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+/*!
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+ * decimal.js v10.4.3
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+ * An arbitrary-precision Decimal type for JavaScript.
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+ * https://github.com/MikeMcl/decimal.js
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+ * Copyright (c) 2022 Michael Mclaughlin <M8ch88l@gmail.com>
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+ * MIT Licence
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+ */
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+
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+
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+// ----------------------------------- EDITABLE DEFAULTS ------------------------------------ //
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+
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+
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+ // The maximum exponent magnitude.
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+ // The limit on the value of `toExpNeg`, `toExpPos`, `minE` and `maxE`.
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+ var EXP_LIMIT = 9e15, // 0 to 9e15
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+
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+ // The limit on the value of `precision`, and on the value of the first argument to
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+ // `toDecimalPlaces`, `toExponential`, `toFixed`, `toPrecision` and `toSignificantDigits`.
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+ MAX_DIGITS = 1e9, // 0 to 1e9
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+
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+ // Base conversion alphabet.
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+ NUMERALS = '0123456789abcdef',
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+
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+ // The natural logarithm of 10 (1025 digits).
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+ LN10 =
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+ '2.3025850929940456840179914546843642076011014886287729760333279009675726096773524802359972050895982983419677840422862486334095254650828067566662873690987816894829072083255546808437998948262331985283935053089653777326288461633662222876982198867465436674744042432743651550489343149393914796194044002221051017141748003688084012647080685567743216228355220114804663715659121373450747856947683463616792101806445070648000277502684916746550586856935673420670581136429224554405758925724208241314695689016758940256776311356919292033376587141660230105703089634572075440370847469940168269282808481184289314848524948644871927809676271275775397027668605952496716674183485704422507197965004714951050492214776567636938662976979522110718264549734772662425709429322582798502585509785265383207606726317164309505995087807523710333101197857547331541421808427543863591778117054309827482385045648019095610299291824318237525357709750539565187697510374970888692180205189339507238539205144634197265287286965110862571492198849978748873771345686209167058',
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+
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+ // Pi (1025 digits).
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+ PI =
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+ '3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461284756482337867831652712019091456485669234603486104543266482133936072602491412737245870066063155881748815209209628292540917153643678925903600113305305488204665213841469519415116094330572703657595919530921861173819326117931051185480744623799627495673518857527248912279381830119491298336733624406566430860213949463952247371907021798609437027705392171762931767523846748184676694051320005681271452635608277857713427577896091736371787214684409012249534301465495853710507922796892589235420199561121290219608640344181598136297747713099605187072113499999983729780499510597317328160963185950244594553469083026425223082533446850352619311881710100031378387528865875332083814206171776691473035982534904287554687311595628638823537875937519577818577805321712268066130019278766111959092164201989380952572010654858632789',
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+
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+
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+ // The initial configuration properties of the Decimal constructor.
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+ DEFAULTS = {
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+
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+ // These values must be integers within the stated ranges (inclusive).
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+ // Most of these values can be changed at run-time using the `Decimal.config` method.
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+
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+ // The maximum number of significant digits of the result of a calculation or base conversion.
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+ // E.g. `Decimal.config({ precision: 20 });`
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+ precision: 20, // 1 to MAX_DIGITS
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+
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+ // The rounding mode used when rounding to `precision`.
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+ //
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+ // ROUND_UP 0 Away from zero.
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+ // ROUND_DOWN 1 Towards zero.
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+ // ROUND_CEIL 2 Towards +Infinity.
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+ // ROUND_FLOOR 3 Towards -Infinity.
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+ // ROUND_HALF_UP 4 Towards nearest neighbour. If equidistant, up.
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+ // ROUND_HALF_DOWN 5 Towards nearest neighbour. If equidistant, down.
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+ // ROUND_HALF_EVEN 6 Towards nearest neighbour. If equidistant, towards even neighbour.
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+ // ROUND_HALF_CEIL 7 Towards nearest neighbour. If equidistant, towards +Infinity.
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+ // ROUND_HALF_FLOOR 8 Towards nearest neighbour. If equidistant, towards -Infinity.
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+ //
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+ // E.g.
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+ // `Decimal.rounding = 4;`
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+ // `Decimal.rounding = Decimal.ROUND_HALF_UP;`
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+ rounding: 4, // 0 to 8
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+
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+ // The modulo mode used when calculating the modulus: a mod n.
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+ // The quotient (q = a / n) is calculated according to the corresponding rounding mode.
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+ // The remainder (r) is calculated as: r = a - n * q.
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+ //
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+ // UP 0 The remainder is positive if the dividend is negative, else is negative.
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+ // DOWN 1 The remainder has the same sign as the dividend (JavaScript %).
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+ // FLOOR 3 The remainder has the same sign as the divisor (Python %).
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+ // HALF_EVEN 6 The IEEE 754 remainder function.
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+ // EUCLID 9 Euclidian division. q = sign(n) * floor(a / abs(n)). Always positive.
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+ //
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+ // Truncated division (1), floored division (3), the IEEE 754 remainder (6), and Euclidian
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+ // division (9) are commonly used for the modulus operation. The other rounding modes can also
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+ // be used, but they may not give useful results.
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+ modulo: 1, // 0 to 9
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+
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+ // The exponent value at and beneath which `toString` returns exponential notation.
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+ // JavaScript numbers: -7
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+ toExpNeg: -7, // 0 to -EXP_LIMIT
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+
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+ // The exponent value at and above which `toString` returns exponential notation.
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+ // JavaScript numbers: 21
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+ toExpPos: 21, // 0 to EXP_LIMIT
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+
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+ // The minimum exponent value, beneath which underflow to zero occurs.
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+ // JavaScript numbers: -324 (5e-324)
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+ minE: -EXP_LIMIT, // -1 to -EXP_LIMIT
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+
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+ // The maximum exponent value, above which overflow to Infinity occurs.
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+ // JavaScript numbers: 308 (1.7976931348623157e+308)
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+ maxE: EXP_LIMIT, // 1 to EXP_LIMIT
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+
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+ // Whether to use cryptographically-secure random number generation, if available.
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+ crypto: false // true/false
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+ },
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+
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+
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+ // ----------------------------------- END OF EDITABLE DEFAULTS ------------------------------- //
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+
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+
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+ Decimal, inexact, noConflict, quadrant,
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+ external = true,
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+
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+ decimalError = '[DecimalError] ',
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+ invalidArgument = decimalError + 'Invalid argument: ',
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+ precisionLimitExceeded = decimalError + 'Precision limit exceeded',
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+ cryptoUnavailable = decimalError + 'crypto unavailable',
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+ tag = '[object Decimal]',
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+
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+ mathfloor = Math.floor,
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+ mathpow = Math.pow,
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+
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+ isBinary = /^0b([01]+(\.[01]*)?|\.[01]+)(p[+-]?\d+)?$/i,
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+ isHex = /^0x([0-9a-f]+(\.[0-9a-f]*)?|\.[0-9a-f]+)(p[+-]?\d+)?$/i,
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+ isOctal = /^0o([0-7]+(\.[0-7]*)?|\.[0-7]+)(p[+-]?\d+)?$/i,
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+ isDecimal = /^(\d+(\.\d*)?|\.\d+)(e[+-]?\d+)?$/i,
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+
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+ BASE = 1e7,
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+ LOG_BASE = 7,
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+ MAX_SAFE_INTEGER = 9007199254740991,
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+
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+ LN10_PRECISION = LN10.length - 1,
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+ PI_PRECISION = PI.length - 1,
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+
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+ // Decimal.prototype object
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+ P = {
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+ toStringTag: tag
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+ };
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+
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+
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+// Decimal prototype methods
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+
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+
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+/*
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+ * absoluteValue abs
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+ * ceil
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+ * clampedTo clamp
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+ * comparedTo cmp
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+ * cosine cos
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+ * cubeRoot cbrt
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+ * decimalPlaces dp
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+ * dividedBy div
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+ * dividedToIntegerBy divToInt
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+ * equals eq
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+ * floor
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+ * greaterThan gt
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+ * greaterThanOrEqualTo gte
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+ * hyperbolicCosine cosh
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+ * hyperbolicSine sinh
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+ * hyperbolicTangent tanh
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+ * inverseCosine acos
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+ * inverseHyperbolicCosine acosh
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+ * inverseHyperbolicSine asinh
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+ * inverseHyperbolicTangent atanh
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+ * inverseSine asin
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+ * inverseTangent atan
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+ * isFinite
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+ * isInteger isInt
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+ * isNaN
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+ * isNegative isNeg
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+ * isPositive isPos
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+ * isZero
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+ * lessThan lt
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+ * lessThanOrEqualTo lte
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+ * logarithm log
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+ * [maximum] [max]
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+ * [minimum] [min]
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+ * minus sub
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+ * modulo mod
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+ * naturalExponential exp
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+ * naturalLogarithm ln
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+ * negated neg
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+ * plus add
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+ * precision sd
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+ * round
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+ * sine sin
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+ * squareRoot sqrt
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+ * tangent tan
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+ * times mul
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+ * toBinary
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+ * toDecimalPlaces toDP
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+ * toExponential
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+ * toFixed
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+ * toFraction
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+ * toHexadecimal toHex
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+ * toNearest
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+ * toNumber
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+ * toOctal
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+ * toPower pow
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+ * toPrecision
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+ * toSignificantDigits toSD
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+ * toString
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+ * truncated trunc
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+ * valueOf toJSON
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+ */
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+
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+
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+/*
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+ * Return a new Decimal whose value is the absolute value of this Decimal.
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+ *
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+ */
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+P.absoluteValue = P.abs = function() {
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+ var x = new this.constructor(this);
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+ if (x.s < 0) x.s = 1;
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+ return finalise(x);
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+};
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+
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+
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+/*
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+ * Return a new Decimal whose value is the value of this Decimal rounded to a whole number in the
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+ * direction of positive Infinity.
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+ *
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+ */
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+P.ceil = function() {
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+ return finalise(new this.constructor(this), this.e + 1, 2);
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+};
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+
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+
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+/*
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+ * Return a new Decimal whose value is the value of this Decimal clamped to the range
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+ * delineated by `min` and `max`.
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+ *
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+ * min {number|string|Decimal}
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+ * max {number|string|Decimal}
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+ *
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+ */
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+P.clampedTo = P.clamp = function(min, max) {
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+ var k,
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+ x = this,
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+ Ctor = x.constructor;
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+ min = new Ctor(min);
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+ max = new Ctor(max);
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+ if (!min.s || !max.s) return new Ctor(NaN);
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+ if (min.gt(max)) throw Error(invalidArgument + max);
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+ k = x.cmp(min);
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+ return k < 0 ? min : x.cmp(max) > 0 ? max : new Ctor(x);
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+};
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+
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+
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+/*
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+ * Return
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+ * 1 if the value of this Decimal is greater than the value of `y`,
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+ * -1 if the value of this Decimal is less than the value of `y`,
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+ * 0 if they have the same value,
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+ * NaN if the value of either Decimal is NaN.
