decimal.mjs

/*
 *  decimal.js v10.2.1
 *  An arbitrary-precision Decimal type for JavaScript.
 *  https://github.com/MikeMcl/decimal.js
 *  Copyright (c) 2020 Michael Mclaughlin <M8ch88l@gmail.com>
 *  MIT Licence
 */


// -----------------------------------  EDITABLE DEFAULTS  ------------------------------------ //


  // The maximum exponent magnitude.
  // The limit on the value of `toExpNeg`, `toExpPos`, `minE` and `maxE`.
var EXP_LIMIT = 9e15,                      // 0 to 9e15

  // The limit on the value of `precision`, and on the value of the first argument to
  // `toDecimalPlaces`, `toExponential`, `toFixed`, `toPrecision` and `toSignificantDigits`.
  MAX_DIGITS = 1e9,                        // 0 to 1e9

  // Base conversion alphabet.
  NUMERALS = '0123456789abcdef',

  // The natural logarithm of 10 (1025 digits).
  LN10 = '2.3025850929940456840179914546843642076011014886287729760333279009675726096773524802359972050895982983419677840422862486334095254650828067566662873690987816894829072083255546808437998948262331985283935053089653777326288461633662222876982198867465436674744042432743651550489343149393914796194044002221051017141748003688084012647080685567743216228355220114804663715659121373450747856947683463616792101806445070648000277502684916746550586856935673420670581136429224554405758925724208241314695689016758940256776311356919292033376587141660230105703089634572075440370847469940168269282808481184289314848524948644871927809676271275775397027668605952496716674183485704422507197965004714951050492214776567636938662976979522110718264549734772662425709429322582798502585509785265383207606726317164309505995087807523710333101197857547331541421808427543863591778117054309827482385045648019095610299291824318237525357709750539565187697510374970888692180205189339507238539205144634197265287286965110862571492198849978748873771345686209167058',

  // Pi (1025 digits).
  PI = '3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461284756482337867831652712019091456485669234603486104543266482133936072602491412737245870066063155881748815209209628292540917153643678925903600113305305488204665213841469519415116094330572703657595919530921861173819326117931051185480744623799627495673518857527248912279381830119491298336733624406566430860213949463952247371907021798609437027705392171762931767523846748184676694051320005681271452635608277857713427577896091736371787214684409012249534301465495853710507922796892589235420199561121290219608640344181598136297747713099605187072113499999983729780499510597317328160963185950244594553469083026425223082533446850352619311881710100031378387528865875332083814206171776691473035982534904287554687311595628638823537875937519577818577805321712268066130019278766111959092164201989380952572010654858632789',


  // The initial configuration properties of the Decimal constructor.
  DEFAULTS = {

    // These values must be integers within the stated ranges (inclusive).
    // Most of these values can be changed at run-time using the `Decimal.config` method.

    // The maximum number of significant digits of the result of a calculation or base conversion.
    // E.g. `Decimal.config({ precision: 20 });`
    precision: 20,                         // 1 to MAX_DIGITS

    // The rounding mode used when rounding to `precision`.
    //
    // ROUND_UP         0 Away from zero.
    // ROUND_DOWN       1 Towards zero.
    // ROUND_CEIL       2 Towards +Infinity.
    // ROUND_FLOOR      3 Towards -Infinity.
    // ROUND_HALF_UP    4 Towards nearest neighbour. If equidistant, up.
    // ROUND_HALF_DOWN  5 Towards nearest neighbour. If equidistant, down.
    // ROUND_HALF_EVEN  6 Towards nearest neighbour. If equidistant, towards even neighbour.
    // ROUND_HALF_CEIL  7 Towards nearest neighbour. If equidistant, towards +Infinity.
    // ROUND_HALF_FLOOR 8 Towards nearest neighbour. If equidistant, towards -Infinity.
    //
    // E.g.
    // `Decimal.rounding = 4;`
    // `Decimal.rounding = Decimal.ROUND_HALF_UP;`
    rounding: 4,                           // 0 to 8

    // The modulo mode used when calculating the modulus: a mod n.
    // The quotient (q = a / n) is calculated according to the corresponding rounding mode.
    // The remainder (r) is calculated as: r = a - n * q.
    //
    // UP         0 The remainder is positive if the dividend is negative, else is negative.
    // DOWN       1 The remainder has the same sign as the dividend (JavaScript %).
    // FLOOR      3 The remainder has the same sign as the divisor (Python %).
    // HALF_EVEN  6 The IEEE 754 remainder function.
    // EUCLID     9 Euclidian division. q = sign(n) * floor(a / abs(n)). Always positive.
    //
    // Truncated division (1), floored division (3), the IEEE 754 remainder (6), and Euclidian
    // division (9) are commonly used for the modulus operation. The other rounding modes can also
    // be used, but they may not give useful results.
    modulo: 1,                             // 0 to 9

    // The exponent value at and beneath which `toString` returns exponential notation.
    // JavaScript numbers: -7
    toExpNeg: -7,                          // 0 to -EXP_LIMIT

    // The exponent value at and above which `toString` returns exponential notation.
    // JavaScript numbers: 21
    toExpPos:  21,                         // 0 to EXP_LIMIT

    // The minimum exponent value, beneath which underflow to zero occurs.
    // JavaScript numbers: -324  (5e-324)
    minE: -EXP_LIMIT,                      // -1 to -EXP_LIMIT

    // The maximum exponent value, above which overflow to Infinity occurs.
    // JavaScript numbers: 308  (1.7976931348623157e+308)
    maxE: EXP_LIMIT,                       // 1 to EXP_LIMIT

    // Whether to use cryptographically-secure random number generation, if available.
    crypto: false                          // true/false
  },


// ----------------------------------- END OF EDITABLE DEFAULTS ------------------------------- //


  inexact, quadrant,
  external = true,

  decimalError = '[DecimalError] ',
  invalidArgument = decimalError + 'Invalid argument: ',
  precisionLimitExceeded = decimalError + 'Precision limit exceeded',
  cryptoUnavailable = decimalError + 'crypto unavailable',

  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 = { name: '[object Decimal]' };


// 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 * mathpow(x.s * x, 1 / 3);

   // Math.cbrt underflow/overflow?
   // Pass x to Math.pow as integer, then adjust the exponent of the result.
  if (!s || Math.abs(s) == 1 / 0) {
    n = digitsToString(x.d);
    e = x.e;

    // Adjust n exponent so it is a multiple of 3 away from x exponent.
    if (s = (e - n.length + 1) % 3) n += (s == 1 || s == -2 ? '0' : '00');
    s = mathpow(n, 1 / 3);

    // Rarely, e may be one less than the result exponent value.
    e = mathfloor((e + 1) / 3) - (e % 3 == (e < 0 ? -1 : 2));

    if (s == 1 / 0) {
      n = '5e' + e;
    } else {
      n = s.toExponential();
      n = n.slice(0, n.indexOf('e') + 1) + e;
    }

    r = new Ctor(n);
    r.s = x.s;
  } else {
    r = new Ctor(s.toString());
  }

  sd = (e = Ctor.precision) + 3;

  // Halley's method.
  // TODO? Compare Newton's method.
  for (;;) {
    t = r;
    t3 = t.times(t).times(t);
    t3plusx = t3.plus(x);
    r = divide(t3plusx.plus(x).times(t), t3plusx.plus(t3), sd + 2, 1);

    // TODO? Replace with for-loop and checkRoundingDigits.
    if (digitsToString(t.d).slice(0, sd) === (n = digitsToString(r.d)).slice(0, sd)) {
      n = n.slice(sd - 3, sd + 1);

      // The 4th rounding digit may be in error by -1 so if the 4 rounding digits are 9999 or 4999
      // , i.e. approaching a rounding boundary, continue the iteration.
      if (n == '9999' || !rep && n == '4999') {

        // On the first iteration only, check to see if rounding up gives the exact result as the
        // nines may infinitely repeat.
        if (!rep) {
          finalise(t, e + 1, 0);

          if (t.times(t).times(t).eq(x)) {
            r = t;
            break;
          }
        }

        sd += 4;
        rep = 1;
      } else {

        // If the rounding digits are null, 0{0,4} or 50{0,3}, check for an exact result.
        // If not, then there are further digits and m will be truthy.
        if (!+n || !+n.slice(1) && n.charAt(0) == '5') {

          // Truncate to the first rounding digit.
          finalise(r, e + 1, 1);
          m = !r.times(r).times(r).eq(x);
        }

        break;
      }
    }
  }

  external = true;

  return finalise(r, e, Ctor.rounding, m);
};


/*
 * Return the number of decimal places of the value of this Decimal.
 *
 */
P.decimalPlaces = P.dp = function () {
  var w,
    d = this.d,
    n = NaN;

  if (d) {
    w = d.length - 1;
    n = (w - mathfloor(this.e / LOG_BASE)) * LOG_BASE;

    // Subtract the number of trailing zeros of the last word.
    w = d[w];
    if (w) for (; w % 10 == 0; w /= 10) n--;
    if (n < 0) n = 0;
  }

  return n;
};


/*
 *  n / 0 = I
 *  n / N = N
 *  n / I = 0
 *  0 / n = 0
 *  0 / 0 = N
 *  0 / N = N
 *  0 / I = 0
 *  N / n = N
 *  N / 0 = N
 *  N / N = N
 *  N / I = N
 *  I / n = I
 *  I / 0 = I
 *  I / N = N
 *  I / I = N
 *
 * Return a new Decimal whose value is the value of this Decimal divided by `y`, rounded to
 * `precision` significant digits using rounding mode `rounding`.
 *
 */
P.dividedBy = P.div = function (y) {
  return divide(this, new this.constructor(y));
};


/*
 * Return a new Decimal whose value is the integer part of dividing the value of this Decimal
 * by the value of `y`, rounded to `precision` significant digits using rounding mode `rounding`.
 *
 */
P.dividedToIntegerBy = P.divToInt = function (y) {
  var x = this,
    Ctor = x.constructor;
  return finalise(divide(x, new Ctor(y), 0, 1, 1), Ctor.precision, Ctor.rounding);
};


/*
 * Return true if the value of this Decimal is equal to the value of `y`, otherwise return false.
 *
 */
P.equals = P.eq = function (y) {
  return this.cmp(y) === 0;
};


/*
 * Return a new Decimal whose value is the value of this Decimal rounded to a whole number in the
 * direction of negative Infinity.
 *
 */
P.floor = function () {
  return finalise(new this.constructor(this), this.e + 1, 3);
};


/*
 * Return true if the value of this Decimal is greater than the value of `y`, otherwise return
 * false.
 *
 */
P.greaterThan = P.gt = function (y) {
  return this.cmp(y) > 0;
};


/*
 * Return true if the value of this Decimal is greater than or equal to the value of `y`,
 * otherwise return false.
 *
 */
P.greaterThanOrEqualTo = P.gte = function (y) {
  var k = this.cmp(y);
  return k == 1 || k === 0;
};


/*
 * Return a new Decimal whose value is the hyperbolic cosine of the value in radians of this
 * Decimal.
 *
 * Domain: [-Infinity, Infinity]
 * Range: [1, Infinity]
 *
 * cosh(x) = 1 + x^2/2! + x^4/4! + x^6/6! + ...
 *
 * cosh(0)         = 1
 * cosh(-0)        = 1
 * cosh(Infinity)  = Infinity
 * cosh(-Infinity) = Infinity
 * cosh(NaN)       = NaN
 *
 *  x        time taken (ms)   result
 * 1000      9                 9.8503555700852349694e+433
 * 10000     25                4.4034091128314607936e+4342
 * 100000    171               1.4033316802130615897e+43429
 * 1000000   3817              1.5166076984010437725e+434294
 * 10000000  abandoned after 2 minute wait
 *
 * TODO? Compare performance of cosh(x) = 0.5 * (exp(x) + exp(-x))
 *
 */
P.hyperbolicCosine = P.cosh = function () {
  var k, n, pr, rm, len,
    x = this,
    Ctor = x.constructor,
    one = new Ctor(1);

  if (!x.isFinite()) return new Ctor(x.s ? 1 / 0 : NaN);
  if (x.isZero()) return one;

  pr = Ctor.precision;
  rm = Ctor.rounding;
  Ctor.precision = pr + Math.max(x.e, x.sd()) + 4;
  Ctor.rounding = 1;
  len = x.d.length;

  // Argument reduction: cos(4x) = 1 - 8cos^2(x) + 8cos^4(x) + 1
  // i.e. cos(x) = 1 - cos^2(x/4)(8 - 8cos^2(x/4))

  // Estimate the optimum number of times to use the argument reduction.
  // TODO? Estimation reused from cosine() and may not be optimal here.
  if (len < 32) {
    k = Math.ceil(len / 3);
    n = (1 / tinyPow(4, k)).toString();
  } else {
    k = 16;
    n = '2.3283064365386962890625e-10';
  }

  x = taylorSeries(Ctor, 1, x.times(n), new Ctor(1), true);

  // Reverse argument reduction
  var cosh2_x,
    i = k,
    d8 = new Ctor(8);
  for (; i--;) {
    cosh2_x = x.times(x);
    x = one.minus(cosh2_x.times(d8.minus(cosh2_x.times(d8))));
  }

  return finalise(x, Ctor.precision = pr, Ctor.rounding = rm, true);
};


/*
 * Return a new Decimal whose value is the hyperbolic sine of the value in radians of this
 * Decimal.
 *
 * Domain: [-Infinity, Infinity]
 * Range: [-Infinity, Infinity]
 *
 * sinh(x) = x + x^3/3! + x^5/5! + x^7/7! + ...
 *
 * sinh(0)         = 0
 * sinh(-0)        = -0
 * sinh(Infinity)  = Infinity
 * sinh(-Infinity) = -Infinity
 * sinh(NaN)       = NaN
 *
 * x        time taken (ms)
 * 10       2 ms
 * 100      5 ms
 * 1000     14 ms
 * 10000    82 ms
 * 100000   886 ms            1.4033316802130615897e+43429
 * 200000   2613 ms
 * 300000   5407 ms
 * 400000   8824 ms
 * 500000   13026 ms          8.7080643612718084129e+217146
 * 1000000  48543 ms
 *
 * TODO? Compare performance of sinh(x) = 0.5 * (exp(x) - exp(-x))
 *
 */
P.hyperbolicSine = P.sinh = function () {
  var k, pr, rm, len,
    x = this,
    Ctor = x.constructor;

  if (!x.isFinite() || x.isZero()) return new Ctor(x);

  pr = Ctor.precision;
  rm = Ctor.rounding;
  Ctor.precision = pr + Math.max(x.e, x.sd()) + 4;
  Ctor.rounding = 1;
  len = x.d.length;

  if (len < 3) {
    x = taylorSeries(Ctor, 2, x, x, true);
  } else {

    // Alternative argument reduction: sinh(3x) = sinh(x)(3 + 4sinh^2(x))
    // i.e. sinh(x) = sinh(x/3)(3 + 4sinh^2(x/3))
    // 3 multiplications and 1 addition

    // Argument reduction: sinh(5x) = sinh(x)(5 + sinh^2(x)(20 + 16sinh^2(x)))
    // i.e. sinh(x) = sinh(x/5)(5 + sinh^2(x/5)(20 + 16sinh^2(x/5)))
    // 4 multiplications and 2 additions

