/*
 *      bignumber.js v9.0.1
 *      A JavaScript library for arbitrary-precision arithmetic.
 *      https://github.com/MikeMcl/bignumber.js
 *      Copyright (c) 2020 Michael Mclaughlin <M8ch88l@gmail.com>
 *      MIT Licensed.
 *
 *      BigNumber.prototype methods     |  BigNumber methods
 *                                      |
 *      absoluteValue            abs    |  clone
 *      comparedTo                      |  config               set
 *      decimalPlaces            dp     |      DECIMAL_PLACES
 *      dividedBy                div    |      ROUNDING_MODE
 *      dividedToIntegerBy       idiv   |      EXPONENTIAL_AT
 *      exponentiatedBy          pow    |      RANGE
 *      integerValue                    |      CRYPTO
 *      isEqualTo                eq     |      MODULO_MODE
 *      isFinite                        |      POW_PRECISION
 *      isGreaterThan            gt     |      FORMAT
 *      isGreaterThanOrEqualTo   gte    |      ALPHABET
 *      isInteger                       |  isBigNumber
 *      isLessThan               lt     |  maximum              max
 *      isLessThanOrEqualTo      lte    |  minimum              min
 *      isNaN                           |  random
 *      isNegative                      |  sum
 *      isPositive                      |
 *      isZero                          |
 *      minus                           |
 *      modulo                   mod    |
 *      multipliedBy             times  |
 *      negated                         |
 *      plus                            |
 *      precision                sd     |
 *      shiftedBy                       |
 *      squareRoot               sqrt   |
 *      toExponential                   |
 *      toFixed                         |
 *      toFormat                        |
 *      toFraction                      |
 *      toJSON                          |
 *      toNumber                        |
 *      toPrecision                     |
 *      toString                        |
 *      valueOf                         |
 *
 */


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

  mathceil = Math.ceil,
  mathfloor = Math.floor,

  bignumberError = '[BigNumber Error] ',
  tooManyDigits = bignumberError + 'Number primitive has more than 15 significant digits: ',

  BASE = 1e14,
  LOG_BASE = 14,
  MAX_SAFE_INTEGER = 0x1fffffffffffff,         // 2^53 - 1
  // MAX_INT32 = 0x7fffffff,                   // 2^31 - 1
  POWS_TEN = [1, 10, 100, 1e3, 1e4, 1e5, 1e6, 1e7, 1e8, 1e9, 1e10, 1e11, 1e12, 1e13],
  SQRT_BASE = 1e7,

  // EDITABLE
  // The limit on the value of DECIMAL_PLACES, TO_EXP_NEG, TO_EXP_POS, MIN_EXP, MAX_EXP, and
  // the arguments to toExponential, toFixed, toFormat, and toPrecision.
  MAX = 1E9;                                   // 0 to MAX_INT32


/*
 * Create and return a BigNumber constructor.
 */
function clone(configObject) {
  var div, convertBase, parseNumeric,
    P = BigNumber.prototype = { constructor: BigNumber, toString: null, valueOf: null },
    ONE = new BigNumber(1),


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


    // The default values below must be integers within the inclusive ranges stated.
    // The values can also be changed at run-time using BigNumber.set.

    // The maximum number of decimal places for operations involving division.
    DECIMAL_PLACES = 20,                     // 0 to MAX

    // The rounding mode used when rounding to the above decimal places, and when using
    // toExponential, toFixed, toFormat and toPrecision, and round (default value).
    // UP         0 Away from zero.
    // DOWN       1 Towards zero.
    // CEIL       2 Towards +Infinity.
    // FLOOR      3 Towards -Infinity.
    // HALF_UP    4 Towards nearest neighbour. If equidistant, up.
    // HALF_DOWN  5 Towards nearest neighbour. If equidistant, down.
    // HALF_EVEN  6 Towards nearest neighbour. If equidistant, towards even neighbour.
    // HALF_CEIL  7 Towards nearest neighbour. If equidistant, towards +Infinity.
    // HALF_FLOOR 8 Towards nearest neighbour. If equidistant, towards -Infinity.
    ROUNDING_MODE = 4,                       // 0 to 8

    // EXPONENTIAL_AT : [TO_EXP_NEG , TO_EXP_POS]

    // The exponent value at and beneath which toString returns exponential notation.
    // Number type: -7
    TO_EXP_NEG = -7,                         // 0 to -MAX

    // The exponent value at and above which toString returns exponential notation.
    // Number type: 21
    TO_EXP_POS = 21,                         // 0 to MAX

    // RANGE : [MIN_EXP, MAX_EXP]

    // The minimum exponent value, beneath which underflow to zero occurs.
    // Number type: -324  (5e-324)
    MIN_EXP = -1e7,                          // -1 to -MAX

    // The maximum exponent value, above which overflow to Infinity occurs.
    // Number type:  308  (1.7976931348623157e+308)
    // For MAX_EXP > 1e7, e.g. new BigNumber('1e100000000').plus(1) may be slow.
    MAX_EXP = 1e7,                           // 1 to MAX

    // Whether to use cryptographically-secure random number generation, if available.
    CRYPTO = false,                          // true or false

    // 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.
    //             This modulo mode is commonly known as 'truncated division' and is
    //             equivalent to (a % n) in JavaScript.
    // FLOOR     3 The remainder has the same sign as the divisor (Python %).
    // HALF_EVEN 6 This modulo mode implements the IEEE 754 remainder function.
    // EUCLID    9 Euclidian division. q = sign(n) * floor(a / abs(n)).
    //             The remainder is always positive.
    //
    // The truncated division, floored division, Euclidian division and IEEE 754 remainder
    // modes are commonly used for the modulus operation.
    // Although the other rounding modes can also be used, they may not give useful results.
    MODULO_MODE = 1,                         // 0 to 9

    // The maximum number of significant digits of the result of the exponentiatedBy operation.
    // If POW_PRECISION is 0, there will be unlimited significant digits.
    POW_PRECISION = 0,                    // 0 to MAX

    // The format specification used by the BigNumber.prototype.toFormat method.
    FORMAT = {
      prefix: '',
      groupSize: 3,
      secondaryGroupSize: 0,
      groupSeparator: ',',
      decimalSeparator: '.',
      fractionGroupSize: 0,
      fractionGroupSeparator: '\xA0',      // non-breaking space
      suffix: ''
    },

    // The alphabet used for base conversion. It must be at least 2 characters long, with no '+',
    // '-', '.', whitespace, or repeated character.
    // '0123456789abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ$_'
    ALPHABET = '0123456789abcdefghijklmnopqrstuvwxyz';


  //------------------------------------------------------------------------------------------


  // CONSTRUCTOR


  /*
   * The BigNumber constructor and exported function.
   * Create and return a new instance of a BigNumber object.
   *
   * v {number|string|BigNumber} A numeric value.
   * [b] {number} The base of v. Integer, 2 to ALPHABET.length inclusive.
   */
  function BigNumber(v, b) {
    var alphabet, c, caseChanged, e, i, isNum, len, str,
      x = this;

    // Enable constructor call without `new`.
    if (!(x instanceof BigNumber)) return new BigNumber(v, b);

    if (b == null) {

      if (v && v._isBigNumber === true) {
        x.s = v.s;

        if (!v.c || v.e > MAX_EXP) {
          x.c = x.e = null;
        } else if (v.e < MIN_EXP) {
          x.c = [x.e = 0];
        } else {
          x.e = v.e;
          x.c = v.c.slice();
        }

        return;
      }

      if ((isNum = typeof v == 'number') && v * 0 == 0) {

        // Use `1 / n` to handle minus zero also.
        x.s = 1 / v < 0 ? (v = -v, -1) : 1;

        // Fast path for integers, where n < 2147483648 (2**31).
        if (v === ~~v) {
          for (e = 0, i = v; i >= 10; i /= 10, e++);

          if (e > MAX_EXP) {
            x.c = x.e = null;
          } else {
            x.e = e;
            x.c = [v];
          }

          return;
        }

        str = String(v);
      } else {

        if (!isNumeric.test(str = String(v))) return parseNumeric(x, str, isNum);

        x.s = str.charCodeAt(0) == 45 ? (str = str.slice(1), -1) : 1;
      }

      // 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;
      }

    } else {

      // '[BigNumber Error] Base {not a primitive number|not an integer|out of range}: {b}'
      intCheck(b, 2, ALPHABET.length, 'Base');

      // Allow exponential notation to be used with base 10 argument, while
      // also rounding to DECIMAL_PLACES as with other bases.
      if (b == 10) {
        x = new BigNumber(v);
        return round(x, DECIMAL_PLACES + x.e + 1, ROUNDING_MODE);
      }

      str = String(v);

      if (isNum = typeof v == 'number') {

        // Avoid potential interpretation of Infinity and NaN as base 44+ values.
        if (v * 0 != 0) return parseNumeric(x, str, isNum, b);

        x.s = 1 / v < 0 ? (str = str.slice(1), -1) : 1;

        // '[BigNumber Error] Number primitive has more than 15 significant digits: {n}'
        if (BigNumber.DEBUG && str.replace(/^0\.0*|\./, '').length > 15) {
          throw Error
           (tooManyDigits + v);
        }
      } else {
        x.s = str.charCodeAt(0) === 45 ? (str = str.slice(1), -1) : 1;
      }

      alphabet = ALPHABET.slice(0, b);
      e = i = 0;

      // Check that str is a valid base b number.
      // Don't use RegExp, so alphabet can contain special characters.
      for (len = str.length; i < len; i++) {
        if (alphabet.indexOf(c = str.charAt(i)) < 0) {
          if (c == '.') {

            // If '.' is not the first character and it has not be found before.
            if (i > e) {
              e = len;
              continue;
            }
          } else if (!caseChanged) {