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+ *
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+ */
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+P.comparedTo = P.cmp = function(y) {
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+ var i, j, xdL, ydL,
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+ x = this,
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+ xd = x.d,
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+ yd = (y = new x.constructor(y)).d,
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+ xs = x.s,
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+ ys = y.s;
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+
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+ // Either NaN or ±Infinity?
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+ if (!xd || !yd) {
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+ return !xs || !ys ? NaN : xs !== ys ? xs : xd === yd ? 0 : !xd ^ xs < 0 ? 1 : -1;
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+ }
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+
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+ // Either zero?
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+ if (!xd[0] || !yd[0]) return xd[0] ? xs : yd[0] ? -ys : 0;
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+
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+ // Signs differ?
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+ if (xs !== ys) return xs;
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+
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+ // Compare exponents.
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+ if (x.e !== y.e) return x.e > y.e ^ xs < 0 ? 1 : -1;
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+
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+ xdL = xd.length;
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+ ydL = yd.length;
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+
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+ // Compare digit by digit.
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+ for (i = 0, j = xdL < ydL ? xdL : ydL; i < j; ++i) {
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+ if (xd[i] !== yd[i]) return xd[i] > yd[i] ^ xs < 0 ? 1 : -1;
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+ }
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+
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+ // Compare lengths.
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+ return xdL === ydL ? 0 : xdL > ydL ^ xs < 0 ? 1 : -1;
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+};
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+
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+
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+/*
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+ * Return a new Decimal whose value is the cosine of the value in radians of this Decimal.
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+ *
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+ * Domain: [-Infinity, Infinity]
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+ * Range: [-1, 1]
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+ *
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+ * cos(0) = 1
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+ * cos(-0) = 1
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+ * cos(Infinity) = NaN
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+ * cos(-Infinity) = NaN
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+ * cos(NaN) = NaN
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+ *
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+ */
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+P.cosine = P.cos = function() {
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+ var pr, rm,
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+ x = this,
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+ Ctor = x.constructor;
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+
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+ if (!x.d) return new Ctor(NaN);
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+
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+ // cos(0) = cos(-0) = 1
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+ if (!x.d[0]) return new Ctor(1);
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+
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+ pr = Ctor.precision;
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+ rm = Ctor.rounding;
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+ Ctor.precision = pr + Math.max(x.e, x.sd()) + LOG_BASE;
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+ Ctor.rounding = 1;
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+
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+ x = cosine(Ctor, toLessThanHalfPi(Ctor, x));
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+
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+ Ctor.precision = pr;
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+ Ctor.rounding = rm;
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+
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+ return finalise(quadrant == 2 || quadrant == 3 ? x.neg() : x, pr, rm, true);
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+};
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+
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+
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+/*
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+ *
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+ * Return a new Decimal whose value is the cube root of the value of this Decimal, rounded to
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+ * `precision` significant digits using rounding mode `rounding`.
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+ *
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+ * cbrt(0) = 0
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+ * cbrt(-0) = -0
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+ * cbrt(1) = 1
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+ * cbrt(-1) = -1
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+ * cbrt(N) = N
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+ * cbrt(-I) = -I
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+ * cbrt(I) = I
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+ *
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+ * Math.cbrt(x) = (x < 0 ? -Math.pow(-x, 1/3) : Math.pow(x, 1/3))
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+ *
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+ */
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+P.cubeRoot = P.cbrt = function() {
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+ var e, m, n, r, rep, s, sd, t, t3, t3plusx,
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+ x = this,
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+ Ctor = x.constructor;
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+
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+ if (!x.isFinite() || x.isZero()) return new Ctor(x);
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+ external = false;
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+
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+ // Initial estimate.
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+ s = x.s * mathpow(x.s * x, 1 / 3);
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+
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+ // Math.cbrt underflow/overflow?
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+ // Pass x to Math.pow as integer, then adjust the exponent of the result.
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+ if (!s || Math.abs(s) == 1 / 0) {
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+ n = digitsToString(x.d);
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+ e = x.e;
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+
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+ // Adjust n exponent so it is a multiple of 3 away from x exponent.
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+ if (s = (e - n.length + 1) % 3) n += (s == 1 || s == -2 ? '0' : '00');
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+ s = mathpow(n, 1 / 3);
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+
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+ // Rarely, e may be one less than the result exponent value.
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+ e = mathfloor((e + 1) / 3) - (e % 3 == (e < 0 ? -1 : 2));
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+
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+ if (s == 1 / 0) {
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+ n = '5e' + e;
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+ } else {
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+ n = s.toExponential();
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+ n = n.slice(0, n.indexOf('e') + 1) + e;
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+ }
|
|
|
+
|
|
|
+ r = new Ctor(n);
|
|
|
+ r.s = x.s;
|
|
|
+ } else {
|
|
|
+ r = new Ctor(s.toString());
|
|
|
+ }
|
|
|
+
|
|
|
+ sd = (e = Ctor.precision) + 3;
|
|
|
+
|
|
|
+ // Halley's method.
|
|
|
+ // TODO? Compare Newton's method.
|
|
|
+ for (;;) {
|
|
|
+ t = r;
|
|
|
+ t3 = t.times(t).times(t);
|
|
|
+ t3plusx = t3.plus(x);
|
|
|
+ r = divide(t3plusx.plus(x).times(t), t3plusx.plus(t3), sd + 2, 1);
|
|
|
+
|
|
|
+ // TODO? Replace with for-loop and checkRoundingDigits.
|
|
|
+ if (digitsToString(t.d).slice(0, sd) === (n = digitsToString(r.d)).slice(0, sd)) {
|
|
|
+ n = n.slice(sd - 3, sd + 1);
|
|
|
+
|
|
|
+ // The 4th rounding digit may be in error by -1 so if the 4 rounding digits are 9999 or 4999
|
|
|
+ // , i.e. approaching a rounding boundary, continue the iteration.
|
|
|
+ if (n == '9999' || !rep && n == '4999') {
|
|
|
+
|
|
|
+ // On the first iteration only, check to see if rounding up gives the exact result as the
|
|
|
+ // nines may infinitely repeat.
|
|
|
+ if (!rep) {
|
|
|
+ finalise(t, e + 1, 0);
|
|
|
+
|
|
|
+ if (t.times(t).times(t).eq(x)) {
|
|
|
+ r = t;
|
|
|
+ break;
|
|
|
+ }
|
|
|
+ }
|
|
|
+
|
|
|
+ sd += 4;
|
|
|
+ rep = 1;
|
|
|
+ } else {
|
|
|
+
|
|
|
+ // If the rounding digits are null, 0{0,4} or 50{0,3}, check for an exact result.
|
|
|
+ // If not, then there are further digits and m will be truthy.
|
|
|
+ if (!+n || !+n.slice(1) && n.charAt(0) == '5') {
|
|
|
+
|
|
|
+ // Truncate to the first rounding digit.
|
|
|
+ finalise(r, e + 1, 1);
|
|
|
+ m = !r.times(r).times(r).eq(x);
|
|
|
+ }
|
|
|
+
|
|
|
+ break;
|
|
|
+ }
|
|
|
+ }
|
|
|
+ }
|
|
|
+
|
|
|
+ external = true;
|
|
|
+
|
|
|
+ return finalise(r, e, Ctor.rounding, m);
|
|
|
+};
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return the number of decimal places of the value of this Decimal.
|
|
|
+ *
|
|
|
+ */
|
|
|
+P.decimalPlaces = P.dp = function() {
|
|
|
+ var w,
|
|
|
+ d = this.d,
|
|
|
+ n = NaN;
|
|
|
+
|
|
|
+ if (d) {
|
|
|
+ w = d.length - 1;
|
|
|
+ n = (w - mathfloor(this.e / LOG_BASE)) * LOG_BASE;
|
|
|
+
|
|
|
+ // Subtract the number of trailing zeros of the last word.
|
|
|
+ w = d[w];
|
|
|
+ if (w)
|
|
|
+ for (; w % 10 == 0; w /= 10) n--;
|
|
|
+ if (n < 0) n = 0;
|
|
|
+ }
|
|
|
+
|
|
|
+ return n;
|
|
|
+};
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * n / 0 = I
|
|
|
+ * n / N = N
|
|
|
+ * n / I = 0
|
|
|
+ * 0 / n = 0
|
|
|
+ * 0 / 0 = N
|
|
|
+ * 0 / N = N
|
|
|
+ * 0 / I = 0
|
|
|
+ * N / n = N
|
|
|
+ * N / 0 = N
|
|
|
+ * N / N = N
|
|
|
+ * N / I = N
|
|
|
+ * I / n = I
|
|
|
+ * I / 0 = I
|
|
|
+ * I / N = N
|
|
|
+ * I / I = N
|
|
|
+ *
|
|
|
+ * Return a new Decimal whose value is the value of this Decimal divided by `y`, rounded to
|
|
|
+ * `precision` significant digits using rounding mode `rounding`.
|
|
|
+ *
|
|
|
+ */
|
|
|
+P.dividedBy = P.div = function(y) {
|
|
|
+ return divide(this, new this.constructor(y));
|
|
|
+};
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is the integer part of dividing the value of this Decimal
|
|
|
+ * by the value of `y`, rounded to `precision` significant digits using rounding mode `rounding`.
|
|
|
+ *
|
|
|
+ */
|
|
|
+P.dividedToIntegerBy = P.divToInt = function(y) {
|
|
|
+ var x = this,
|
|
|
+ Ctor = x.constructor;
|
|
|
+ return finalise(divide(x, new Ctor(y), 0, 1, 1), Ctor.precision, Ctor.rounding);
|
|
|
+};
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return true if the value of this Decimal is equal to the value of `y`, otherwise return false.
|
|
|
+ *
|
|
|
+ */
|
|
|
+P.equals = P.eq = function(y) {
|
|
|
+ return this.cmp(y) === 0;
|
|
|
+};
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is the value of this Decimal rounded to a whole number in the
|
|
|
+ * direction of negative Infinity.
|
|
|
+ *
|
|
|
+ */
|
|
|
+P.floor = function() {
|
|
|
+ return finalise(new this.constructor(this), this.e + 1, 3);
|
|
|
+};
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return true if the value of this Decimal is greater than the value of `y`, otherwise return
|
|
|
+ * false.