    // Estimate the optimum number of times to use the argument reduction.
    k = 1.4 * Math.sqrt(len);
    k = k > 16 ? 16 : k | 0;

    x = x.times(1 / tinyPow(5, k));
    x = taylorSeries(Ctor, 2, x, x, true);

    // Reverse argument reduction
    var sinh2_x,
      d5 = new Ctor(5),
      d16 = new Ctor(16),
      d20 = new Ctor(20);
    for (; k--;) {
      sinh2_x = x.times(x);
      x = x.times(d5.plus(sinh2_x.times(d16.times(sinh2_x).plus(d20))));
    }
  }

  Ctor.precision = pr;
  Ctor.rounding = rm;

  return finalise(x, pr, rm, true);
};


/*
 * Return a new Decimal whose value is the hyperbolic tangent of the value in radians of this
 * Decimal.
 *
 * Domain: [-Infinity, Infinity]
 * Range: [-1, 1]
 *
 * tanh(x) = sinh(x) / cosh(x)
 *
 * tanh(0)         = 0
 * tanh(-0)        = -0
 * tanh(Infinity)  = 1
 * tanh(-Infinity) = -1
 * tanh(NaN)       = NaN
 *
 */
P.hyperbolicTangent = P.tanh = function () {
  var pr, rm,
    x = this,
    Ctor = x.constructor;

  if (!x.isFinite()) return new Ctor(x.s);
  if (x.isZero()) return new Ctor(x);

  pr = Ctor.precision;
  rm = Ctor.rounding;
  Ctor.precision = pr + 7;
  Ctor.rounding = 1;

  return divide(x.sinh(), x.cosh(), Ctor.precision = pr, Ctor.rounding = rm);
};


/*
 * Return a new Decimal whose value is the arccosine (inverse cosine) in radians of the value of
 * this Decimal.
 *
 * Domain: [-1, 1]
 * Range: [0, pi]
 *
 * acos(x) = pi/2 - asin(x)
 *
 * acos(0)       = pi/2
 * acos(-0)      = pi/2
 * acos(1)       = 0
 * acos(-1)      = pi
 * acos(1/2)     = pi/3
 * acos(-1/2)    = 2*pi/3
 * acos(|x| > 1) = NaN
 * acos(NaN)     = NaN
 *
 */
P.inverseCosine = P.acos = function () {
  var halfPi,
    x = this,
    Ctor = x.constructor,
    k = x.abs().cmp(1),
    pr = Ctor.precision,
    rm = Ctor.rounding;

  if (k !== -1) {
    return k === 0
      // |x| is 1
      ? x.isNeg() ? getPi(Ctor, pr, rm) : new Ctor(0)
      // |x| > 1 or x is NaN
      : new Ctor(NaN);
  }

  if (x.isZero()) return getPi(Ctor, pr + 4, rm).times(0.5);

  // TODO? Special case acos(0.5) = pi/3 and acos(-0.5) = 2*pi/3

  Ctor.precision = pr + 6;
  Ctor.rounding = 1;

  x = x.asin();
  halfPi = getPi(Ctor, pr + 4, rm).times(0.5);

  Ctor.precision = pr;
  Ctor.rounding = rm;

  return halfPi.minus(x);
};


/*
 * Return a new Decimal whose value is the inverse of the hyperbolic cosine in radians of the
 * value of this Decimal.
 *
 * Domain: [1, Infinity]
 * Range: [0, Infinity]
 *
 * acosh(x) = ln(x + sqrt(x^2 - 1))
 *
 * acosh(x < 1)     = NaN
 * acosh(NaN)       = NaN
 * acosh(Infinity)  = Infinity
 * acosh(-Infinity) = NaN
 * acosh(0)         = NaN
 * acosh(-0)        = NaN
 * acosh(1)         = 0
 * acosh(-1)        = NaN
 *
 */
P.inverseHyperbolicCosine = P.acosh = function () {
  var pr, rm,
    x = this,
    Ctor = x.constructor;

  if (x.lte(1)) return new Ctor(x.eq(1) ? 0 : NaN);
  if (!x.isFinite()) return new Ctor(x);

  pr = Ctor.precision;
  rm = Ctor.rounding;
  Ctor.precision = pr + Math.max(Math.abs(x.e), x.sd()) + 4;
  Ctor.rounding = 1;
  external = false;

  x = x.times(x).minus(1).sqrt().plus(x);

  external = true;
  Ctor.precision = pr;
  Ctor.rounding = rm;

  return x.ln();
};


/*
 * Return a new Decimal whose value is the inverse of the hyperbolic sine in radians of the value
 * of this Decimal.
 *
 * Domain: [-Infinity, Infinity]
 * Range: [-Infinity, Infinity]
 *
 * asinh(x) = ln(x + sqrt(x^2 + 1))
 *
 * asinh(NaN)       = NaN
 * asinh(Infinity)  = Infinity
 * asinh(-Infinity) = -Infinity
 * asinh(0)         = 0
 * asinh(-0)        = -0
 *
 */
P.inverseHyperbolicSine = P.asinh = function () {
  var pr, rm,
    x = this,
    Ctor = x.constructor;

  if (!x.isFinite() || x.isZero()) return new Ctor(x);

  pr = Ctor.precision;
  rm = Ctor.rounding;
  Ctor.precision = pr + 2 * Math.max(Math.abs(x.e), x.sd()) + 6;
  Ctor.rounding = 1;
  external = false;

  x = x.times(x).plus(1).sqrt().plus(x);

  external = true;
  Ctor.precision = pr;
  Ctor.rounding = rm;

  return x.ln();
};


/*
 * Return a new Decimal whose value is the inverse of the hyperbolic tangent in radians of the
 * value of this Decimal.
 *
 * Domain: [-1, 1]
 * Range: [-Infinity, Infinity]
 *
 * atanh(x) = 0.5 * ln((1 + x) / (1 - x))
 *
 * atanh(|x| > 1)   = NaN
 * atanh(NaN)       = NaN
 * atanh(Infinity)  = NaN
 * atanh(-Infinity) = NaN
 * atanh(0)         = 0
 * atanh(-0)        = -0
 * atanh(1)         = Infinity
 * atanh(-1)        = -Infinity
 *
 */
P.inverseHyperbolicTangent = P.atanh = function () {
  var pr, rm, wpr, xsd,
    x = this,
    Ctor = x.constructor;

  if (!x.isFinite()) return new Ctor(NaN);
  if (x.e >= 0) return new Ctor(x.abs().eq(1) ? x.s / 0 : x.isZero() ? x : NaN);

  pr = Ctor.precision;
  rm = Ctor.rounding;
  xsd = x.sd();

  if (Math.max(xsd, pr) < 2 * -x.e - 1) return finalise(new Ctor(x), pr, rm, true);

  Ctor.precision = wpr = xsd - x.e;

  x = divide(x.plus(1), new Ctor(1).minus(x), wpr + pr, 1);

  Ctor.precision = pr + 4;
  Ctor.rounding = 1;

  x = x.ln();

  Ctor.precision = pr;
  Ctor.rounding = rm;

  return x.times(0.5);
};


/*
 * Return a new Decimal whose value is the arcsine (inverse sine) in radians of the value of this
 * Decimal.
 *
 * Domain: [-Infinity, Infinity]
 * Range: [-pi/2, pi/2]
 *
 * asin(x) = 2*atan(x/(1 + sqrt(1 - x^2)))
 *
 * asin(0)       = 0
 * asin(-0)      = -0
 * asin(1/2)     = pi/6
 * asin(-1/2)    = -pi/6
 * asin(1)       = pi/2
 * asin(-1)      = -pi/2
 * asin(|x| > 1) = NaN
 * asin(NaN)     = NaN
 *
 * TODO? Compare performance of Taylor series.
 *
 */
P.inverseSine = P.asin = function () {
  var halfPi, k,
    pr, rm,
    x = this,
    Ctor = x.constructor;

  if (x.isZero()) return new Ctor(x);

  k = x.abs().cmp(1);
  pr = Ctor.precision;
  rm = Ctor.rounding;

  if (k !== -1) {

    // |x| is 1
    if (k === 0) {
      halfPi = getPi(Ctor, pr + 4, rm).times(0.5);
      halfPi.s = x.s;
      return halfPi;
    }

    // |x| > 1 or x is NaN
    return new Ctor(NaN);
  }

  // TODO? Special case asin(1/2) = pi/6 and asin(-1/2) = -pi/6

  Ctor.precision = pr + 6;
  Ctor.rounding = 1;

  x = x.div(new Ctor(1).minus(x.times(x)).sqrt().plus(1)).atan();

  Ctor.precision = pr;
  Ctor.rounding = rm;

  return x.times(2);
};


/*
 * Return a new Decimal whose value is the arctangent (inverse tangent) in radians of the value
 * of this Decimal.
 *
 * Domain: [-Infinity, Infinity]
 * Range: [-pi/2, pi/2]
 *
 * atan(x) = x - x^3/3 + x^5/5 - x^7/7 + ...
 *
 * atan(0)         = 0
 * atan(-0)        = -0
 * atan(1)         = pi/4
 * atan(-1)        = -pi/4
 * atan(Infinity)  = pi/2
 * atan(-Infinity) = -pi/2
 * atan(NaN)       = NaN
 *
 */
P.inverseTangent = P.atan = function () {
  var i, j, k, n, px, t, r, wpr, x2,
    x = this,
    Ctor = x.constructor,
    pr = Ctor.precision,
    rm = Ctor.rounding;

  if (!x.isFinite()) {
    if (!x.s) return new Ctor(NaN);
    if (pr + 4 <= PI_PRECISION) {
      r = getPi(Ctor, pr + 4, rm).times(0.5);
      r.s = x.s;
      return r;
    }
  } else if (x.isZero()) {
    return new Ctor(x);
  } else if (x.abs().eq(1) && pr + 4 <= PI_PRECISION) {
    r = getPi(Ctor, pr + 4, rm).times(0.25);
    r.s = x.s;
    return r;
  }

  Ctor.precision = wpr = pr + 10;
  Ctor.rounding = 1;

  // TODO? if (x >= 1 && pr <= PI_PRECISION) atan(x) = halfPi * x.s - atan(1 / x);

  // Argument reduction
  // Ensure |x| < 0.42
  // atan(x) = 2 * atan(x / (1 + sqrt(1 + x^2)))

  k = Math.min(28, wpr / LOG_BASE + 2 | 0);

  for (i = k; i; --i) x = x.div(x.times(x).plus(1).sqrt().plus(1));

  external = false;

  j = Math.ceil(wpr / LOG_BASE);
  n = 1;
  x2 = x.times(x);
  r = new Ctor(x);
  px = x;

  // atan(x) = x - x^3/3 + x^5/5 - x^7/7 + ...
  for (; i !== -1;) {
    px = px.times(x2);
    t = r.minus(px.div(n += 2));

    px = px.times(x2);
    r = t.plus(px.div(n += 2));

    if (r.d[j] !== void 0) for (i = j; r.d[i] === t.d[i] && i--;);
  }

  if (k) r = r.times(2 << (k - 1));

  external = true;

  return finalise(r, Ctor.precision = pr, Ctor.rounding = rm, true);
};


/*
 * Return true if the value of this Decimal is a finite number, otherwise return false.
 *
 */
P.isFinite = function () {
  return !!this.d;
};


/*
 * Return true if the value of this Decimal is an integer, otherwise return false.
 *
 */
P.isInteger = P.isInt = function () {
  return !!this.d && mathfloor(this.e / LOG_BASE) > this.d.length - 2;
};


/*
 * Return true if the value of this Decimal is NaN, otherwise return false.
 *
 */
P.isNaN = function () {
  return !this.s;
};


/*
 * Return true if the value of this Decimal is negative, otherwise return false.
 *
 */
P.isNegative = P.isNeg = function () {
  return this.s < 0;
};


/*
 * Return true if the value of this Decimal is positive, otherwise return false.
 *
 */
P.isPositive = P.isPos = function () {
  return this.s > 0;
};


/*
 * Return true if the value of this Decimal is 0 or -0, otherwise return false.
 *
 */
P.isZero = function () {
  return !!this.d && this.d[0] === 0;
};


/*
 * Return true if the value of this Decimal is less than `y`, otherwise return false.
 *
 */
P.lessThan = P.lt = function (y) {
  return this.cmp(y) < 0;
};


/*
 * Return true if the value of this Decimal is less than or equal to `y`, otherwise return false.
 *
 */
P.lessThanOrEqualTo = P.lte = function (y) {
  return this.cmp(y) < 1;
};


/*
 * Return the logarithm of the value of this Decimal to the specified base, rounded to `precision`
 * significant digits using rounding mode `rounding`.
 *
 * If no base is specified, return log[10](arg).
 *
 * log[base](arg) = ln(arg) / ln(base)
 *
 * The result will always be correctly rounded if the base of the log is 10, and 'almost always'
 * otherwise:
 *
 * Depending on the rounding mode, the result may be incorrectly rounded if the first fifteen
 * rounding digits are [49]99999999999999 or [50]00000000000000. In that case, the maximum error
 * between the result and the correctly rounded result will be one ulp (unit in the last place).
 *
 * log[-b](a)       = NaN
 * log[0](a)        = NaN
 * log[1](a)        = NaN
 * log[NaN](a)      = NaN
 * log[Infinity](a) = NaN
 * log[b](0)        = -Infinity
 * log[b](-0)       = -Infinity
 * log[b](-a)       = NaN
 * log[b](1)        = 0
 * log[b](Infinity) = Infinity
 * log[b](NaN)      = NaN
 *
 * [base] {number|string|Decimal} The base of the logarithm.
 *
 */
P.logarithm = P.log = function (base) {
  var isBase10, d, denominator, k, inf, num, sd, r,
    arg = this,
    Ctor = arg.constructor,
    pr = Ctor.precision,
    rm = Ctor.rounding,
    guard = 5;

  // Default base is 10.
  if (base == null) {
    base = new Ctor(10);
    isBase10 = true;
  } else {
    base = new Ctor(base);
    d = base.d;

    // Return NaN if base is negative, or non-finite, or is 0 or 1.
    if (base.s < 0 || !d || !d[0] || base.eq(1)) return new Ctor(NaN);

    isBase10 = base.eq(10);
  }

  d = arg.d;

  // Is arg negative, non-finite, 0 or 1?
  if (arg.s < 0 || !d || !d[0] || arg.eq(1)) {
    return new Ctor(d && !d[0] ? -1 / 0 : arg.s != 1 ? NaN : d ? 0 : 1 / 0);
  }

  // The result will have a non-terminating decimal expansion if base is 10 and arg is not an
  // integer power of 10.
  if (isBase10) {
    if (d.length > 1) {
      inf = true;
    } else {
      for (k = d[0]; k % 10 === 0;) k /= 10;
      inf = k !== 1;
    }
  }

  external = false;
  sd = pr + guard;
  num = naturalLogarithm(arg, sd);
  denominator = isBase10 ? getLn10(Ctor, sd + 10) : naturalLogarithm(base, sd);

  // The result will have 5 rounding digits.
  r = divide(num, denominator, sd, 1);

