decimal.js

/*! decimal.js-light v2.5.1 https://github.com/MikeMcl/decimal.js-light/LICENCE */
;(function (globalScope) {
  'use strict';


  /*
   *  decimal.js-light v2.5.1
   *  An arbitrary-precision Decimal type for JavaScript.
   *  https://github.com/MikeMcl/decimal.js-light
   *  Copyright (c) 2020 Michael Mclaughlin <M8ch88l@gmail.com>
   *  MIT Expat Licence
   */


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


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


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

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

      // 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 by default by `toInteger`, `toDecimalPlaces`, `toExponential`,
      // `toFixed`, `toPrecision` and `toSignificantDigits`.
      //
      // 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 exponent value at and beneath which `toString` returns exponential notation.
      // JavaScript numbers: -7
      toExpNeg: -7,                          // 0 to -MAX_E

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

      // The natural logarithm of 10.
      // 115 digits
      LN10: '2.302585092994045684017991454684364207601101488628772976033327900967572609677352480235997205089598298341967784042286'
    },


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


    external = true,

    decimalError = '[DecimalError] ',
    invalidArgument = decimalError + 'Invalid argument: ',
    exponentOutOfRange = decimalError + 'Exponent out of range: ',

    mathfloor = Math.floor,
    mathpow = Math.pow,

    isDecimal = /^(\d+(\.\d*)?|\.\d+)(e[+-]?\d+)?$/i,

    ONE,
    BASE = 1e7,
    LOG_BASE = 7,
    MAX_SAFE_INTEGER = 9007199254740991,
    MAX_E = mathfloor(MAX_SAFE_INTEGER / LOG_BASE),    // 1286742750677284

    // Decimal.prototype object
    P = {};


  // Decimal prototype methods


  /*
   *  absoluteValue                       abs
   *  comparedTo                          cmp
   *  decimalPlaces                       dp
   *  dividedBy                           div
   *  dividedToIntegerBy                  idiv
   *  equals                              eq
   *  exponent
   *  greaterThan                         gt
   *  greaterThanOrEqualTo                gte
   *  isInteger                           isint
   *  isNegative                          isneg
   *  isPositive                          ispos
   *  isZero
   *  lessThan                            lt
   *  lessThanOrEqualTo                   lte
   *  logarithm                           log
   *  minus                               sub
   *  modulo                              mod
   *  naturalExponential                  exp
   *  naturalLogarithm                    ln
   *  negated                             neg
   *  plus                                add
   *  precision                           sd
   *  squareRoot                          sqrt
   *  times                               mul
   *  toDecimalPlaces                     todp
   *  toExponential
   *  toFixed
   *  toInteger                           toint
   *  toNumber
   *  toPower                             pow
   *  toPrecision
   *  toSignificantDigits                 tosd
   *  toString
   *  valueOf                             val
   */


  /*
   * 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) x.s = 1;
    return x;
  };


  /*
   * Return
   *   1    if the value of this Decimal is greater than the value of `y`,
   *  -1    if the value of this Decimal is less than the value of `y`,
   *   0    if they have the same value
   *
   */
  P.comparedTo = P.cmp = function (y) {
    var i, j, xdL, ydL,
      x = this;

    y = new x.constructor(y);

    // Signs differ?
    if (x.s !== y.s) return x.s || -y.s;

    // Compare exponents.
    if (x.e !== y.e) return x.e > y.e ^ x.s < 0 ? 1 : -1;

    xdL = x.d.length;
    ydL = y.d.length;

    // Compare digit by digit.
    for (i = 0, j = xdL < ydL ? xdL : ydL; i < j; ++i) {
      if (x.d[i] !== y.d[i]) return x.d[i] > y.d[i] ^ x.s < 0 ? 1 : -1;
    }

    // Compare lengths.
    return xdL === ydL ? 0 : xdL > ydL ^ x.s < 0 ? 1 : -1;
  };


  /*
   * Return the number of decimal places of the value of this Decimal.
   *
   */
  P.decimalPlaces = P.dp = function () {
    var x = this,
      w = x.d.length - 1,
      dp = (w - x.e) * LOG_BASE;

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

    return dp < 0 ? 0 : dp;
  };