|
|
|
+ *
|
|
|
+ */
|
|
|
+P.greaterThan = P.gt = function(y) {
|
|
|
+ return this.cmp(y) > 0;
|
|
|
+};
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return true if the value of this Decimal is greater than or equal to the value of `y`,
|
|
|
+ * otherwise return false.
|
|
|
+ *
|
|
|
+ */
|
|
|
+P.greaterThanOrEqualTo = P.gte = function(y) {
|
|
|
+ var k = this.cmp(y);
|
|
|
+ return k == 1 || k === 0;
|
|
|
+};
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is the hyperbolic cosine of the value in radians of this
|
|
|
+ * Decimal.
|
|
|
+ *
|
|
|
+ * Domain: [-Infinity, Infinity]
|
|
|
+ * Range: [1, Infinity]
|
|
|
+ *
|
|
|
+ * cosh(x) = 1 + x^2/2! + x^4/4! + x^6/6! + ...
|
|
|
+ *
|
|
|
+ * cosh(0) = 1
|
|
|
+ * cosh(-0) = 1
|
|
|
+ * cosh(Infinity) = Infinity
|
|
|
+ * cosh(-Infinity) = Infinity
|
|
|
+ * cosh(NaN) = NaN
|
|
|
+ *
|
|
|
+ * x time taken (ms) result
|
|
|
+ * 1000 9 9.8503555700852349694e+433
|
|
|
+ * 10000 25 4.4034091128314607936e+4342
|
|
|
+ * 100000 171 1.4033316802130615897e+43429
|
|
|
+ * 1000000 3817 1.5166076984010437725e+434294
|
|
|
+ * 10000000 abandoned after 2 minute wait
|
|
|
+ *
|
|
|
+ * TODO? Compare performance of cosh(x) = 0.5 * (exp(x) + exp(-x))
|
|
|
+ *
|
|
|
+ */
|
|
|
+P.hyperbolicCosine = P.cosh = function() {
|
|
|
+ var k, n, pr, rm, len,
|
|
|
+ x = this,
|
|
|
+ Ctor = x.constructor,
|
|
|
+ one = new Ctor(1);
|
|
|
+
|
|
|
+ if (!x.isFinite()) return new Ctor(x.s ? 1 / 0 : NaN);
|
|
|
+ if (x.isZero()) return one;
|
|
|
+
|
|
|
+ pr = Ctor.precision;
|
|
|
+ rm = Ctor.rounding;
|
|
|
+ Ctor.precision = pr + Math.max(x.e, x.sd()) + 4;
|
|
|
+ Ctor.rounding = 1;
|
|
|
+ len = x.d.length;
|
|
|
+
|
|
|
+ // Argument reduction: cos(4x) = 1 - 8cos^2(x) + 8cos^4(x) + 1
|
|
|
+ // i.e. cos(x) = 1 - cos^2(x/4)(8 - 8cos^2(x/4))
|
|
|
+
|
|
|
+ // Estimate the optimum number of times to use the argument reduction.
|
|
|
+ // TODO? Estimation reused from cosine() and may not be optimal here.
|
|
|
+ if (len < 32) {
|
|
|
+ k = Math.ceil(len / 3);
|
|
|
+ n = (1 / tinyPow(4, k)).toString();
|
|
|
+ } else {
|
|
|
+ k = 16;
|
|
|
+ n = '2.3283064365386962890625e-10';
|
|
|
+ }
|
|
|
+
|
|
|
+ x = taylorSeries(Ctor, 1, x.times(n), new Ctor(1), true);
|
|
|
+
|
|
|
+ // Reverse argument reduction
|
|
|
+ var cosh2_x,
|
|
|
+ i = k,
|
|
|
+ d8 = new Ctor(8);
|
|
|
+ for (; i--;) {
|
|
|
+ cosh2_x = x.times(x);
|
|
|
+ x = one.minus(cosh2_x.times(d8.minus(cosh2_x.times(d8))));
|
|
|
+ }
|
|
|
+
|
|
|
+ return finalise(x, Ctor.precision = pr, Ctor.rounding = rm, true);
|
|
|
+};
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is the hyperbolic sine of the value in radians of this
|
|
|
+ * Decimal.
|
|
|
+ *
|
|
|
+ * Domain: [-Infinity, Infinity]
|
|
|
+ * Range: [-Infinity, Infinity]
|
|
|
+ *
|
|
|
+ * sinh(x) = x + x^3/3! + x^5/5! + x^7/7! + ...
|
|
|
+ *
|
|
|
+ * sinh(0) = 0
|
|
|
+ * sinh(-0) = -0
|
|
|
+ * sinh(Infinity) = Infinity
|
|
|
+ * sinh(-Infinity) = -Infinity
|
|
|
+ * sinh(NaN) = NaN
|
|
|
+ *
|
|
|
+ * x time taken (ms)
|
|
|
+ * 10 2 ms
|
|
|
+ * 100 5 ms
|
|
|
+ * 1000 14 ms
|
|
|
+ * 10000 82 ms
|
|
|
+ * 100000 886 ms 1.4033316802130615897e+43429
|
|
|
+ * 200000 2613 ms
|
|
|
+ * 300000 5407 ms
|
|
|
+ * 400000 8824 ms
|
|
|
+ * 500000 13026 ms 8.7080643612718084129e+217146
|
|
|
+ * 1000000 48543 ms
|
|
|
+ *
|
|
|
+ * TODO? Compare performance of sinh(x) = 0.5 * (exp(x) - exp(-x))
|
|
|
+ *
|
|
|
+ */
|
|
|
+P.hyperbolicSine = P.sinh = function() {
|
|
|
+ var k, pr, rm, len,
|
|
|
+ x = this,
|
|
|
+ Ctor = x.constructor;
|
|
|
+
|
|
|
+ if (!x.isFinite() || x.isZero()) return new Ctor(x);
|
|
|
+
|
|
|
+ pr = Ctor.precision;
|
|
|
+ rm = Ctor.rounding;
|
|
|
+ Ctor.precision = pr + Math.max(x.e, x.sd()) + 4;
|
|
|
+ Ctor.rounding = 1;
|
|
|
+ len = x.d.length;
|
|
|
+
|
|
|
+ if (len < 3) {
|
|
|
+ x = taylorSeries(Ctor, 2, x, x, true);
|
|
|
+ } else {
|
|
|
+
|
|
|
+ // Alternative argument reduction: sinh(3x) = sinh(x)(3 + 4sinh^2(x))
|
|
|
+ // i.e. sinh(x) = sinh(x/3)(3 + 4sinh^2(x/3))
|
|
|
+ // 3 multiplications and 1 addition
|
|
|
+
|
|
|
+ // Argument reduction: sinh(5x) = sinh(x)(5 + sinh^2(x)(20 + 16sinh^2(x)))
|
|
|
+ // i.e. sinh(x) = sinh(x/5)(5 + sinh^2(x/5)(20 + 16sinh^2(x/5)))
|
|
|
+ // 4 multiplications and 2 additions
|
|
|
+
|
|
|
+ // Estimate the optimum number of times to use the argument reduction.
|
|
|
+ k = 1.4 * Math.sqrt(len);
|
|
|
+ k = k > 16 ? 16 : k | 0;
|
|
|
+
|
|
|
+ x = x.times(1 / tinyPow(5, k));
|
|
|
+ x = taylorSeries(Ctor, 2, x, x, true);
|
|
|
+
|
|
|
+ // Reverse argument reduction
|
|
|
+ var sinh2_x,
|
|
|
+ d5 = new Ctor(5),
|
|
|
+ d16 = new Ctor(16),
|
|
|
+ d20 = new Ctor(20);
|
|
|
+ for (; k--;) {
|
|
|
+ sinh2_x = x.times(x);
|
|
|
+ x = x.times(d5.plus(sinh2_x.times(d16.times(sinh2_x).plus(d20))));
|
|
|
+ }
|
|
|
+ }
|
|
|
+
|
|
|
+ Ctor.precision = pr;
|
|
|
+ Ctor.rounding = rm;
|
|
|
+
|
|
|
+ return finalise(x, pr, rm, true);
|
|
|
+};
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is the hyperbolic tangent of the value in radians of this
|
|
|
+ * Decimal.
|
|
|
+ *
|
|
|
+ * Domain: [-Infinity, Infinity]
|
|
|
+ * Range: [-1, 1]
|
|
|
+ *
|
|
|
+ * tanh(x) = sinh(x) / cosh(x)
|
|
|
+ *
|
|
|
+ * tanh(0) = 0
|
|
|
+ * tanh(-0) = -0
|
|
|
+ * tanh(Infinity) = 1
|
|
|
+ * tanh(-Infinity) = -1
|
|
|
+ * tanh(NaN) = NaN
|
|
|
+ *
|
|
|
+ */
|
|
|
+P.hyperbolicTangent = P.tanh = function() {
|
|
|
+ var pr, rm,
|
|
|
+ x = this,
|
|
|
+ Ctor = x.constructor;
|
|
|
+
|
|
|
+ if (!x.isFinite()) return new Ctor(x.s);
|
|
|
+ if (x.isZero()) return new Ctor(x);
|
|
|
+
|
|
|
+ pr = Ctor.precision;
|
|
|
+ rm = Ctor.rounding;
|
|
|
+ Ctor.precision = pr + 7;
|
|
|
+ Ctor.rounding = 1;
|
|
|
+
|
|
|
+ return divide(x.sinh(), x.cosh(), Ctor.precision = pr, Ctor.rounding = rm);
|
|
|
+};
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is the arccosine (inverse cosine) in radians of the value of
|
|
|
+ * this Decimal.
|
|
|
+ *
|
|
|
+ * Domain: [-1, 1]
|
|
|
+ * Range: [0, pi]
|
|
|
+ *
|
|
|
+ * acos(x) = pi/2 - asin(x)
|
|
|
+ *
|
|
|
+ * acos(0) = pi/2
|
|
|
+ * acos(-0) = pi/2
|
|
|
+ * acos(1) = 0
|
|
|
+ * acos(-1) = pi
|
|
|
+ * acos(1/2) = pi/3
|
|
|
+ * acos(-1/2) = 2*pi/3
|
|
|
+ * acos(|x| > 1) = NaN
|
|
|
+ * acos(NaN) = NaN
|
|
|
+ *
|
|
|
+ */
|
|
|
+P.inverseCosine = P.acos = function() {
|
|
|
+ var halfPi,
|
|
|
+ x = this,
|
|
|
+ Ctor = x.constructor,
|
|
|
+ k = x.abs().cmp(1),
|
|
|
+ pr = Ctor.precision,
|
|
|
+ rm = Ctor.rounding;
|
|
|
+
|
|
|
+ if (k !== -1) {
|
|
|
+ return k === 0
|
|
|
+ // |x| is 1
|
|
|
+ ?