  // If at a rounding boundary, i.e. the result's rounding digits are [49]9999 or [50]0000,
  // calculate 10 further digits.
  //
  // If the result is known to have an infinite decimal expansion, repeat this until it is clear
  // that the result is above or below the boundary. Otherwise, if after calculating the 10
  // further digits, the last 14 are nines, round up and assume the result is exact.
  // Also assume the result is exact if the last 14 are zero.
  //
  // Example of a result that will be incorrectly rounded:
  // log[1048576](4503599627370502) = 2.60000000000000009610279511444746...
  // The above result correctly rounded using ROUND_CEIL to 1 decimal place should be 2.7, but it
  // will be given as 2.6 as there are 15 zeros immediately after the requested decimal place, so
  // the exact result would be assumed to be 2.6, which rounded using ROUND_CEIL to 1 decimal
  // place is still 2.6.
  if (checkRoundingDigits(r.d, k = pr, rm)) {

    do {
      sd += 10;
      num = naturalLogarithm(arg, sd);
      denominator = isBase10 ? getLn10(Ctor, sd + 10) : naturalLogarithm(base, sd);
      r = divide(num, denominator, sd, 1);

      if (!inf) {

        // Check for 14 nines from the 2nd rounding digit, as the first may be 4.
        if (+digitsToString(r.d).slice(k + 1, k + 15) + 1 == 1e14) {
          r = finalise(r, pr + 1, 0);
        }

        break;
      }
    } while (checkRoundingDigits(r.d, k += 10, rm));
  }

  external = true;

  return finalise(r, pr, rm);
};


/*
 * Return a new Decimal whose value is the maximum of the arguments and the value of this Decimal.
 *
 * arguments {number|string|Decimal}
 *
P.max = function () {
  Array.prototype.push.call(arguments, this);
  return maxOrMin(this.constructor, arguments, 'lt');
};
 */


/*
 * Return a new Decimal whose value is the minimum of the arguments and the value of this Decimal.
 *
 * arguments {number|string|Decimal}
 *
P.min = function () {
  Array.prototype.push.call(arguments, this);
  return maxOrMin(this.constructor, arguments, 'gt');
};
 */


/*
 *  n - 0 = n
 *  n - N = N
 *  n - I = -I
 *  0 - n = -n
 *  0 - 0 = 0
 *  0 - N = N
 *  0 - I = -I
 *  N - n = N
 *  N - 0 = N
 *  N - N = N
 *  N - I = N
 *  I - n = I
 *  I - 0 = I
 *  I - N = N
 *  I - I = N
 *
 * Return a new Decimal whose value is the value of this Decimal minus `y`, rounded to `precision`
 * significant digits using rounding mode `rounding`.
 *
 */
P.minus = P.sub = function (y) {
  var d, e, i, j, k, len, pr, rm, xd, xe, xLTy, yd,
    x = this,
    Ctor = x.constructor;

  y = new Ctor(y);

  // If either is not finite...
  if (!x.d || !y.d) {

    // Return NaN if either is NaN.
    if (!x.s || !y.s) y = new Ctor(NaN);

    // Return y negated if x is finite and y is ±Infinity.
    else if (x.d) y.s = -y.s;

    // Return x if y is finite and x is ±Infinity.
    // Return x if both are ±Infinity with different signs.
    // Return NaN if both are ±Infinity with the same sign.
    else y = new Ctor(y.d || x.s !== y.s ? x : NaN);

    return y;
  }

  // If signs differ...
  if (x.s != y.s) {
    y.s = -y.s;
    return x.plus(y);
  }

  xd = x.d;
  yd = y.d;
  pr = Ctor.precision;
  rm = Ctor.rounding;

  // If either is zero...
  if (!xd[0] || !yd[0]) {

    // Return y negated if x is zero and y is non-zero.
    if (yd[0]) y.s = -y.s;

    // Return x if y is zero and x is non-zero.
    else if (xd[0]) y = new Ctor(x);

    // Return zero if both are zero.
    // From IEEE 754 (2008) 6.3: 0 - 0 = -0 - -0 = -0 when rounding to -Infinity.
    else return new Ctor(rm === 3 ? -0 : 0);

    return external ? finalise(y, pr, rm) : y;
  }

  // x and y are finite, non-zero numbers with the same sign.

  // Calculate base 1e7 exponents.
  e = mathfloor(y.e / LOG_BASE);
  xe = mathfloor(x.e / LOG_BASE);

  xd = xd.slice();
  k = xe - e;

  // If base 1e7 exponents differ...
  if (k) {
    xLTy = k < 0;

    if (xLTy) {
      d = xd;
      k = -k;
      len = yd.length;
    } else {
      d = yd;
      e = xe;
      len = xd.length;
    }

    // Numbers with massively different exponents would result in a very high number of
    // zeros needing to be prepended, but this can be avoided while still ensuring correct
    // rounding by limiting the number of zeros to `Math.ceil(pr / LOG_BASE) + 2`.
    i = Math.max(Math.ceil(pr / LOG_BASE), len) + 2;

    if (k > i) {
      k = i;
      d.length = 1;
    }

    // Prepend zeros to equalise exponents.
    d.reverse();
    for (i = k; i--;) d.push(0);
    d.reverse();

  // Base 1e7 exponents equal.
  } else {

    // Check digits to determine which is the bigger number.

    i = xd.length;
    len = yd.length;
    xLTy = i < len;
    if (xLTy) len = i;

    for (i = 0; i < len; i++) {
      if (xd[i] != yd[i]) {
        xLTy = xd[i] < yd[i];
        break;
      }
    }

    k = 0;
  }

  if (xLTy) {
    d = xd;
    xd = yd;
    yd = d;
    y.s = -y.s;
  }

  len = xd.length;

  // Append zeros to `xd` if shorter.
  // Don't add zeros to `yd` if shorter as subtraction only needs to start at `yd` length.
  for (i = yd.length - len; i > 0; --i) xd[len++] = 0;

  // Subtract yd from xd.
  for (i = yd.length; i > k;) {

    if (xd[--i] < yd[i]) {
      for (j = i; j && xd[--j] === 0;) xd[j] = BASE - 1;
      --xd[j];
      xd[i] += BASE;
    }

    xd[i] -= yd[i];
  }

  // Remove trailing zeros.
  for (; xd[--len] === 0;) xd.pop();

  // Remove leading zeros and adjust exponent accordingly.
  for (; xd[0] === 0; xd.shift()) --e;

  // Zero?
  if (!xd[0]) return new Ctor(rm === 3 ? -0 : 0);

  y.d = xd;
  y.e = getBase10Exponent(xd, e);

  return external ? finalise(y, pr, rm) : y;
};


/*
 *   n % 0 =  N
 *   n % N =  N
 *   n % I =  n
 *   0 % n =  0
 *  -0 % n = -0
 *   0 % 0 =  N
 *   0 % N =  N
 *   0 % I =  0
 *   N % n =  N
 *   N % 0 =  N
 *   N % N =  N
 *   N % I =  N
 *   I % n =  N
 *   I % 0 =  N
 *   I % N =  N
 *   I % I =  N
 *
 * Return a new Decimal whose value is the value of this Decimal modulo `y`, rounded to
 * `precision` significant digits using rounding mode `rounding`.
 *
 * The result depends on the modulo mode.
 *
 */
P.modulo = P.mod = function (y) {
  var q,
    x = this,
    Ctor = x.constructor;

  y = new Ctor(y);

  // Return NaN if x is ±Infinity or NaN, or y is NaN or ±0.
  if (!x.d || !y.s || y.d && !y.d[0]) return new Ctor(NaN);

  // Return x if y is ±Infinity or x is ±0.
  if (!y.d || x.d && !x.d[0]) {
    return finalise(new Ctor(x), Ctor.precision, Ctor.rounding);
  }

  // Prevent rounding of intermediate calculations.
  external = false;

  if (Ctor.modulo == 9) {

    // Euclidian division: q = sign(y) * floor(x / abs(y))
    // result = x - q * y    where  0 <= result < abs(y)
    q = divide(x, y.abs(), 0, 3, 1);
    q.s *= y.s;
  } else {
    q = divide(x, y, 0, Ctor.modulo, 1);
  }

  q = q.times(y);

  external = true;

  return x.minus(q);
};


/*
 * Return a new Decimal whose value is the natural exponential of the value of this Decimal,
 * i.e. the base e raised to the power the value of this Decimal, rounded to `precision`
 * significant digits using rounding mode `rounding`.
 *
 */
P.naturalExponential = P.exp = function () {
  return naturalExponential(this);
};


/*
 * Return a new Decimal whose value is the natural logarithm of the value of this Decimal,
 * rounded to `precision` significant digits using rounding mode `rounding`.
 *
 */
P.naturalLogarithm = P.ln = function () {
  return naturalLogarithm(this);
};


/*
 * Return a new Decimal whose value is the value of this Decimal negated, i.e. as if multiplied by
 * -1.
 *
 */
P.negated = P.neg = function () {
  var x = new this.constructor(this);
  x.s = -x.s;
  return finalise(x);
};


/*
 *  n + 0 = n
 *  n + N = N
 *  n + I = I
 *  0 + n = n
 *  0 + 0 = 0
 *  0 + N = N
 *  0 + I = I
 *  N + n = N
 *  N + 0 = N
 *  N + N = N
 *  N + I = N
 *  I + n = I
 *  I + 0 = I
 *  I + N = N
 *  I + I = I
 *
 * Return a new Decimal whose value is the value of this Decimal plus `y`, rounded to `precision`
 * significant digits using rounding mode `rounding`.
 *
 */
P.plus = P.add = function (y) {
  var carry, d, e, i, k, len, pr, rm, xd, yd,
    x = this,
    Ctor = x.constructor;

  y = new Ctor(y);

  // If either is not finite...
  if (!x.d || !y.d) {

    // Return NaN if either is NaN.
    if (!x.s || !y.s) y = new Ctor(NaN);

    // Return x if y is finite and x is ±Infinity.
    // Return x if both are ±Infinity with the same sign.
    // Return NaN if both are ±Infinity with different signs.
    // Return y if x is finite and y is ±Infinity.
    else if (!x.d) y = new Ctor(y.d || x.s === y.s ? x : NaN);

    return y;
  }

   // If signs differ...
  if (x.s != y.s) {
    y.s = -y.s;
    return x.minus(y);
  }

  xd = x.d;
  yd = y.d;
  pr = Ctor.precision;
  rm = Ctor.rounding;

  // If either is zero...
  if (!xd[0] || !yd[0]) {

    // Return x if y is zero.
    // Return y if y is non-zero.
    if (!yd[0]) y = new Ctor(x);

    return external ? finalise(y, pr, rm) : y;
  }

  // x and y are finite, non-zero numbers with the same sign.

  // Calculate base 1e7 exponents.
  k = mathfloor(x.e / LOG_BASE);
  e = mathfloor(y.e / LOG_BASE);

  xd = xd.slice();
  i = k - e;

  // If base 1e7 exponents differ...
  if (i) {

    if (i < 0) {
      d = xd;
      i = -i;
      len = yd.length;
    } else {
      d = yd;
      e = k;
      len = xd.length;
    }

    // Limit number of zeros prepended to max(ceil(pr / LOG_BASE), len) + 1.
    k = Math.ceil(pr / LOG_BASE);
    len = k > len ? k + 1 : len + 1;

    if (i > len) {
      i = len;
      d.length = 1;
    }

    // Prepend zeros to equalise exponents. Note: Faster to use reverse then do unshifts.
    d.reverse();
    for (; i--;) d.push(0);
    d.reverse();
  }

  len = xd.length;
  i = yd.length;

  // If yd is longer than xd, swap xd and yd so xd points to the longer array.
  if (len - i < 0) {
    i = len;
    d = yd;
    yd = xd;
    xd = d;
  }

  // Only start adding at yd.length - 1 as the further digits of xd can be left as they are.
  for (carry = 0; i;) {
    carry = (xd[--i] = xd[i] + yd[i] + carry) / BASE | 0;
    xd[i] %= BASE;
  }

  if (carry) {
    xd.unshift(carry);
    ++e;
  }

  // Remove trailing zeros.
  // No need to check for zero, as +x + +y != 0 && -x + -y != 0
  for (len = xd.length; xd[--len] == 0;) xd.pop();

  y.d = xd;
  y.e = getBase10Exponent(xd, e);

  return external ? finalise(y, pr, rm) : y;
};


/*
 * Return the number of significant digits of the value of this Decimal.
 *
 * [z] {boolean|number} Whether to count integer-part trailing zeros: true, false, 1 or 0.
 *
 */
P.precision = P.sd = function (z) {
  var k,
    x = this;

  if (z !== void 0 && z !== !!z && z !== 1 && z !== 0) throw Error(invalidArgument + z);

  if (x.d) {
    k = getPrecision(x.d);
    if (z && x.e + 1 > k) k = x.e + 1;
  } else {
    k = NaN;
  }

  return k;
};


/*
 * Return a new Decimal whose value is the value of this Decimal rounded to a whole number using
 * rounding mode `rounding`.
 *
 */
P.round = function () {
  var x = this,
    Ctor = x.constructor;

  return finalise(new Ctor(x), x.e + 1, Ctor.rounding);
};


/*
 * Return a new Decimal whose value is the sine of the value in radians of this Decimal.
 *
 * Domain: [-Infinity, Infinity]
 * Range: [-1, 1]
 *
 * sin(x) = x - x^3/3! + x^5/5! - ...
 *
 * sin(0)         = 0
 * sin(-0)        = -0
 * sin(Infinity)  = NaN
 * sin(-Infinity) = NaN
 * sin(NaN)       = NaN
 *
 */
P.sine = P.sin = function () {
  var pr, rm,
    x = this,
    Ctor = x.constructor;

  if (!x.isFinite()) return new Ctor(NaN);
  if (x.isZero()) return new Ctor(x);

  pr = Ctor.precision;
  rm = Ctor.rounding;
  Ctor.precision = pr + Math.max(x.e, x.sd()) + LOG_BASE;
  Ctor.rounding = 1;

  x = sine(Ctor, toLessThanHalfPi(Ctor, x));

  Ctor.precision = pr;
  Ctor.rounding = rm;

  return finalise(quadrant > 2 ? x.neg() : x, pr, rm, true);
};


/*
 * Return a new Decimal whose value is the square root of this Decimal, rounded to `precision`
 * significant digits using rounding mode `rounding`.
 *
 *  sqrt(-n) =  N
 *  sqrt(N)  =  N
 *  sqrt(-I) =  N
 *  sqrt(I)  =  I
 *  sqrt(0)  =  0
 *  sqrt(-0) = -0
 *
 */
P.squareRoot = P.sqrt = function () {
  var m, n, sd, r, rep, t,
    x = this,
    d = x.d,
    e = x.e,
    s = x.s,
    Ctor = x.constructor;

  // Negative/NaN/Infinity/zero?
  if (s !== 1 || !d || !d[0]) {
    return new Ctor(!s || s < 0 && (!d || d[0]) ? NaN : d ? x : 1 / 0);
  }

  external = false;

  // Initial estimate.
  s = Math.sqrt(+x);

