/*
 *
 * bignumber.js v4.1.0
 * A JavaScript library for arbitrary-precision arithmetic.
 * https://github.com/MikeMcl/bignumber.js
 * Copyright (c) 2017 Michael Mclaughlin <M8ch88l@gmail.com>
 * MIT Expat Licence
 *
 */


var BigNumber,
    isNumeric = /^-?(\d+(\.\d*)?|\.\d+)(e[+-]?\d+)?$/i,
    mathceil = Math.ceil,
    mathfloor = Math.floor,
    notBool = ' not a boolean or binary digit',
    roundingMode = 'rounding mode',
    tooManyDigits = 'number type has more than 15 significant digits',
    ALPHABET = '0123456789abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ$_',
    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,

    /*
     * 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, beyond which an
     * exception is thrown (if ERRORS is true).
     */
    MAX = 1E9;                                   // 0 to MAX_INT32


/*
 * Create and return a BigNumber constructor.
 */
function constructorFactory(config) {
    var div, parseNumeric,

        // id tracks the caller function, so its name can be included in error messages.
        id = 0,
        P = BigNumber.prototype,
        ONE = new BigNumber(1),


/*************************************** EDITABLE DEFAULTS ****************************************/


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

        // 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 BigNumber Errors are ever thrown.
        ERRORS = true,                           // true or false

        // Change to intValidatorNoErrors if ERRORS is false.
        isValidInt = intValidatorWithErrors,     // intValidatorWithErrors/intValidatorNoErrors

        // 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 toPower 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 = {
            decimalSeparator: '.',
            groupSeparator: ',',
            groupSize: 3,
            secondaryGroupSize: 0,
            fractionGroupSeparator: '\xA0',      // non-breaking space
            fractionGroupSize: 0
        };


/**************************************************************************************************/


    // CONSTRUCTOR


    /*
     * The BigNumber constructor and exported function.
     * Create and return a new instance of a BigNumber object.
     *
     * n {number|string|BigNumber} A numeric value.
     * [b] {number} The base of n. Integer, 2 to 64 inclusive.
     */
    function BigNumber( n, b ) {
        var c, e, i, num, len, str,
            x = this;

        // Enable constructor usage without new.
        if ( !( x instanceof BigNumber ) ) {

            // 'BigNumber() constructor call without new: {n}'
            if (ERRORS) raise( 26, 'constructor call without new', n );
            return new BigNumber( n, b );
        }

        // 'new BigNumber() base not an integer: {b}'
        // 'new BigNumber() base out of range: {b}'
        if ( b == null || !isValidInt( b, 2, 64, id, 'base' ) ) {

            // Duplicate.
            if ( n instanceof BigNumber ) {
                x.s = n.s;
                x.e = n.e;
                x.c = ( n = n.c ) ? n.slice() : n;
                id = 0;
                return;
            }

            if ( ( num = typeof n == 'number' ) && n * 0 == 0 ) {
                x.s = 1 / n < 0 ? ( n = -n, -1 ) : 1;

                // Fast path for integers.
                if ( n === ~~n ) {
                    for ( e = 0, i = n; i >= 10; i /= 10, e++ );
                    x.e = e;
                    x.c = [n];
                    id = 0;
                    return;
                }

                str = n + '';
            } else {
                if ( !isNumeric.test( str = n + '' ) ) return parseNumeric( x, str, num );
                x.s = str.charCodeAt(0) === 45 ? ( str = str.slice(1), -1 ) : 1;
            }
        } else {
            b = b | 0;
            str = n + '';

            // Ensure return value is rounded to DECIMAL_PLACES as with other bases.
            // Allow exponential notation to be used with base 10 argument.
            if ( b == 10 ) {
                x = new BigNumber( n instanceof BigNumber ? n : str );
                return round( x, DECIMAL_PLACES + x.e + 1, ROUNDING_MODE );
            }

            // Avoid potential interpretation of Infinity and NaN as base 44+ values.
            // Any number in exponential form will fail due to the [Ee][+-].
            if ( ( num = typeof n == 'number' ) && n * 0 != 0 ||
              !( new RegExp( '^-?' + ( c = '[' + ALPHABET.slice( 0, b ) + ']+' ) +
                '(?:\\.' + c + ')?$',b < 37 ? 'i' : '' ) ).test(str) ) {
                return parseNumeric( x, str, num, b );
            }

            if (num) {
                x.s = 1 / n < 0 ? ( str = str.slice(1), -1 ) : 1;

                if ( ERRORS && str.replace( /^0\.0*|\./, '' ).length > 15 ) {

                    // 'new BigNumber() number type has more than 15 significant digits: {n}'
                    raise( id, tooManyDigits, n );
                }