|
|
|
+ x.isNeg() ? getPi(Ctor, pr, rm) : new Ctor(0)
|
|
|
+ // |x| > 1 or x is NaN
|
|
|
+ :
|
|
|
+ new Ctor(NaN);
|
|
|
+ }
|
|
|
+
|
|
|
+ if (x.isZero()) return getPi(Ctor, pr + 4, rm).times(0.5);
|
|
|
+
|
|
|
+ // TODO? Special case acos(0.5) = pi/3 and acos(-0.5) = 2*pi/3
|
|
|
+
|
|
|
+ Ctor.precision = pr + 6;
|
|
|
+ Ctor.rounding = 1;
|
|
|
+
|
|
|
+ x = x.asin();
|
|
|
+ halfPi = getPi(Ctor, pr + 4, rm).times(0.5);
|
|
|
+
|
|
|
+ Ctor.precision = pr;
|
|
|
+ Ctor.rounding = rm;
|
|
|
+
|
|
|
+ return halfPi.minus(x);
|
|
|
+};
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is the inverse of the hyperbolic cosine in radians of the
|
|
|
+ * value of this Decimal.
|
|
|
+ *
|
|
|
+ * Domain: [1, Infinity]
|
|
|
+ * Range: [0, Infinity]
|
|
|
+ *
|
|
|
+ * acosh(x) = ln(x + sqrt(x^2 - 1))
|
|
|
+ *
|
|
|
+ * acosh(x < 1) = NaN
|
|
|
+ * acosh(NaN) = NaN
|
|
|
+ * acosh(Infinity) = Infinity
|
|
|
+ * acosh(-Infinity) = NaN
|
|
|
+ * acosh(0) = NaN
|
|
|
+ * acosh(-0) = NaN
|
|
|
+ * acosh(1) = 0
|
|
|
+ * acosh(-1) = NaN
|
|
|
+ *
|
|
|
+ */
|
|
|
+P.inverseHyperbolicCosine = P.acosh = function() {
|
|
|
+ var pr, rm,
|
|
|
+ x = this,
|
|
|
+ Ctor = x.constructor;
|
|
|
+
|
|
|
+ if (x.lte(1)) return new Ctor(x.eq(1) ? 0 : NaN);
|
|
|
+ if (!x.isFinite()) return new Ctor(x);
|
|
|
+
|
|
|
+ pr = Ctor.precision;
|
|
|
+ rm = Ctor.rounding;
|
|
|
+ Ctor.precision = pr + Math.max(Math.abs(x.e), x.sd()) + 4;
|
|
|
+ Ctor.rounding = 1;
|
|
|
+ external = false;
|
|
|
+
|
|
|
+ x = x.times(x).minus(1).sqrt().plus(x);
|
|
|
+
|
|
|
+ external = true;
|
|
|
+ Ctor.precision = pr;
|
|
|
+ Ctor.rounding = rm;
|
|
|
+
|
|
|
+ return x.ln();
|
|
|
+};
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is the inverse of the hyperbolic sine in radians of the value
|
|
|
+ * of this Decimal.
|
|
|
+ *
|
|
|
+ * Domain: [-Infinity, Infinity]
|
|
|
+ * Range: [-Infinity, Infinity]
|
|
|
+ *
|
|
|
+ * asinh(x) = ln(x + sqrt(x^2 + 1))
|
|
|
+ *
|
|
|
+ * asinh(NaN) = NaN
|
|
|
+ * asinh(Infinity) = Infinity
|
|
|
+ * asinh(-Infinity) = -Infinity
|
|
|
+ * asinh(0) = 0
|
|
|
+ * asinh(-0) = -0
|
|
|
+ *
|
|
|
+ */
|
|
|
+P.inverseHyperbolicSine = P.asinh = function() {
|
|
|
+ var pr, rm,
|
|
|
+ x = this,
|
|
|
+ Ctor = x.constructor;
|
|
|
+
|
|
|
+ if (!x.isFinite() || x.isZero()) return new Ctor(x);
|
|
|
+
|
|
|
+ pr = Ctor.precision;
|
|
|
+ rm = Ctor.rounding;
|
|
|
+ Ctor.precision = pr + 2 * Math.max(Math.abs(x.e), x.sd()) + 6;
|
|
|
+ Ctor.rounding = 1;
|
|
|
+ external = false;
|
|
|
+
|
|
|
+ x = x.times(x).plus(1).sqrt().plus(x);
|
|
|
+
|
|
|
+ external = true;
|
|
|
+ Ctor.precision = pr;
|
|
|
+ Ctor.rounding = rm;
|
|
|
+
|
|
|
+ return x.ln();
|
|
|
+};
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is the inverse of the hyperbolic tangent in radians of the
|
|
|
+ * value of this Decimal.
|
|
|
+ *
|
|
|
+ * Domain: [-1, 1]
|
|
|
+ * Range: [-Infinity, Infinity]
|
|
|
+ *
|
|
|
+ * atanh(x) = 0.5 * ln((1 + x) / (1 - x))
|
|
|
+ *
|
|
|
+ * atanh(|x| > 1) = NaN
|
|
|
+ * atanh(NaN) = NaN
|
|
|
+ * atanh(Infinity) = NaN
|
|
|
+ * atanh(-Infinity) = NaN
|
|
|
+ * atanh(0) = 0
|
|
|
+ * atanh(-0) = -0
|
|
|
+ * atanh(1) = Infinity
|
|
|
+ * atanh(-1) = -Infinity
|
|
|
+ *
|
|
|
+ */
|
|
|
+P.inverseHyperbolicTangent = P.atanh = function() {
|
|
|
+ var pr, rm, wpr, xsd,
|
|
|
+ x = this,
|
|
|
+ Ctor = x.constructor;
|
|
|
+
|
|
|
+ if (!x.isFinite()) return new Ctor(NaN);
|
|
|
+ if (x.e >= 0) return new Ctor(x.abs().eq(1) ? x.s / 0 : x.isZero() ? x : NaN);
|
|
|
+
|
|
|
+ pr = Ctor.precision;
|
|
|
+ rm = Ctor.rounding;
|
|
|
+ xsd = x.sd();
|
|
|
+
|
|
|
+ if (Math.max(xsd, pr) < 2 * -x.e - 1) return finalise(new Ctor(x), pr, rm, true);
|
|
|
+
|
|
|
+ Ctor.precision = wpr = xsd - x.e;
|
|
|
+
|
|
|
+ x = divide(x.plus(1), new Ctor(1).minus(x), wpr + pr, 1);
|
|
|
+
|
|
|
+ Ctor.precision = pr + 4;
|
|
|
+ Ctor.rounding = 1;
|
|
|
+
|
|
|
+ x = x.ln();
|
|
|
+
|
|
|
+ Ctor.precision = pr;
|
|
|
+ Ctor.rounding = rm;
|
|
|
+
|
|
|
+ return x.times(0.5);
|
|
|
+};
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is the arcsine (inverse sine) in radians of the value of this
|
|
|
+ * Decimal.
|
|
|
+ *
|
|
|
+ * Domain: [-Infinity, Infinity]
|
|
|
+ * Range: [-pi/2, pi/2]
|
|
|
+ *
|
|
|
+ * asin(x) = 2*atan(x/(1 + sqrt(1 - x^2)))
|
|
|
+ *
|
|
|
+ * asin(0) = 0
|
|
|
+ * asin(-0) = -0
|
|
|
+ * asin(1/2) = pi/6
|
|
|
+ * asin(-1/2) = -pi/6
|
|
|
+ * asin(1) = pi/2
|
|
|
+ * asin(-1) = -pi/2
|
|
|
+ * asin(|x| > 1) = NaN
|
|
|
+ * asin(NaN) = NaN
|
|
|
+ *
|
|
|
+ * TODO? Compare performance of Taylor series.
|
|
|
+ *
|
|
|
+ */
|
|
|
+P.inverseSine = P.asin = function() {
|
|
|
+ var halfPi, k,
|
|
|
+ pr, rm,
|
|
|
+ x = this,
|
|
|
+ Ctor = x.constructor;
|
|
|
+
|
|
|
+ if (x.isZero()) return new Ctor(x);
|
|
|
+
|
|
|
+ k = x.abs().cmp(1);
|
|
|
+ pr = Ctor.precision;
|
|
|
+ rm = Ctor.rounding;
|
|
|
+
|
|
|
+ if (k !== -1) {
|
|
|
+
|
|
|
+ // |x| is 1
|
|
|
+ if (k === 0) {
|
|
|
+ halfPi = getPi(Ctor, pr + 4, rm).times(0.5);
|
|
|
+ halfPi.s = x.s;
|
|
|
+ return halfPi;
|
|
|
+ }
|
|
|
+
|
|
|
+ // |x| > 1 or x is NaN
|
|
|
+ return new Ctor(NaN);
|
|
|
+ }
|
|
|
+
|
|
|
+ // TODO? Special case asin(1/2) = pi/6 and asin(-1/2) = -pi/6
|
|
|
+
|
|
|
+ Ctor.precision = pr + 6;
|
|
|
+ Ctor.rounding = 1;
|
|
|
+
|
|
|
+ x = x.div(new Ctor(1).minus(x.times(x)).sqrt().plus(1)).atan();
|
|
|
+
|
|
|
+ Ctor.precision = pr;
|
|
|
+ Ctor.rounding = rm;
|
|
|
+
|
|
|
+ return x.times(2);
|
|
|
+};
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is the arctangent (inverse tangent) in radians of the value
|
|
|
+ * of this Decimal.
|
|
|
+ *
|
|
|
+ * Domain: [-Infinity, Infinity]
|
|
|
+ * Range: [-pi/2, pi/2]
|
|
|
+ *
|
|
|
+ * atan(x) = x - x^3/3 + x^5/5 - x^7/7 + ...