  // Math.sqrt underflow/overflow?
  // Pass x to Math.sqrt as integer, then adjust the exponent of the result.
  if (s == 0 || s == 1 / 0) {
    n = digitsToString(d);

    if ((n.length + e) % 2 == 0) n += '0';
    s = Math.sqrt(n);
    e = mathfloor((e + 1) / 2) - (e < 0 || e % 2);

    if (s == 1 / 0) {
      n = '5e' + e;
    } else {
      n = s.toExponential();
      n = n.slice(0, n.indexOf('e') + 1) + e;
    }

    r = new Ctor(n);
  } else {
    r = new Ctor(s.toString());
  }

  sd = (e = Ctor.precision) + 3;

  // Newton-Raphson iteration.
  for (;;) {
    t = r;
    r = t.plus(divide(x, t, sd + 2, 1)).times(0.5);

    // TODO? Replace with for-loop and checkRoundingDigits.
    if (digitsToString(t.d).slice(0, sd) === (n = digitsToString(r.d)).slice(0, sd)) {
      n = n.slice(sd - 3, sd + 1);

      // The 4th rounding digit may be in error by -1 so if the 4 rounding digits are 9999 or
      // 4999, i.e. approaching a rounding boundary, continue the iteration.
      if (n == '9999' || !rep && n == '4999') {

        // On the first iteration only, check to see if rounding up gives the exact result as the
        // nines may infinitely repeat.
        if (!rep) {
          finalise(t, e + 1, 0);

          if (t.times(t).eq(x)) {
            r = t;
            break;
          }
        }

        sd += 4;
        rep = 1;
      } else {

        // If the rounding digits are null, 0{0,4} or 50{0,3}, check for an exact result.
        // If not, then there are further digits and m will be truthy.
        if (!+n || !+n.slice(1) && n.charAt(0) == '5') {

          // Truncate to the first rounding digit.
          finalise(r, e + 1, 1);
          m = !r.times(r).eq(x);
        }

        break;
      }
    }
  }

  external = true;

  return finalise(r, e, Ctor.rounding, m);
};


/*
 * Return a new Decimal whose value is the tangent of the value in radians of this Decimal.
 *
 * Domain: [-Infinity, Infinity]
 * Range: [-Infinity, Infinity]
 *
 * tan(0)         = 0
 * tan(-0)        = -0
 * tan(Infinity)  = NaN
 * tan(-Infinity) = NaN
 * tan(NaN)       = NaN
 *
 */
P.tangent = P.tan = function () {
  var pr, rm,
    x = this,
    Ctor = x.constructor;

  if (!x.isFinite()) return new Ctor(NaN);
  if (x.isZero()) return new Ctor(x);

  pr = Ctor.precision;
  rm = Ctor.rounding;
  Ctor.precision = pr + 10;
  Ctor.rounding = 1;

  x = x.sin();
  x.s = 1;
  x = divide(x, new Ctor(1).minus(x.times(x)).sqrt(), pr + 10, 0);

  Ctor.precision = pr;
  Ctor.rounding = rm;

  return finalise(quadrant == 2 || quadrant == 4 ? x.neg() : x, pr, rm, true);
};


/*
 *  n * 0 = 0
 *  n * N = N
 *  n * I = I
 *  0 * n = 0
 *  0 * 0 = 0
 *  0 * N = N
 *  0 * I = N
 *  N * n = N
 *  N * 0 = N
 *  N * N = N
 *  N * I = N
 *  I * n = I
 *  I * 0 = N
 *  I * N = N
 *  I * I = I
 *
 * Return a new Decimal whose value is this Decimal times `y`, rounded to `precision` significant
 * digits using rounding mode `rounding`.
 *
 */
P.times = P.mul = function (y) {
  var carry, e, i, k, r, rL, t, xdL, ydL,
    x = this,
    Ctor = x.constructor,
    xd = x.d,
    yd = (y = new Ctor(y)).d;

  y.s *= x.s;

   // If either is NaN, ±Infinity or ±0...
  if (!xd || !xd[0] || !yd || !yd[0]) {

    return new Ctor(!y.s || xd && !xd[0] && !yd || yd && !yd[0] && !xd

      // Return NaN if either is NaN.
      // Return NaN if x is ±0 and y is ±Infinity, or y is ±0 and x is ±Infinity.
      ? NaN

      // Return ±Infinity if either is ±Infinity.
      // Return ±0 if either is ±0.
      : !xd || !yd ? y.s / 0 : y.s * 0);
  }

  e = mathfloor(x.e / LOG_BASE) + mathfloor(y.e / LOG_BASE);
  xdL = xd.length;
  ydL = yd.length;

  // Ensure xd points to the longer array.
  if (xdL < ydL) {
    r = xd;
    xd = yd;
    yd = r;
    rL = xdL;
    xdL = ydL;
    ydL = rL;
  }

  // Initialise the result array with zeros.
  r = [];
  rL = xdL + ydL;
  for (i = rL; i--;) r.push(0);

  // Multiply!
  for (i = ydL; --i >= 0;) {
    carry = 0;
    for (k = xdL + i; k > i;) {
      t = r[k] + yd[i] * xd[k - i - 1] + carry;
      r[k--] = t % BASE | 0;
      carry = t / BASE | 0;
    }

    r[k] = (r[k] + carry) % BASE | 0;
  }

  // Remove trailing zeros.
  for (; !r[--rL];) r.pop();

  if (carry) ++e;
  else r.shift();

  y.d = r;
  y.e = getBase10Exponent(r, e);

  return external ? finalise(y, Ctor.precision, Ctor.rounding) : y;
};


/*
 * Return a string representing the value of this Decimal in base 2, round to `sd` significant
 * digits using rounding mode `rm`.
 *
 * If the optional `sd` argument is present then return binary exponential notation.
 *
 * [sd] {number} Significant digits. Integer, 1 to MAX_DIGITS inclusive.
 * [rm] {number} Rounding mode. Integer, 0 to 8 inclusive.
 *
 */
P.toBinary = function (sd, rm) {
  return toStringBinary(this, 2, sd, rm);
};


/*
 * Return a new Decimal whose value is the value of this Decimal rounded to a maximum of `dp`
 * decimal places using rounding mode `rm` or `rounding` if `rm` is omitted.
 *
 * If `dp` is omitted, return a new Decimal whose value is the value of this Decimal.
 *
 * [dp] {number} Decimal places. Integer, 0 to MAX_DIGITS inclusive.
 * [rm] {number} Rounding mode. Integer, 0 to 8 inclusive.
 *
 */
P.toDecimalPlaces = P.toDP = function (dp, rm) {
  var x = this,
    Ctor = x.constructor;

  x = new Ctor(x);
  if (dp === void 0) return x;

  checkInt32(dp, 0, MAX_DIGITS);

  if (rm === void 0) rm = Ctor.rounding;
  else checkInt32(rm, 0, 8);

  return finalise(x, dp + x.e + 1, rm);
};


/*
 * Return a string representing the value of this Decimal in exponential notation rounded to
 * `dp` fixed decimal places using rounding mode `rounding`.
 *
 * [dp] {number} Decimal places. Integer, 0 to MAX_DIGITS inclusive.
 * [rm] {number} Rounding mode. Integer, 0 to 8 inclusive.
 *
 */
P.toExponential = function (dp, rm) {
  var str,
    x = this,
    Ctor = x.constructor;

  if (dp === void 0) {
    str = finiteToString(x, true);
  } else {
    checkInt32(dp, 0, MAX_DIGITS);

    if (rm === void 0) rm = Ctor.rounding;
    else checkInt32(rm, 0, 8);

    x = finalise(new Ctor(x), dp + 1, rm);
    str = finiteToString(x, true, dp + 1);
  }

  return x.isNeg() && !x.isZero() ? '-' + str : str;
};


/*
 * Return a string representing the value of this Decimal in normal (fixed-point) notation to
 * `dp` fixed decimal places and rounded using rounding mode `rm` or `rounding` if `rm` is
 * omitted.
 *
 * As with JavaScript numbers, (-0).toFixed(0) is '0', but e.g. (-0.00001).toFixed(0) is '-0'.
 *
 * [dp] {number} Decimal places. Integer, 0 to MAX_DIGITS inclusive.
 * [rm] {number} Rounding mode. Integer, 0 to 8 inclusive.
 *
 * (-0).toFixed(0) is '0', but (-0.1).toFixed(0) is '-0'.
 * (-0).toFixed(1) is '0.0', but (-0.01).toFixed(1) is '-0.0'.
 * (-0).toFixed(3) is '0.000'.
 * (-0.5).toFixed(0) is '-0'.
 *
 */
P.toFixed = function (dp, rm) {
  var str, y,
    x = this,
    Ctor = x.constructor;

  if (dp === void 0) {
    str = finiteToString(x);
  } else {
    checkInt32(dp, 0, MAX_DIGITS);

    if (rm === void 0) rm = Ctor.rounding;
    else checkInt32(rm, 0, 8);

    y = finalise(new Ctor(x), dp + x.e + 1, rm);
    str = finiteToString(y, false, dp + y.e + 1);
  }

  // To determine whether to add the minus sign look at the value before it was rounded,
  // i.e. look at `x` rather than `y`.
  return x.isNeg() && !x.isZero() ? '-' + str : str;
};


/*
 * Return an array representing the value of this Decimal as a simple fraction with an integer
 * numerator and an integer denominator.
 *
 * The denominator will be a positive non-zero value less than or equal to the specified maximum
 * denominator. If a maximum denominator is not specified, the denominator will be the lowest
 * value necessary to represent the number exactly.
 *
 * [maxD] {number|string|Decimal} Maximum denominator. Integer >= 1 and < Infinity.
 *
 */
P.toFraction = function (maxD) {
  var d, d0, d1, d2, e, k, n, n0, n1, pr, q, r,
    x = this,
    xd = x.d,
    Ctor = x.constructor;

  if (!xd) return new Ctor(x);

  n1 = d0 = new Ctor(1);
  d1 = n0 = new Ctor(0);

  d = new Ctor(d1);
  e = d.e = getPrecision(xd) - x.e - 1;
  k = e % LOG_BASE;
  d.d[0] = mathpow(10, k < 0 ? LOG_BASE + k : k);

  if (maxD == null) {

    // d is 10**e, the minimum max-denominator needed.
    maxD = e > 0 ? d : n1;
  } else {
    n = new Ctor(maxD);
    if (!n.isInt() || n.lt(n1)) throw Error(invalidArgument + n);
    maxD = n.gt(d) ? (e > 0 ? d : n1) : n;
  }

  external = false;
  n = new Ctor(digitsToString(xd));
  pr = Ctor.precision;
  Ctor.precision = e = xd.length * LOG_BASE * 2;

  for (;;)  {
    q = divide(n, d, 0, 1, 1);
    d2 = d0.plus(q.times(d1));
    if (d2.cmp(maxD) == 1) break;
    d0 = d1;
    d1 = d2;
    d2 = n1;
    n1 = n0.plus(q.times(d2));
    n0 = d2;
    d2 = d;
    d = n.minus(q.times(d2));
    n = d2;
  }

  d2 = divide(maxD.minus(d0), d1, 0, 1, 1);
  n0 = n0.plus(d2.times(n1));
  d0 = d0.plus(d2.times(d1));
  n0.s = n1.s = x.s;

  // Determine which fraction is closer to x, n0/d0 or n1/d1?
  r = divide(n1, d1, e, 1).minus(x).abs().cmp(divide(n0, d0, e, 1).minus(x).abs()) < 1
      ? [n1, d1] : [n0, d0];

  Ctor.precision = pr;
  external = true;

  return r;
};


/*
 * Return a string representing the value of this Decimal in base 16, round to `sd` significant
 * digits using rounding mode `rm`.
 *
 * If the optional `sd` argument is present then return binary exponential notation.
 *
 * [sd] {number} Significant digits. Integer, 1 to MAX_DIGITS inclusive.
 * [rm] {number} Rounding mode. Integer, 0 to 8 inclusive.
 *
 */
P.toHexadecimal = P.toHex = function (sd, rm) {
  return toStringBinary(this, 16, sd, rm);
};


/*
 * Returns a new Decimal whose value is the nearest multiple of `y` in the direction of rounding
 * mode `rm`, or `Decimal.rounding` if `rm` is omitted, to the value of this Decimal.
 *
 * The return value will always have the same sign as this Decimal, unless either this Decimal
 * or `y` is NaN, in which case the return value will be also be NaN.
 *
 * The return value is not affected by the value of `precision`.
 *
 * y {number|string|Decimal} The magnitude to round to a multiple of.
 * [rm] {number} Rounding mode. Integer, 0 to 8 inclusive.
 *
 * 'toNearest() rounding mode not an integer: {rm}'
 * 'toNearest() rounding mode out of range: {rm}'
 *
 */
P.toNearest = function (y, rm) {
  var x = this,
    Ctor = x.constructor;

  x = new Ctor(x);

  if (y == null) {

    // If x is not finite, return x.
    if (!x.d) return x;

    y = new Ctor(1);
    rm = Ctor.rounding;
  } else {
    y = new Ctor(y);
    if (rm === void 0) {
      rm = Ctor.rounding;
    } else {
      checkInt32(rm, 0, 8);
    }

    // If x is not finite, return x if y is not NaN, else NaN.
    if (!x.d) return y.s ? x : y;

    // If y is not finite, return Infinity with the sign of x if y is Infinity, else NaN.
    if (!y.d) {
      if (y.s) y.s = x.s;
      return y;
    }
  }

  // If y is not zero, calculate the nearest multiple of y to x.
  if (y.d[0]) {
    external = false;
    x = divide(x, y, 0, rm, 1).times(y);
    external = true;
    finalise(x);

  // If y is zero, return zero with the sign of x.
  } else {
    y.s = x.s;
    x = y;
  }

  return x;
};


/*
 * Return the value of this Decimal converted to a number primitive.
 * Zero keeps its sign.
 *
 */
P.toNumber = function () {
  return +this;
};


/*
 * Return a string representing the value of this Decimal in base 8, round to `sd` significant
 * digits using rounding mode `rm`.
 *
 * If the optional `sd` argument is present then return binary exponential notation.
 *
 * [sd] {number} Significant digits. Integer, 1 to MAX_DIGITS inclusive.
 * [rm] {number} Rounding mode. Integer, 0 to 8 inclusive.
 *
 */
P.toOctal = function (sd, rm) {
  return toStringBinary(this, 8, sd, rm);
};


/*
 * Return a new Decimal whose value is the value of this Decimal raised to the power `y`, rounded
 * to `precision` significant digits using rounding mode `rounding`.
 *
 * ECMAScript compliant.
 *
 *   pow(x, NaN)                           = NaN
 *   pow(x, ±0)                            = 1