                // Prevent later check for length on converted number.
                num = false;
            } else {
                x.s = str.charCodeAt(0) === 45 ? ( str = str.slice(1), -1 ) : 1;
            }

            str = convertBase( str, 10, b, x.s );
        }

        // 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) === 48; );
        str = str.slice( i, len + 1 );

        if (str) {
            len = str.length;

            // Disallow numbers with over 15 significant digits if number type.
            // 'new BigNumber() number type has more than 15 significant digits: {n}'
            if ( num && ERRORS && len > 15 && ( n > MAX_SAFE_INTEGER || n !== mathfloor(n) ) ) {
                raise( id, tooManyDigits, x.s * n );
            }

            e = e - i - 1;

             // Overflow?
            if ( e > 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;

                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 ) );
                    }

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

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

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

        id = 0;
    }


    // CONSTRUCTOR PROPERTIES


    BigNumber.another = constructorFactory;

    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 or an argument list, with one or many of the following properties or
     * parameters respectively:
     *
     *   DECIMAL_PLACES  {number}  Integer, 0 to MAX inclusive
     *   ROUNDING_MODE   {number}  Integer, 0 to 8 inclusive
     *   EXPONENTIAL_AT  {number|number[]}  Integer, -MAX to MAX inclusive or
     *                                      [integer -MAX to 0 incl., 0 to MAX incl.]
     *   RANGE           {number|number[]}  Non-zero integer, -MAX to MAX inclusive or
     *                                      [integer -MAX to -1 incl., integer 1 to MAX incl.]
     *   ERRORS          {boolean|number}   true, false, 1 or 0
     *   CRYPTO          {boolean|number}   true, false, 1 or 0
     *   MODULO_MODE     {number}           0 to 9 inclusive
     *   POW_PRECISION   {number}           0 to MAX inclusive
     *   FORMAT          {object}           See BigNumber.prototype.toFormat
     *      decimalSeparator       {string}
     *      groupSeparator         {string}
     *      groupSize              {number}
     *      secondaryGroupSize     {number}
     *      fractionGroupSeparator {string}
     *      fractionGroupSize      {number}
     *
     * (The values assigned to the above FORMAT object properties are not checked for validity.)
     *
     * E.g.
     * BigNumber.config(20, 4) is equivalent to
     * BigNumber.config({ DECIMAL_PLACES : 20, ROUNDING_MODE : 4 })
     *
     * Ignore properties/parameters set to null or undefined.
     * Return an object with the properties current values.
     */
    BigNumber.config = BigNumber.set = function () {
        var v, p,
            i = 0,
            r = {},
            a = arguments,
            o = a[0],
            has = o && typeof o == 'object'
              ? function () { if ( o.hasOwnProperty(p) ) return ( v = o[p] ) != null; }
              : function () { if ( a.length > i ) return ( v = a[i++] ) != null; };

        // DECIMAL_PLACES {number} Integer, 0 to MAX inclusive.
        // 'config() DECIMAL_PLACES not an integer: {v}'
        // 'config() DECIMAL_PLACES out of range: {v}'
        if ( has( p = 'DECIMAL_PLACES' ) && isValidInt( v, 0, MAX, 2, p ) ) {
            DECIMAL_PLACES = v | 0;
        }
        r[p] = DECIMAL_PLACES;

        // ROUNDING_MODE {number} Integer, 0 to 8 inclusive.
        // 'config() ROUNDING_MODE not an integer: {v}'
        // 'config() ROUNDING_MODE out of range: {v}'
        if ( has( p = 'ROUNDING_MODE' ) && isValidInt( v, 0, 8, 2, p ) ) {
            ROUNDING_MODE = v | 0;
        }
        r[p] = ROUNDING_MODE;