|
|
|
+ *
|
|
|
+ * atan(0) = 0
|
|
|
+ * atan(-0) = -0
|
|
|
+ * atan(1) = pi/4
|
|
|
+ * atan(-1) = -pi/4
|
|
|
+ * atan(Infinity) = pi/2
|
|
|
+ * atan(-Infinity) = -pi/2
|
|
|
+ * atan(NaN) = NaN
|
|
|
+ *
|
|
|
+ */
|
|
|
+P.inverseTangent = P.atan = function() {
|
|
|
+ var i, j, k, n, px, t, r, wpr, x2,
|
|
|
+ x = this,
|
|
|
+ Ctor = x.constructor,
|
|
|
+ pr = Ctor.precision,
|
|
|
+ rm = Ctor.rounding;
|
|
|
+
|
|
|
+ if (!x.isFinite()) {
|
|
|
+ if (!x.s) return new Ctor(NaN);
|
|
|
+ if (pr + 4 <= PI_PRECISION) {
|
|
|
+ r = getPi(Ctor, pr + 4, rm).times(0.5);
|
|
|
+ r.s = x.s;
|
|
|
+ return r;
|
|
|
+ }
|
|
|
+ } else if (x.isZero()) {
|
|
|
+ return new Ctor(x);
|
|
|
+ } else if (x.abs().eq(1) && pr + 4 <= PI_PRECISION) {
|
|
|
+ r = getPi(Ctor, pr + 4, rm).times(0.25);
|
|
|
+ r.s = x.s;
|
|
|
+ return r;
|
|
|
+ }
|
|
|
+
|
|
|
+ Ctor.precision = wpr = pr + 10;
|
|
|
+ Ctor.rounding = 1;
|
|
|
+
|
|
|
+ // TODO? if (x >= 1 && pr <= PI_PRECISION) atan(x) = halfPi * x.s - atan(1 / x);
|
|
|
+
|
|
|
+ // Argument reduction
|
|
|
+ // Ensure |x| < 0.42
|
|
|
+ // atan(x) = 2 * atan(x / (1 + sqrt(1 + x^2)))
|
|
|
+
|
|
|
+ k = Math.min(28, wpr / LOG_BASE + 2 | 0);
|
|
|
+
|
|
|
+ for (i = k; i; --i) x = x.div(x.times(x).plus(1).sqrt().plus(1));
|
|
|
+
|
|
|
+ external = false;
|
|
|
+
|
|
|
+ j = Math.ceil(wpr / LOG_BASE);
|
|
|
+ n = 1;
|
|
|
+ x2 = x.times(x);
|
|
|
+ r = new Ctor(x);
|
|
|
+ px = x;
|
|
|
+
|
|
|
+ // atan(x) = x - x^3/3 + x^5/5 - x^7/7 + ...
|
|
|
+ for (; i !== -1;) {
|
|
|
+ px = px.times(x2);
|
|
|
+ t = r.minus(px.div(n += 2));
|
|
|
+
|
|
|
+ px = px.times(x2);
|
|
|
+ r = t.plus(px.div(n += 2));
|
|
|
+
|
|
|
+ if (r.d[j] !== void 0)
|
|
|
+ for (i = j; r.d[i] === t.d[i] && i--;);
|
|
|
+ }
|
|
|
+
|
|
|
+ if (k) r = r.times(2 << (k - 1));
|
|
|
+
|
|
|
+ external = true;
|
|
|
+
|
|
|
+ return finalise(r, Ctor.precision = pr, Ctor.rounding = rm, true);
|
|
|
+};
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return true if the value of this Decimal is a finite number, otherwise return false.
|
|
|
+ *
|
|
|
+ */
|
|
|
+P.isFinite = function() {
|
|
|
+ return !!this.d;
|
|
|
+};
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return true if the value of this Decimal is an integer, otherwise return false.
|
|
|
+ *
|
|
|
+ */
|
|
|
+P.isInteger = P.isInt = function() {
|
|
|
+ return !!this.d && mathfloor(this.e / LOG_BASE) > this.d.length - 2;
|
|
|
+};
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return true if the value of this Decimal is NaN, otherwise return false.
|
|
|
+ *
|
|
|
+ */
|
|
|
+P.isNaN = function() {
|
|
|
+ return !this.s;
|
|
|
+};
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return true if the value of this Decimal is negative, otherwise return false.
|
|
|
+ *
|
|
|
+ */
|
|
|
+P.isNegative = P.isNeg = function() {
|
|
|
+ return this.s < 0;
|
|
|
+};
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return true if the value of this Decimal is positive, otherwise return false.
|
|
|
+ *
|
|
|
+ */
|
|
|
+P.isPositive = P.isPos = function() {
|
|
|
+ return this.s > 0;
|
|
|
+};
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return true if the value of this Decimal is 0 or -0, otherwise return false.
|
|
|
+ *
|
|
|
+ */
|
|
|
+P.isZero = function() {
|
|
|
+ return !!this.d && this.d[0] === 0;
|
|
|
+};
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return true if the value of this Decimal is less than `y`, otherwise return false.
|
|
|
+ *
|
|
|
+ */
|
|
|
+P.lessThan = P.lt = function(y) {
|
|
|
+ return this.cmp(y) < 0;
|
|
|
+};
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return true if the value of this Decimal is less than or equal to `y`, otherwise return false.
|
|
|
+ *
|
|
|
+ */
|
|
|
+P.lessThanOrEqualTo = P.lte = function(y) {
|
|
|
+ return this.cmp(y) < 1;
|
|
|
+};
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return the logarithm of the value of this Decimal to the specified base, rounded to `precision`
|
|
|
+ * significant digits using rounding mode `rounding`.
|
|
|
+ *
|
|
|
+ * If no base is specified, return log[10](arg).
|
|
|
+ *
|
|
|
+ * log[base](arg) = ln(arg) / ln(base)
|
|
|
+ *
|
|
|
+ * The result will always be correctly rounded if the base of the log is 10, and 'almost always'
|
|
|
+ * otherwise:
|
|
|
+ *
|
|
|
+ * Depending on the rounding mode, the result may be incorrectly rounded if the first fifteen
|
|
|
+ * rounding digits are [49]99999999999999 or [50]00000000000000. In that case, the maximum error
|
|
|
+ * between the result and the correctly rounded result will be one ulp (unit in the last place).
|
|
|
+ *
|
|
|
+ * log[-b](a) = NaN
|
|
|
+ * log[0](a) = NaN
|
|
|
+ * log[1](a) = NaN
|
|
|
+ * log[NaN](a) = NaN
|
|
|
+ * log[Infinity](a) = NaN
|
|
|
+ * log[b](0) = -Infinity
|
|
|
+ * log[b](-0) = -Infinity
|
|
|
+ * log[b](-a) = NaN
|
|
|
+ * log[b](1) = 0
|
|
|
+ * log[b](Infinity) = Infinity
|
|
|
+ * log[b](NaN) = NaN
|
|
|
+ *
|
|
|
+ * [base] {number|string|Decimal} The base of the logarithm.
|
|
|
+ *
|
|
|
+ */
|
|
|
+P.logarithm = P.log = function(base) {
|
|
|
+ var isBase10, d, denominator, k, inf, num, sd, r,
|
|
|
+ arg = this,
|
|
|
+ Ctor = arg.constructor,
|
|
|
+ pr = Ctor.precision,
|
|
|
+ rm = Ctor.rounding,
|
|
|
+ guard = 5;
|
|
|
+
|
|
|
+ // Default base is 10.
|
|
|
+ if (base == null) {
|
|
|
+ base = new Ctor(10);
|
|
|
+ isBase10 = true;
|
|
|
+ } else {
|
|
|
+ base = new Ctor(base);
|
|
|
+ d = base.d;
|
|
|
+
|
|
|
+ // Return NaN if base is negative, or non-finite, or is 0 or 1.
|
|
|
+ if (base.s < 0 || !d || !d[0] || base.eq(1)) return new Ctor(NaN);
|
|
|
+
|
|
|
+ isBase10 = base.eq(10);
|
|
|
+ }
|
|
|
+
|
|
|
+ d = arg.d;
|
|
|
+
|
|
|
+ // Is arg negative, non-finite, 0 or 1?
|
|
|
+ if (arg.s < 0 || !d || !d[0] || arg.eq(1)) {
|
|
|
+ return new Ctor(d && !d[0] ? -1 / 0 : arg.s != 1 ? NaN : d ? 0 : 1 / 0);
|
|
|
+ }
|
|
|
+
|
|
|
+ // The result will have a non-terminating decimal expansion if base is 10 and arg is not an
|
|
|
+ // integer power of 10.
|
|
|
+ if (isBase10) {
|
|
|
+ if (d.length > 1) {
|
|
|
+ inf = true;
|
|
|
+ } else {
|
|
|
+ for (k = d[0]; k % 10 === 0;) k /= 10;
|
|
|
+ inf = k !== 1;
|
|
|
+ }
|
|
|
+ }
|
|
|
+
|
|
|
+ external = false;
|
|
|
+ sd = pr + guard;
|
|
|
+ num = naturalLogarithm(arg, sd);
|
|
|
+ denominator = isBase10 ? getLn10(Ctor, sd + 10) : naturalLogarithm(base, sd);
|
|
|
+
|
|
|
+ // The result will have 5 rounding digits.
|
|
|
+ r = divide(num, denominator, sd, 1);
|
|
|
+
|
|
|
+ // If at a rounding boundary, i.e. the result's rounding digits are [49]9999 or [50]0000,
|
|
|
+ // calculate 10 further digits.
|
|
|
+ //
|
|
|
+ // If the result is known to have an infinite decimal expansion, repeat this until it is clear
|
|
|
+ // that the result is above or below the boundary. Otherwise, if after calculating the 10
|
|
|
+ // further digits, the last 14 are nines, round up and assume the result is exact.
|
|
|
+ // Also assume the result is exact if the last 14 are zero.
|
|
|
+ //
|
|
|
+ // Example of a result that will be incorrectly rounded:
|
|
|
+ // log[1048576](4503599627370502) = 2.60000000000000009610279511444746...
|
|
|
+ // The above result correctly rounded using ROUND_CEIL to 1 decimal place should be 2.7, but it
|
|
|
+ // will be given as 2.6 as there are 15 zeros immediately after the requested decimal place, so
|
|
|
+ // the exact result would be assumed to be 2.6, which rounded using ROUND_CEIL to 1 decimal
|
|
|
+ // place is still 2.6.
|
|
|
+ if (checkRoundingDigits(r.d, k = pr, rm)) {
|
|
|
+
|
|
|
+ do {
|
|
|
+ sd += 10;
|
|
|
+ num = naturalLogarithm(arg, sd);
|
|
|
+ denominator = isBase10 ? getLn10(Ctor, sd + 10) : naturalLogarithm(base, sd);
|
|
|
+ r = divide(num, denominator, sd, 1);
|
|
|
+
|
|
|
+ if (!inf) {
|
|
|
+
|
|
|
+ // Check for 14 nines from the 2nd rounding digit, as the first may be 4.
|
|
|
+ if (+digitsToString(r.d).slice(k + 1, k + 15) + 1 == 1e14) {
|
|
|
+ r = finalise(r, pr + 1, 0);
|
|
|
+ }
|
|
|
+
|
|
|
+ break;
|
|
|
+ }
|
|
|
+ } while (checkRoundingDigits(r.d, k += 10, rm));
|
|
|
+ }
|
|
|
+
|
|
|
+ external = true;
|
|
|
+
|
|
|
+ return finalise(r, pr, rm);
|
|
|
+};
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is the maximum of the arguments and the value of this Decimal.