  /*
   * Return a new Decimal whose value is the value of this Decimal divided by `y`, truncated to
   * `precision` significant digits.
   *
   */
  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`, truncated to `precision` significant digits.
   *
   */
  P.dividedToIntegerBy = P.idiv = function (y) {
    var x = this,
      Ctor = x.constructor;
    return round(divide(x, new Ctor(y), 0, 1), Ctor.precision);
  };


  /*
   * 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);
  };


  /*
   * Return the (base 10) exponent value of this Decimal (this.e is the base 10000000 exponent).
   *
   */
  P.exponent = function () {
    return getBase10Exponent(this);
  };


  /*
   * 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) {
    return this.cmp(y) >= 0;
  };


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


  /*
   * 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, otherwise return false.
   *
   */
  P.isZero = function () {
    return this.s === 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, truncated to
   * `precision` significant digits.
   *
   * If no base is specified, return log[10](x).
   *
   * log[base](x) = ln(x) / ln(base)
   *
   * The maximum error of the result is 1 ulp (unit in the last place).
   *
   * [base] {number|string|Decimal} The base of the logarithm.
   *
   */
  P.logarithm = P.log = function (base) {
    var r,
      x = this,
      Ctor = x.constructor,
      pr = Ctor.precision,
      wpr = pr + 5;

    // Default base is 10.
    if (base === void 0) {
      base = new Ctor(10);
    } else {
      base = new Ctor(base);

      // log[-b](x) = NaN
      // log[0](x)  = NaN
      // log[1](x)  = NaN
      if (base.s < 1 || base.eq(ONE)) throw Error(decimalError + 'NaN');
    }

    // log[b](-x) = NaN
    // log[b](0) = -Infinity
    if (x.s < 1) throw Error(decimalError + (x.s ? 'NaN' : '-Infinity'));

    // log[b](1) = 0
    if (x.eq(ONE)) return new Ctor(0);

    external = false;
    r = divide(ln(x, wpr), ln(base, wpr), wpr);
    external = true;

    return round(r, pr);
  };


  /*
   * Return a new Decimal whose value is the value of this Decimal minus `y`, truncated to
   * `precision` significant digits.
   *
   */
  P.minus = P.sub = function (y) {
    var x = this;
    y = new x.constructor(y);
    return x.s == y.s ? subtract(x, y) : add(x, (y.s = -y.s, y));
  };


  /*
   * Return a new Decimal whose value is the value of this Decimal modulo `y`, truncated to
   * `precision` significant digits.
   *
   */
  P.modulo = P.mod = function (y) {
    var q,
      x = this,
      Ctor = x.constructor,
      pr = Ctor.precision;

    y = new Ctor(y);

    // x % 0 = NaN
    if (!y.s) throw Error(decimalError + 'NaN');

    // Return x if x is 0.
    if (!x.s) return round(new Ctor(x), pr);

    // Prevent rounding of intermediate calculations.
    external = false;
    q = divide(x, y, 0, 1).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, truncated to `precision`
   * significant digits.
   *
   */
  P.naturalExponential = P.exp = function () {
    return exp(this);
  };


  /*
   * Return a new Decimal whose value is the natural logarithm of the value of this Decimal,
   * truncated to `precision` significant digits.
   *
   */
  P.naturalLogarithm = P.ln = function () {
    return ln(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 || 0;
    return x;
  };


  /*
   * Return a new Decimal whose value is the value of this Decimal plus `y`, truncated to
   * `precision` significant digits.
   *
   */
  P.plus = P.add = function (y) {
    var x = this;
    y = new x.constructor(y);
    return x.s == y.s ? add(x, y) : subtract(x, (y.s = -y.s, 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 e, sd, w,
      x = this;

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

    e = getBase10Exponent(x) + 1;
    w = x.d.length - 1;
    sd = w * LOG_BASE + 1;
    w = x.d[w];

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

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

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

    return z && e > sd ? e : sd;
  };


  /*
   * Return a new Decimal whose value is the square root of this Decimal, truncated to `precision`
   * significant digits.
   *
   */
  P.squareRoot = P.sqrt = function () {
    var e, n, pr, r, s, t, wpr,
      x = this,
      Ctor = x.constructor;

    // Negative or zero?
    if (x.s < 1) {
      if (!x.s) return new Ctor(0);