        // EXPONENTIAL_AT {number|number[]}
        // Integer, -MAX to MAX inclusive or [integer -MAX to 0 inclusive, 0 to MAX inclusive].
        // 'config() EXPONENTIAL_AT not an integer: {v}'
        // 'config() EXPONENTIAL_AT out of range: {v}'
        if ( has( p = 'EXPONENTIAL_AT' ) ) {

            if ( isArray(v) ) {
                if ( isValidInt( v[0], -MAX, 0, 2, p ) && isValidInt( v[1], 0, MAX, 2, p ) ) {
                    TO_EXP_NEG = v[0] | 0;
                    TO_EXP_POS = v[1] | 0;
                }
            } else if ( isValidInt( v, -MAX, MAX, 2, p ) ) {
                TO_EXP_NEG = -( TO_EXP_POS = ( v < 0 ? -v : v ) | 0 );
            }
        }
        r[p] = [ TO_EXP_NEG, TO_EXP_POS ];

        // RANGE {number|number[]} Non-zero integer, -MAX to MAX inclusive or
        // [integer -MAX to -1 inclusive, integer 1 to MAX inclusive].
        // 'config() RANGE not an integer: {v}'
        // 'config() RANGE cannot be zero: {v}'
        // 'config() RANGE out of range: {v}'
        if ( has( p = 'RANGE' ) ) {

            if ( isArray(v) ) {
                if ( isValidInt( v[0], -MAX, -1, 2, p ) && isValidInt( v[1], 1, MAX, 2, p ) ) {
                    MIN_EXP = v[0] | 0;
                    MAX_EXP = v[1] | 0;
                }
            } else if ( isValidInt( v, -MAX, MAX, 2, p ) ) {
                if ( v | 0 ) MIN_EXP = -( MAX_EXP = ( v < 0 ? -v : v ) | 0 );
                else if (ERRORS) raise( 2, p + ' cannot be zero', v );
            }
        }
        r[p] = [ MIN_EXP, MAX_EXP ];

        // ERRORS {boolean|number} true, false, 1 or 0.
        // 'config() ERRORS not a boolean or binary digit: {v}'
        if ( has( p = 'ERRORS' ) ) {

            if ( v === !!v || v === 1 || v === 0 ) {
                id = 0;
                isValidInt = ( ERRORS = !!v ) ? intValidatorWithErrors : intValidatorNoErrors;
            } else if (ERRORS) {
                raise( 2, p + notBool, v );
            }
        }
        r[p] = ERRORS;

        // CRYPTO {boolean|number} true, false, 1 or 0.
        // 'config() CRYPTO not a boolean or binary digit: {v}'
        // 'config() crypto unavailable: {crypto}'
        if ( has( p = 'CRYPTO' ) ) {

            if ( v === true || v === false || v === 1 || v === 0 ) {
                if (v) {
                    v = typeof crypto == 'undefined';
                    if ( !v && crypto && (crypto.getRandomValues || crypto.randomBytes)) {
                        CRYPTO = true;
                    } else if (ERRORS) {
                        raise( 2, 'crypto unavailable', v ? void 0 : crypto );
                    } else {
                        CRYPTO = false;
                    }
                } else {
                    CRYPTO = false;
                }
            } else if (ERRORS) {
                raise( 2, p + notBool, v );
            }
        }
        r[p] = CRYPTO;

        // MODULO_MODE {number} Integer, 0 to 9 inclusive.
        // 'config() MODULO_MODE not an integer: {v}'
        // 'config() MODULO_MODE out of range: {v}'
        if ( has( p = 'MODULO_MODE' ) && isValidInt( v, 0, 9, 2, p ) ) {
            MODULO_MODE = v | 0;
        }
        r[p] = MODULO_MODE;

        // POW_PRECISION {number} Integer, 0 to MAX inclusive.
        // 'config() POW_PRECISION not an integer: {v}'
        // 'config() POW_PRECISION out of range: {v}'
        if ( has( p = 'POW_PRECISION' ) && isValidInt( v, 0, MAX, 2, p ) ) {
            POW_PRECISION = v | 0;
        }
        r[p] = POW_PRECISION;

        // FORMAT {object}
        // 'config() FORMAT not an object: {v}'
        if ( has( p = 'FORMAT' ) ) {

            if ( typeof v == 'object' ) {
                FORMAT = v;
            } else if (ERRORS) {
                raise( 2, p + ' not an object', v );
            }
        }
        r[p] = FORMAT;

        return r;
    };


    /*
     * Return a new BigNumber whose value is the maximum of the arguments.
     *
     * arguments {number|string|BigNumber}
     */
    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.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.
     *
     * 'random() decimal places not an integer: {dp}'
     * 'random() decimal places out of range: {dp}'
     * 'random() crypto unavailable: {crypto}'
     */
    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);

            dp = dp == null || !isValidInt( dp, 0, MAX, 14 ) ? DECIMAL_PLACES : dp | 0;
            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;
                    if (ERRORS) raise( 14, 'crypto unavailable', crypto );
                }
            }