|
|
|
+ *
|
|
|
+ * arguments {number|string|Decimal}
|
|
|
+ *
|
|
|
+P.max = function () {
|
|
|
+ Array.prototype.push.call(arguments, this);
|
|
|
+ return maxOrMin(this.constructor, arguments, 'lt');
|
|
|
+};
|
|
|
+ */
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is the minimum of the arguments and the value of this Decimal.
|
|
|
+ *
|
|
|
+ * arguments {number|string|Decimal}
|
|
|
+ *
|
|
|
+P.min = function () {
|
|
|
+ Array.prototype.push.call(arguments, this);
|
|
|
+ return maxOrMin(this.constructor, arguments, 'gt');
|
|
|
+};
|
|
|
+ */
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * n - 0 = n
|
|
|
+ * n - N = N
|
|
|
+ * n - I = -I
|
|
|
+ * 0 - n = -n
|
|
|
+ * 0 - 0 = 0
|
|
|
+ * 0 - N = N
|
|
|
+ * 0 - I = -I
|
|
|
+ * N - n = N
|
|
|
+ * N - 0 = N
|
|
|
+ * N - N = N
|
|
|
+ * N - I = N
|
|
|
+ * I - n = I
|
|
|
+ * I - 0 = I
|
|
|
+ * I - N = N
|
|
|
+ * I - I = N
|
|
|
+ *
|
|
|
+ * Return a new Decimal whose value is the value of this Decimal minus `y`, rounded to `precision`
|
|
|
+ * significant digits using rounding mode `rounding`.
|
|
|
+ *
|
|
|
+ */
|
|
|
+P.minus = P.sub = function(y) {
|
|
|
+ var d, e, i, j, k, len, pr, rm, xd, xe, xLTy, yd,
|
|
|
+ x = this,
|
|
|
+ Ctor = x.constructor;
|
|
|
+
|
|
|
+ y = new Ctor(y);
|
|
|
+
|
|
|
+ // If either is not finite...
|
|
|
+ if (!x.d || !y.d) {
|
|
|
+
|
|
|
+ // Return NaN if either is NaN.
|
|
|
+ if (!x.s || !y.s) y = new Ctor(NaN);
|
|
|
+
|
|
|
+ // Return y negated if x is finite and y is ±Infinity.
|
|
|
+ else if (x.d) y.s = -y.s;
|
|
|
+
|
|
|
+ // Return x if y is finite and x is ±Infinity.
|
|
|
+ // Return x if both are ±Infinity with different signs.
|
|
|
+ // Return NaN if both are ±Infinity with the same sign.
|
|
|
+ else y = new Ctor(y.d || x.s !== y.s ? x : NaN);
|
|
|
+
|
|
|
+ return y;
|
|
|
+ }
|
|
|
+
|
|
|
+ // If signs differ...
|
|
|
+ if (x.s != y.s) {
|
|
|
+ y.s = -y.s;
|
|
|
+ return x.plus(y);
|
|
|
+ }
|
|
|
+
|
|
|
+ xd = x.d;
|
|
|
+ yd = y.d;
|
|
|
+ pr = Ctor.precision;
|
|
|
+ rm = Ctor.rounding;
|
|
|
+
|
|
|
+ // If either is zero...
|
|
|
+ if (!xd[0] || !yd[0]) {
|
|
|
+
|
|
|
+ // Return y negated if x is zero and y is non-zero.
|
|
|
+ if (yd[0]) y.s = -y.s;
|
|
|
+
|
|
|
+ // Return x if y is zero and x is non-zero.
|
|
|
+ else if (xd[0]) y = new Ctor(x);
|
|
|
+
|
|
|
+ // Return zero if both are zero.
|
|
|
+ // From IEEE 754 (2008) 6.3: 0 - 0 = -0 - -0 = -0 when rounding to -Infinity.
|
|
|
+ else return new Ctor(rm === 3 ? -0 : 0);
|
|
|
+
|
|
|
+ return external ? finalise(y, pr, rm) : y;
|
|
|
+ }
|
|
|
+
|
|
|
+ // x and y are finite, non-zero numbers with the same sign.
|
|
|
+
|
|
|
+ // Calculate base 1e7 exponents.
|
|
|
+ e = mathfloor(y.e / LOG_BASE);
|
|
|
+ xe = mathfloor(x.e / LOG_BASE);
|
|
|
+
|
|
|
+ xd = xd.slice();
|
|
|
+ k = xe - e;
|
|
|
+
|
|
|
+ // If base 1e7 exponents differ...
|
|
|
+ if (k) {
|
|
|
+ xLTy = k < 0;
|
|
|
+
|
|
|
+ if (xLTy) {
|
|
|
+ d = xd;
|
|
|
+ k = -k;
|
|
|
+ len = yd.length;
|
|
|
+ } else {
|
|
|
+ d = yd;
|
|
|
+ e = xe;
|
|
|
+ len = xd.length;
|
|
|
+ }
|
|
|
+
|
|
|
+ // Numbers with massively different exponents would result in a very high number of
|
|
|
+ // zeros needing to be prepended, but this can be avoided while still ensuring correct
|
|
|
+ // rounding by limiting the number of zeros to `Math.ceil(pr / LOG_BASE) + 2`.
|
|
|
+ i = Math.max(Math.ceil(pr / LOG_BASE), len) + 2;
|
|
|
+
|
|
|
+ if (k > i) {
|
|
|
+ k = i;
|
|
|
+ d.length = 1;
|
|
|
+ }
|
|
|
+
|
|
|
+ // Prepend zeros to equalise exponents.
|
|
|
+ d.reverse();
|
|
|
+ for (i = k; i--;) d.push(0);
|
|
|
+ d.reverse();
|
|
|
+
|
|
|
+ // Base 1e7 exponents equal.
|
|
|
+ } else {
|
|
|
+
|
|
|
+ // Check digits to determine which is the bigger number.
|
|
|
+
|
|
|
+ i = xd.length;
|
|
|
+ len = yd.length;
|
|
|
+ xLTy = i < len;
|
|
|
+ if (xLTy) len = i;
|
|
|
+
|
|
|
+ for (i = 0; i < len; i++) {
|
|
|
+ if (xd[i] != yd[i]) {
|
|
|
+ xLTy = xd[i] < yd[i];
|
|
|
+ break;
|
|
|
+ }
|
|
|
+ }
|
|
|
+
|
|
|
+ k = 0;
|
|
|
+ }
|
|
|
+
|
|
|
+ if (xLTy) {
|
|
|
+ d = xd;
|
|
|
+ xd = yd;
|
|
|
+ yd = d;
|
|
|
+ y.s = -y.s;
|
|
|
+ }
|
|
|
+
|
|
|
+ len = xd.length;
|
|
|
+
|
|
|
+ // Append zeros to `xd` if shorter.
|
|
|
+ // Don't add zeros to `yd` if shorter as subtraction only needs to start at `yd` length.
|
|
|
+ for (i = yd.length - len; i > 0; --i) xd[len++] = 0;
|
|
|
+
|
|
|
+ // Subtract yd from xd.
|
|
|
+ for (i = yd.length; i > k;) {
|
|
|
+
|
|
|
+ if (xd[--i] < yd[i]) {
|
|
|
+ for (j = i; j && xd[--j] === 0;) xd[j] = BASE - 1;
|
|
|
+ --xd[j];
|
|
|
+ xd[i] += BASE;
|
|
|
+ }
|
|
|
+
|
|
|
+ xd[i] -= yd[i];
|
|
|
+ }
|
|
|
+
|
|
|
+ // Remove trailing zeros.
|
|
|
+ for (; xd[--len] === 0;) xd.pop();
|
|
|
+
|
|
|
+ // Remove leading zeros and adjust exponent accordingly.
|
|
|
+ for (; xd[0] === 0; xd.shift()) --e;
|
|
|
+
|
|
|
+ // Zero?
|
|
|
+ if (!xd[0]) return new Ctor(rm === 3 ? -0 : 0);
|
|
|
+
|
|
|
+ y.d = xd;
|
|
|
+ y.e = getBase10Exponent(xd, e);
|
|
|
+
|
|
|
+ return external ? finalise(y, pr, rm) : y;
|
|
|
+};
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * n % 0 = N
|
|
|
+ * n % N = N
|
|
|
+ * n % I = n
|
|
|
+ * 0 % n = 0
|
|
|
+ * -0 % n = -0
|
|
|
+ * 0 % 0 = N
|
|
|
+ * 0 % N = N
|
|
|
+ * 0 % I = 0
|
|
|
+ * N % n = N
|
|
|
+ * N % 0 = N
|
|
|
+ * N % N = N
|
|
|
+ * N % I = N
|
|
|
+ * I % n = N
|
|
|
+ * I % 0 = N
|
|
|
+ * I % N = N
|
|
|
+ * I % I = N
|
|
|
+ *
|
|
|
+ * Return a new Decimal whose value is the value of this Decimal modulo `y`, rounded to
|
|
|
+ * `precision` significant digits using rounding mode `rounding`.
|
|
|
+ *
|
|
|
+ * The result depends on the modulo mode.
|
|
|
+ *
|
|
|
+ */
|
|
|
+P.modulo = P.mod = function(y) {
|
|
|
+ var q,
|
|
|
+ x = this,
|
|
|
+ Ctor = x.constructor;
|
|
|
+
|
|
|
+ y = new Ctor(y);
|
|
|
+
|
|
|
+ // Return NaN if x is ±Infinity or NaN, or y is NaN or ±0.
|
|
|
+ if (!x.d || !y.s || y.d && !y.d[0]) return new Ctor(NaN);
|
|
|
+
|
|
|
+ // Return x if y is ±Infinity or x is ±0.
|
|
|
+ if (!y.d || x.d && !x.d[0]) {
|
|
|
+ return finalise(new Ctor(x), Ctor.precision, Ctor.rounding);
|
|
|
+ }
|
|
|
+
|
|
|
+ // Prevent rounding of intermediate calculations.
|
|
|
+ external = false;
|
|
|
+
|
|
|
+ if (Ctor.modulo == 9) {
|
|
|
+
|
|
|
+ // Euclidian division: q = sign(y) * floor(x / abs(y))
|
|
|
+ // result = x - q * y where 0 <= result < abs(y)
|
|
|
+ q = divide(x, y.abs(), 0, 3, 1);
|
|
|
+ q.s *= y.s;
|
|
|
+ } else {
|
|
|
+ q = divide(x, y, 0, Ctor.modulo, 1);
|
|
|
+ }
|
|
|
+
|
|
|
+ q = q.times(y);
|
|
|
+
|
|
|
+ external = true;
|
|
|
+
|
|
|
+ return x.minus(q);
|
|
|
+};
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is the natural exponential of the value of this Decimal,
|
|
|
+ * i.e. the base e raised to the power the value of this Decimal, rounded to `precision`
|
|
|
+ * significant digits using rounding mode `rounding`.