      // sqrt(-x) = NaN
      throw Error(decimalError + 'NaN');
    }

    e = getBase10Exponent(x);
    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(x.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());
    }

    pr = Ctor.precision;
    s = wpr = pr + 3;

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

      if (digitsToString(t.d).slice(0, wpr) === (n = digitsToString(r.d)).slice(0, wpr)) {
        n = n.slice(wpr - 3, wpr + 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 (s == wpr && n == '4999') {

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

          if (t.times(t).eq(x)) {
            r = t;
            break;
          }
        } else if (n != '9999') {
          break;
        }

        wpr += 4;
      }
    }

    external = true;

    return round(r, pr);
  };


  /*
   * Return a new Decimal whose value is the value of this Decimal times `y`, truncated to
   * `precision` significant digits.
   *
   */
  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;

    // Return 0 if either is 0.
    if (!x.s || !y.s) return new Ctor(0);

    y.s *= x.s;
    e = x.e + y.e;
    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 = e;

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


  /*
   * 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 round(x, dp + getBase10Exponent(x) + 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 = toString(x, true);
    } else {
      checkInt32(dp, 0, MAX_DIGITS);

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

      x = round(new Ctor(x), dp + 1, rm);
      str = toString(x, true, dp + 1);
    }

    return 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) return toString(x);

    checkInt32(dp, 0, MAX_DIGITS);

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

    y = round(new Ctor(x), dp + getBase10Exponent(x) + 1, rm);
    str = toString(y.abs(), false, dp + getBase10Exponent(y) + 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 a new Decimal whose value is the value of this Decimal rounded to a whole number using
   * rounding mode `rounding`.
   *
   */
  P.toInteger = P.toint = function () {
    var x = this,
      Ctor = x.constructor;
    return round(new Ctor(x), getBase10Exponent(x) + 1, Ctor.rounding);
  };


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


  /*
   * Return a new Decimal whose value is the value of this Decimal raised to the power `y`,
   * truncated to `precision` significant digits.
   *
   * For non-integer or very large exponents pow(x, y) is calculated using
   *
   *   x^y = exp(y*ln(x))
   *
   * The maximum error is 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, sign, yIsInt,
      x = this,
      Ctor = x.constructor,
      guard = 12,
      yn = +(y = new Ctor(y));

    // pow(x, 0) = 1
    if (!y.s) return new Ctor(ONE);

    x = new Ctor(x);

    // pow(0, y > 0) = 0
    // pow(0, y < 0) = Infinity
    if (!x.s) {
      if (y.s < 1) throw Error(decimalError + 'Infinity');
      return x;
    }

    // pow(1, y) = 1
    if (x.eq(ONE)) return x;

    pr = Ctor.precision;

    // pow(x, 1) = x
    if (y.eq(ONE)) return round(x, pr);

    e = y.e;
    k = y.d.length - 1;
    yIsInt = e >= k;
    sign = x.s;

    if (!yIsInt) {

      // pow(x < 0, y non-integer) = NaN
      if (sign < 0) throw Error(decimalError + 'NaN');

    // If y is a small integer use the 'exponentiation by squaring' algorithm.
    } else if ((k = yn < 0 ? -yn : yn) <= MAX_SAFE_INTEGER) {
      r = new Ctor(ONE);

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

      external = false;

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

        k = mathfloor(k / 2);
        if (k === 0) break;

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

      external = true;

      return y.s < 0 ? new Ctor(ONE).div(r) : round(r, pr);
    }

    // Result is negative if x is negative and the last digit of integer y is odd.
    sign = sign < 0 && y.d[Math.max(e, k)] & 1 ? -1 : 1;

    x.s = 1;
    external = false;
    r = y.times(ln(x, pr + guard));
    external = true;
    r = exp(r);
    r.s = sign;

    return r;
  };


  /*
   * 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 e, str,
      x = this,
      Ctor = x.constructor;

    if (sd === void 0) {
      e = getBase10Exponent(x);
      str = toString(x, e <= Ctor.toExpNeg || e >= Ctor.toExpPos);
    } else {
      checkInt32(sd, 1, MAX_DIGITS);

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

      x = round(new Ctor(x), sd, rm);
      e = getBase10Exponent(x);
      str = toString(x, sd <= e || e <= Ctor.toExpNeg, sd);
    }

    return 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.
   *
   */
  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 round(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 = P.valueOf = P.val = P.toJSON = function () {
    var x = this,
      e = getBase10Exponent(x),
      Ctor = x.constructor;

    return toString(x, e <= Ctor.toExpNeg || e >= Ctor.toExpPos);
  };


  // Helper functions for Decimal.prototype (P) and/or Decimal methods, and their callers.