            // Allow e.g. hexadecimal 'FF' as well as 'ff'.
            if (str == str.toUpperCase() && (str = str.toLowerCase()) ||
                str == str.toLowerCase() && (str = str.toUpperCase())) {
              caseChanged = true;
              i = -1;
              e = 0;
              continue;
            }
          }

          return parseNumeric(x, String(v), isNum, b);
        }
      }

      // Prevent later check for length on converted number.
      isNum = false;
      str = convertBase(str, b, 10, x.s);

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

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

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

    if (str = str.slice(i, ++len)) {
      len -= i;

      // '[BigNumber Error] Number primitive has more than 15 significant digits: {n}'
      if (isNum && BigNumber.DEBUG &&
        len > 15 && (v > MAX_SAFE_INTEGER || v !== mathfloor(v))) {
          throw Error
           (tooManyDigits + (x.s * v));
      }

       // Overflow?
      if ((e = e - i - 1) > MAX_EXP) {

        // Infinity.
        x.c = x.e = null;

      // Underflow?
      } else if (e < MIN_EXP) {

        // Zero.
        x.c = [x.e = 0];
      } else {
        x.e = e;
        x.c = [];

        // Transform base

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

        if (i < len) {
          if (i) x.c.push(+str.slice(0, i));

          for (len -= LOG_BASE; i < len;) {
            x.c.push(+str.slice(i, i += LOG_BASE));
          }

          i = LOG_BASE - (str = str.slice(i)).length;
        } else {
          i -= len;
        }

        for (; i--; str += '0');
        x.c.push(+str);
      }
    } else {

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


  // CONSTRUCTOR PROPERTIES


  BigNumber.clone = clone;

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


  /*
   * Configure infrequently-changing library-wide settings.
   *
   * Accept an object with the following optional properties (if the value of a property is
   * a number, it must be an integer within the inclusive range stated):
   *
   *   DECIMAL_PLACES   {number}           0 to MAX
   *   ROUNDING_MODE    {number}           0 to 8
   *   EXPONENTIAL_AT   {number|number[]}  -MAX to MAX  or  [-MAX to 0, 0 to MAX]
   *   RANGE            {number|number[]}  -MAX to MAX (not zero)  or  [-MAX to -1, 1 to MAX]
   *   CRYPTO           {boolean}          true or false
   *   MODULO_MODE      {number}           0 to 9
   *   POW_PRECISION       {number}           0 to MAX
   *   ALPHABET         {string}           A string of two or more unique characters which does
   *                                     not contain '.'.
   *   FORMAT           {object}           An object with some of the following properties:
   *     prefix                 {string}
   *     groupSize              {number}
   *     secondaryGroupSize     {number}
   *     groupSeparator         {string}
   *     decimalSeparator       {string}
   *     fractionGroupSize      {number}
   *     fractionGroupSeparator {string}
   *     suffix                 {string}
   *
   * (The values assigned to the above FORMAT object properties are not checked for validity.)
   *
   * E.g.
   * BigNumber.config({ DECIMAL_PLACES : 20, ROUNDING_MODE : 4 })
   *
   * Ignore properties/parameters set to null or undefined, except for ALPHABET.
   *
   * Return an object with the properties current values.
   */
  BigNumber.config = BigNumber.set = function (obj) {
    var p, v;

    if (obj != null) {

      if (typeof obj == 'object') {

        // DECIMAL_PLACES {number} Integer, 0 to MAX inclusive.
        // '[BigNumber Error] DECIMAL_PLACES {not a primitive number|not an integer|out of range}: {v}'
        if (obj.hasOwnProperty(p = 'DECIMAL_PLACES')) {
          v = obj[p];
          intCheck(v, 0, MAX, p);
          DECIMAL_PLACES = v;
        }

        // ROUNDING_MODE {number} Integer, 0 to 8 inclusive.
        // '[BigNumber Error] ROUNDING_MODE {not a primitive number|not an integer|out of range}: {v}'
        if (obj.hasOwnProperty(p = 'ROUNDING_MODE')) {
          v = obj[p];
          intCheck(v, 0, 8, p);
          ROUNDING_MODE = v;
        }

        // EXPONENTIAL_AT {number|number[]}
        // Integer, -MAX to MAX inclusive or
        // [integer -MAX to 0 inclusive, 0 to MAX inclusive].
        // '[BigNumber Error] EXPONENTIAL_AT {not a primitive number|not an integer|out of range}: {v}'
        if (obj.hasOwnProperty(p = 'EXPONENTIAL_AT')) {
          v = obj[p];
          if (v && v.pop) {
            intCheck(v[0], -MAX, 0, p);
            intCheck(v[1], 0, MAX, p);
            TO_EXP_NEG = v[0];
            TO_EXP_POS = v[1];
          } else {
            intCheck(v, -MAX, MAX, p);
            TO_EXP_NEG = -(TO_EXP_POS = v < 0 ? -v : v);
          }
        }

        // RANGE {number|number[]} Non-zero integer, -MAX to MAX inclusive or
        // [integer -MAX to -1 inclusive, integer 1 to MAX inclusive].
        // '[BigNumber Error] RANGE {not a primitive number|not an integer|out of range|cannot be zero}: {v}'
        if (obj.hasOwnProperty(p = 'RANGE')) {
          v = obj[p];
          if (v && v.pop) {
            intCheck(v[0], -MAX, -1, p);
            intCheck(v[1], 1, MAX, p);
            MIN_EXP = v[0];
            MAX_EXP = v[1];
          } else {
            intCheck(v, -MAX, MAX, p);
            if (v) {
              MIN_EXP = -(MAX_EXP = v < 0 ? -v : v);
            } else {
              throw Error
               (bignumberError + p + ' cannot be zero: ' + v);
            }
          }
        }

        // CRYPTO {boolean} true or false.
        // '[BigNumber Error] CRYPTO not true or false: {v}'
        // '[BigNumber Error] crypto unavailable'
        if (obj.hasOwnProperty(p = 'CRYPTO')) {
          v = obj[p];
          if (v === !!v) {
            if (v) {
              if (typeof crypto != 'undefined' && crypto &&
               (crypto.getRandomValues || crypto.randomBytes)) {
                CRYPTO = v;
              } else {
                CRYPTO = !v;
                throw Error
                 (bignumberError + 'crypto unavailable');
              }
            } else {
              CRYPTO = v;
            }
          } else {
            throw Error
             (bignumberError + p + ' not true or false: ' + v);
          }
        }

        // MODULO_MODE {number} Integer, 0 to 9 inclusive.
        // '[BigNumber Error] MODULO_MODE {not a primitive number|not an integer|out of range}: {v}'
        if (obj.hasOwnProperty(p = 'MODULO_MODE')) {
          v = obj[p];
          intCheck(v, 0, 9, p);
          MODULO_MODE = v;
        }

        // POW_PRECISION {number} Integer, 0 to MAX inclusive.
        // '[BigNumber Error] POW_PRECISION {not a primitive number|not an integer|out of range}: {v}'
        if (obj.hasOwnProperty(p = 'POW_PRECISION')) {
          v = obj[p];
          intCheck(v, 0, MAX, p);
          POW_PRECISION = v;
        }

        // FORMAT {object}
        // '[BigNumber Error] FORMAT not an object: {v}'
        if (obj.hasOwnProperty(p = 'FORMAT')) {
          v = obj[p];
          if (typeof v == 'object') FORMAT = v;
          else throw Error
           (bignumberError + p + ' not an object: ' + v);
        }

        // ALPHABET {string}
        // '[BigNumber Error] ALPHABET invalid: {v}'
        if (obj.hasOwnProperty(p = 'ALPHABET')) {
          v = obj[p];

          // Disallow if only one character,
          // or if it contains '+', '-', '.', whitespace, or a repeated character.
          if (typeof v == 'string' && !/^.$|[+-.\s]|(.).*\1/.test(v)) {
            ALPHABET = v;
          } else {
            throw Error
             (bignumberError + p + ' invalid: ' + v);
          }
        }

      } else {

        // '[BigNumber Error] Object expected: {v}'
        throw Error
         (bignumberError + 'Object expected: ' + obj);
      }
    }

    return {
      DECIMAL_PLACES: DECIMAL_PLACES,
      ROUNDING_MODE: ROUNDING_MODE,
      EXPONENTIAL_AT: [TO_EXP_NEG, TO_EXP_POS],
      RANGE: [MIN_EXP, MAX_EXP],
      CRYPTO: CRYPTO,
      MODULO_MODE: MODULO_MODE,
      POW_PRECISION: POW_PRECISION,
      FORMAT: FORMAT,
      ALPHABET: ALPHABET
    };
  };


  /*
   * Return true if v is a BigNumber instance, otherwise return false.
   *
   * If BigNumber.DEBUG is true, throw if a BigNumber instance is not well-formed.
   *
   * v {any}
   *
   * '[BigNumber Error] Invalid BigNumber: {v}'
   */
  BigNumber.isBigNumber = function (v) {
    if (!v || v._isBigNumber !== true) return false;
    if (!BigNumber.DEBUG) return true;

    var i, n,
      c = v.c,
      e = v.e,
      s = v.s;

    out: if ({}.toString.call(c) == '[object Array]') {

      if ((s === 1 || s === -1) && e >= -MAX && e <= MAX && e === mathfloor(e)) {

        // If the first element is zero, the BigNumber value must be zero.
        if (c[0] === 0) {
          if (e === 0 && c.length === 1) return true;
          break out;
        }

        // Calculate number of digits that c[0] should have, based on the exponent.
        i = (e + 1) % LOG_BASE;
        if (i < 1) i += LOG_BASE;

        // Calculate number of digits of c[0].
        //if (Math.ceil(Math.log(c[0] + 1) / Math.LN10) == i) {
        if (String(c[0]).length == i) {

          for (i = 0; i < c.length; i++) {
            n = c[i];
            if (n < 0 || n >= BASE || n !== mathfloor(n)) break out;
          }