            // 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;
        };
    })();


    // PRIVATE FUNCTIONS


    // Convert a numeric string of baseIn to a numeric string of baseOut.
    function convertBase( str, baseOut, baseIn, sign ) {
        var d, e, k, r, x, xc, y,
            i = str.indexOf( '.' ),
            dp = DECIMAL_PLACES,
            rm = ROUNDING_MODE;

        if ( baseIn < 37 ) str = str.toLowerCase();

        // 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 ), 10, baseOut );
            y.e = y.c.length;
        }

        // Convert the number as integer.
        xc = toBaseOut( str, baseIn, baseOut );
        e = k = xc.length;

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

        if ( i < 0 ) {
            --e;
        } else {
            x.c = xc;
            x.e = e;

            // 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;
        }

        d = e + dp + 1;

        // The rounding digit, i.e. the digit to the right of the digit that may be rounded up.
        i = xc[d];
        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 ( d < 1 || !xc[0] ) {

            // 1^-dp or 0.
            str = r ? toFixedPoint( '1', -dp ) : '0';
        } else {
            xc.length = d;

            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++] ) );
            str = toFixedPoint( str, e );
        }

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

                        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.
                            // 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 is a BigNumber.
     * i is the index of the last digit required (i.e. the digit that may be rounded up).
     * rm is the rounding mode.
     * caller is caller id: toExponential 19, toFixed 20, toFormat 21, toPrecision 24.
     */
    function format( n, i, rm, caller ) {
        var c0, e, ne, len, str;

        rm = rm != null && isValidInt( rm, 0, 8, caller, roundingMode )
          ? rm | 0 : ROUNDING_MODE;

        if ( !n.c ) return n.toString();
        c0 = n.c[0];
        ne = n.e;

        if ( i == null ) {
            str = coeffToString( n.c );
            str = caller == 19 || caller == 24 && ne <= TO_EXP_NEG
              ? toExponential( str, ne )
              : toFixedPoint( str, ne );
        } 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 ( caller == 19 || caller == 24 && ( 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 );

                // 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 m, n,
            i = 0;

        if ( isArray( args[0] ) ) args = args[0];
        m = new BigNumber( args[0] );

        for ( ; ++i < args.length; ) {
            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;
    }


    /*
     * Return true if n is an integer in range, otherwise throw.
     * Use for argument validation when ERRORS is true.
     */
    function intValidatorWithErrors( n, min, max, caller, name ) {
        if ( n < min || n > max || n != truncate(n) ) {
            raise( caller, ( name || 'decimal places' ) +
              ( n < min || n > max ? ' out of range' : ' not an integer' ), n );
        }

        return true;
    }


    /*
     * 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, num, b ) {
            var base,
                s = num ? 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 ( !num ) {

                    // 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 );
                }

                // 'new BigNumber() not a number: {n}'
                // 'new BigNumber() not a base {b} number: {n}'
                if (ERRORS) raise( id, 'not a' + ( b ? ' base ' + b : '' ) + ' number', str );
                x.s = null;
            }

            x.c = x.e = null;
            id = 0;
        }
    })();


    // Throw a BigNumber Error.
    function raise( caller, msg, val ) {
        var error = new Error( [
            'new BigNumber',     // 0
            'cmp',               // 1
            'config',            // 2
            'div',               // 3
            'divToInt',          // 4
            'eq',                // 5
            'gt',                // 6
            'gte',               // 7
            'lt',                // 8
            'lte',               // 9
            'minus',             // 10
            'mod',               // 11
            'plus',              // 12
            'precision',         // 13
            'random',            // 14
            'round',             // 15
            'shift',             // 16
            'times',             // 17
            'toDigits',          // 18
            'toExponential',     // 19
            'toFixed',           // 20
            'toFormat',          // 21
            'toFraction',        // 22
            'pow',               // 23
            'toPrecision',       // 24
            'toString',          // 25
            'BigNumber'          // 26
        ][caller] + '() ' + msg + ': ' + val );

        error.name = 'BigNumber Error';
        id = 0;
        throw error;
    }


    /*
     * 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;
    }


    // 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 a new BigNumber whose value is the value of this BigNumber rounded to a whole
     * number in the direction of Infinity.
     */
    P.ceil = function () {
        return round( new BigNumber(this), this.e + 1, 2 );
    };