  /*
   *  add                 P.minus, P.plus
   *  checkInt32          P.todp, P.toExponential, P.toFixed, P.toPrecision, P.tosd
   *  digitsToString      P.log, P.sqrt, P.pow, toString, exp, ln
   *  divide              P.div, P.idiv, P.log, P.mod, P.sqrt, exp, ln
   *  exp                 P.exp, P.pow
   *  getBase10Exponent   P.exponent, P.sd, P.toint, P.sqrt, P.todp, P.toFixed, P.toPrecision,
   *                      P.toString, divide, round, toString, exp, ln
   *  getLn10             P.log, ln
   *  getZeroString       digitsToString, toString
   *  ln                  P.log, P.ln, P.pow, exp
   *  parseDecimal        Decimal
   *  round               P.abs, P.idiv, P.log, P.minus, P.mod, P.neg, P.plus, P.toint, P.sqrt,
   *                      P.times, P.todp, P.toExponential, P.toFixed, P.pow, P.toPrecision, P.tosd,
   *                      divide, getLn10, exp, ln
   *  subtract            P.minus, P.plus
   *  toString            P.toExponential, P.toFixed, P.toPrecision, P.toString, P.valueOf
   *  truncate            P.pow
   *
   *  Throws:             P.log, P.mod, P.sd, P.sqrt, P.pow,  checkInt32, divide, round,
   *                      getLn10, exp, ln, parseDecimal, Decimal, config
   */


  function add(x, y) {
    var carry, d, e, i, k, len, xd, yd,
      Ctor = x.constructor,
      pr = Ctor.precision;

    // If either is zero...
    if (!x.s || !y.s) {

      // Return x if y is zero.
      // Return y if y is non-zero.
      if (!y.s) y = new Ctor(x);
      return external ? round(y, pr) : y;
    }

    xd = x.d;
    yd = y.d;

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

    k = x.e;
    e = y.e;
    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 = e;

    return external ? round(y, pr) : y;
  }


  function checkInt32(i, min, max) {
    if (i !== ~~i || i < min || i > max) {
      throw Error(invalidArgument + i);
    }
  }


  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;
  }


  var divide = (function () {

    // Assumes non-zero x and k, and hence non-zero result.
    function multiplyInteger(x, k) {
      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) {
      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, dp) {
      var cmp, e, i, k, 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 0?
      if (!x.s) return new Ctor(x);
      if (!y.s) throw Error(decimalError + 'Division by zero');

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

      // Result exponent may be one less than e.
      for (i = 0; yd[i] == (xd[i] || 0); ) ++i;
      if (yd[i] > (xd[i] || 0)) --e;

      if (pr == null) {
        sd = pr = Ctor.precision;
      } else if (dp) {
        sd = pr + (getBase10Exponent(x) - getBase10Exponent(y)) + 1;
      } else {
        sd = pr;
      }

      if (sd < 0) return new Ctor(0);

      // Convert precision in number of base 10 digits to base 1e7 digits.
      sd = sd / LOG_BASE + 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;
        }

      // 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);
          xd = multiplyInteger(xd, k);
          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);
              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);
              }
            } 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);

            // 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);
              }
            }

            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--);
      }

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

      q.e = e;

      return round(q, dp ? pr + getBase10Exponent(q) + 1 : pr);
    };
  })();


  /*
   * Return a new Decimal whose value is the natural exponential of `x` truncated 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.
   *
   * (Math object integer min/max: Math.exp(709) = 8.2e+307, Math.exp(-745) = 5e-324)
   *
   *  exp(x) is non-terminating for any finite, non-zero x.
   *
   */
  function exp(x, sd) {
    var denominator, guard, pow, sum, t, wpr,
      i = 0,
      k = 0,
      Ctor = x.constructor,
      pr = Ctor.precision;

    if (getBase10Exponent(x) > 16) throw Error(exponentOutOfRange + getBase10Exponent(x));