          // Last element cannot be zero, unless it is the only element.
          if (n !== 0) return true;
        }
      }

    // Infinity/NaN
    } else if (c === null && e === null && (s === null || s === 1 || s === -1)) {
      return true;
    }

    throw Error
      (bignumberError + 'Invalid BigNumber: ' + v);
  };


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


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


  /*
   * Return a new BigNumber with a random value equal to or greater than 0 and less than 1,
   * and with dp, or DECIMAL_PLACES if dp is omitted, decimal places (or less if trailing
   * zeros are produced).
   *
   * [dp] {number} Decimal places. Integer, 0 to MAX inclusive.
   *
   * '[BigNumber Error] Argument {not a primitive number|not an integer|out of range}: {dp}'
   * '[BigNumber Error] crypto unavailable'
   */
  BigNumber.random = (function () {
    var pow2_53 = 0x20000000000000;

    // Return a 53 bit integer n, where 0 <= n < 9007199254740992.
    // Check if Math.random() produces more than 32 bits of randomness.
    // If it does, assume at least 53 bits are produced, otherwise assume at least 30 bits.
    // 0x40000000 is 2^30, 0x800000 is 2^23, 0x1fffff is 2^21 - 1.
    var random53bitInt = (Math.random() * pow2_53) & 0x1fffff
     ? function () { return mathfloor(Math.random() * pow2_53); }
     : function () { return ((Math.random() * 0x40000000 | 0) * 0x800000) +
       (Math.random() * 0x800000 | 0); };

    return function (dp) {
      var a, b, e, k, v,
        i = 0,
        c = [],
        rand = new BigNumber(ONE);

      if (dp == null) dp = DECIMAL_PLACES;
      else intCheck(dp, 0, MAX);

      k = mathceil(dp / LOG_BASE);

      if (CRYPTO) {

        // Browsers supporting crypto.getRandomValues.
        if (crypto.getRandomValues) {

          a = crypto.getRandomValues(new Uint32Array(k *= 2));

          for (; i < k;) {

            // 53 bits:
            // ((Math.pow(2, 32) - 1) * Math.pow(2, 21)).toString(2)
            // 11111 11111111 11111111 11111111 11100000 00000000 00000000
            // ((Math.pow(2, 32) - 1) >>> 11).toString(2)
            //                                     11111 11111111 11111111
            // 0x20000 is 2^21.
            v = a[i] * 0x20000 + (a[i + 1] >>> 11);

            // Rejection sampling:
            // 0 <= v < 9007199254740992
            // Probability that v >= 9e15, is
            // 7199254740992 / 9007199254740992 ~= 0.0008, i.e. 1 in 1251
            if (v >= 9e15) {
              b = crypto.getRandomValues(new Uint32Array(2));
              a[i] = b[0];
              a[i + 1] = b[1];
            } else {

              // 0 <= v <= 8999999999999999
              // 0 <= (v % 1e14) <= 99999999999999
              c.push(v % 1e14);
              i += 2;
            }
          }
          i = k / 2;

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

          // buffer
          a = crypto.randomBytes(k *= 7);

          for (; i < k;) {

            // 0x1000000000000 is 2^48, 0x10000000000 is 2^40
            // 0x100000000 is 2^32, 0x1000000 is 2^24
            // 11111 11111111 11111111 11111111 11111111 11111111 11111111
            // 0 <= v < 9007199254740992
            v = ((a[i] & 31) * 0x1000000000000) + (a[i + 1] * 0x10000000000) +
               (a[i + 2] * 0x100000000) + (a[i + 3] * 0x1000000) +
               (a[i + 4] << 16) + (a[i + 5] << 8) + a[i + 6];

            if (v >= 9e15) {
              crypto.randomBytes(7).copy(a, i);
            } else {

              // 0 <= (v % 1e14) <= 99999999999999
              c.push(v % 1e14);
              i += 7;
            }
          }
          i = k / 7;
        } else {
          CRYPTO = false;
          throw Error
           (bignumberError + 'crypto unavailable');
        }
      }

      // Use Math.random.
      if (!CRYPTO) {

        for (; i < k;) {
          v = random53bitInt();
          if (v < 9e15) c[i++] = v % 1e14;
        }
      }

      k = c[--i];
      dp %= LOG_BASE;

      // Convert trailing digits to zeros according to dp.
      if (k && dp) {
        v = POWS_TEN[LOG_BASE - dp];
        c[i] = mathfloor(k / v) * v;
      }

      // Remove trailing elements which are zero.
      for (; c[i] === 0; c.pop(), i--);

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

        // Remove leading elements which are zero and adjust exponent accordingly.
        for (e = -1 ; c[0] === 0; c.splice(0, 1), e -= LOG_BASE);

        // Count the digits of the first element of c to determine leading zeros, and...
        for (i = 1, v = c[0]; v >= 10; v /= 10, i++);

        // adjust the exponent accordingly.
        if (i < LOG_BASE) e -= LOG_BASE - i;
      }

      rand.e = e;
      rand.c = c;
      return rand;
    };
  })();


   /*
   * Return a BigNumber whose value is the sum of the arguments.
   *
   * arguments {number|string|BigNumber}
   */
  BigNumber.sum = function () {
    var i = 1,
      args = arguments,
      sum = new BigNumber(args[0]);
    for (; i < args.length;) sum = sum.plus(args[i++]);
    return sum;
  };


  // PRIVATE FUNCTIONS


  // Called by BigNumber and BigNumber.prototype.toString.
  convertBase = (function () {
    var decimal = '0123456789';

    /*
     * Convert string of baseIn to an array of numbers of baseOut.
     * Eg. toBaseOut('255', 10, 16) returns [15, 15].
     * Eg. toBaseOut('ff', 16, 10) returns [2, 5, 5].
     */
    function toBaseOut(str, baseIn, baseOut, alphabet) {
      var j,
        arr = [0],
        arrL,
        i = 0,
        len = str.length;

      for (; i < len;) {
        for (arrL = arr.length; arrL--; arr[arrL] *= baseIn);

        arr[0] += alphabet.indexOf(str.charAt(i++));

        for (j = 0; j < arr.length; j++) {

          if (arr[j] > baseOut - 1) {
            if (arr[j + 1] == null) arr[j + 1] = 0;
            arr[j + 1] += arr[j] / baseOut | 0;
            arr[j] %= baseOut;
          }
        }
      }

      return arr.reverse();
    }

    // Convert a numeric string of baseIn to a numeric string of baseOut.
    // If the caller is toString, we are converting from base 10 to baseOut.
    // If the caller is BigNumber, we are converting from baseIn to base 10.
    return function (str, baseIn, baseOut, sign, callerIsToString) {
      var alphabet, d, e, k, r, x, xc, y,
        i = str.indexOf('.'),
        dp = DECIMAL_PLACES,
        rm = ROUNDING_MODE;

      // Non-integer.
      if (i >= 0) {
        k = POW_PRECISION;

        // Unlimited precision.
        POW_PRECISION = 0;
        str = str.replace('.', '');
        y = new BigNumber(baseIn);
        x = y.pow(str.length - i);
        POW_PRECISION = k;

        // Convert str as if an integer, then restore the fraction part by dividing the
        // result by its base raised to a power.

        y.c = toBaseOut(toFixedPoint(coeffToString(x.c), x.e, '0'),
         10, baseOut, decimal);
        y.e = y.c.length;
      }

      // Convert the number as integer.

      xc = toBaseOut(str, baseIn, baseOut, callerIsToString
       ? (alphabet = ALPHABET, decimal)
       : (alphabet = decimal, ALPHABET));

      // xc now represents str as an integer and converted to baseOut. e is the exponent.
      e = k = xc.length;

      // Remove trailing zeros.
      for (; xc[--k] == 0; xc.pop());

      // Zero?
      if (!xc[0]) return alphabet.charAt(0);

      // Does str represent an integer? If so, no need for the division.
      if (i < 0) {
        --e;
      } else {
        x.c = xc;
        x.e = e;

        // The sign is needed for correct rounding.
        x.s = sign;
        x = div(x, y, dp, rm, baseOut);
        xc = x.c;
        r = x.r;
        e = x.e;
      }

      // xc now represents str converted to baseOut.

      // THe index of the rounding digit.
      d = e + dp + 1;

      // The rounding digit: the digit to the right of the digit that may be rounded up.
      i = xc[d];

      // Look at the rounding digits and mode to determine whether to round up.

      k = baseOut / 2;
      r = r || d < 0 || xc[d + 1] != null;

      r = rm < 4 ? (i != null || r) && (rm == 0 || rm == (x.s < 0 ? 3 : 2))
            : i > k || i == k &&(rm == 4 || r || rm == 6 && xc[d - 1] & 1 ||
             rm == (x.s < 0 ? 8 : 7));

      // If the index of the rounding digit is not greater than zero, or xc represents
      // zero, then the result of the base conversion is zero or, if rounding up, a value
      // such as 0.00001.
      if (d < 1 || !xc[0]) {

        // 1^-dp or 0
        str = r ? toFixedPoint(alphabet.charAt(1), -dp, alphabet.charAt(0)) : alphabet.charAt(0);
      } else {

        // Truncate xc to the required number of decimal places.
        xc.length = d;

        // Round up?
        if (r) {

          // Rounding up may mean the previous digit has to be rounded up and so on.
          for (--baseOut; ++xc[--d] > baseOut;) {
            xc[d] = 0;

            if (!d) {
              ++e;
              xc = [1].concat(xc);
            }
          }
        }

        // Determine trailing zeros.
        for (k = xc.length; !xc[--k];);

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

        // Add leading zeros, decimal point and trailing zeros as required.
        str = toFixedPoint(str, e, alphabet.charAt(0));
      }

      // The caller will add the sign.
      return str;
    };
  })();


  // Perform division in the specified base. Called by div and convertBase.
  div = (function () {