    /*
     * 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 = P.cmp = function ( y, b ) {
        id = 1;
        return compare( this, new BigNumber( y, b ) );
    };


    /*
     * Return the number of decimal places of the value of this BigNumber, or null if the value
     * of this BigNumber is ±Infinity or NaN.
     */
    P.decimalPlaces = P.dp = function () {
        var n, v,
            c = this.c;

        if ( !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 ) {
        id = 3;
        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.divToInt = function ( y, b ) {
        id = 4;
        return div( this, new BigNumber( y, b ), 0, 1 );
    };


    /*
     * Return true if the value of this BigNumber is equal to the value of BigNumber(y, b),
     * otherwise returns false.
     */
    P.equals = P.eq = function ( y, b ) {
        id = 5;
        return compare( this, new BigNumber( y, b ) ) === 0;
    };


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


    /*
     * Return true if the value of this BigNumber is greater than the value of BigNumber(y, b),
     * otherwise returns false.
     */
    P.greaterThan = P.gt = function ( y, b ) {
        id = 6;
        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 returns false.
     */
    P.greaterThanOrEqualTo = P.gte = function ( y, b ) {
        id = 7;
        return ( b = compare( this, new BigNumber( y, b ) ) ) === 1 || b === 0;

    };


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


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


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


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


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


    /*
     * Return true if the value of this BigNumber is less than the value of BigNumber(y, b),
     * otherwise returns false.
     */
    P.lessThan = P.lt = function ( y, b ) {
        id = 8;
        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 returns false.
     */
    P.lessThanOrEqualTo = P.lte = function ( y, b ) {
        id = 9;
        return ( b = compare( this, new BigNumber( y, b ) ) ) === -1 || b === 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 = P.sub = function ( y, b ) {
        var i, j, t, xLTy,
            x = this,
            a = x.s;

        id = 10;
        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;

        id = 11;
        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 );
        }

        return x.minus( q.times(y) );
    };


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

        id = 12;
        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 );
    };


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

        // 'precision() argument not a boolean or binary digit: {z}'
        if ( z != null && z !== !!z && z !== 1 && z !== 0 ) {
            if (ERRORS) raise( 13, 'argument' + notBool, z );
            if ( z != !!z ) z = null;
        }

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

        return n;
    };


    /*
     * 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 to 0 and ROUNDING_MODE respectively if
     * omitted.
     *
     * [dp] {number} Decimal places. Integer, 0 to MAX inclusive.
     * [rm] {number} Rounding mode. Integer, 0 to 8 inclusive.
     *
     * 'round() decimal places out of range: {dp}'
     * 'round() decimal places not an integer: {dp}'
     * 'round() rounding mode not an integer: {rm}'
     * 'round() rounding mode out of range: {rm}'
     */
    P.round = function ( dp, rm ) {
        var n = new BigNumber(this);

        if ( dp == null || isValidInt( dp, 0, MAX, 15 ) ) {
            round( n, ~~dp + this.e + 1, rm == null ||
              !isValidInt( rm, 0, 8, 15, roundingMode ) ? ROUNDING_MODE : rm | 0 );
        }

        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.
     *
     * If k is out of range and ERRORS is false, the result will be ±0 if k < 0, or ±Infinity
     * otherwise.
     *
     * 'shift() argument not an integer: {k}'
     * 'shift() argument out of range: {k}'
     */
    P.shift = function (k) {
        var n = this;
        return isValidInt( k, -MAX_SAFE_INTEGER, MAX_SAFE_INTEGER, 16, 'argument' )

          // k < 1e+21, or truncate(k) will produce exponential notation.
          ? n.times( '1e' + truncate(k) )
          : new BigNumber( n.c && n.c[0] && ( k < -MAX_SAFE_INTEGER || k > MAX_SAFE_INTEGER )
            ? n.s * ( k < 0 ? 0 : 1 / 0 )
            : n );
    };


    /*
     *  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( +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 = '1e' + 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 );
    };