    // exp(0) = 1
    if (!x.s) return new Ctor(ONE);

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

    t = new Ctor(0.03125);

    while (x.abs().gte(0.1)) {
      x = x.times(t);    // x = x / 2^5
      k += 5;
    }

    // Estimate the precision increase 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(ONE);
    Ctor.precision = wpr;

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

      if (digitsToString(t.d).slice(0, wpr) === digitsToString(sum.d).slice(0, wpr)) {
        while (k--) sum = round(sum.times(sum), wpr);
        Ctor.precision = pr;
        return sd == null ? (external = true, round(sum, pr)) : sum;
      }

      sum = t;
    }
  }


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

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


  function getLn10(Ctor, sd, pr) {

    if (sd > Ctor.LN10.sd()) {


      // Reset global state in case the exception is caught.
      external = true;
      if (pr) Ctor.precision = pr;
      throw Error(decimalError + 'LN10 precision limit exceeded');
    }

    return round(new Ctor(Ctor.LN10), sd);
  }


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


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

    // ln(-x) = NaN
    // ln(0) = -Infinity
    if (x.s < 1) throw Error(decimalError + (x.s ? 'NaN' : '-Infinity'));

    // ln(1) = 0
    if (x.eq(ONE)) return new Ctor(0);

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

    if (x.eq(10)) {
      if (sd == null) external = true;
      return getLn10(Ctor, wpr);
    }

    wpr += guard;
    Ctor.precision = wpr;
    c = digitsToString(xd);
    c0 = c.charAt(0);
    e = getBase10Exponent(x);

    if (Math.abs(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 = getBase10Exponent(x);

      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 = ln(new Ctor(c0 + '.' + c.slice(1)), wpr - guard).plus(t);

      Ctor.precision = pr;
      return sd == null ? (external = true, round(x, pr)) : x;
    }

    // x is reduced to a value near 1.

    // 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(ONE), x.plus(ONE), wpr);
    x2 = round(x.times(x), wpr);
    denominator = 3;

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

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

        // Reverse the argument reduction.
        if (e !== 0) sum = sum.plus(getLn10(Ctor, wpr + 2, pr).times(e + ''));
        sum = divide(sum, new Ctor(n), wpr);

        Ctor.precision = pr;
        return sd == null ? (external = true, round(sum, pr)) : sum;
      }

      sum = t;
      denominator += 2;
    }
  }


  /*
   * 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;
      e = e - i - 1;
      x.e = mathfloor(e / LOG_BASE);
      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 && (x.e > MAX_E || x.e < -MAX_E)) throw Error(exponentOutOfRange + e);
    } else {

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

    return x;
  }


  /*
   * Round `x` to `sd` significant digits, using rounding mode `rm` if present (truncate otherwise).
   */
   function round(x, sd, rm) {
    var i, j, k, n, rd, doRound, w, xdi,
      xd = x.d;

    // rd: the rounding digit, i.e. the digit after the digit that may be rounded up.
    // w: the word of xd which contains the rounding digit, a base 1e7 number.
    // xdi: the index of w within xd.
    // n: 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 (n = 1, k = xd[0]; k >= 10; k /= 10) n++;
    i = sd - n;

    // Is the rounding digit in the first word of xd?
    if (i < 0) {
      i += LOG_BASE;
      j = sd;
      w = xd[xdi = 0];
    } else {
      xdi = Math.ceil((i + 1) / LOG_BASE);
      k = xd.length;
      if (xdi >= k) return x;
      w = k = xd[xdi];

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

      // 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 - n.
      j = i - LOG_BASE + n;
    }

    if (rm !== void 0) {
      k = mathpow(10, n - j - 1);

      // Get the rounding digit at index j of w.
      rd = w / k % 10 | 0;

      // Are there any non-zero digits after the rounding digit?
      doRound = sd < 0 || xd[xdi + 1] !== void 0 || w % k;

      // The expression `w % mathpow(10, n - 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.