    // Assume non-zero x and k.
    function multiply(x, k, base) {
      var m, temp, xlo, xhi,
        carry = 0,
        i = x.length,
        klo = k % SQRT_BASE,
        khi = k / SQRT_BASE | 0;

      for (x = x.slice(); i--;) {
        xlo = x[i] % SQRT_BASE;
        xhi = x[i] / SQRT_BASE | 0;
        m = khi * xlo + xhi * klo;
        temp = klo * xlo + ((m % SQRT_BASE) * SQRT_BASE) + carry;
        carry = (temp / base | 0) + (m / SQRT_BASE | 0) + khi * xhi;
        x[i] = temp % base;
      }

      if (carry) x = [carry].concat(x);

      return x;
    }

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

      if (aL != bL) {
        cmp = aL > bL ? 1 : -1;
      } else {

        for (i = cmp = 0; i < aL; i++) {

          if (a[i] != b[i]) {
            cmp = a[i] > b[i] ? 1 : -1;
            break;
          }
        }
      }

      return cmp;
    }

    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.splice(0, 1));
    }

    // x: dividend, y: divisor.
    return function (x, y, dp, rm, base) {
      var cmp, e, i, more, n, prod, prodL, q, qc, rem, remL, rem0, xi, xL, yc0,
        yL, yz,
        s = x.s == y.s ? 1 : -1,
        xc = x.c,
        yc = y.c;

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

        return new BigNumber(

         // Return NaN if either NaN, or both Infinity or 0.
         !x.s || !y.s || (xc ? yc && xc[0] == yc[0] : !yc) ? NaN :

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

      q = new BigNumber(s);
      qc = q.c = [];
      e = x.e - y.e;
      s = dp + e + 1;

      if (!base) {
        base = BASE;
        e = bitFloor(x.e / LOG_BASE) - bitFloor(y.e / LOG_BASE);
        s = s / LOG_BASE | 0;
      }

      // Result exponent may be one less then the current value of e.
      // The coefficients of the BigNumbers from convertBase may have trailing zeros.
      for (i = 0; yc[i] == (xc[i] || 0); i++);

      if (yc[i] > (xc[i] || 0)) e--;

      if (s < 0) {
        qc.push(1);
        more = true;
      } else {
        xL = xc.length;
        yL = yc.length;
        i = 0;
        s += 2;

        // Normalise xc and yc so highest order digit of yc is >= base / 2.

        n = mathfloor(base / (yc[0] + 1));

        // Not necessary, but to handle odd bases where yc[0] == (base / 2) - 1.
        // if (n > 1 || n++ == 1 && yc[0] < base / 2) {
        if (n > 1) {
          yc = multiply(yc, n, base);
          xc = multiply(xc, n, base);
          yL = yc.length;
          xL = xc.length;
        }

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

        // Add zeros to make remainder as long as divisor.
        for (; remL < yL; rem[remL++] = 0);
        yz = yc.slice();
        yz = [0].concat(yz);
        yc0 = yc[0];
        if (yc[1] >= base / 2) yc0++;
        // Not necessary, but to prevent trial digit n > base, when using base 3.
        // else if (base == 3 && yc0 == 1) yc0 = 1 + 1e-15;

        do {
          n = 0;

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

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

            // Calculate trial digit, n.

            rem0 = rem[0];
            if (yL != remL) rem0 = rem0 * base + (rem[1] || 0);

            // n is how many times the divisor goes into the current remainder.
            n = mathfloor(rem0 / yc0);

            //  Algorithm:
            //  product = divisor multiplied by trial digit (n).
            //  Compare product and remainder.
            //  If product is greater than remainder:
            //    Subtract divisor from product, decrement trial digit.
            //  Subtract product from remainder.
            //  If product was less than remainder at the last compare:
            //    Compare new remainder and divisor.
            //    If remainder is greater than divisor:
            //      Subtract divisor from remainder, increment trial digit.

            if (n > 1) {

              // n may be > base only when base is 3.
              if (n >= base) n = base - 1;

              // product = divisor * trial digit.
              prod = multiply(yc, n, base);
              prodL = prod.length;
              remL = rem.length;

              // Compare product and remainder.
              // If product > remainder then trial digit n too high.
              // n is 1 too high about 5% of the time, and is not known to have
              // ever been more than 1 too high.
              while (compare(prod, rem, prodL, remL) == 1) {
                n--;

                // Subtract divisor from product.
                subtract(prod, yL < prodL ? yz : yc, prodL, base);
                prodL = prod.length;
                cmp = 1;
              }
            } else {

              // n is 0 or 1, cmp is -1.
              // If n is 0, there is no need to compare yc and rem again below,
              // so change cmp to 1 to avoid it.
              // If n is 1, leave cmp as -1, so yc and rem are compared again.
              if (n == 0) {

                // divisor < remainder, so n must be at least 1.
                cmp = n = 1;
              }

              // product = divisor
              prod = yc.slice();
              prodL = prod.length;
            }

            if (prodL < remL) prod = [0].concat(prod);

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

             // If product was < remainder.
            if (cmp == -1) {

              // Compare divisor and new remainder.
              // If divisor < new remainder, subtract divisor from remainder.
              // Trial digit n too low.
              // n is 1 too low about 5% of the time, and very rarely 2 too low.
              while (compare(yc, rem, yL, remL) < 1) {
                n++;

                // Subtract divisor from remainder.
                subtract(rem, yL < remL ? yz : yc, remL, base);
                remL = rem.length;
              }
            }
          } else if (cmp === 0) {
            n++;
            rem = [0];
          } // else cmp === 1 and n will be 0

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

          // Update the remainder.
          if (rem[0]) {
            rem[remL++] = xc[xi] || 0;
          } else {
            rem = [xc[xi]];
            remL = 1;
          }
        } while ((xi++ < xL || rem[0] != null) && s--);

        more = rem[0] != null;

        // Leading zero?
        if (!qc[0]) qc.splice(0, 1);
      }

      if (base == BASE) {

        // To calculate q.e, first get the number of digits of qc[0].
        for (i = 1, s = qc[0]; s >= 10; s /= 10, i++);

        round(q, dp + (q.e = i + e * LOG_BASE - 1) + 1, rm, more);

      // Caller is convertBase.
      } else {
        q.e = e;
        q.r = +more;
      }

      return q;
    };
  })();


  /*
   * Return a string representing the value of BigNumber n in fixed-point or exponential
   * notation rounded to the specified decimal places or significant digits.
   *
   * n: a BigNumber.
   * i: the index of the last digit required (i.e. the digit that may be rounded up).
   * rm: the rounding mode.
   * id: 1 (toExponential) or 2 (toPrecision).
   */
  function format(n, i, rm, id) {
    var c0, e, ne, len, str;

    if (rm == null) rm = ROUNDING_MODE;
    else intCheck(rm, 0, 8);

    if (!n.c) return n.toString();

    c0 = n.c[0];
    ne = n.e;

    if (i == null) {
      str = coeffToString(n.c);
      str = id == 1 || id == 2 && (ne <= TO_EXP_NEG || ne >= TO_EXP_POS)
       ? toExponential(str, ne)
       : toFixedPoint(str, ne, '0');
    } else {
      n = round(new BigNumber(n), i, rm);

      // n.e may have changed if the value was rounded up.
      e = n.e;

      str = coeffToString(n.c);
      len = str.length;

      // toPrecision returns exponential notation if the number of significant digits
      // specified is less than the number of digits necessary to represent the integer
      // part of the value in fixed-point notation.

      // Exponential notation.
      if (id == 1 || id == 2 && (i <= e || e <= TO_EXP_NEG)) {

        // Append zeros?
        for (; len < i; str += '0', len++);
        str = toExponential(str, e);

      // Fixed-point notation.
      } else {
        i -= ne;
        str = toFixedPoint(str, e, '0');

        // Append zeros?
        if (e + 1 > len) {
          if (--i > 0) for (str += '.'; i--; str += '0');
        } else {
          i += e - len;
          if (i > 0) {
            if (e + 1 == len) str += '.';
            for (; i--; str += '0');
          }
        }
      }
    }

    return n.s < 0 && c0 ? '-' + str : str;
  }


  // Handle BigNumber.max and BigNumber.min.
  function maxOrMin(args, method) {
    var n,
      i = 1,
      m = new BigNumber(args[0]);

    for (; i < args.length; i++) {
      n = new BigNumber(args[i]);

      // If any number is NaN, return NaN.
      if (!n.s) {
        m = n;
        break;
      } else if (method.call(m, n)) {
        m = n;
      }
    }

    return m;
  }


  /*
   * Strip trailing zeros, calculate base 10 exponent and check against MIN_EXP and MAX_EXP.
   * Called by minus, plus and times.
   */
  function normalise(n, c, e) {
    var i = 1,
      j = c.length;

     // Remove trailing zeros.
    for (; !c[--j]; c.pop());

    // Calculate the base 10 exponent. First get the number of digits of c[0].
    for (j = c[0]; j >= 10; j /= 10, i++);

    // Overflow?
    if ((e = i + e * LOG_BASE - 1) > MAX_EXP) {

      // Infinity.
      n.c = n.e = null;

    // Underflow?
    } else if (e < MIN_EXP) {

      // Zero.
      n.c = [n.e = 0];
    } else {
      n.e = e;
      n.c = c;
    }

    return n;
  }


  // Handle values that fail the validity test in BigNumber.
  parseNumeric = (function () {
    var basePrefix = /^(-?)0([xbo])(?=\w[\w.]*$)/i,
      dotAfter = /^([^.]+)\.$/,
      dotBefore = /^\.([^.]+)$/,
      isInfinityOrNaN = /^-?(Infinity|NaN)$/,
      whitespaceOrPlus = /^\s*\+(?=[\w.])|^\s+|\s+$/g;

    return function (x, str, isNum, b) {
      var base,
        s = isNum ? str : str.replace(whitespaceOrPlus, '');

