text: /*! decimal.js v7.2.3 https://github.com/MikeMcl/decimal.js/LICENCE */
  ;(function (globalScope) {
    'use strict';
  
  
    /*
     *  decimal.js v7.2.3
     *  An arbitrary-precision Decimal type for JavaScript.
     *  https://github.com/MikeMcl/decimal.js
     *  Copyright (c) 2017 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 = '2.30258509299404568401799145468436420760110148862877297603332790096757260967735248023599',
  
      // Pi (1025 digits).
      PI = '3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461284756482337867831652712019091456485669234603486104543266482133936072602491412737245870066063155881748815209209628292540917153643678925903600113305305488204665213841469519415116094330572703657595919530921861173819326117931051185480744623799627495673518857527248912279381830119491298336733624406566430860213949463952247371907021798609437027705392171762931767523846748184676694051320005681271452635608277857713427577896091736371787214684409012249534301465495853710507922796892589235420199561121290219608640344181598136297747713099605187072113499999983729780499510597317328160963185950244594553469083026425223082533446850352619311881710100031378387528865875332083814206171776691473035982534904287554687311595628638823537875937519577818577805321712268066130019278766111959092164201989380952572010654858632789',
  
  
      // The initial configuration properties of the Decimal constructor.
      Decimal = {
  
        // 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: 8,                         // 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, noConflict, quadrant,
      external = true,
  
      decimalError = '[DecimalError] ',
      invalidArgument = decimalError + 'Invalid argument: ',
      precisionLimitExceeded = decimalError + 'Precision limit exceeded',
      cryptoUnavailable = decimalError + 'crypto unavailable',
  
      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 = {};
  
  
    // Decimal prototype methods
  
  
    /*
     *  absoluteValue             abs
     *  ceil
     *  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
     *   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 * Math.pow(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 = Math.pow(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 = Math.pow(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(Math.pow(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);
    };
  
  
  
    P.digitsTo0Mantissa = function () {
      var string = "";
      for (var i = 0; i < Math.min(this.d.length, 3); ++i)
      {
        string = string.concat(this.d[i]);
      }
      return parseFloat("0." + string);
      }
      
      //exponent and mantissa in base 10 format, with 0 < mantissa < 1. from https://math.stackexchange.com/a/1425113/72291
    P.fastlog10 = P.flog10 = function () {
      return this.e+1 + Math.log10(this.digitsTo0Mantissa());
    }
  
  
    /*
     * 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 = '1e' + 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 the magnitude of `y` to the value
     * of this Decimal.
     *
     * If the value of this Decimal is equidistant from two multiples of `y`, the rounding mode `rm`,
     * or `Decimal.rounding` if `rm` is omitted, determines the direction of the nearest multiple.
     *
     * In the context of this method, rounding mode 4 (ROUND_HALF_UP) is the same as rounding mode 0
     * (ROUND_UP), and so on.
     *
     * 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) 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;
        if (rm < 4) rm = [4, 5, 7, 8][rm];
        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;
    };
  
  
    /*
    // Add aliases to match BigDecimal method names.
    // P.add = P.plus;
    P.subtract = P.minus;
    P.multiply = P.times;
    P.divide = P.div;
    P.remainder = P.mod;
    P.compareTo = P.cmp;
    P.negate = P.neg;
     */
  
  
    // 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, y,
        len = x.d.length;
  
      // 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.
      if (len < 32) {
        k = Math.ceil(len / 3);
        y = Math.pow(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 === '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 ? Math.pow(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 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;
  
      // Max k before Math.pow precision loss is 22
      x = x.times(Math.pow(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;
    }
  
  
    // 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
     *  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
     *  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);
    }
  
  
    /*
     * 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}
     *
     * 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,
        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 ((v = obj[p = ps[i]]) !== void 0) {
          if (mathfloor(v) === v && v >= ps[i + 1] && v <= ps[i + 2]) this[p] = v;
          else throw Error(invalidArgument + p + ': ' + v);
        }
      }
  
      if ((v = obj[p = 'crypto']) !== 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 (v instanceof Decimal) {
          x.s = v.s;
          x.e = v.e;
          x.d = (v = v.d) ? v.slice() : v;
          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++;
            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 (v.charCodeAt(0) === 45) {
          v = v.slice(1);
          x.s = -1;
        } else {
          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.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.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.tan = tan;
      Decimal.tanh = tanh;          // ES6
      Decimal.trunc = trunc;        // ES6
  
      if (obj === void 0) obj = {};
      if (obj) {
        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 + ...)
     *
     */
    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 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
     *
     */
    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 tangent of `x`, rounded to `precision` significant
     * digits using rounding mode `rounding`.
     *
     * x {number|string|Decimal} A value in radians.
     *
     */
    function tan(x) {
      return new this(x).tan();
    }
  
  
    /*
     * Return a new Decimal whose value is the hyperbolic tangent of `x`, rounded to `precision`
     * significant digits using rounding mode `rounding`.
     *
     * x {number|string|Decimal} A value in radians.
     *
     */
    function tanh(x) {
      return new this(x).tanh();
    }
  
  
    /*
     * Return a new Decimal whose value is `x` truncated to an integer.
     *
     * x {number|string|Decimal}
     *
     */
    function trunc(x) {
      return finalise(x = new this(x), x.e + 1, 1);
    }
  
  
    // Create and configure initial Decimal constructor.
    Decimal = clone(Decimal);
  
    // Create the internal constants from their string values.
    LN10 = new Decimal(LN10);
    PI = new Decimal(PI);
  
  
    // Export.
  
  
    // AMD.
    if (typeof define == 'function' && define.amd) {
      define(function () {
        return Decimal;
      });
  
    // Node and other environments that support module.exports.
    } else if (typeof module != 'undefined' && module.exports) {
      module.exports = Decimal['default'] = Decimal.Decimal = Decimal;
  
    // Browser.
    } else {
      if (!globalScope) {
        globalScope = typeof self != 'undefined' && self && self.self == self
          ? self : Function('return this')();
      }
  
      noConflict = globalScope.Decimal;
      Decimal.noConflict = function () {
        globalScope.Decimal = noConflict;
        return Decimal;
      };
  
      globalScope.Decimal = Decimal;
    }
  })(this);
  