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

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

    if (sd < 1 || !xd[0]) {
      if (doRound) {
        k = getBase10Exponent(x);
        xd.length = 1;

        // Convert sd to decimal places.
        sd = sd - k - 1;

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

        // Zero.
        xd[0] = x.e = x.s = 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, n - j) % mathpow(10, j) | 0) * k : 0;
    }

    if (doRound) {
      for (;;) {

        // Is the digit to be rounded up in the first word of xd?
        if (xdi == 0) {
          if ((xd[0] += k) == BASE) {
            xd[0] = 1;
            ++x.e;
          }

          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 && (x.e > MAX_E || x.e < -MAX_E)) {
      throw Error(exponentOutOfRange + getBase10Exponent(x));
    }

    return x;
  }


  function subtract(x, y) {
    var d, e, i, j, k, len, xd, xe, xLTy, yd,
      Ctor = x.constructor,
      pr = Ctor.precision;

    // Return y negated if x is zero.
    // Return x if y is zero and x is non-zero.
    if (!x.s || !y.s) {
      if (y.s) y.s = -y.s;
      else y = new Ctor(x);
      return external ? round(y, pr) : y;
    }

    xd = x.d;
    yd = y.d;

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

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

    // If 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(0);

    y.d = xd;
    y.e = e;

    //return external && xd.length >= pr / LOG_BASE ? round(y, pr) : y;
    return external ? round(y, pr) : y;
  }


  function toString(x, isExp, sd) {
    var k,
      e = getBase10Exponent(x),
      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 + (e < 0 ? 'e' : 'e+') + 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 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


  /*
   *  clone
   *  config/set
   */


  /*
   * 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.
     *
     * value {number|string|Decimal} A numeric value.
     *
     */
    function Decimal(value) {
      var x = this;

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

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

      // Duplicate.
      if (value instanceof Decimal) {
        x.s = value.s;
        x.e = value.e;
        x.d = (value = value.d) ? value.slice() : value;
        return;
      }

      if (typeof value === 'number') {

        // Reject Infinity/NaN.
        if (value * 0 !== 0) {
          throw Error(invalidArgument + value);
        }

        if (value > 0) {
          x.s = 1;
        } else if (value < 0) {
          value = -value;
          x.s = -1;
        } else {
          x.s = 0;
          x.e = 0;
          x.d = [0];
          return;
        }

        // Fast path for small integers.
        if (value === ~~value && value < 1e7) {
          x.e = 0;
          x.d = [value];
          return;
        }

        return parseDecimal(x, value.toString());
      } else if (typeof value !== 'string') {
        throw Error(invalidArgument + value);
      }

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

      if (isDecimal.test(value)) parseDecimal(x, value);
      else throw Error(invalidArgument + value);
    }

    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.clone = clone;
    Decimal.config = Decimal.set = config;

    if (obj === void 0) obj = {};
    if (obj) {
      ps = ['precision', 'rounding', 'toExpNeg', 'toExpPos', 'LN10'];
      for (i = 0; i < ps.length;) if (!obj.hasOwnProperty(p = ps[i++])) obj[p] = this[p];
    }

    Decimal.config(obj);

    return Decimal;
  }


  /*
   * 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}
   *
   * E.g. Decimal.config({ precision: 20, rounding: 4 })
   *
   */
  function config(obj) {
    if (!obj || typeof obj !== 'object') {
      throw Error(decimalError + 'Object expected');
    }
    var i, p, v,
      ps = [
        'precision', 1, MAX_DIGITS,
        'rounding', 0, 8,
        'toExpNeg', -1 / 0, 0,
        'toExpPos', 0, 1 / 0
      ];

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

    if ((v = obj[p = 'LN10']) !== void 0) {
        if (v == Math.LN10) this[p] = new this(v);
        else throw Error(invalidArgument + p + ': ' + v);
    }

    return this;
  }


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

  Decimal['default'] = Decimal.Decimal = Decimal;

  // Internal constant.
  ONE = new Decimal(1);


  // Export.


  // AMD.
  if (typeof define == 'function' && define.amd) {
    define(function () {
      return Decimal;
    });

  // Node and other environments that support module.exports.
  } else if (typeof module != 'undefined' && module.exports) {
    module.exports = Decimal;

    // Browser.
  } else {
    if (!globalScope) {
      globalScope = typeof self != 'undefined' && self && self.self == self
        ? self : Function('return this')();
    }

    globalScope.Decimal = Decimal;
  }
})(this);