      // No exception on ±Infinity or NaN.
      if (isInfinityOrNaN.test(s)) {
        x.s = isNaN(s) ? null : s < 0 ? -1 : 1;
      } else {
        if (!isNum) {

          // basePrefix = /^(-?)0([xbo])(?=\w[\w.]*$)/i
          s = s.replace(basePrefix, function (m, p1, p2) {
            base = (p2 = p2.toLowerCase()) == 'x' ? 16 : p2 == 'b' ? 2 : 8;
            return !b || b == base ? p1 : m;
          });

          if (b) {
            base = b;

            // E.g. '1.' to '1', '.1' to '0.1'
            s = s.replace(dotAfter, '$1').replace(dotBefore, '0.$1');
          }

          if (str != s) return new BigNumber(s, base);
        }

        // '[BigNumber Error] Not a number: {n}'
        // '[BigNumber Error] Not a base {b} number: {n}'
        if (BigNumber.DEBUG) {
          throw Error
            (bignumberError + 'Not a' + (b ? ' base ' + b : '') + ' number: ' + str);
        }

        // NaN
        x.s = null;
      }

      x.c = x.e = null;
    }
  })();


  /*
   * Round x to sd significant digits using rounding mode rm. Check for over/under-flow.
   * If r is truthy, it is known that there are more digits after the rounding digit.
   */
  function round(x, sd, rm, r) {
    var d, i, j, k, n, ni, rd,
      xc = x.c,
      pows10 = POWS_TEN;

    // if x is not Infinity or NaN...
    if (xc) {

      // rd is the rounding digit, i.e. the digit after the digit that may be rounded up.
      // n is a base 1e14 number, the value of the element of array x.c containing rd.
      // ni is the index of n within x.c.
      // d is the number of digits of n.
      // i is the index of rd within n including leading zeros.
      // j is the actual index of rd within n (if < 0, rd is a leading zero).
      out: {

        // Get the number of digits of the first element of xc.
        for (d = 1, k = xc[0]; k >= 10; k /= 10, d++);
        i = sd - d;

        // If the rounding digit is in the first element of xc...
        if (i < 0) {
          i += LOG_BASE;
          j = sd;
          n = xc[ni = 0];

          // Get the rounding digit at index j of n.
          rd = n / pows10[d - j - 1] % 10 | 0;
        } else {
          ni = mathceil((i + 1) / LOG_BASE);

          if (ni >= xc.length) {

            if (r) {

              // Needed by sqrt.
              for (; xc.length <= ni; xc.push(0));
              n = rd = 0;
              d = 1;
              i %= LOG_BASE;
              j = i - LOG_BASE + 1;
            } else {
              break out;
            }
          } else {
            n = k = xc[ni];

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

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

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

            // Get the rounding digit at index j of n.
            rd = j < 0 ? 0 : n / pows10[d - j - 1] % 10 | 0;
          }
        }

        r = r || sd < 0 ||

        // Are there any non-zero digits after the rounding digit?
        // The expression  n % pows10[d - j - 1]  returns all digits of n to the right
        // of the digit at j, e.g. if n is 908714 and j is 2, the expression gives 714.
         xc[ni + 1] != null || (j < 0 ? n : n % pows10[d - j - 1]);

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

          // Check whether the digit to the left of the rounding digit is odd.
          ((i > 0 ? j > 0 ? n / pows10[d - j] : 0 : xc[ni - 1]) % 10) & 1 ||
           rm == (x.s < 0 ? 8 : 7));

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

          if (r) {

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

            // 1, 0.1, 0.01, 0.001, 0.0001 etc.
            xc[0] = pows10[(LOG_BASE - sd % LOG_BASE) % LOG_BASE];
            x.e = -sd || 0;
          } else {

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

          return x;
        }

        // Remove excess digits.
        if (i == 0) {
          xc.length = ni;
          k = 1;
          ni--;
        } else {
          xc.length = ni + 1;
          k = pows10[LOG_BASE - i];

          // E.g. 56700 becomes 56000 if 7 is the rounding digit.
          // j > 0 means i > number of leading zeros of n.
          xc[ni] = j > 0 ? mathfloor(n / pows10[d - j] % pows10[j]) * k : 0;
        }

        // Round up?
        if (r) {

          for (; ;) {

            // If the digit to be rounded up is in the first element of xc...
            if (ni == 0) {

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

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

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

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

      // Overflow? Infinity.
      if (x.e > MAX_EXP) {
        x.c = x.e = null;

      // Underflow? Zero.
      } else if (x.e < MIN_EXP) {
        x.c = [x.e = 0];
      }
    }

    return x;
  }


  function valueOf(n) {
    var str,
      e = n.e;

    if (e === null) return n.toString();

    str = coeffToString(n.c);

    str = e <= TO_EXP_NEG || e >= TO_EXP_POS
      ? toExponential(str, e)
      : toFixedPoint(str, e, '0');

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


  // PROTOTYPE/INSTANCE METHODS


  /*
   * Return a new BigNumber whose value is the absolute value of this BigNumber.
   */
  P.absoluteValue = P.abs = function () {
    var x = new BigNumber(this);
    if (x.s < 0) x.s = 1;
    return x;
  };


  /*
   * Return
   *   1 if the value of this BigNumber is greater than the value of BigNumber(y, b),
   *   -1 if the value of this BigNumber is less than the value of BigNumber(y, b),
   *   0 if they have the same value,
   *   or null if the value of either is NaN.
   */
  P.comparedTo = function (y, b) {
    return compare(this, new BigNumber(y, b));
  };


  /*
   * If dp is undefined or null or true or false, return the number of decimal places of the
   * value of this BigNumber, or null if the value of this BigNumber is ±Infinity or NaN.
   *
   * Otherwise, if dp is a number, return a new BigNumber whose value is the value of this
   * BigNumber rounded to a maximum of dp decimal places using rounding mode rm, or
   * ROUNDING_MODE if rm is omitted.
   *
   * [dp] {number} Decimal places: integer, 0 to MAX inclusive.
   * [rm] {number} Rounding mode. Integer, 0 to 8 inclusive.
   *
   * '[BigNumber Error] Argument {not a primitive number|not an integer|out of range}: {dp|rm}'
   */
  P.decimalPlaces = P.dp = function (dp, rm) {
    var c, n, v,
      x = this;

    if (dp != null) {
      intCheck(dp, 0, MAX);
      if (rm == null) rm = ROUNDING_MODE;
      else intCheck(rm, 0, 8);

      return round(new BigNumber(x), dp + x.e + 1, rm);
    }

    if (!(c = x.c)) return null;
    n = ((v = c.length - 1) - bitFloor(this.e / LOG_BASE)) * LOG_BASE;

    // Subtract the number of trailing zeros of the last number.
    if (v = c[v]) for (; v % 10 == 0; v /= 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 BigNumber whose value is the value of this BigNumber divided by the value of
   * BigNumber(y, b), rounded according to DECIMAL_PLACES and ROUNDING_MODE.
   */
  P.dividedBy = P.div = function (y, b) {
    return div(this, new BigNumber(y, b), DECIMAL_PLACES, ROUNDING_MODE);
  };


  /*
   * Return a new BigNumber whose value is the integer part of dividing the value of this
   * BigNumber by the value of BigNumber(y, b).
   */
  P.dividedToIntegerBy = P.idiv = function (y, b) {
    return div(this, new BigNumber(y, b), 0, 1);
  };


  /*
   * Return a BigNumber whose value is the value of this BigNumber exponentiated by n.
   *
   * If m is present, return the result modulo m.
   * If n is negative round according to DECIMAL_PLACES and ROUNDING_MODE.
   * If POW_PRECISION is non-zero and m is not present, round to POW_PRECISION using ROUNDING_MODE.
   *
   * The modular power operation works efficiently when x, n, and m are integers, otherwise it
   * is equivalent to calculating x.exponentiatedBy(n).modulo(m) with a POW_PRECISION of 0.
   *
   * n {number|string|BigNumber} The exponent. An integer.
   * [m] {number|string|BigNumber} The modulus.
   *
   * '[BigNumber Error] Exponent not an integer: {n}'
   */
  P.exponentiatedBy = P.pow = function (n, m) {
    var half, isModExp, i, k, more, nIsBig, nIsNeg, nIsOdd, y,
      x = this;

    n = new BigNumber(n);

    // Allow NaN and ±Infinity, but not other non-integers.
    if (n.c && !n.isInteger()) {
      throw Error
        (bignumberError + 'Exponent not an integer: ' + valueOf(n));
    }

    if (m != null) m = new BigNumber(m);

    // Exponent of MAX_SAFE_INTEGER is 15.
    nIsBig = n.e > 14;

    // If x is NaN, ±Infinity, ±0 or ±1, or n is ±Infinity, NaN or ±0.
    if (!x.c || !x.c[0] || x.c[0] == 1 && !x.e && x.c.length == 1 || !n.c || !n.c[0]) {

      // The sign of the result of pow when x is negative depends on the evenness of n.
      // If +n overflows to ±Infinity, the evenness of n would be not be known.
      y = new BigNumber(Math.pow(+valueOf(x), nIsBig ? 2 - isOdd(n) : +valueOf(n)));
      return m ? y.mod(m) : y;
    }

    nIsNeg = n.s < 0;

    if (m) {

      // x % m returns NaN if abs(m) is zero, or m is NaN.
      if (m.c ? !m.c[0] : !m.s) return new BigNumber(NaN);

      isModExp = !nIsNeg && x.isInteger() && m.isInteger();

      if (isModExp) x = x.mod(m);