 *   pow(NaN, non-zero)                    = NaN
 *   pow(abs(x) > 1, +Infinity)            = +Infinity
 *   pow(abs(x) > 1, -Infinity)            = +0
 *   pow(abs(x) == 1, ±Infinity)           = NaN
 *   pow(abs(x) < 1, +Infinity)            = +0
 *   pow(abs(x) < 1, -Infinity)            = +Infinity
 *   pow(+Infinity, y > 0)                 = +Infinity
 *   pow(+Infinity, y < 0)                 = +0
 *   pow(-Infinity, odd integer > 0)       = -Infinity
 *   pow(-Infinity, even integer > 0)      = +Infinity
 *   pow(-Infinity, odd integer < 0)       = -0
 *   pow(-Infinity, even integer < 0)      = +0
 *   pow(+0, y > 0)                        = +0
 *   pow(+0, y < 0)                        = +Infinity
 *   pow(-0, odd integer > 0)              = -0
 *   pow(-0, even integer > 0)             = +0
 *   pow(-0, odd integer < 0)              = -Infinity
 *   pow(-0, even integer < 0)             = +Infinity
 *   pow(finite x < 0, finite non-integer) = NaN
 *
 * For non-integer or very large exponents pow(x, y) is calculated using
 *
 *   x^y = exp(y*ln(x))
 *
 * Assuming the first 15 rounding digits are each equally likely to be any digit 0-9, the
 * probability of an incorrectly rounded result
 * P([49]9{14} | [50]0{14}) = 2 * 0.2 * 10^-14 = 4e-15 = 1/2.5e+14
 * i.e. 1 in 250,000,000,000,000
 *
 * If a result is incorrectly rounded the maximum error will be 1 ulp (unit in last place).
 *
 * y {number|string|Decimal} The power to which to raise this Decimal.
 *
 */
P.toPower = P.pow = function (y) {
  var e, k, pr, r, rm, s,
    x = this,
    Ctor = x.constructor,
    yn = +(y = new Ctor(y));

  // Either ±Infinity, NaN or ±0?
  if (!x.d || !y.d || !x.d[0] || !y.d[0]) return new Ctor(mathpow(+x, yn));

  x = new Ctor(x);

  if (x.eq(1)) return x;

  pr = Ctor.precision;
  rm = Ctor.rounding;

  if (y.eq(1)) return finalise(x, pr, rm);

  // y exponent
  e = mathfloor(y.e / LOG_BASE);

  // If y is a small integer use the 'exponentiation by squaring' algorithm.
  if (e >= y.d.length - 1 && (k = yn < 0 ? -yn : yn) <= MAX_SAFE_INTEGER) {
    r = intPow(Ctor, x, k, pr);
    return y.s < 0 ? new Ctor(1).div(r) : finalise(r, pr, rm);
  }

  s = x.s;

  // if x is negative
  if (s < 0) {

    // if y is not an integer
    if (e < y.d.length - 1) return new Ctor(NaN);

    // Result is positive if x is negative and the last digit of integer y is even.
    if ((y.d[e] & 1) == 0) s = 1;

    // if x.eq(-1)
    if (x.e == 0 && x.d[0] == 1 && x.d.length == 1) {
      x.s = s;
      return x;
    }
  }

  // Estimate result exponent.
  // x^y = 10^e,  where e = y * log10(x)
  // log10(x) = log10(x_significand) + x_exponent
  // log10(x_significand) = ln(x_significand) / ln(10)
  k = mathpow(+x, yn);
  e = k == 0 || !isFinite(k)
    ? mathfloor(yn * (Math.log('0.' + digitsToString(x.d)) / Math.LN10 + x.e + 1))
    : new Ctor(k + '').e;

  // Exponent estimate may be incorrect e.g. x: 0.999999999999999999, y: 2.29, e: 0, r.e: -1.

  // Overflow/underflow?
  if (e > Ctor.maxE + 1 || e < Ctor.minE - 1) return new Ctor(e > 0 ? s / 0 : 0);

  external = false;
  Ctor.rounding = x.s = 1;

  // Estimate the extra guard digits needed to ensure five correct rounding digits from
  // naturalLogarithm(x). Example of failure without these extra digits (precision: 10):
  // new Decimal(2.32456).pow('2087987436534566.46411')
  // should be 1.162377823e+764914905173815, but is 1.162355823e+764914905173815
  k = Math.min(12, (e + '').length);

  // r = x^y = exp(y*ln(x))
  r = naturalExponential(y.times(naturalLogarithm(x, pr + k)), pr);

  // r may be Infinity, e.g. (0.9999999999999999).pow(-1e+40)
  if (r.d) {

    // Truncate to the required precision plus five rounding digits.
    r = finalise(r, pr + 5, 1);

    // If the rounding digits are [49]9999 or [50]0000 increase the precision by 10 and recalculate
    // the result.
    if (checkRoundingDigits(r.d, pr, rm)) {
      e = pr + 10;

      // Truncate to the increased precision plus five rounding digits.
      r = finalise(naturalExponential(y.times(naturalLogarithm(x, e + k)), e), e + 5, 1);

      // Check for 14 nines from the 2nd rounding digit (the first rounding digit may be 4 or 9).
      if (+digitsToString(r.d).slice(pr + 1, pr + 15) + 1 == 1e14) {
        r = finalise(r, pr + 1, 0);
      }
    }
  }

  r.s = s;
  external = true;
  Ctor.rounding = rm;

  return finalise(r, pr, rm);
};


/*
 * Return a string representing the value of this Decimal rounded to `sd` significant digits
 * using rounding mode `rounding`.
 *
 * Return exponential notation if `sd` is less than the number of digits necessary to represent
 * the integer part of the value in normal notation.
 *
 * [sd] {number} Significant digits. Integer, 1 to MAX_DIGITS inclusive.
 * [rm] {number} Rounding mode. Integer, 0 to 8 inclusive.
 *
 */
P.toPrecision = function (sd, rm) {
  var str,
    x = this,
    Ctor = x.constructor;

  if (sd === void 0) {
    str = finiteToString(x, x.e <= Ctor.toExpNeg || x.e >= Ctor.toExpPos);
  } else {
    checkInt32(sd, 1, MAX_DIGITS);

    if (rm === void 0) rm = Ctor.rounding;
    else checkInt32(rm, 0, 8);

    x = finalise(new Ctor(x), sd, rm);
    str = finiteToString(x, sd <= x.e || x.e <= Ctor.toExpNeg, sd);
  }

  return x.isNeg() && !x.isZero() ? '-' + str : str;
};


/*
 * Return a new Decimal whose value is the value of this Decimal rounded to a maximum of `sd`
 * significant digits using rounding mode `rm`, or to `precision` and `rounding` respectively if
 * omitted.
 *
 * [sd] {number} Significant digits. Integer, 1 to MAX_DIGITS inclusive.
 * [rm] {number} Rounding mode. Integer, 0 to 8 inclusive.
 *
 * 'toSD() digits out of range: {sd}'
 * 'toSD() digits not an integer: {sd}'
 * 'toSD() rounding mode not an integer: {rm}'
 * 'toSD() rounding mode out of range: {rm}'
 *
 */
P.toSignificantDigits = P.toSD = function (sd, rm) {
  var x = this,
    Ctor = x.constructor;

  if (sd === void 0) {
    sd = Ctor.precision;
    rm = Ctor.rounding;
  } else {
    checkInt32(sd, 1, MAX_DIGITS);

    if (rm === void 0) rm = Ctor.rounding;
    else checkInt32(rm, 0, 8);
  }

  return finalise(new Ctor(x), sd, rm);
};


/*
 * Return a string representing the value of this Decimal.
 *
 * Return exponential notation if this Decimal has a positive exponent equal to or greater than
 * `toExpPos`, or a negative exponent equal to or less than `toExpNeg`.
 *
 */
P.toString = function () {
  var x = this,
    Ctor = x.constructor,
    str = finiteToString(x, x.e <= Ctor.toExpNeg || x.e >= Ctor.toExpPos);

  return x.isNeg() && !x.isZero() ? '-' + str : str;
};


/*
 * Return a new Decimal whose value is the value of this Decimal truncated to a whole number.
 *
 */
P.truncated = P.trunc = function () {
  return finalise(new this.constructor(this), this.e + 1, 1);
};


/*
 * Return a string representing the value of this Decimal.
 * Unlike `toString`, negative zero will include the minus sign.
 *
 */
P.valueOf = P.toJSON = function () {
  var x = this,
    Ctor = x.constructor,
    str = finiteToString(x, x.e <= Ctor.toExpNeg || x.e >= Ctor.toExpPos);

  return x.isNeg() ? '-' + str : str;
};


/*
// 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 = (1 / tinyPow(4, k)).toString();
  } else {
    k = 16;
    y = '2.3283064365386962890625e-10';
  }

  Ctor.precision += k;

  x = taylorSeries(Ctor, 1, x.times(y), new Ctor(1));

  // Reverse argument reduction
  for (var i = k; i--;) {
    var cos2x = x.times(x);
    x = cos2x.times(cos2x).minus(cos2x).times(8).plus(1);
  }

  Ctor.precision -= k;

  return x;
}


/*
 * Perform division in the specified base.
 */
var divide = (function () {

  // Assumes non-zero x and k, and hence non-zero result.
  function multiplyInteger(x, k, base) {
    var temp,
      carry = 0,
      i = x.length;

    for (x = x.slice(); i--;) {
      temp = x[i] * k + carry;
      x[i] = temp % base | 0;
      carry = temp / base | 0;
    }

    if (carry) x.unshift(carry);

    return x;
  }

  function compare(a, b, aL, bL) {
    var i, r;

    if (aL != bL) {
      r = aL > bL ? 1 : -1;
    } else {
      for (i = r = 0; i < aL; i++) {
        if (a[i] != b[i]) {
          r = a[i] > b[i] ? 1 : -1;
          break;
        }
      }
    }

    return r;
  }

  function subtract(a, b, aL, base) {
    var i = 0;

    // Subtract b from a.
    for (; aL--;) {
      a[aL] -= i;
      i = a[aL] < b[aL] ? 1 : 0;
      a[aL] = i * base + a[aL] - b[aL];
    }

    // Remove leading zeros.
    for (; !a[0] && a.length > 1;) a.shift();
  }

  return function (x, y, pr, rm, dp, base) {
    var cmp, e, i, k, logBase, more, prod, prodL, q, qd, rem, remL, rem0, sd, t, xi, xL, yd0,
      yL, yz,
      Ctor = x.constructor,
      sign = x.s == y.s ? 1 : -1,
      xd = x.d,
      yd = y.d;

    // Either NaN, Infinity or 0?
    if (!xd || !xd[0] || !yd || !yd[0]) {

      return new Ctor(// Return NaN if either NaN, or both Infinity or 0.
        !x.s || !y.s || (xd ? yd && xd[0] == yd[0] : !yd) ? NaN :

        // Return ±0 if x is 0 or y is ±Infinity, or return ±Infinity as y is 0.
        xd && xd[0] == 0 || !yd ? sign * 0 : sign / 0);
    }

    if (base) {
      logBase = 1;
      e = x.e - y.e;
    } else {
      base = BASE;
      logBase = LOG_BASE;
      e = mathfloor(x.e / logBase) - mathfloor(y.e / logBase);
    }

    yL = yd.length;
    xL = xd.length;
    q = new Ctor(sign);
    qd = q.d = [];

    // Result exponent may be one less than e.
    // The digit array of a Decimal from toStringBinary may have trailing zeros.
    for (i = 0; yd[i] == (xd[i] || 0); i++);

    if (yd[i] > (xd[i] || 0)) e--;

    if (pr == null) {
      sd = pr = Ctor.precision;
      rm = Ctor.rounding;
    } else if (dp) {
      sd = pr + (x.e - y.e) + 1;
    } else {
      sd = pr;
    }

    if (sd < 0) {
      qd.push(1);
      more = true;
    } else {

      // Convert precision in number of base 10 digits to base 1e7 digits.
      sd = sd / logBase + 2 | 0;
      i = 0;

      // divisor < 1e7
      if (yL == 1) {
        k = 0;
        yd = yd[0];
        sd++;

        // k is the carry.
        for (; (i < xL || k) && sd--; i++) {
          t = k * base + (xd[i] || 0);
          qd[i] = t / yd | 0;
          k = t % yd | 0;
        }

        more = k || i < xL;

      // divisor >= 1e7
      } else {

        // Normalise xd and yd so highest order digit of yd is >= base/2
        k = base / (yd[0] + 1) | 0;

        if (k > 1) {
          yd = multiplyInteger(yd, k, base);
          xd = multiplyInteger(xd, k, base);
          yL = yd.length;
          xL = xd.length;
        }

        xi = yL;
        rem = xd.slice(0, yL);
        remL = rem.length;

        // Add zeros to make remainder as long as divisor.
        for (; remL < yL;) rem[remL++] = 0;

        yz = yd.slice();
        yz.unshift(0);
        yd0 = yd[0];

        if (yd[1] >= base / 2) ++yd0;

        do {
          k = 0;

          // Compare divisor and remainder.
          cmp = compare(yd, rem, yL, remL);

          // If divisor < remainder.
          if (cmp < 0) {

            // Calculate trial digit, k.
            rem0 = rem[0];
            if (yL != remL) rem0 = rem0 * base + (rem[1] || 0);

            // k will be how many times the divisor goes into the current remainder.
            k = rem0 / yd0 | 0;

            //  Algorithm:
            //  1. product = divisor * trial digit (k)
            //  2. if product > remainder: product -= divisor, k--
            //  3. remainder -= product
            //  4. if product was < remainder at 2:
            //    5. compare new remainder and divisor
            //    6. If remainder > divisor: remainder -= divisor, k++

            if (k > 1) {
              if (k >= base) k = base - 1;

              // product = divisor * trial digit.
              prod = multiplyInteger(yd, k, base);
              prodL = prod.length;
              remL = rem.length;

              // Compare product and remainder.
              cmp = compare(prod, rem, prodL, remL);

              // product > remainder.
              if (cmp == 1) {
                k--;

                // Subtract divisor from product.
                subtract(prod, yL < prodL ? yz : yd, prodL, base);
              }
            } else {

              // cmp is -1.
              // If k is 0, there is no need to compare yd and rem again below, so change cmp to 1
              // to avoid it. If k is 1 there is a need to compare yd and rem again below.
              if (k == 0) cmp = k = 1;
              prod = yd.slice();
            }

            prodL = prod.length;
            if (prodL < remL) prod.unshift(0);

            // Subtract product from remainder.
            subtract(rem, prod, remL, base);

            // If product was < previous remainder.
            if (cmp == -1) {
              remL = rem.length;

              // Compare divisor and new remainder.
              cmp = compare(yd, rem, yL, remL);

              // If divisor < new remainder, subtract divisor from remainder.
              if (cmp < 1) {
                k++;

                // Subtract divisor from remainder.
                subtract(rem, yL < remL ? yz : yd, remL, base);
              }
            }

            remL = rem.length;
          } else if (cmp === 0) {
            k++;
            rem = [0];
          }    // if cmp === 1, k will be 0

          // Add the next digit, k, to the result array.
          qd[i++] = k;

          // Update the remainder.
          if (cmp && rem[0]) {
            rem[remL++] = xd[xi] || 0;
          } else {
            rem = [xd[xi]];
            remL = 1;
          }

        } while ((xi++ < xL || rem[0] !== void 0) && sd--);

        more = rem[0] !== void 0;
      }

      // Leading zero?
      if (!qd[0]) qd.shift();
    }

    // logBase is 1 when divide is being used for base conversion.
    if (logBase == 1) {
      q.e = e;
      inexact = more;
    } else {

      // To calculate q.e, first get the number of digits of qd[0].
      for (i = 1, k = qd[0]; k >= 10; k /= 10) i++;
      q.e = i + e * logBase - 1;

      finalise(q, dp ? pr + q.e + 1 : pr, rm, more);
    }

    return q;
  };
})();


/*
 * Round `x` to `sd` significant digits using rounding mode `rm`.
 * Check for over/under-flow.
 */
 function finalise(x, sd, rm, isTruncated) {
  var digits, i, j, k, rd, roundUp, w, xd, xdi,
    Ctor = x.constructor;

  // Don't round if sd is null or undefined.
  out: if (sd != null) {
    xd = x.d;

    // Infinity/NaN.
    if (!xd) return x;

    // rd: the rounding digit, i.e. the digit after the digit that may be rounded up.
    // w: the word of xd containing rd, a base 1e7 number.
    // xdi: the index of w within xd.
    // digits: the number of digits of w.
    // i: what would be the index of rd within w if all the numbers were 7 digits long (i.e. if
    // they had leading zeros)
    // j: if > 0, the actual index of rd within w (if < 0, rd is a leading zero).