|
|
|
+ *
|
|
|
+ */
|
|
|
+P.naturalExponential = P.exp = function() {
|
|
|
+ return naturalExponential(this);
|
|
|
+};
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is the natural logarithm of the value of this Decimal,
|
|
|
+ * rounded to `precision` significant digits using rounding mode `rounding`.
|
|
|
+ *
|
|
|
+ */
|
|
|
+P.naturalLogarithm = P.ln = function() {
|
|
|
+ return naturalLogarithm(this);
|
|
|
+};
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is the value of this Decimal negated, i.e. as if multiplied by
|
|
|
+ * -1.
|
|
|
+ *
|
|
|
+ */
|
|
|
+P.negated = P.neg = function() {
|
|
|
+ var x = new this.constructor(this);
|
|
|
+ x.s = -x.s;
|
|
|
+ return finalise(x);
|
|
|
+};
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * n + 0 = n
|
|
|
+ * n + N = N
|
|
|
+ * n + I = I
|
|
|
+ * 0 + n = n
|
|
|
+ * 0 + 0 = 0
|
|
|
+ * 0 + N = N
|
|
|
+ * 0 + I = I
|
|
|
+ * N + n = N
|
|
|
+ * N + 0 = N
|
|
|
+ * N + N = N
|
|
|
+ * N + I = N
|
|
|
+ * I + n = I
|
|
|
+ * I + 0 = I
|
|
|
+ * I + N = N
|
|
|
+ * I + I = I
|
|
|
+ *
|
|
|
+ * Return a new Decimal whose value is the value of this Decimal plus `y`, rounded to `precision`
|
|
|
+ * significant digits using rounding mode `rounding`.
|
|
|
+ *
|
|
|
+ */
|
|
|
+P.plus = P.add = function(y) {
|
|
|
+ var carry, d, e, i, k, len, pr, rm, xd, yd,
|
|
|
+ x = this,
|
|
|
+ Ctor = x.constructor;
|
|
|
+
|
|
|
+ y = new Ctor(y);
|
|
|
+
|
|
|
+ // If either is not finite...
|
|
|
+ if (!x.d || !y.d) {
|
|
|
+
|
|
|
+ // Return NaN if either is NaN.
|
|
|
+ if (!x.s || !y.s) y = new Ctor(NaN);
|
|
|
+
|
|
|
+ // Return x if y is finite and x is ±Infinity.
|
|
|
+ // Return x if both are ±Infinity with the same sign.
|
|
|
+ // Return NaN if both are ±Infinity with different signs.
|
|
|
+ // Return y if x is finite and y is ±Infinity.
|
|
|
+ else if (!x.d) y = new Ctor(y.d || x.s === y.s ? x : NaN);
|
|
|
+
|
|
|
+ return y;
|
|
|
+ }
|
|
|
+
|
|
|
+ // If signs differ...
|
|
|
+ if (x.s != y.s) {
|
|
|
+ y.s = -y.s;
|
|
|
+ return x.minus(y);
|
|
|
+ }
|
|
|
+
|
|
|
+ xd = x.d;
|
|
|
+ yd = y.d;
|
|
|
+ pr = Ctor.precision;
|
|
|
+ rm = Ctor.rounding;
|
|
|
+
|
|
|
+ // If either is zero...
|
|
|
+ if (!xd[0] || !yd[0]) {
|
|
|
+
|
|
|
+ // Return x if y is zero.
|
|
|
+ // Return y if y is non-zero.
|
|
|
+ if (!yd[0]) y = new Ctor(x);
|
|
|
+
|
|
|
+ return external ? finalise(y, pr, rm) : y;
|
|
|
+ }
|
|
|
+
|
|
|
+ // x and y are finite, non-zero numbers with the same sign.
|
|
|
+
|
|
|
+ // Calculate base 1e7 exponents.
|
|
|
+ k = mathfloor(x.e / LOG_BASE);
|
|
|
+ e = mathfloor(y.e / LOG_BASE);
|
|
|
+
|
|
|
+ xd = xd.slice();
|
|
|
+ i = k - e;
|
|
|
+
|
|
|
+ // If base 1e7 exponents differ...
|
|
|
+ if (i) {
|
|
|
+
|
|
|
+ if (i < 0) {
|
|
|
+ d = xd;
|
|
|
+ i = -i;
|
|
|
+ len = yd.length;
|
|
|
+ } else {
|
|
|
+ d = yd;
|
|
|
+ e = k;
|
|
|
+ len = xd.length;
|
|
|
+ }
|
|
|
+
|
|
|
+ // Limit number of zeros prepended to max(ceil(pr / LOG_BASE), len) + 1.
|
|
|
+ k = Math.ceil(pr / LOG_BASE);
|
|
|
+ len = k > len ? k + 1 : len + 1;
|
|
|
+
|
|
|
+ if (i > len) {
|
|
|
+ i = len;
|
|
|
+ d.length = 1;
|
|
|
+ }
|
|
|
+
|
|
|
+ // Prepend zeros to equalise exponents. Note: Faster to use reverse then do unshifts.
|
|
|
+ d.reverse();
|
|
|
+ for (; i--;) d.push(0);
|
|
|
+ d.reverse();
|
|
|
+ }
|
|
|
+
|
|
|
+ len = xd.length;
|
|
|
+ i = yd.length;
|
|
|
+
|
|
|
+ // If yd is longer than xd, swap xd and yd so xd points to the longer array.
|
|
|
+ if (len - i < 0) {
|
|
|
+ i = len;
|
|
|
+ d = yd;
|
|
|
+ yd = xd;
|
|
|
+ xd = d;
|
|
|
+ }
|
|
|
+
|
|
|
+ // Only start adding at yd.length - 1 as the further digits of xd can be left as they are.
|
|
|
+ for (carry = 0; i;) {
|
|
|
+ carry = (xd[--i] = xd[i] + yd[i] + carry) / BASE | 0;
|
|
|
+ xd[i] %= BASE;
|
|
|
+ }
|
|
|
+
|
|
|
+ if (carry) {
|
|
|
+ xd.unshift(carry);
|
|
|
+ ++e;
|
|
|
+ }
|
|
|
+
|
|
|
+ // Remove trailing zeros.
|
|
|
+ // No need to check for zero, as +x + +y != 0 && -x + -y != 0
|
|
|
+ for (len = xd.length; xd[--len] == 0;) xd.pop();
|
|
|
+
|
|
|
+ y.d = xd;
|
|
|
+ y.e = getBase10Exponent(xd, e);
|
|
|
+
|
|
|
+ return external ? finalise(y, pr, rm) : y;
|
|
|
+};
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return the number of significant digits of the value of this Decimal.
|
|
|
+ *
|
|
|
+ * [z] {boolean|number} Whether to count integer-part trailing zeros: true, false, 1 or 0.
|
|
|
+ *
|
|
|
+ */
|
|
|
+P.precision = P.sd = function(z) {
|
|
|
+ var k,
|
|
|
+ x = this;
|
|
|
+
|
|
|
+ if (z !== void 0 && z !== !!z && z !== 1 && z !== 0) throw Error(invalidArgument + z);
|
|
|
+
|
|
|
+ if (x.d) {
|
|
|
+ k = getPrecision(x.d);
|
|
|
+ if (z && x.e + 1 > k) k = x.e + 1;
|
|
|
+ } else {
|
|
|
+ k = NaN;
|
|
|
+ }
|
|
|
+
|
|
|
+ return k;
|
|
|
+};
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is the value of this Decimal rounded to a whole number using
|
|
|
+ * rounding mode `rounding`.
|
|
|
+ *
|
|
|
+ */
|
|
|
+P.round = function() {
|
|
|
+ var x = this,
|
|
|
+ Ctor = x.constructor;
|
|
|
+
|
|
|
+ return finalise(new Ctor(x), x.e + 1, Ctor.rounding);
|
|
|
+};
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is the sine of the value in radians of this Decimal.
|
|
|
+ *
|
|
|
+ * Domain: [-Infinity, Infinity]
|
|
|
+ * Range: [-1, 1]
|
|
|
+ *
|
|
|
+ * sin(x) = x - x^3/3! + x^5/5! - ...
|
|
|
+ *
|
|
|
+ * sin(0) = 0
|
|
|
+ * sin(-0) = -0
|
|
|
+ * sin(Infinity) = NaN
|
|
|
+ * sin(-Infinity) = NaN
|
|
|
+ * sin(NaN) = NaN
|
|
|
+ *
|
|
|
+ */
|
|
|
+P.sine = P.sin = function() {
|
|
|
+ var pr, rm,
|
|
|
+ x = this,
|
|
|
+ Ctor = x.constructor;
|
|
|
+
|
|
|
+ if (!x.isFinite()) return new Ctor(NaN);
|
|
|
+ if (x.isZero()) return new Ctor(x);
|
|
|
+
|
|
|
+ pr = Ctor.precision;
|
|
|
+ rm = Ctor.rounding;
|
|
|
+ Ctor.precision = pr + Math.max(x.e, x.sd()) + LOG_BASE;
|
|
|
+ Ctor.rounding = 1;
|
|
|
+
|
|
|
+ x = sine(Ctor, toLessThanHalfPi(Ctor, x));
|
|
|
+
|
|
|
+ Ctor.precision = pr;
|
|
|
+ Ctor.rounding = rm;
|
|
|
+
|
|
|
+ return finalise(quadrant > 2 ? x.neg() : x, pr, rm, true);
|
|
|
+};
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is the square root of this Decimal, rounded to `precision`
|
|
|
+ * significant digits using rounding mode `rounding`.
|
|
|
+ *
|
|
|
+ * sqrt(-n) = N
|
|
|
+ * sqrt(N) = N
|
|
|
+ * sqrt(-I) = N
|
|
|
+ * sqrt(I) = I
|
|
|
+ * sqrt(0) = 0
|
|
|
+ * sqrt(-0) = -0
|
|
|
+ *
|
|
|
+ */
|
|
|
+P.squareRoot = P.sqrt = function() {
|
|
|
+ var m, n, sd, r, rep, t,
|
|
|
+ x = this,
|
|
|
+ d = x.d,
|
|
|
+ e = x.e,
|
|
|
+ s = x.s,
|
|
|
+ Ctor = x.constructor;
|
|
|
+
|
|
|
+ // Negative/NaN/Infinity/zero?