    // Overflow to ±Infinity: >=2**1e10 or >=1.0000024**1e15.
    // Underflow to ±0: <=0.79**1e10 or <=0.9999975**1e15.
    } else if (n.e > 9 && (x.e > 0 || x.e < -1 || (x.e == 0
      // [1, 240000000]
      ? x.c[0] > 1 || nIsBig && x.c[1] >= 24e7
      // [80000000000000]  [99999750000000]
      : x.c[0] < 8e13 || nIsBig && x.c[0] <= 9999975e7))) {

      // If x is negative and n is odd, k = -0, else k = 0.
      k = x.s < 0 && isOdd(n) ? -0 : 0;

      // If x >= 1, k = ±Infinity.
      if (x.e > -1) k = 1 / k;

      // If n is negative return ±0, else return ±Infinity.
      return new BigNumber(nIsNeg ? 1 / k : k);

    } else if (POW_PRECISION) {

      // Truncating each coefficient array to a length of k after each multiplication
      // equates to truncating significant digits to POW_PRECISION + [28, 41],
      // i.e. there will be a minimum of 28 guard digits retained.
      k = mathceil(POW_PRECISION / LOG_BASE + 2);
    }

    if (nIsBig) {
      half = new BigNumber(0.5);
      if (nIsNeg) n.s = 1;
      nIsOdd = isOdd(n);
    } else {
      i = Math.abs(+valueOf(n));
      nIsOdd = i % 2;
    }

    y = new BigNumber(ONE);

    // Performs 54 loop iterations for n of 9007199254740991.
    for (; ;) {

      if (nIsOdd) {
        y = y.times(x);
        if (!y.c) break;

        if (k) {
          if (y.c.length > k) y.c.length = k;
        } else if (isModExp) {
          y = y.mod(m);    //y = y.minus(div(y, m, 0, MODULO_MODE).times(m));
        }
      }

      if (i) {
        i = mathfloor(i / 2);
        if (i === 0) break;
        nIsOdd = i % 2;
      } else {
        n = n.times(half);
        round(n, n.e + 1, 1);

        if (n.e > 14) {
          nIsOdd = isOdd(n);
        } else {
          i = +valueOf(n);
          if (i === 0) break;
          nIsOdd = i % 2;
        }
      }

      x = x.times(x);

      if (k) {
        if (x.c && x.c.length > k) x.c.length = k;
      } else if (isModExp) {
        x = x.mod(m);    //x = x.minus(div(x, m, 0, MODULO_MODE).times(m));
      }
    }

    if (isModExp) return y;
    if (nIsNeg) y = ONE.div(y);

    return m ? y.mod(m) : k ? round(y, POW_PRECISION, ROUNDING_MODE, more) : y;
  };


  /*
   * Return a new BigNumber whose value is the value of this BigNumber rounded to an integer
   * using rounding mode rm, or ROUNDING_MODE if rm is omitted.
   *
   * [rm] {number} Rounding mode. Integer, 0 to 8 inclusive.
   *
   * '[BigNumber Error] Argument {not a primitive number|not an integer|out of range}: {rm}'
   */
  P.integerValue = function (rm) {
    var n = new BigNumber(this);
    if (rm == null) rm = ROUNDING_MODE;
    else intCheck(rm, 0, 8);
    return round(n, n.e + 1, rm);
  };


  /*
   * Return true if the value of this BigNumber is equal to the value of BigNumber(y, b),
   * otherwise return false.
   */
  P.isEqualTo = P.eq = function (y, b) {
    return compare(this, new BigNumber(y, b)) === 0;
  };


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


  /*
   * Return true if the value of this BigNumber is greater than the value of BigNumber(y, b),
   * otherwise return false.
   */
  P.isGreaterThan = P.gt = function (y, b) {
    return compare(this, new BigNumber(y, b)) > 0;
  };


  /*
   * Return true if the value of this BigNumber is greater than or equal to the value of
   * BigNumber(y, b), otherwise return false.
   */
  P.isGreaterThanOrEqualTo = P.gte = function (y, b) {
    return (b = compare(this, new BigNumber(y, b))) === 1 || b === 0;

  };


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


  /*
   * Return true if the value of this BigNumber is less than the value of BigNumber(y, b),
   * otherwise return false.
   */
  P.isLessThan = P.lt = function (y, b) {
    return compare(this, new BigNumber(y, b)) < 0;
  };


  /*
   * Return true if the value of this BigNumber is less than or equal to the value of
   * BigNumber(y, b), otherwise return false.
   */
  P.isLessThanOrEqualTo = P.lte = function (y, b) {
    return (b = compare(this, new BigNumber(y, b))) === -1 || b === 0;
  };


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


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


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


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


  /*
   *  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 BigNumber whose value is the value of this BigNumber minus the value of
   * BigNumber(y, b).
   */
  P.minus = function (y, b) {
    var i, j, t, xLTy,
      x = this,
      a = x.s;

    y = new BigNumber(y, b);
    b = y.s;

    // Either NaN?
    if (!a || !b) return new BigNumber(NaN);

    // Signs differ?
    if (a != b) {
      y.s = -b;
      return x.plus(y);
    }

    var xe = x.e / LOG_BASE,
      ye = y.e / LOG_BASE,
      xc = x.c,
      yc = y.c;

    if (!xe || !ye) {

      // Either Infinity?
      if (!xc || !yc) return xc ? (y.s = -b, y) : new BigNumber(yc ? x : NaN);

      // Either zero?
      if (!xc[0] || !yc[0]) {

        // Return y if y is non-zero, x if x is non-zero, or zero if both are zero.
        return yc[0] ? (y.s = -b, y) : new BigNumber(xc[0] ? x :

         // IEEE 754 (2008) 6.3: n - n = -0 when rounding to -Infinity
         ROUNDING_MODE == 3 ? -0 : 0);
      }
    }

    xe = bitFloor(xe);
    ye = bitFloor(ye);
    xc = xc.slice();

    // Determine which is the bigger number.
    if (a = xe - ye) {

      if (xLTy = a < 0) {
        a = -a;
        t = xc;
      } else {
        ye = xe;
        t = yc;
      }

      t.reverse();

      // Prepend zeros to equalise exponents.
      for (b = a; b--; t.push(0));
      t.reverse();
    } else {

      // Exponents equal. Check digit by digit.
      j = (xLTy = (a = xc.length) < (b = yc.length)) ? a : b;

      for (a = b = 0; b < j; b++) {

        if (xc[b] != yc[b]) {
          xLTy = xc[b] < yc[b];
          break;
        }
      }
    }

    // x < y? Point xc to the array of the bigger number.
    if (xLTy) t = xc, xc = yc, yc = t, y.s = -y.s;

    b = (j = yc.length) - (i = xc.length);

    // Append zeros to xc if shorter.
    // No need to add zeros to yc if shorter as subtract only needs to start at yc.length.
    if (b > 0) for (; b--; xc[i++] = 0);
    b = BASE - 1;

    // Subtract yc from xc.
    for (; j > a;) {

      if (xc[--j] < yc[j]) {
        for (i = j; i && !xc[--i]; xc[i] = b);
        --xc[i];
        xc[j] += BASE;
      }

      xc[j] -= yc[j];
    }

    // Remove leading zeros and adjust exponent accordingly.
    for (; xc[0] == 0; xc.splice(0, 1), --ye);

    // Zero?
    if (!xc[0]) {

      // Following IEEE 754 (2008) 6.3,
      // n - n = +0  but  n - n = -0  when rounding towards -Infinity.
      y.s = ROUNDING_MODE == 3 ? -1 : 1;
      y.c = [y.e = 0];
      return y;
    }

    // No need to check for Infinity as +x - +y != Infinity && -x - -y != Infinity
    // for finite x and y.
    return normalise(y, xc, ye);
  };


  /*
   *   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 BigNumber whose value is the value of this BigNumber modulo the value of
   * BigNumber(y, b). The result depends on the value of MODULO_MODE.
   */
  P.modulo = P.mod = function (y, b) {
    var q, s,
      x = this;

    y = new BigNumber(y, b);

    // Return NaN if x is Infinity or NaN, or y is NaN or zero.
    if (!x.c || !y.s || y.c && !y.c[0]) {
      return new BigNumber(NaN);

    // Return x if y is Infinity or x is zero.
    } else if (!y.c || x.c && !x.c[0]) {
      return new BigNumber(x);
    }

    if (MODULO_MODE == 9) {

      // Euclidian division: q = sign(y) * floor(x / abs(y))
      // r = x - qy    where  0 <= r < abs(y)
      s = y.s;
      y.s = 1;
      q = div(x, y, 0, 3);
      y.s = s;
      q.s *= s;
    } else {
      q = div(x, y, 0, MODULO_MODE);
    }

    y = x.minus(q.times(y));

    // To match JavaScript %, ensure sign of zero is sign of dividend.
    if (!y.c[0] && MODULO_MODE == 1) y.s = x.s;

    return y;
  };


  /*
   *  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 BigNumber whose value is the value of this BigNumber multiplied by the value
   * of BigNumber(y, b).
   */
  P.multipliedBy = P.times = function (y, b) {
    var c, e, i, j, k, m, xcL, xlo, xhi, ycL, ylo, yhi, zc,
      base, sqrtBase,
      x = this,
      xc = x.c,
      yc = (y = new BigNumber(y, b)).c;

    // Either NaN, ±Infinity or ±0?
    if (!xc || !yc || !xc[0] || !yc[0]) {

      // Return NaN if either is NaN, or one is 0 and the other is Infinity.
      if (!x.s || !y.s || xc && !xc[0] && !yc || yc && !yc[0] && !xc) {
        y.c = y.e = y.s = null;
      } else {
        y.s *= x.s;

        // Return ±Infinity if either is ±Infinity.
        if (!xc || !yc) {
          y.c = y.e = null;

        // Return ±0 if either is ±0.
        } else {
          y.c = [0];
          y.e = 0;
        }
      }

      return y;
    }

    e = bitFloor(x.e / LOG_BASE) + bitFloor(y.e / LOG_BASE);
    y.s *= x.s;
    xcL = xc.length;
    ycL = yc.length;