    // Get the length of the first word of the digits array xd.
    for (digits = 1, k = xd[0]; k >= 10; k /= 10) digits++;
    i = sd - digits;

    // Is the rounding digit in the first word of xd?
    if (i < 0) {
      i += LOG_BASE;
      j = sd;
      w = xd[xdi = 0];

      // Get the rounding digit at index j of w.
      rd = w / mathpow(10, digits - j - 1) % 10 | 0;
    } else {
      xdi = Math.ceil((i + 1) / LOG_BASE);
      k = xd.length;
      if (xdi >= k) {
        if (isTruncated) {

          // Needed by `naturalExponential`, `naturalLogarithm` and `squareRoot`.
          for (; k++ <= xdi;) xd.push(0);
          w = rd = 0;
          digits = 1;
          i %= LOG_BASE;
          j = i - LOG_BASE + 1;
        } else {
          break out;
        }
      } else {
        w = k = xd[xdi];

        // Get the number of digits of w.
        for (digits = 1; k >= 10; k /= 10) digits++;

        // Get the index of rd within w.
        i %= LOG_BASE;

        // Get the index of rd within w, adjusted for leading zeros.
        // The number of leading zeros of w is given by LOG_BASE - digits.
        j = i - LOG_BASE + digits;

        // Get the rounding digit at index j of w.
        rd = j < 0 ? 0 : w / mathpow(10, digits - j - 1) % 10 | 0;
      }
    }

    // Are there any non-zero digits after the rounding digit?
    isTruncated = isTruncated || sd < 0 ||
      xd[xdi + 1] !== void 0 || (j < 0 ? w : w % mathpow(10, digits - j - 1));

    // The expression `w % mathpow(10, digits - j - 1)` returns all the digits of w to the right
    // of the digit at (left-to-right) index j, e.g. if w is 908714 and j is 2, the expression
    // will give 714.

    roundUp = rm < 4
      ? (rd || isTruncated) && (rm == 0 || rm == (x.s < 0 ? 3 : 2))
      : rd > 5 || rd == 5 && (rm == 4 || isTruncated || rm == 6 &&

        // Check whether the digit to the left of the rounding digit is odd.
        ((i > 0 ? j > 0 ? w / mathpow(10, digits - j) : 0 : xd[xdi - 1]) % 10) & 1 ||
          rm == (x.s < 0 ? 8 : 7));

    if (sd < 1 || !xd[0]) {
      xd.length = 0;
      if (roundUp) {

        // Convert sd to decimal places.
        sd -= x.e + 1;

        // 1, 0.1, 0.01, 0.001, 0.0001 etc.
        xd[0] = mathpow(10, (LOG_BASE - sd % LOG_BASE) % LOG_BASE);
        x.e = -sd || 0;
      } else {

        // Zero.
        xd[0] = x.e = 0;
      }

      return x;
    }

    // Remove excess digits.
    if (i == 0) {
      xd.length = xdi;
      k = 1;
      xdi--;
    } else {
      xd.length = xdi + 1;
      k = mathpow(10, LOG_BASE - i);

      // E.g. 56700 becomes 56000 if 7 is the rounding digit.
      // j > 0 means i > number of leading zeros of w.
      xd[xdi] = j > 0 ? (w / mathpow(10, digits - j) % mathpow(10, j) | 0) * k : 0;
    }

    if (roundUp) {
      for (;;) {

        // Is the digit to be rounded up in the first word of xd?
        if (xdi == 0) {

          // i will be the length of xd[0] before k is added.
          for (i = 1, j = xd[0]; j >= 10; j /= 10) i++;
          j = xd[0] += k;
          for (k = 1; j >= 10; j /= 10) k++;

          // if i != k the length has increased.
          if (i != k) {
            x.e++;
            if (xd[0] == BASE) xd[0] = 1;
          }

          break;
        } else {
          xd[xdi] += k;
          if (xd[xdi] != BASE) break;
          xd[xdi--] = 0;
          k = 1;
        }
      }
    }

    // Remove trailing zeros.
    for (i = xd.length; xd[--i] === 0;) xd.pop();
  }

  if (external) {

    // Overflow?
    if (x.e > Ctor.maxE) {

      // Infinity.
      x.d = null;
      x.e = NaN;

    // Underflow?
    } else if (x.e < Ctor.minE) {

      // Zero.
      x.e = 0;
      x.d = [0];
      // Ctor.underflow = true;
    } // else Ctor.underflow = false;
  }

  return x;
}


function finiteToString(x, isExp, sd) {
  if (!x.isFinite()) return nonFiniteToString(x);
  var k,
    e = x.e,
    str = digitsToString(x.d),
    len = str.length;

  if (isExp) {
    if (sd && (k = sd - len) > 0) {
      str = str.charAt(0) + '.' + str.slice(1) + getZeroString(k);
    } else if (len > 1) {
      str = str.charAt(0) + '.' + str.slice(1);
    }

    str = str + (x.e < 0 ? 'e' : 'e+') + x.e;
  } else if (e < 0) {
    str = '0.' + getZeroString(-e - 1) + str;
    if (sd && (k = sd - len) > 0) str += getZeroString(k);
  } else if (e >= len) {
    str += getZeroString(e + 1 - len);
    if (sd && (k = sd - e - 1) > 0) str = str + '.' + getZeroString(k);
  } else {
    if ((k = e + 1) < len) str = str.slice(0, k) + '.' + str.slice(k);
    if (sd && (k = sd - len) > 0) {
      if (e + 1 === len) str += '.';
      str += getZeroString(k);
    }
  }

  return str;
}


// Calculate the base 10 exponent from the base 1e7 exponent.
function getBase10Exponent(digits, e) {
  var w = digits[0];

  // Add the number of digits of the first word of the digits array.
  for ( e *= LOG_BASE; w >= 10; w /= 10) e++;
  return e;
}


function getLn10(Ctor, sd, pr) {
  if (sd > LN10_PRECISION) {

    // Reset global state in case the exception is caught.
    external = true;
    if (pr) Ctor.precision = pr;
    throw Error(precisionLimitExceeded);
  }
  return finalise(new Ctor(LN10), sd, 1, true);
}


function getPi(Ctor, sd, rm) {
  if (sd > PI_PRECISION) throw Error(precisionLimitExceeded);
  return finalise(new Ctor(PI), sd, rm, true);
}


function getPrecision(digits) {
  var w = digits.length - 1,
    len = w * LOG_BASE + 1;

  w = digits[w];

  // If non-zero...
  if (w) {

    // Subtract the number of trailing zeros of the last word.
    for (; w % 10 == 0; w /= 10) len--;

    // Add the number of digits of the first word.
    for (w = digits[0]; w >= 10; w /= 10) len++;
  }

  return len;
}


function getZeroString(k) {
  var zs = '';
  for (; k--;) zs += '0';
  return zs;
}


/*
 * Return a new Decimal whose value is the value of Decimal `x` to the power `n`, where `n` is an
 * integer of type number.
 *
 * Implements 'exponentiation by squaring'. Called by `pow` and `parseOther`.
 *
 */
function intPow(Ctor, x, n, pr) {
  var isTruncated,
    r = new Ctor(1),

    // Max n of 9007199254740991 takes 53 loop iterations.
    // Maximum digits array length; leaves [28, 34] guard digits.
    k = Math.ceil(pr / LOG_BASE + 4);

  external = false;

  for (;;) {
    if (n % 2) {
      r = r.times(x);
      if (truncate(r.d, k)) isTruncated = true;
    }

    n = mathfloor(n / 2);
    if (n === 0) {

      // To ensure correct rounding when r.d is truncated, increment the last word if it is zero.
      n = r.d.length - 1;
      if (isTruncated && r.d[n] === 0) ++r.d[n];
      break;
    }

    x = x.times(x);
    truncate(x.d, k);
  }

  external = true;

  return r;
}


function isOdd(n) {
  return n.d[n.d.length - 1] & 1;
}


/*
 * Handle `max` and `min`. `ltgt` is 'lt' or 'gt'.
 */
function maxOrMin(Ctor, args, ltgt) {
  var y,
    x = new Ctor(args[0]),
    i = 0;

  for (; ++i < args.length;) {
    y = new Ctor(args[i]);
    if (!y.s) {
      x = y;
      break;
    } else if (x[ltgt](y)) {
      x = y;
    }
  }

  return x;
}


/*
 * Return a new Decimal whose value is the natural exponential of `x` rounded to `sd` significant
 * digits.
 *
 * Taylor/Maclaurin series.
 *
 * exp(x) = x^0/0! + x^1/1! + x^2/2! + x^3/3! + ...
 *
 * Argument reduction:
 *   Repeat x = x / 32, k += 5, until |x| < 0.1
 *   exp(x) = exp(x / 2^k)^(2^k)
 *
 * Previously, the argument was initially reduced by
 * exp(x) = exp(r) * 10^k  where r = x - k * ln10, k = floor(x / ln10)
 * to first put r in the range [0, ln10], before dividing by 32 until |x| < 0.1, but this was
 * found to be slower than just dividing repeatedly by 32 as above.
 *
 * Max integer argument: exp('20723265836946413') = 6.3e+9000000000000000
 * Min integer argument: exp('-20723265836946411') = 1.2e-9000000000000000
 * (Math object integer min/max: Math.exp(709) = 8.2e+307, Math.exp(-745) = 5e-324)
 *
 *  exp(Infinity)  = Infinity
 *  exp(-Infinity) = 0
 *  exp(NaN)       = NaN
 *  exp(±0)        = 1
 *
 *  exp(x) is non-terminating for any finite, non-zero x.
 *
 *  The result will always be correctly rounded.
 *
 */
function naturalExponential(x, sd) {
  var denominator, guard, j, pow, sum, t, wpr,
    rep = 0,
    i = 0,
    k = 0,
    Ctor = x.constructor,
    rm = Ctor.rounding,
    pr = Ctor.precision;

  // 0/NaN/Infinity?
  if (!x.d || !x.d[0] || x.e > 17) {

    return new Ctor(x.d
      ? !x.d[0] ? 1 : x.s < 0 ? 0 : 1 / 0
      : x.s ? x.s < 0 ? 0 : x : 0 / 0);
  }

  if (sd == null) {
    external = false;
    wpr = pr;
  } else {
    wpr = sd;
  }

  t = new Ctor(0.03125);

  // while abs(x) >= 0.1
  while (x.e > -2) {

    // x = x / 2^5
    x = x.times(t);
    k += 5;
  }

  // Use 2 * log10(2^k) + 5 (empirically derived) to estimate the increase in precision
  // necessary to ensure the first 4 rounding digits are correct.
  guard = Math.log(mathpow(2, k)) / Math.LN10 * 2 + 5 | 0;
  wpr += guard;
  denominator = pow = sum = new Ctor(1);
  Ctor.precision = wpr;

  for (;;) {
    pow = finalise(pow.times(x), wpr, 1);
    denominator = denominator.times(++i);
    t = sum.plus(divide(pow, denominator, wpr, 1));

    if (digitsToString(t.d).slice(0, wpr) === digitsToString(sum.d).slice(0, wpr)) {
      j = k;
      while (j--) sum = finalise(sum.times(sum), wpr, 1);

      // Check to see if the first 4 rounding digits are [49]999.
      // If so, repeat the summation with a higher precision, otherwise
      // e.g. with precision: 18, rounding: 1
      // exp(18.404272462595034083567793919843761) = 98372560.1229999999 (should be 98372560.123)
      // `wpr - guard` is the index of first rounding digit.
      if (sd == null) {

        if (rep < 3 && checkRoundingDigits(sum.d, wpr - guard, rm, rep)) {
          Ctor.precision = wpr += 10;
          denominator = pow = t = new Ctor(1);
          i = 0;
          rep++;
        } else {
          return finalise(sum, Ctor.precision = pr, rm, external = true);
        }
      } else {
        Ctor.precision = pr;
        return sum;
      }
    }

    sum = t;
  }
}


/*
 * Return a new Decimal whose value is the natural logarithm of `x` rounded to `sd` significant
 * digits.
 *
 *  ln(-n)        = NaN
 *  ln(0)         = -Infinity
 *  ln(-0)        = -Infinity
 *  ln(1)         = 0
 *  ln(Infinity)  = Infinity
 *  ln(-Infinity) = NaN
 *  ln(NaN)       = NaN
 *
 *  ln(n) (n != 1) is non-terminating.
 *
 */
function naturalLogarithm(y, sd) {
  var c, c0, denominator, e, numerator, rep, sum, t, wpr, x1, x2,
    n = 1,
    guard = 10,
    x = y,
    xd = x.d,
    Ctor = x.constructor,
    rm = Ctor.rounding,
    pr = Ctor.precision;

  // Is x negative or Infinity, NaN, 0 or 1?
  if (x.s < 0 || !xd || !xd[0] || !x.e && xd[0] == 1 && xd.length == 1) {
    return new Ctor(xd && !xd[0] ? -1 / 0 : x.s != 1 ? NaN : xd ? 0 : x);
  }

  if (sd == null) {
    external = false;
    wpr = pr;
  } else {
    wpr = sd;
  }

  Ctor.precision = wpr += guard;
  c = digitsToString(xd);
  c0 = c.charAt(0);

  if (Math.abs(e = x.e) < 1.5e15) {

    // Argument reduction.
    // The series converges faster the closer the argument is to 1, so using
    // ln(a^b) = b * ln(a),   ln(a) = ln(a^b) / b
    // multiply the argument by itself until the leading digits of the significand are 7, 8, 9,
    // 10, 11, 12 or 13, recording the number of multiplications so the sum of the series can
    // later be divided by this number, then separate out the power of 10 using
    // ln(a*10^b) = ln(a) + b*ln(10).