    /*
     *  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 times the value of
     * BigNumber(y, b).
     */
    P.times = P.mul = 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 = ( id = 17, 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 rounded to a maximum of
     * sd significant digits using rounding mode rm, or ROUNDING_MODE if rm is omitted.
     *
     * [sd] {number} Significant digits. Integer, 1 to MAX inclusive.
     * [rm] {number} Rounding mode. Integer, 0 to 8 inclusive.
     *
     * 'toDigits() precision out of range: {sd}'
     * 'toDigits() precision not an integer: {sd}'
     * 'toDigits() rounding mode not an integer: {rm}'
     * 'toDigits() rounding mode out of range: {rm}'
     */
    P.toDigits = function ( sd, rm ) {
        var n = new BigNumber(this);
        sd = sd == null || !isValidInt( sd, 1, MAX, 18, 'precision' ) ? null : sd | 0;
        rm = rm == null || !isValidInt( rm, 0, 8, 18, roundingMode ) ? ROUNDING_MODE : rm | 0;
        return sd ? round( n, sd, rm ) : n;
    };


    /*
     * 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.
     *
     * 'toExponential() decimal places not an integer: {dp}'
     * 'toExponential() decimal places out of range: {dp}'
     * 'toExponential() rounding mode not an integer: {rm}'
     * 'toExponential() rounding mode out of range: {rm}'
     */
    P.toExponential = function ( dp, rm ) {
        return format( this,
          dp != null && isValidInt( dp, 0, MAX, 19 ) ? ~~dp + 1 : null, rm, 19 );
    };


    /*
     * 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.
     *
     * 'toFixed() decimal places not an integer: {dp}'
     * 'toFixed() decimal places out of range: {dp}'
     * 'toFixed() rounding mode not an integer: {rm}'
     * 'toFixed() rounding mode out of range: {rm}'
     */
    P.toFixed = function ( dp, rm ) {
        return format( this, dp != null && isValidInt( dp, 0, MAX, 20 )
          ? ~~dp + this.e + 1 : null, rm, 20 );
    };


    /*
     * 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 object (see BigNumber.config).
     *
     * FORMAT = {
     *      decimalSeparator : '.',
     *      groupSeparator : ',',
     *      groupSize : 3,
     *      secondaryGroupSize : 0,
     *      fractionGroupSeparator : '\xA0',    // non-breaking space
     *      fractionGroupSize : 0
     * };
     *
     * [dp] {number} Decimal places. Integer, 0 to MAX inclusive.
     * [rm] {number} Rounding mode. Integer, 0 to 8 inclusive.
     *
     * 'toFormat() decimal places not an integer: {dp}'
     * 'toFormat() decimal places out of range: {dp}'
     * 'toFormat() rounding mode not an integer: {rm}'
     * 'toFormat() rounding mode out of range: {rm}'
     */
    P.toFormat = function ( dp, rm ) {
        var str = format( this, dp != null && isValidInt( dp, 0, MAX, 21 )
          ? ~~dp + this.e + 1 : null, rm, 21 );

        if ( this.c ) {
            var i,
                arr = str.split('.'),
                g1 = +FORMAT.groupSize,
                g2 = +FORMAT.secondaryGroupSize,
                groupSeparator = FORMAT.groupSeparator,
                intPart = arr[0],
                fractionPart = arr[1],
                isNeg = this.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 str;
    };


    /*
     * Return a string array 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 and < Infinity. The maximum denominator.
     *
     * 'toFraction() max denominator not an integer: {md}'
     * 'toFraction() max denominator out of range: {md}'
     */
    P.toFraction = function (md) {
        var arr, d0, d2, e, exp, n, n0, q, s,
            k = ERRORS,
            x = this,
            xc = x.c,
            d = new BigNumber(ONE),
            n1 = d0 = new BigNumber(ONE),
            d1 = n0 = new BigNumber(ONE);

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

            if ( !( k = n.isInt() ) || n.lt(ONE) ) {

                if (ERRORS) {
                    raise( 22,
                      'max denominator ' + ( k ? 'out of range' : 'not an integer' ), md );
                }

                // ERRORS is false:
                // If md is a finite non-integer >= 1, round it to an integer and use it.
                md = !k && n.c && round( n, n.e + 1, 1 ).gte(ONE) ? n : null;
            }
        }

        if ( !xc ) return x.toString();
        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.cmp(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.cmp(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 *= 2;

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

        MAX_EXP = exp;
        return arr;
    };