    // Ensure xc points to longer array and xcL to its length.
    if (xcL < ycL) zc = xc, xc = yc, yc = zc, i = xcL, xcL = ycL, ycL = i;

    // Initialise the result array with zeros.
    for (i = xcL + ycL, zc = []; i--; zc.push(0));

    base = BASE;
    sqrtBase = SQRT_BASE;

    for (i = ycL; --i >= 0;) {
      c = 0;
      ylo = yc[i] % sqrtBase;
      yhi = yc[i] / sqrtBase | 0;

      for (k = xcL, j = i + k; j > i;) {
        xlo = xc[--k] % sqrtBase;
        xhi = xc[k] / sqrtBase | 0;
        m = yhi * xlo + xhi * ylo;
        xlo = ylo * xlo + ((m % sqrtBase) * sqrtBase) + zc[j] + c;
        c = (xlo / base | 0) + (m / sqrtBase | 0) + yhi * xhi;
        zc[j--] = xlo % base;
      }

      zc[j] = c;
    }

    if (c) {
      ++e;
    } else {
      zc.splice(0, 1);
    }

    return normalise(y, zc, e);
  };


  /*
   * Return a new BigNumber whose value is the value of this BigNumber negated,
   * i.e. multiplied by -1.
   */
  P.negated = function () {
    var x = new BigNumber(this);
    x.s = -x.s || null;
    return 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 BigNumber whose value is the value of this BigNumber plus the value of
   * BigNumber(y, b).
   */
  P.plus = function (y, b) {
    var t,
      x = this,
      a = x.s;

    y = new BigNumber(y, b);
    b = y.s;

    // Either NaN?
    if (!a || !b) return new BigNumber(NaN);

    // Signs differ?
     if (a != b) {
      y.s = -b;
      return x.minus(y);
    }

    var xe = x.e / LOG_BASE,
      ye = y.e / LOG_BASE,
      xc = x.c,
      yc = y.c;

    if (!xe || !ye) {

      // Return ±Infinity if either ±Infinity.
      if (!xc || !yc) return new BigNumber(a / 0);

      // Either zero?
      // Return y if y is non-zero, x if x is non-zero, or zero if both are zero.
      if (!xc[0] || !yc[0]) return yc[0] ? y : new BigNumber(xc[0] ? x : a * 0);
    }

    xe = bitFloor(xe);
    ye = bitFloor(ye);
    xc = xc.slice();

    // Prepend zeros to equalise exponents. Faster to use reverse then do unshifts.
    if (a = xe - ye) {
      if (a > 0) {
        ye = xe;
        t = yc;
      } else {
        a = -a;
        t = xc;
      }

      t.reverse();
      for (; a--; t.push(0));
      t.reverse();
    }

    a = xc.length;
    b = yc.length;

    // Point xc to the longer array, and b to the shorter length.
    if (a - b < 0) t = yc, yc = xc, xc = t, b = a;

    // Only start adding at yc.length - 1 as the further digits of xc can be ignored.
    for (a = 0; b;) {
      a = (xc[--b] = xc[b] + yc[b] + a) / BASE | 0;
      xc[b] = BASE === xc[b] ? 0 : xc[b] % BASE;
    }

    if (a) {
      xc = [a].concat(xc);
      ++ye;
    }

    // No need to check for zero, as +x + +y != 0 && -x + -y != 0
    // ye = MAX_EXP + 1 possible
    return normalise(y, xc, ye);
  };


  /*
   * If sd is undefined or null or true or false, return the number of significant digits of
   * the value of this BigNumber, or null if the value of this BigNumber is ±Infinity or NaN.
   * If sd is true include integer-part trailing zeros in the count.
   *
   * Otherwise, if sd is a number, return a new BigNumber whose value is the value of this
   * BigNumber rounded to a maximum of sd significant digits using rounding mode rm, or
   * ROUNDING_MODE if rm is omitted.
   *
   * sd {number|boolean} number: significant digits: integer, 1 to MAX inclusive.
   *                     boolean: whether to count integer-part trailing zeros: true or false.
   * [rm] {number} Rounding mode. Integer, 0 to 8 inclusive.
   *
   * '[BigNumber Error] Argument {not a primitive number|not an integer|out of range}: {sd|rm}'
   */
  P.precision = P.sd = function (sd, rm) {
    var c, n, v,
      x = this;

    if (sd != null && sd !== !!sd) {
      intCheck(sd, 1, MAX);
      if (rm == null) rm = ROUNDING_MODE;
      else intCheck(rm, 0, 8);

      return round(new BigNumber(x), sd, rm);
    }

    if (!(c = x.c)) return null;
    v = c.length - 1;
    n = v * LOG_BASE + 1;

    if (v = c[v]) {

      // Subtract the number of trailing zeros of the last element.
      for (; v % 10 == 0; v /= 10, n--);

      // Add the number of digits of the first element.
      for (v = c[0]; v >= 10; v /= 10, n++);
    }

    if (sd && x.e + 1 > n) n = x.e + 1;

    return n;
  };


  /*
   * Return a new BigNumber whose value is the value of this BigNumber shifted by k places
   * (powers of 10). Shift to the right if n > 0, and to the left if n < 0.
   *
   * k {number} Integer, -MAX_SAFE_INTEGER to MAX_SAFE_INTEGER inclusive.
   *
   * '[BigNumber Error] Argument {not a primitive number|not an integer|out of range}: {k}'
   */
  P.shiftedBy = function (k) {
    intCheck(k, -MAX_SAFE_INTEGER, MAX_SAFE_INTEGER);
    return this.times('1e' + k);
  };


  /*
   *  sqrt(-n) =  N
   *  sqrt(N) =  N
   *  sqrt(-I) =  N
   *  sqrt(I) =  I
   *  sqrt(0) =  0
   *  sqrt(-0) = -0
   *
   * Return a new BigNumber whose value is the square root of the value of this BigNumber,
   * rounded according to DECIMAL_PLACES and ROUNDING_MODE.
   */
  P.squareRoot = P.sqrt = function () {
    var m, n, r, rep, t,
      x = this,
      c = x.c,
      s = x.s,
      e = x.e,
      dp = DECIMAL_PLACES + 4,
      half = new BigNumber('0.5');

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

    // Initial estimate.
    s = Math.sqrt(+valueOf(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 = coeffToString(c);
      if ((n.length + e) % 2 == 0) n += '0';
      s = Math.sqrt(+n);
      e = bitFloor((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 BigNumber(n);
    } else {
      r = new BigNumber(s + '');
    }

    // Check for zero.
    // r could be zero if MIN_EXP is changed after the this value was created.
    // This would cause a division by zero (x/t) and hence Infinity below, which would cause
    // coeffToString to throw.
    if (r.c[0]) {
      e = r.e;
      s = e + dp;
      if (s < 3) s = 0;

      // Newton-Raphson iteration.
      for (; ;) {
        t = r;
        r = half.times(t.plus(div(x, t, dp, 1)));

        if (coeffToString(t.c).slice(0, s) === (n = coeffToString(r.c)).slice(0, s)) {

          // The exponent of r may here be one less than the final result exponent,
          // e.g 0.0009999 (e-4) --> 0.001 (e-3), so adjust s so the rounding digits
          // are indexed correctly.
          if (r.e < e) --s;
          n = n.slice(s - 3, s + 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) {
              round(t, t.e + DECIMAL_PLACES + 2, 0);

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

            dp += 4;
            s += 4;
            rep = 1;
          } else {

            // If rounding digits are null, 0{0,4} or 50{0,3}, check for 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.
              round(r, r.e + DECIMAL_PLACES + 2, 1);
              m = !r.times(r).eq(x);
            }

            break;
          }
        }
      }
    }

    return round(r, r.e + DECIMAL_PLACES + 1, ROUNDING_MODE, m);
  };


  /*
   * Return a string representing the value of this BigNumber in exponential notation and
   * rounded using ROUNDING_MODE to dp fixed decimal places.
   *
   * [dp] {number} Decimal places. Integer, 0 to MAX inclusive.
   * [rm] {number} Rounding mode. Integer, 0 to 8 inclusive.
   *
   * '[BigNumber Error] Argument {not a primitive number|not an integer|out of range}: {dp|rm}'
   */
  P.toExponential = function (dp, rm) {
    if (dp != null) {
      intCheck(dp, 0, MAX);
      dp++;
    }
    return format(this, dp, rm, 1);
  };


  /*
   * Return a string representing the value of this BigNumber in fixed-point notation rounding
   * to dp fixed decimal places using rounding mode rm, or ROUNDING_MODE if rm is omitted.
   *
   * Note: as with JavaScript's number type, (-0).toFixed(0) is '0',
   * but e.g. (-0.00001).toFixed(0) is '-0'.
   *
   * [dp] {number} Decimal places. Integer, 0 to MAX inclusive.
   * [rm] {number} Rounding mode. Integer, 0 to 8 inclusive.
   *
   * '[BigNumber Error] Argument {not a primitive number|not an integer|out of range}: {dp|rm}'
   */
  P.toFixed = function (dp, rm) {
    if (dp != null) {
      intCheck(dp, 0, MAX);
      dp = dp + this.e + 1;
    }
    return format(this, dp, rm);
  };