    // max n is 21 (gives 0.9, 1.0 or 1.1) (9e15 / 21 = 4.2e14).
    //while (c0 < 9 && c0 != 1 || c0 == 1 && c.charAt(1) > 1) {
    // max n is 6 (gives 0.7 - 1.3)
    while (c0 < 7 && c0 != 1 || c0 == 1 && c.charAt(1) > 3) {
      x = x.times(y);
      c = digitsToString(x.d);
      c0 = c.charAt(0);
      n++;
    }

    e = x.e;

    if (c0 > 1) {
      x = new Ctor('0.' + c);
      e++;
    } else {
      x = new Ctor(c0 + '.' + c.slice(1));
    }
  } else {

    // The argument reduction method above may result in overflow if the argument y is a massive
    // number with exponent >= 1500000000000000 (9e15 / 6 = 1.5e15), so instead recall this
    // function using ln(x*10^e) = ln(x) + e*ln(10).
    t = getLn10(Ctor, wpr + 2, pr).times(e + '');
    x = naturalLogarithm(new Ctor(c0 + '.' + c.slice(1)), wpr - guard).plus(t);
    Ctor.precision = pr;

    return sd == null ? finalise(x, pr, rm, external = true) : x;
  }

  // x1 is x reduced to a value near 1.
  x1 = x;

  // Taylor series.
  // ln(y) = ln((1 + x)/(1 - x)) = 2(x + x^3/3 + x^5/5 + x^7/7 + ...)
  // where x = (y - 1)/(y + 1)    (|x| < 1)
  sum = numerator = x = divide(x.minus(1), x.plus(1), wpr, 1);
  x2 = finalise(x.times(x), wpr, 1);
  denominator = 3;

  for (;;) {
    numerator = finalise(numerator.times(x2), wpr, 1);
    t = sum.plus(divide(numerator, new Ctor(denominator), wpr, 1));

    if (digitsToString(t.d).slice(0, wpr) === digitsToString(sum.d).slice(0, wpr)) {
      sum = sum.times(2);

      // Reverse the argument reduction. Check that e is not 0 because, besides preventing an
      // unnecessary calculation, -0 + 0 = +0 and to ensure correct rounding -0 needs to stay -0.
      if (e !== 0) sum = sum.plus(getLn10(Ctor, wpr + 2, pr).times(e + ''));
      sum = divide(sum, new Ctor(n), wpr, 1);

      // Is rm > 3 and the first 4 rounding digits 4999, or rm < 4 (or the summation has
      // been repeated previously) and the first 4 rounding digits 9999?
      // If so, restart the summation with a higher precision, otherwise
      // e.g. with precision: 12, rounding: 1
      // ln(135520028.6126091714265381533) = 18.7246299999 when it should be 18.72463.
      // `wpr - guard` is the index of first rounding digit.
      if (sd == null) {
        if (checkRoundingDigits(sum.d, wpr - guard, rm, rep)) {
          Ctor.precision = wpr += guard;
          t = numerator = x = divide(x1.minus(1), x1.plus(1), wpr, 1);
          x2 = finalise(x.times(x), wpr, 1);
          denominator = rep = 1;
        } else {
          return finalise(sum, Ctor.precision = pr, rm, external = true);
        }
      } else {
        Ctor.precision = pr;
        return sum;
      }
    }

    sum = t;
    denominator += 2;
  }
}


// ±Infinity, NaN.
function nonFiniteToString(x) {
  // Unsigned.
  return String(x.s * x.s / 0);
}


/*
 * Parse the value of a new Decimal `x` from string `str`.
 */
function parseDecimal(x, str) {
  var e, i, len;

  // Decimal point?
  if ((e = str.indexOf('.')) > -1) str = str.replace('.', '');

  // Exponential form?
  if ((i = str.search(/e/i)) > 0) {

    // Determine exponent.
    if (e < 0) e = i;
    e += +str.slice(i + 1);
    str = str.substring(0, i);
  } else if (e < 0) {

    // Integer.
    e = str.length;
  }

  // Determine leading zeros.
  for (i = 0; str.charCodeAt(i) === 48; i++);

  // Determine trailing zeros.
  for (len = str.length; str.charCodeAt(len - 1) === 48; --len);
  str = str.slice(i, len);

  if (str) {
    len -= i;
    x.e = e = e - i - 1;
    x.d = [];

    // Transform base

    // e is the base 10 exponent.
    // i is where to slice str to get the first word of the digits array.
    i = (e + 1) % LOG_BASE;
    if (e < 0) i += LOG_BASE;

    if (i < len) {
      if (i) x.d.push(+str.slice(0, i));
      for (len -= LOG_BASE; i < len;) x.d.push(+str.slice(i, i += LOG_BASE));
      str = str.slice(i);
      i = LOG_BASE - str.length;
    } else {
      i -= len;
    }

    for (; i--;) str += '0';
    x.d.push(+str);

    if (external) {

      // Overflow?
      if (x.e > x.constructor.maxE) {

        // Infinity.
        x.d = null;
        x.e = NaN;

      // Underflow?
      } else if (x.e < x.constructor.minE) {

        // Zero.
        x.e = 0;
        x.d = [0];
        // x.constructor.underflow = true;
      } // else x.constructor.underflow = false;
    }
  } else {

    // Zero.
    x.e = 0;
    x.d = [0];
  }

  return x;
}


/*
 * Parse the value of a new Decimal `x` from a string `str`, which is not a decimal value.
 */
function parseOther(x, str) {
  var base, Ctor, divisor, i, isFloat, len, p, xd, xe;

  if (str === 'Infinity' || str === 'NaN') {
    if (!+str) x.s = NaN;
    x.e = NaN;
    x.d = null;
    return x;
  }

  if (isHex.test(str))  {
    base = 16;
    str = str.toLowerCase();
  } else if (isBinary.test(str))  {
    base = 2;
  } else if (isOctal.test(str))  {
    base = 8;
  } else {
    throw Error(invalidArgument + str);
  }

  // Is there a binary exponent part?
  i = str.search(/p/i);

  if (i > 0) {
    p = +str.slice(i + 1);
    str = str.substring(2, i);
  } else {
    str = str.slice(2);
  }

  // Convert `str` as an integer then divide the result by `base` raised to a power such that the
  // fraction part will be restored.
  i = str.indexOf('.');
  isFloat = i >= 0;
  Ctor = x.constructor;

  if (isFloat) {
    str = str.replace('.', '');
    len = str.length;
    i = len - i;

    // log[10](16) = 1.2041... , log[10](88) = 1.9444....
    divisor = intPow(Ctor, new Ctor(base), i, i * 2);
  }

  xd = convertBase(str, base, BASE);
  xe = xd.length - 1;

  // Remove trailing zeros.
  for (i = xe; xd[i] === 0; --i) xd.pop();
  if (i < 0) return new Ctor(x.s * 0);
  x.e = getBase10Exponent(xd, xe);
  x.d = xd;
  external = false;

  // At what precision to perform the division to ensure exact conversion?
  // maxDecimalIntegerPartDigitCount = ceil(log[10](b) * otherBaseIntegerPartDigitCount)
  // log[10](2) = 0.30103, log[10](8) = 0.90309, log[10](16) = 1.20412
  // E.g. ceil(1.2 * 3) = 4, so up to 4 decimal digits are needed to represent 3 hex int digits.
  // maxDecimalFractionPartDigitCount = {Hex:4|Oct:3|Bin:1} * otherBaseFractionPartDigitCount
  // Therefore using 4 * the number of digits of str will always be enough.
  if (isFloat) x = divide(x, divisor, len * 4);

  // Multiply by the binary exponent part if present.
  if (p) x = x.times(Math.abs(p) < 54 ? mathpow(2, p) : Decimal.pow(2, p));
  external = true;

  return x;
}


/*
 * sin(x) = x - x^3/3! + x^5/5! - ...
 * |x| < pi/2
 *
 */
function sine(Ctor, x) {
  var k,
    len = x.d.length;

  if (len < 3) return taylorSeries(Ctor, 2, x, x);

  // Argument reduction: sin(5x) = 16*sin^5(x) - 20*sin^3(x) + 5*sin(x)
  // i.e. sin(x) = 16*sin^5(x/5) - 20*sin^3(x/5) + 5*sin(x/5)
  // and  sin(x) = sin(x/5)(5 + sin^2(x/5)(16sin^2(x/5) - 20))

  // Estimate the optimum number of times to use the argument reduction.
  k = 1.4 * Math.sqrt(len);
  k = k > 16 ? 16 : k | 0;

  x = x.times(1 / tinyPow(5, k));
  x = taylorSeries(Ctor, 2, x, x);

  // Reverse argument reduction
  var sin2_x,
    d5 = new Ctor(5),
    d16 = new Ctor(16),
    d20 = new Ctor(20);
  for (; k--;) {
    sin2_x = x.times(x);
    x = x.times(d5.plus(sin2_x.times(d16.times(sin2_x).minus(d20))));
  }

  return x;
}


// Calculate Taylor series for `cos`, `cosh`, `sin` and `sinh`.
function taylorSeries(Ctor, n, x, y, isHyperbolic) {
  var j, t, u, x2,
    i = 1,
    pr = Ctor.precision,
    k = Math.ceil(pr / LOG_BASE);

  external = false;
  x2 = x.times(x);
  u = new Ctor(y);

  for (;;) {
    t = divide(u.times(x2), new Ctor(n++ * n++), pr, 1);
    u = isHyperbolic ? y.plus(t) : y.minus(t);
    y = divide(t.times(x2), new Ctor(n++ * n++), pr, 1);
    t = u.plus(y);

    if (t.d[k] !== void 0) {
      for (j = k; t.d[j] === u.d[j] && j--;);
      if (j == -1) break;
    }

    j = u;
    u = y;
    y = t;
    t = j;
    i++;
  }

  external = true;
  t.d.length = k + 1;

  return t;
}


// Exponent e must be positive and non-zero.
function tinyPow(b, e) {
  var n = b;
  while (--e) n *= b;
  return n;
}


// Return the absolute value of `x` reduced to less than or equal to half pi.
function toLessThanHalfPi(Ctor, x) {
  var t,
    isNeg = x.s < 0,
    pi = getPi(Ctor, Ctor.precision, 1),
    halfPi = pi.times(0.5);

  x = x.abs();

  if (x.lte(halfPi)) {
    quadrant = isNeg ? 4 : 1;
    return x;
  }

  t = x.divToInt(pi);

  if (t.isZero()) {
    quadrant = isNeg ? 3 : 2;
  } else {
    x = x.minus(t.times(pi));

    // 0 <= x < pi
    if (x.lte(halfPi)) {
      quadrant = isOdd(t) ? (isNeg ? 2 : 3) : (isNeg ? 4 : 1);
      return x;
    }

    quadrant = isOdd(t) ? (isNeg ? 1 : 4) : (isNeg ? 3 : 2);
  }

  return x.minus(pi).abs();
}


/*
 * Return the value of Decimal `x` as a string in base `baseOut`.
 *
 * If the optional `sd` argument is present include a binary exponent suffix.
 */
function toStringBinary(x, baseOut, sd, rm) {
  var base, e, i, k, len, roundUp, str, xd, y,
    Ctor = x.constructor,
    isExp = sd !== void 0;

  if (isExp) {
    checkInt32(sd, 1, MAX_DIGITS);
    if (rm === void 0) rm = Ctor.rounding;
    else checkInt32(rm, 0, 8);
  } else {
    sd = Ctor.precision;
    rm = Ctor.rounding;
  }

  if (!x.isFinite()) {
    str = nonFiniteToString(x);
  } else {
    str = finiteToString(x);
    i = str.indexOf('.');

    // Use exponential notation according to `toExpPos` and `toExpNeg`? No, but if required:
    // maxBinaryExponent = floor((decimalExponent + 1) * log[2](10))
    // minBinaryExponent = floor(decimalExponent * log[2](10))
    // log[2](10) = 3.321928094887362347870319429489390175864

    if (isExp) {
      base = 2;
      if (baseOut == 16) {
        sd = sd * 4 - 3;
      } else if (baseOut == 8) {
        sd = sd * 3 - 2;
      }
    } else {
      base = baseOut;
    }

    // Convert the number as an integer then divide the result by its base raised to a power such
    // that the fraction part will be restored.

    // Non-integer.
    if (i >= 0) {
      str = str.replace('.', '');
      y = new Ctor(1);
      y.e = str.length - i;
      y.d = convertBase(finiteToString(y), 10, base);
      y.e = y.d.length;
    }

    xd = convertBase(str, 10, base);
    e = len = xd.length;

    // Remove trailing zeros.
    for (; xd[--len] == 0;) xd.pop();

    if (!xd[0]) {
      str = isExp ? '0p+0' : '0';
    } else {
      if (i < 0) {
        e--;
      } else {
        x = new Ctor(x);
        x.d = xd;
        x.e = e;
        x = divide(x, y, sd, rm, 0, base);
        xd = x.d;
        e = x.e;
        roundUp = inexact;
      }

      // The rounding digit, i.e. the digit after the digit that may be rounded up.
      i = xd[sd];
      k = base / 2;
      roundUp = roundUp || xd[sd + 1] !== void 0;

      roundUp = rm < 4
        ? (i !== void 0 || roundUp) && (rm === 0 || rm === (x.s < 0 ? 3 : 2))
        : i > k || i === k && (rm === 4 || roundUp || rm === 6 && xd[sd - 1] & 1 ||
          rm === (x.s < 0 ? 8 : 7));

      xd.length = sd;

      if (roundUp) {

        // Rounding up may mean the previous digit has to be rounded up and so on.
        for (; ++xd[--sd] > base - 1;) {
          xd[sd] = 0;
          if (!sd) {
            ++e;
            xd.unshift(1);
          }
        }
      }

      // Determine trailing zeros.
      for (len = xd.length; !xd[len - 1]; --len);

      // E.g. [4, 11, 15] becomes 4bf.
      for (i = 0, str = ''; i < len; i++) str += NUMERALS.charAt(xd[i]);

      // Add binary exponent suffix?
      if (isExp) {
        if (len > 1) {
          if (baseOut == 16 || baseOut == 8) {
            i = baseOut == 16 ? 4 : 3;
            for (--len; len % i; len++) str += '0';
            xd = convertBase(str, base, baseOut);
            for (len = xd.length; !xd[len - 1]; --len);

            // xd[0] will always be be 1
            for (i = 1, str = '1.'; i < len; i++) str += NUMERALS.charAt(xd[i]);
          } else {
            str = str.charAt(0) + '.' + str.slice(1);
          }
        }

        str =  str + (e < 0 ? 'p' : 'p+') + e;
      } else if (e < 0) {
        for (; ++e;) str = '0' + str;
        str = '0.' + str;
      } else {
        if (++e > len) for (e -= len; e-- ;) str += '0';
        else if (e < len) str = str.slice(0, e) + '.' + str.slice(e);
      }
    }

    str = (baseOut == 16 ? '0x' : baseOut == 2 ? '0b' : baseOut == 8 ? '0o' : '') + str;
  }

  return x.s < 0 ? '-' + str : str;
}


// Does not strip trailing zeros.
function truncate(arr, len) {
  if (arr.length > len) {
    arr.length = len;
    return true;
  }
}