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


    /*
     * Return a BigNumber whose value is the value of this BigNumber raised to the power 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 positive integers,
     * otherwise it is equivalent to calculating x.toPower(n).modulo(m) (with POW_PRECISION 0).
     *
     * n {number} Integer, -MAX_SAFE_INTEGER to MAX_SAFE_INTEGER inclusive.
     * [m] {number|string|BigNumber} The modulus.
     *
     * 'pow() exponent not an integer: {n}'
     * 'pow() exponent out of range: {n}'
     *
     * Performs 54 loop iterations for n of 9007199254740991.
     */
    P.toPower = P.pow = function ( n, m ) {
        var k, y, z,
            i = mathfloor( n < 0 ? -n : +n ),
            x = this;

        if ( m != null ) {
            id = 23;
            m = new BigNumber(m);
        }

        // Pass ±Infinity to Math.pow if exponent is out of range.
        if ( !isValidInt( n, -MAX_SAFE_INTEGER, MAX_SAFE_INTEGER, 23, 'exponent' ) &&
          ( !isFinite(n) || i > MAX_SAFE_INTEGER && ( n /= 0 ) ||
            parseFloat(n) != n && !( n = NaN ) ) || n == 0 ) {
            k = Math.pow( +x, n );
            return new BigNumber( m ? k % m : k );
        }

        if (m) {
            if ( n > 1 && x.gt(ONE) && x.isInt() && m.gt(ONE) && m.isInt() ) {
                x = x.mod(m);
            } else {
                z = m;

                // Nullify m so only a single mod operation is performed at the end.
                m = null;
            }
        } 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.
            // (Using + 1.5 would give [9, 21] guard digits.)
            k = mathceil( POW_PRECISION / LOG_BASE + 2 );
        }

        y = new BigNumber(ONE);

        for ( ; ; ) {
            if ( i % 2 ) {
                y = y.times(x);
                if ( !y.c ) break;
                if (k) {
                    if ( y.c.length > k ) y.c.length = k;
                } else if (m) {
                    y = y.mod(m);
                }
            }

            i = mathfloor( i / 2 );
            if ( !i ) break;
            x = x.times(x);
            if (k) {
                if ( x.c && x.c.length > k ) x.c.length = k;
            } else if (m) {
                x = x.mod(m);
            }
        }

        if (m) return y;
        if ( n < 0 ) y = ONE.div(y);

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


    /*
     * 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.
     *
     * 'toPrecision() precision not an integer: {sd}'
     * 'toPrecision() precision out of range: {sd}'
     * 'toPrecision() rounding mode not an integer: {rm}'
     * 'toPrecision() rounding mode out of range: {rm}'
     */
    P.toPrecision = function ( sd, rm ) {
        return format( this, sd != null && isValidInt( sd, 1, MAX, 24, 'precision' )
          ? sd | 0 : null, rm, 24 );
    };


    /*
     * 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 64 inclusive.
     *
     * 'toString() base not an integer: {b}'
     * 'toString() base 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 {
            str = coeffToString( n.c );

            if ( b == null || !isValidInt( b, 2, 64, 25, 'base' ) ) {
                str = e <= TO_EXP_NEG || e >= TO_EXP_POS
                  ? toExponential( str, e )
                  : toFixedPoint( str, e );
            } else {
                str = convertBase( toFixedPoint( str, e ), b | 0, 10, s );
            }

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

        return str;
    };


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


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

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


    P.isBigNumber = true;

    if ( config != null ) BigNumber.config(config);

    return BigNumber;
}


// PRIVATE HELPER FUNCTIONS


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;
}


/*
 * Return true if n is a valid number in range, otherwise false.
 * Use for argument validation when ERRORS is false.
 * Note: parseInt('1e+1') == 1 but parseFloat('1e+1') == 10.
 */
function intValidatorNoErrors( n, min, max ) {
    return ( n = truncate(n) ) >= min && n <= max;
}


function isArray(obj) {
    return Object.prototype.toString.call(obj) == '[object Array]';
}


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

    for ( ; i < len; ) {
        for ( arrL = arr.length; arrL--; arr[arrL] *= baseIn );
        arr[ j = 0 ] += ALPHABET.indexOf( str.charAt( i++ ) );

        for ( ; 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();
}


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


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

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

        // Prepend zeros.
        for ( z = '0.'; ++e; z += '0' );
        str = z + str;

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

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

    return str;
}


function truncate(n) {
    n = parseFloat(n);
    return n < 0 ? mathceil(n) : mathfloor(n);
}


// EXPORT


BigNumber = constructorFactory();
BigNumber['default'] = BigNumber.BigNumber = BigNumber;

export default BigNumber;