  /*
   * Return a string representing the value of this BigNumber in fixed-point notation rounded
   * using rm or ROUNDING_MODE to dp decimal places, and formatted according to the properties
   * of the format or FORMAT object (see BigNumber.set).
   *
   * The formatting object may contain some or all of the properties shown below.
   *
   * FORMAT = {
   *   prefix: '',
   *   groupSize: 3,
   *   secondaryGroupSize: 0,
   *   groupSeparator: ',',
   *   decimalSeparator: '.',
   *   fractionGroupSize: 0,
   *   fractionGroupSeparator: '\xA0',      // non-breaking space
   *   suffix: ''
   * };
   *
   * [dp] {number} Decimal places. Integer, 0 to MAX inclusive.
   * [rm] {number} Rounding mode. Integer, 0 to 8 inclusive.
   * [format] {object} Formatting options. See FORMAT pbject above.
   *
   * '[BigNumber Error] Argument {not a primitive number|not an integer|out of range}: {dp|rm}'
   * '[BigNumber Error] Argument not an object: {format}'
   */
  P.toFormat = function (dp, rm, format) {
    var str,
      x = this;

    if (format == null) {
      if (dp != null && rm && typeof rm == 'object') {
        format = rm;
        rm = null;
      } else if (dp && typeof dp == 'object') {
        format = dp;
        dp = rm = null;
      } else {
        format = FORMAT;
      }
    } else if (typeof format != 'object') {
      throw Error
        (bignumberError + 'Argument not an object: ' + format);
    }

    str = x.toFixed(dp, rm);

    if (x.c) {
      var i,
        arr = str.split('.'),
        g1 = +format.groupSize,
        g2 = +format.secondaryGroupSize,
        groupSeparator = format.groupSeparator || '',
        intPart = arr[0],
        fractionPart = arr[1],
        isNeg = x.s < 0,
        intDigits = isNeg ? intPart.slice(1) : intPart,
        len = intDigits.length;

      if (g2) i = g1, g1 = g2, g2 = i, len -= i;

      if (g1 > 0 && len > 0) {
        i = len % g1 || g1;
        intPart = intDigits.substr(0, i);
        for (; i < len; i += g1) intPart += groupSeparator + intDigits.substr(i, g1);
        if (g2 > 0) intPart += groupSeparator + intDigits.slice(i);
        if (isNeg) intPart = '-' + intPart;
      }

      str = fractionPart
       ? intPart + (format.decimalSeparator || '') + ((g2 = +format.fractionGroupSize)
        ? fractionPart.replace(new RegExp('\\d{' + g2 + '}\\B', 'g'),
         '$&' + (format.fractionGroupSeparator || ''))
        : fractionPart)
       : intPart;
    }

    return (format.prefix || '') + str + (format.suffix || '');
  };


  /*
   * Return an array of two BigNumbers representing the value of this BigNumber 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.
   *
   * [md] {number|string|BigNumber} Integer >= 1, or Infinity. The maximum denominator.
   *
   * '[BigNumber Error] Argument {not an integer|out of range} : {md}'
   */
  P.toFraction = function (md) {
    var d, d0, d1, d2, e, exp, n, n0, n1, q, r, s,
      x = this,
      xc = x.c;

    if (md != null) {
      n = new BigNumber(md);

      // Throw if md is less than one or is not an integer, unless it is Infinity.
      if (!n.isInteger() && (n.c || n.s !== 1) || n.lt(ONE)) {
        throw Error
          (bignumberError + 'Argument ' +
            (n.isInteger() ? 'out of range: ' : 'not an integer: ') + valueOf(n));
      }
    }

    if (!xc) return new BigNumber(x);

    d = new BigNumber(ONE);
    n1 = d0 = new BigNumber(ONE);
    d1 = n0 = new BigNumber(ONE);
    s = coeffToString(xc);

    // Determine initial denominator.
    // d is a power of 10 and the minimum max denominator that specifies the value exactly.
    e = d.e = s.length - x.e - 1;
    d.c[0] = POWS_TEN[(exp = e % LOG_BASE) < 0 ? LOG_BASE + exp : exp];
    md = !md || n.comparedTo(d) > 0 ? (e > 0 ? d : n1) : n;

    exp = MAX_EXP;
    MAX_EXP = 1 / 0;
    n = new BigNumber(s);

    // n0 = d1 = 0
    n0.c[0] = 0;

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

    d2 = div(md.minus(d0), d1, 0, 1);
    n0 = n0.plus(d2.times(n1));
    d0 = d0.plus(d2.times(d1));
    n0.s = n1.s = x.s;
    e = e * 2;

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

    MAX_EXP = exp;

    return r;
  };


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


  /*
   * Return a string representing the value of this BigNumber rounded to sd significant digits
   * using rounding mode rm or ROUNDING_MODE. If sd is less than the number of digits
   * necessary to represent the integer part of the value in fixed-point notation, then use
   * exponential notation.
   *
   * [sd] {number} Significant digits. Integer, 1 to MAX inclusive.
   * [rm] {number} Rounding mode. Integer, 0 to 8 inclusive.
   *
   * '[BigNumber Error] Argument {not a primitive number|not an integer|out of range}: {sd|rm}'
   */
  P.toPrecision = function (sd, rm) {
    if (sd != null) intCheck(sd, 1, MAX);
    return format(this, sd, rm, 2);
  };


  /*
   * Return a string representing the value of this BigNumber in base b, or base 10 if b is
   * omitted. If a base is specified, including base 10, round according to DECIMAL_PLACES and
   * ROUNDING_MODE. If a base is not specified, and this BigNumber has a positive exponent
   * that is equal to or greater than TO_EXP_POS, or a negative exponent equal to or less than
   * TO_EXP_NEG, return exponential notation.
   *
   * [b] {number} Integer, 2 to ALPHABET.length inclusive.
   *
   * '[BigNumber Error] Base {not a primitive number|not an integer|out of range}: {b}'
   */
  P.toString = function (b) {
    var str,
      n = this,
      s = n.s,
      e = n.e;

    // Infinity or NaN?
    if (e === null) {
      if (s) {
        str = 'Infinity';
        if (s < 0) str = '-' + str;
      } else {
        str = 'NaN';
      }
    } else {
      if (b == null) {
        str = e <= TO_EXP_NEG || e >= TO_EXP_POS
         ? toExponential(coeffToString(n.c), e)
         : toFixedPoint(coeffToString(n.c), e, '0');
      } else if (b === 10) {
        n = round(new BigNumber(n), DECIMAL_PLACES + e + 1, ROUNDING_MODE);
        str = toFixedPoint(coeffToString(n.c), n.e, '0');
      } else {
        intCheck(b, 2, ALPHABET.length, 'Base');
        str = convertBase(toFixedPoint(coeffToString(n.c), e, '0'), 10, b, s, true);
      }

      if (s < 0 && n.c[0]) str = '-' + str;
    }

    return str;
  };


  /*
   * Return as toString, but do not accept a base argument, and include the minus sign for
   * negative zero.
   */
  P.valueOf = P.toJSON = function () {
    return valueOf(this);
  };


  P._isBigNumber = true;

  P[Symbol.toStringTag] = 'BigNumber';

  // Node.js v10.12.0+
  P[Symbol.for('nodejs.util.inspect.custom')] = P.valueOf;

  if (configObject != null) BigNumber.set(configObject);

  return BigNumber;
}


// PRIVATE HELPER FUNCTIONS

// These functions don't need access to variables,
// e.g. DECIMAL_PLACES, in the scope of the `clone` function above.


function bitFloor(n) {
  var i = n | 0;
  return n > 0 || n === i ? i : i - 1;
}


// Return a coefficient array as a string of base 10 digits.
function coeffToString(a) {
  var s, z,
    i = 1,
    j = a.length,
    r = a[0] + '';

  for (; i < j;) {
    s = a[i++] + '';
    z = LOG_BASE - s.length;
    for (; z--; s = '0' + s);
    r += s;
  }

  // Determine trailing zeros.
  for (j = r.length; r.charCodeAt(--j) === 48;);

  return r.slice(0, j + 1 || 1);
}


// Compare the value of BigNumbers x and y.
function compare(x, y) {
  var a, b,
    xc = x.c,
    yc = y.c,
    i = x.s,
    j = y.s,
    k = x.e,
    l = y.e;

  // Either NaN?
  if (!i || !j) return null;

  a = xc && !xc[0];
  b = yc && !yc[0];

  // Either zero?
  if (a || b) return a ? b ? 0 : -j : i;

  // Signs differ?
  if (i != j) return i;

  a = i < 0;
  b = k == l;

  // Either Infinity?
  if (!xc || !yc) return b ? 0 : !xc ^ a ? 1 : -1;

  // Compare exponents.
  if (!b) return k > l ^ a ? 1 : -1;

  j = (k = xc.length) < (l = yc.length) ? k : l;

  // Compare digit by digit.
  for (i = 0; i < j; i++) if (xc[i] != yc[i]) return xc[i] > yc[i] ^ a ? 1 : -1;

  // Compare lengths.
  return k == l ? 0 : k > l ^ a ? 1 : -1;
}


/*
 * Check that n is a primitive number, an integer, and in range, otherwise throw.
 */
function intCheck(n, min, max, name) {
  if (n < min || n > max || n !== mathfloor(n)) {
    throw Error
     (bignumberError + (name || 'Argument') + (typeof n == 'number'
       ? n < min || n > max ? ' out of range: ' : ' not an integer: '
       : ' not a primitive number: ') + String(n));
  }
}


// Assumes finite n.
function isOdd(n) {
  var k = n.c.length - 1;
  return bitFloor(n.e / LOG_BASE) == k && n.c[k] % 2 != 0;
}


function toExponential(str, e) {
  return (str.length > 1 ? str.charAt(0) + '.' + str.slice(1) : str) +
   (e < 0 ? 'e' : 'e+') + e;
}


function toFixedPoint(str, e, z) {
  var len, zs;

  // Negative exponent?
  if (e < 0) {

    // Prepend zeros.
    for (zs = z + '.'; ++e; zs += z);
    str = zs + str;

  // Positive exponent
  } else {
    len = str.length;

    // Append zeros.
    if (++e > len) {
      for (zs = z, e -= len; --e; zs += z);
      str += zs;
    } else if (e < len) {
      str = str.slice(0, e) + '.' + str.slice(e);
    }
  }

  return str;
}


// EXPORT


export var BigNumber = clone();

export default BigNumber;