// Decimal methods


/*
 *  abs
 *  acos
 *  acosh
 *  add
 *  asin
 *  asinh
 *  atan
 *  atanh
 *  atan2
 *  cbrt
 *  ceil
 *  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}
 *   defaults   {true}
 *
 * E.g. Decimal.config({ precision: 20, rounding: 4 })
 *
 */
function config(obj) {
  if (!obj || typeof obj !== 'object') throw Error(decimalError + 'Object expected');
  var i, p, v,
    useDefaults = obj.defaults === true,
    ps = [
      'precision', 1, MAX_DIGITS,
      'rounding', 0, 8,
      'toExpNeg', -EXP_LIMIT, 0,
      'toExpPos', 0, EXP_LIMIT,
      'maxE', 0, EXP_LIMIT,
      'minE', -EXP_LIMIT, 0,
      'modulo', 0, 9
    ];

  for (i = 0; i < ps.length; i += 3) {
    if (p = ps[i], useDefaults) this[p] = DEFAULTS[p];
    if ((v = obj[p]) !== void 0) {
      if (mathfloor(v) === v && v >= ps[i + 1] && v <= ps[i + 2]) this[p] = v;
      else throw Error(invalidArgument + p + ': ' + v);
    }
  }

  if (p = 'crypto', useDefaults) this[p] = DEFAULTS[p];
  if ((v = obj[p]) !== void 0) {
    if (v === true || v === false || v === 0 || v === 1) {
      if (v) {
        if (typeof crypto != 'undefined' && crypto &&
          (crypto.getRandomValues || crypto.randomBytes)) {
          this[p] = true;
        } else {
          throw Error(cryptoUnavailable);
        }
      } else {
        this[p] = false;
      }
    } else {
      throw Error(invalidArgument + p + ': ' + v);
    }
  }

  return this;
}


/*
 * Return a new Decimal whose value is the cosine of `x`, rounded to `precision` significant
 * digits using rounding mode `rounding`.
 *
 * x {number|string|Decimal} A value in radians.
 *
 */
function cos(x) {
  return new this(x).cos();
}


/*
 * Return a new Decimal whose value is the hyperbolic cosine of `x`, rounded to precision
 * significant digits using rounding mode `rounding`.
 *
 * x {number|string|Decimal} A value in radians.
 *
 */
function cosh(x) {
  return new this(x).cosh();
}


/*
 * Create and return a Decimal constructor with the same configuration properties as this Decimal
 * constructor.
 *
 */
function clone(obj) {
  var i, p, ps;

  /*
   * The Decimal constructor and exported function.
   * Return a new Decimal instance.
   *
   * v {number|string|Decimal} A numeric value.
   *
   */
  function Decimal(v) {
    var e, i, t,
      x = this;

    // Decimal called without new.
    if (!(x instanceof Decimal)) return new Decimal(v);

    // Retain a reference to this Decimal constructor, and shadow Decimal.prototype.constructor
    // which points to Object.
    x.constructor = Decimal;

    // Duplicate.
    if (v instanceof Decimal) {
      x.s = v.s;

      if (external) {
        if (!v.d || v.e > Decimal.maxE) {

          // Infinity.
          x.e = NaN;
          x.d = null;
        } else if (v.e < Decimal.minE) {

          // Zero.
          x.e = 0;
          x.d = [0];
        } else {
          x.e = v.e;
          x.d = v.d.slice();
        }
      } else {
        x.e = v.e;
        x.d = v.d ? v.d.slice() : v.d;
      }

      return;
    }

    t = typeof v;

    if (t === 'number') {
      if (v === 0) {
        x.s = 1 / v < 0 ? -1 : 1;
        x.e = 0;
        x.d = [0];
        return;
      }

      if (v < 0) {
        v = -v;
        x.s = -1;
      } else {
        x.s = 1;
      }

      // Fast path for small integers.
      if (v === ~~v && v < 1e7) {
        for (e = 0, i = v; i >= 10; i /= 10) e++;

        if (external) {
          if (e > Decimal.maxE) {
            x.e = NaN;
            x.d = null;
          } else if (e < Decimal.minE) {
            x.e = 0;
            x.d = [0];
          } else {
            x.e = e;
            x.d = [v];
          }
        } else {
          x.e = e;
          x.d = [v];
        }

        return;

      // Infinity, NaN.
      } else if (v * 0 !== 0) {
        if (!v) x.s = NaN;
        x.e = NaN;
        x.d = null;
        return;
      }

      return parseDecimal(x, v.toString());

    } else if (t !== 'string') {
      throw Error(invalidArgument + v);
    }

    // Minus sign?
    if ((i = v.charCodeAt(0)) === 45) {
      v = v.slice(1);
      x.s = -1;
    } else {
      // Plus sign?
      if (i === 43) v = v.slice(1);
      x.s = 1;
    }

    return isDecimal.test(v) ? parseDecimal(x, v) : parseOther(x, v);
  }

  Decimal.prototype = P;

  Decimal.ROUND_UP = 0;
  Decimal.ROUND_DOWN = 1;
  Decimal.ROUND_CEIL = 2;
  Decimal.ROUND_FLOOR = 3;
  Decimal.ROUND_HALF_UP = 4;
  Decimal.ROUND_HALF_DOWN = 5;
  Decimal.ROUND_HALF_EVEN = 6;
  Decimal.ROUND_HALF_CEIL = 7;
  Decimal.ROUND_HALF_FLOOR = 8;
  Decimal.EUCLID = 9;

  Decimal.config = Decimal.set = config;
  Decimal.clone = clone;
  Decimal.isDecimal = isDecimalInstance;

  Decimal.abs = abs;
  Decimal.acos = acos;
  Decimal.acosh = acosh;        // ES6
  Decimal.add = add;
  Decimal.asin = asin;
  Decimal.asinh = asinh;        // ES6
  Decimal.atan = atan;
  Decimal.atanh = atanh;        // ES6
  Decimal.atan2 = atan2;
  Decimal.cbrt = cbrt;          // ES6
  Decimal.ceil = ceil;
  Decimal.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) {
    if (obj.defaults !== true) {
      ps = ['precision', 'rounding', 'toExpNeg', 'toExpPos', 'maxE', 'minE', 'modulo', 'crypto'];
      for (i = 0; i < ps.length;) if (!obj.hasOwnProperty(p = ps[i++])) obj[p] = this[p];
    }
  }

  Decimal.config(obj);

  return Decimal;
}


/*
 * Return a new Decimal whose value is `x` divided by `y`, rounded to `precision` significant
 * digits using rounding mode `rounding`.
 *
 * x {number|string|Decimal}
 * y {number|string|Decimal}
 *
 */
function div(x, y) {
  return new this(x).div(y);
}


/*
 * Return a new Decimal whose value is the natural exponential of `x`, rounded to `precision`
 * significant digits using rounding mode `rounding`.
 *
 * x {number|string|Decimal} The power to which to raise the base of the natural log.
 *
 */
function exp(x) {
  return new this(x).exp();
}


/*
 * Return a new Decimal whose value is `x` round to an integer using `ROUND_FLOOR`.
 *
 * x {number|string|Decimal}
 *
 */
function floor(x) {
  return finalise(x = new this(x), x.e + 1, 3);
}


/*
 * Return a new Decimal whose value is the square root of the sum of the squares of the arguments,
 * rounded to `precision` significant digits using rounding mode `rounding`.
 *
 * hypot(a, b, ...) = sqrt(a^2 + b^2 + ...)
 *
 * arguments {number|string|Decimal}
 *
 */
function hypot() {
  var i, n,
    t = new this(0);

  external = false;

  for (i = 0; i < arguments.length;) {
    n = new this(arguments[i++]);
    if (!n.d) {
      if (n.s) {
        external = true;
        return new this(1 / 0);
      }
      t = n;
    } else if (t.d) {
      t = t.plus(n.times(n));
    }
  }

  external = true;

  return t.sqrt();
}


/*
 * Return true if object is a Decimal instance (where Decimal is any Decimal constructor),
 * otherwise return false.
 *
 */
function isDecimalInstance(obj) {
  return obj instanceof Decimal || obj && obj.name === '[object Decimal]' || false;
}


/*
 * Return a new Decimal whose value is the natural logarithm of `x`, rounded to `precision`
 * significant digits using rounding mode `rounding`.
 *
 * x {number|string|Decimal}
 *
 */
function ln(x) {
  return new this(x).ln();
}


/*
 * Return a new Decimal whose value is the log of `x` to the base `y`, or to base 10 if no base
 * is specified, rounded to `precision` significant digits using rounding mode `rounding`.
 *
 * log[y](x)
 *
 * x {number|string|Decimal} The argument of the logarithm.
 * y {number|string|Decimal} The base of the logarithm.
 *
 */
function log(x, y) {
  return new this(x).log(y);
}


/*
 * Return a new Decimal whose value is the base 2 logarithm of `x`, rounded to `precision`
 * significant digits using rounding mode `rounding`.
 *
 * x {number|string|Decimal}
 *
 */
function log2(x) {
  return new this(x).log(2);
}


/*
 * Return a new Decimal whose value is the base 10 logarithm of `x`, rounded to `precision`
 * significant digits using rounding mode `rounding`.
 *
 * x {number|string|Decimal}
 *
 */
function log10(x) {
  return new this(x).log(10);
}


/*
 * Return a new Decimal whose value is the maximum of the arguments.
 *
 * arguments {number|string|Decimal}
 *
 */
function max() {
  return maxOrMin(this, arguments, 'lt');
}


/*
 * Return a new Decimal whose value is the minimum of the arguments.
 *
 * arguments {number|string|Decimal}
 *
 */
function min() {
  return maxOrMin(this, arguments, 'gt');
}


/*
 * Return a new Decimal whose value is `x` modulo `y`, rounded to `precision` significant digits
 * using rounding mode `rounding`.
 *
 * x {number|string|Decimal}
 * y {number|string|Decimal}
 *
 */
function mod(x, y) {
  return new this(x).mod(y);
}


/*
 * Return a new Decimal whose value is `x` multiplied by `y`, rounded to `precision` significant
 * digits using rounding mode `rounding`.
 *
 * x {number|string|Decimal}
 * y {number|string|Decimal}
 *
 */
function mul(x, y) {
  return new this(x).mul(y);
}


/*
 * Return a new Decimal whose value is `x` raised to the power `y`, rounded to precision
 * significant digits using rounding mode `rounding`.
 *
 * x {number|string|Decimal} The base.
 * y {number|string|Decimal} The exponent.
 *
 */
function pow(x, y) {
  return new this(x).pow(y);
}


/*
 * Returns a new Decimal with a random value equal to or greater than 0 and less than 1, and with
 * `sd`, or `Decimal.precision` if `sd` is omitted, significant digits (or less if trailing zeros
 * are produced).
 *
 * [sd] {number} Significant digits. Integer, 0 to MAX_DIGITS inclusive.
 *
 */
function random(sd) {
  var d, e, k, n,
    i = 0,
    r = new this(1),
    rd = [];

  if (sd === void 0) sd = this.precision;
  else checkInt32(sd, 1, MAX_DIGITS);

  k = Math.ceil(sd / LOG_BASE);

  if (!this.crypto) {
    for (; i < k;) rd[i++] = Math.random() * 1e7 | 0;

  // Browsers supporting crypto.getRandomValues.
  } else if (crypto.getRandomValues) {
    d = crypto.getRandomValues(new Uint32Array(k));

    for (; i < k;) {
      n = d[i];

      // 0 <= n < 4294967296
      // Probability n >= 4.29e9, is 4967296 / 4294967296 = 0.00116 (1 in 865).
      if (n >= 4.29e9) {
        d[i] = crypto.getRandomValues(new Uint32Array(1))[0];
      } else {

        // 0 <= n <= 4289999999
        // 0 <= (n % 1e7) <= 9999999
        rd[i++] = n % 1e7;
      }
    }

  // Node.js supporting crypto.randomBytes.
  } else if (crypto.randomBytes) {

    // buffer
    d = crypto.randomBytes(k *= 4);

    for (; i < k;) {

      // 0 <= n < 2147483648
      n = d[i] + (d[i + 1] << 8) + (d[i + 2] << 16) + ((d[i + 3] & 0x7f) << 24);

      // Probability n >= 2.14e9, is 7483648 / 2147483648 = 0.0035 (1 in 286).
      if (n >= 2.14e9) {
        crypto.randomBytes(4).copy(d, i);
      } else {

        // 0 <= n <= 2139999999
        // 0 <= (n % 1e7) <= 9999999
        rd.push(n % 1e7);
        i += 4;
      }
    }

    i = k / 4;
  } else {
    throw Error(cryptoUnavailable);
  }

  k = rd[--i];
  sd %= LOG_BASE;

  // Convert trailing digits to zeros according to sd.
  if (k && sd) {
    n = mathpow(10, LOG_BASE - sd);
    rd[i] = (k / n | 0) * n;
  }

  // Remove trailing words which are zero.
  for (; rd[i] === 0; i--) rd.pop();

  // Zero?
  if (i < 0) {
    e = 0;
    rd = [0];
  } else {
    e = -1;

    // Remove leading words which are zero and adjust exponent accordingly.
    for (; rd[0] === 0; e -= LOG_BASE) rd.shift();

    // Count the digits of the first word of rd to determine leading zeros.
    for (k = 1, n = rd[0]; n >= 10; n /= 10) k++;

    // Adjust the exponent for leading zeros of the first word of rd.
    if (k < LOG_BASE) e -= LOG_BASE - k;
  }

  r.e = e;
  r.d = rd;

  return r;
}


/*
 * Return a new Decimal whose value is `x` rounded to an integer using rounding mode `rounding`.
 *
 * To emulate `Math.round`, set rounding to 7 (ROUND_HALF_CEIL).
 *
 * x {number|string|Decimal}
 *
 */
function round(x) {
  return finalise(x = new this(x), x.e + 1, this.rounding);
}


/*
 * Return
 *   1    if x > 0,
 *  -1    if x < 0,
 *   0    if x is 0,
 *  -0    if x is -0,
 *   NaN  otherwise
 *
 * x {number|string|Decimal}
 *
 */
function sign(x) {
  x = new this(x);
  return x.d ? (x.d[0] ? x.s : 0 * x.s) : x.s || NaN;
}


/*
 * Return a new Decimal whose value is the sine of `x`, rounded to `precision` significant digits
 * using rounding mode `rounding`.
 *
 * x {number|string|Decimal} A value in radians.
 *
 */
function sin(x) {
  return new this(x).sin();
}


/*
 * Return a new Decimal whose value is the hyperbolic sine of `x`, rounded to `precision`
 * significant digits using rounding mode `rounding`.
 *
 * x {number|string|Decimal} A value in radians.
 *
 */
function sinh(x) {
  return new this(x).sinh();
}


/*
 * Return a new Decimal whose value is the square root of `x`, rounded to `precision` significant
 * digits using rounding mode `rounding`.
 *
 * x {number|string|Decimal}
 *
 */
function sqrt(x) {
  return new this(x).sqrt();
}


/*
 * Return a new Decimal whose value is `x` minus `y`, rounded to `precision` significant digits
 * using rounding mode `rounding`.
 *
 * x {number|string|Decimal}
 * y {number|string|Decimal}
 *
 */
function sub(x, y) {
  return new this(x).sub(y);
}


/*
 * Return a new Decimal whose value is the tangent of `x`, rounded to `precision` significant
 * digits using rounding mode `rounding`.
 *
 * x {number|string|Decimal} A value in radians.
 *
 */
function tan(x) {
  return new this(x).tan();
}


/*
 * Return a new Decimal whose value is the hyperbolic tangent of `x`, rounded to `precision`
 * significant digits using rounding mode `rounding`.
 *
 * x {number|string|Decimal} A value in radians.
 *
 */
function tanh(x) {
  return new this(x).tanh();
}


/*
 * Return a new Decimal whose value is `x` truncated to an integer.
 *
 * x {number|string|Decimal}
 *
 */
function trunc(x) {
  return finalise(x = new this(x), x.e + 1, 1);
}


P[Symbol.for('nodejs.util.inspect.custom')] = P.toString;
P[Symbol.toStringTag] = 'Decimal';

// Create and configure initial Decimal constructor.
export var Decimal = clone(DEFAULTS);

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

export default Decimal;