues(new Uint32Array(1))[0];
       } else {
 
         // 0 <= n <= 4289999999
         // 0 <= (n % 1e7) <= 9999999
         rd[i++] = n % 1e7;
       }
     }
 
   // Node.js supporting crypto.randomBytes.
   } else if (crypto.randomBytes) {
 
     // buffer
     d = crypto.randomBytes(k *= 4);
 
     for (; i < k;) {
 
       // 0 <= n < 2147483648
       n = d[i] + (d[i + 1] << 8) + (d[i + 2] << 16) + ((d[i + 3] & 0x7f) << 24);
 
       // Probability n >= 2.14e9, is 7483648 / 2147483648 = 0.0035 (1 in 286).
       if (n >= 2.14e9) {
         crypto.randomBytes(4).copy(d, i);
       } else {
 
         // 0 <= n <= 2139999999
         // 0 <= (n % 1e7) <= 9999999
         rd.push(n % 1e7);
         i += 4;
       }
     }
 
     i = k / 4;
   } else {
     throw Error(cryptoUnavailable);
   }
 
   k = rd[--i];
   sd %= LOG_BASE;
 
   // Convert trailing digits to zeros according to sd.
   if (k && sd) {
     n = mathpow(10, LOG_BASE - sd);
     rd[i] = (k / n | 0) * n;
   }
 
   // Remove trailing words which are zero.
   for (; rd[i] === 0; i--) rd.pop();
 
   // Zero?
   if (i < 0) {
     e = 0;
     rd = [0];
   } else {
     e = -1;
 
     // Remove leading words which are zero and adjust exponent accordingly.
     for (; rd[0] === 0; e -= LOG_BASE) rd.shift();
 
     // Count the digits of the first word of rd to determine leading zeros.
     for (k = 1, n = rd[0]; n >= 10; n /= 10) k++;
 
     // Adjust the exponent for leading zeros of the first word of rd.
     if (k < LOG_BASE) e -= LOG_BASE - k;
   }
 
   r.e = e;
   r.d = rd;
 
   return r;
 }
 
 
 /*
  * Return a new Decimal whose value is `x` rounded to an integer using rounding mode `rounding`.
  *
  * To emulate `Math.round`, set rounding to 7 (ROUND_HALF_CEIL).
  *
  * x {number|string|Decimal}
  *
  */
 function round(x) {
   return finalise(x = new this(x), x.e + 1, this.rounding);
 }
 
 
 /*
  * Return
  *   1    if x > 0,
  *  -1    if x < 0,
  *   0    if x is 0,
  *  -0    if x is -0,
  *   NaN  otherwise
  *
  * x {number|string|Decimal}
  *
  */
 function sign(x) {
   x = new this(x);
   return x.d ? (x.d[0] ? x.s : 0 * x.s) : x.s || NaN;
 }
 
 
 /*
  * Return a new Decimal whose value is the sine of `x`, rounded to `precision` significant digits
  * using rounding mode `rounding`.
  *
  * x {number|string|Decimal} A value in radians.
  *
  */
 function sin(x) {
   return new this(x).sin();
 }
 
 
 /*
  * Return a new Decimal whose value is the hyperbolic sine of `x`, rounded to `precision`
  * significant digits using rounding mode `rounding`.
  *
  * x {number|string|Decimal} A value in radians.
  *
  */
 function sinh(x) {
   return new this(x).sinh();
 }
 
 
 /*
  * Return a new Decimal whose value is the square root of `x`, rounded to `precision` significant
  * digits using rounding mode `rounding`.
  *
  * x {number|string|Decimal}
  *
  */
 function sqrt(x) {
   return new this(x).sqrt();
 }
 
 
 /*
  * Return a new Decimal whose value is `x` minus `y`, rounded to `precision` significant digits
  * using rounding mode `rounding`.
  *
  * x {number|string|Decimal}
  * y {number|string|Decimal}
  *
  */
 function sub(x, y) {
   return new this(x).sub(y);
 }
 
 
 /*
  * Return a new Decimal whose value is the sum of the arguments, rounded to `precision`
  * significant digits using rounding mode `rounding`.
  *
  * Only the result is rounded, not the intermediate calculations.
  *
  * arguments {number|string|Decimal}
  *
  */
 function sum() {
   var i = 0,
     args = arguments,
     x = new this(args[i]);
 
   external = false;
   for (; x.s && ++i < args.length;) x = x.plus(args[i]);
   external = true;
 
   return finalise(x, this.precision, this.rounding);
 }
 
 
 /*
  * Return a new Decimal whose value is the tangent of `x`, rounded to `precision` significant
  * digits using rounding mode `rounding`.
  *
  * x {number|string|Decimal} A value in radians.
  *
  */
 function tan(x) {
   return new this(x).tan();
 }
 
 
 /*
  * Return a new Decimal whose value is the hyperbolic tangent of `x`, rounded to `precision`
  * significant digits using rounding mode `rounding`.
  *
  * x {number|string|Decimal} A value in radians.
  *
  */
 function tanh(x) {
   return new this(x).tanh();
 }
 
 
 /*
  * Return a new Decimal whose value is `x` truncated to an integer.
  *
  * x {number|string|Decimal}
  *
  */
 function trunc(x) {
   return finalise(x = new this(x), x.e + 1, 1);
 }
 
 
 P[Symbol.for('nodejs.util.inspect.custom')] = P.toString;
 P[Symbol.toStringTag] = 'Decimal';
 
 // Create and configure initial Decimal constructor.
 export var Decimal = P.constructor = clone(DEFAULTS);
 
 // Create the internal constants from their string values.
 LN10 = new Decimal(LN10);
 PI = new Decimal(PI);
 
 export default Decimal;