|
|
|
+ if (s !== 1 || !d || !d[0]) {
|
|
|
+ return new Ctor(!s || s < 0 && (!d || d[0]) ? NaN : d ? x : 1 / 0);
|
|
|
+ }
|
|
|
+
|
|
|
+ external = false;
|
|
|
+
|
|
|
+ // Initial estimate.
|
|
|
+ s = Math.sqrt(+x);
|
|
|
+
|
|
|
+ // Math.sqrt underflow/overflow?
|
|
|
+ // Pass x to Math.sqrt as integer, then adjust the exponent of the result.
|
|
|
+ if (s == 0 || s == 1 / 0) {
|
|
|
+ n = digitsToString(d);
|
|
|
+
|
|
|
+ if ((n.length + e) % 2 == 0) n += '0';
|
|
|
+ s = Math.sqrt(n);
|
|
|
+ e = mathfloor((e + 1) / 2) - (e < 0 || e % 2);
|
|
|
+
|
|
|
+ if (s == 1 / 0) {
|
|
|
+ n = '5e' + e;
|
|
|
+ } else {
|
|
|
+ n = s.toExponential();
|
|
|
+ n = n.slice(0, n.indexOf('e') + 1) + e;
|
|
|
+ }
|
|
|
+
|
|
|
+ r = new Ctor(n);
|
|
|
+ } else {
|
|
|
+ r = new Ctor(s.toString());
|
|
|
+ }
|
|
|
+
|
|
|
+ sd = (e = Ctor.precision) + 3;
|
|
|
+
|
|
|
+ // Newton-Raphson iteration.
|
|
|
+ for (;;) {
|
|
|
+ t = r;
|
|
|
+ r = t.plus(divide(x, t, sd + 2, 1)).times(0.5);
|
|
|
+
|
|
|
+ // TODO? Replace with for-loop and checkRoundingDigits.
|
|
|
+ if (digitsToString(t.d).slice(0, sd) === (n = digitsToString(r.d)).slice(0, sd)) {
|
|
|
+ n = n.slice(sd - 3, sd + 1);
|
|
|
+
|
|
|
+ // The 4th rounding digit may be in error by -1 so if the 4 rounding digits are 9999 or
|
|
|
+ // 4999, i.e. approaching a rounding boundary, continue the iteration.
|
|
|
+ if (n == '9999' || !rep && n == '4999') {
|
|
|
+
|
|
|
+ // On the first iteration only, check to see if rounding up gives the exact result as the
|
|
|
+ // nines may infinitely repeat.
|
|
|
+ if (!rep) {
|
|
|
+ finalise(t, e + 1, 0);
|
|
|
+
|
|
|
+ if (t.times(t).eq(x)) {
|
|
|
+ r = t;
|
|
|
+ break;
|
|
|
+ }
|
|
|
+ }
|
|
|
+
|
|
|
+ sd += 4;
|
|
|
+ rep = 1;
|
|
|
+ } else {
|
|
|
+
|
|
|
+ // If the rounding digits are null, 0{0,4} or 50{0,3}, check for an exact result.
|
|
|
+ // If not, then there are further digits and m will be truthy.
|
|
|
+ if (!+n || !+n.slice(1) && n.charAt(0) == '5') {
|
|
|
+
|
|
|
+ // Truncate to the first rounding digit.
|
|
|
+ finalise(r, e + 1, 1);
|
|
|
+ m = !r.times(r).eq(x);
|
|
|
+ }
|
|
|
+
|
|
|
+ break;
|
|
|
+ }
|
|
|
+ }
|
|
|
+ }
|
|
|
+
|
|
|
+ external = true;
|
|
|
+
|
|
|
+ return finalise(r, e, Ctor.rounding, m);
|
|
|
+};
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is the tangent of the value in radians of this Decimal.
|
|
|
+ *
|
|
|
+ * Domain: [-Infinity, Infinity]
|
|
|
+ * Range: [-Infinity, Infinity]
|
|
|
+ *
|
|
|
+ * tan(0) = 0
|
|
|
+ * tan(-0) = -0
|
|
|
+ * tan(Infinity) = NaN
|
|
|
+ * tan(-Infinity) = NaN
|
|
|
+ * tan(NaN) = NaN
|
|
|
+ *
|
|
|
+ */
|
|
|
+P.tangent = P.tan = function() {
|
|
|
+ var pr, rm,
|
|
|
+ x = this,
|
|
|
+ Ctor = x.constructor;
|
|
|
+
|
|
|
+ if (!x.isFinite()) return new Ctor(NaN);
|
|
|
+ if (x.isZero()) return new Ctor(x);
|
|
|
+
|
|
|
+ pr = Ctor.precision;
|
|
|
+ rm = Ctor.rounding;
|
|
|
+ Ctor.precision = pr + 10;
|
|
|
+ Ctor.rounding = 1;
|
|
|
+
|
|
|
+ x = x.sin();
|
|
|
+ x.s = 1;
|
|
|
+ x = divide(x, new Ctor(1).minus(x.times(x)).sqrt(), pr + 10, 0);
|
|
|
+
|
|
|
+ Ctor.precision = pr;
|
|
|
+ Ctor.rounding = rm;
|
|
|
+
|
|
|
+ return finalise(quadrant == 2 || quadrant == 4 ? x.neg() : x, pr, rm, true);
|
|
|
+};
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * n * 0 = 0
|
|
|
+ * n * N = N
|
|
|
+ * n * I = I
|
|
|
+ * 0 * n = 0
|
|
|
+ * 0 * 0 = 0
|
|
|
+ * 0 * N = N
|
|
|
+ * 0 * I = N
|
|
|
+ * N * n = N
|
|
|
+ * N * 0 = N
|
|
|
+ * N * N = N
|
|
|
+ * N * I = N
|
|
|
+ * I * n = I
|
|
|
+ * I * 0 = N
|
|
|
+ * I * N = N
|
|
|
+ * I * I = I
|
|
|
+ *
|
|
|
+ * Return a new Decimal whose value is this Decimal times `y`, rounded to `precision` significant
|
|
|
+ * digits using rounding mode `rounding`.
|
|
|
+ *
|
|
|
+ */
|
|
|
+P.times = P.mul = function(y) {
|
|
|
+ var carry, e, i, k, r, rL, t, xdL, ydL,
|
|
|
+ x = this,
|
|
|
+ Ctor = x.constructor,
|
|
|
+ xd = x.d,
|
|
|
+ yd = (y = new Ctor(y)).d;
|
|
|
+
|
|
|
+ y.s *= x.s;
|
|
|
+
|
|
|
+ // If either is NaN, ±Infinity or ±0...
|
|
|
+ if (!xd || !xd[0] || !yd || !yd[0]) {
|
|
|
+
|
|
|
+ return new Ctor(!y.s || xd && !xd[0] && !yd || yd && !yd[0] && !xd
|
|
|
+
|
|
|
+ // Return NaN if either is NaN.
|
|
|
+ // Return NaN if x is ±0 and y is ±Infinity, or y is ±0 and x is ±Infinity.
|
|
|
+ ?
|
|
|
+ NaN
|
|
|
+
|
|
|
+ // Return ±Infinity if either is ±Infinity.
|
|
|
+ // Return ±0 if either is ±0.
|
|
|
+ :
|
|
|
+ !xd || !yd ? y.s / 0 : y.s * 0);
|
|
|
+ }
|
|
|
+
|
|
|
+ e = mathfloor(x.e / LOG_BASE) + mathfloor(y.e / LOG_BASE);
|
|
|
+ xdL = xd.length;
|
|
|
+ ydL = yd.length;
|
|
|
+
|
|
|
+ // Ensure xd points to the longer array.
|
|
|
+ if (xdL < ydL) {
|
|
|
+ r = xd;
|
|
|
+ xd = yd;
|
|
|
+ yd = r;
|
|
|
+ rL = xdL;
|
|
|
+ xdL = ydL;
|
|
|
+ ydL = rL;
|
|
|
+ }
|
|
|
+
|
|
|
+ // Initialise the result array with zeros.
|
|
|
+ r = [];
|
|
|
+ rL = xdL + ydL;
|
|
|
+ for (i = rL; i--;) r.push(0);
|
|
|
+
|
|
|
+ // Multiply!
|
|
|
+ for (i = ydL; --i >= 0;) {
|
|
|
+ carry = 0;
|
|
|
+ for (k = xdL + i; k > i;) {
|
|
|
+ t = r[k] + yd[i] * xd[k - i - 1] + carry;
|
|
|
+ r[k--] = t % BASE | 0;
|
|
|
+ carry = t / BASE | 0;
|
|
|
+ }
|
|
|
+
|
|
|
+ r[k] = (r[k] + carry) % BASE | 0;
|
|
|
+ }
|
|
|
+
|
|
|
+ // Remove trailing zeros.
|
|
|
+ for (; !r[--rL];) r.pop();
|
|
|
+
|
|
|
+ if (carry) ++e;
|
|
|
+ else r.shift();
|
|
|
+
|
|
|
+ y.d = r;
|
|
|
+ y.e = getBase10Exponent(r, e);
|
|
|
+
|
|
|
+ return external ? finalise(y, Ctor.precision, Ctor.rounding) : y;
|
|
|
+};
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a string representing the value of this Decimal in base 2, round to `sd` significant
|
|
|
+ * digits using rounding mode `rm`.
|
|
|
+ *
|
|
|
+ * If the optional `sd` argument is present then return binary exponential notation.
|
|
|
+ *
|
|
|
+ * [sd] {number} Significant digits. Integer, 1 to MAX_DIGITS inclusive.
|
|
|
+ * [rm] {number} Rounding mode. Integer, 0 to 8 inclusive.
|
|
|
+ *
|
|
|
+ */
|
|
|
+P.toBinary = function(sd, rm) {
|
|
|
+ return toStringBinary(this, 2, sd, rm);
|
|
|
+};
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is the value of this Decimal rounded to a maximum of `dp`
|
|
|
+ * decimal places using rounding mode `rm` or `rounding` if `rm` is omitted.
|
|
|
+ *
|
|
|
+ * If `dp` is omitted, return a new Decimal whose value is the value of this Decimal.
|
|
|
+ *
|
|
|
+ * [dp] {number} Decimal places. Integer, 0 to MAX_DIGITS inclusive.
|
|
|
+ * [rm] {number} Rounding mode. Integer, 0 to 8 inclusive.
|
|
|
+ *
|
|
|
+ */
|
|
|
+P.toDecimalPlaces = P.toDP = function(dp, rm) {
|
|
|
+ var x = this,
|
|
|
+ Ctor = x.constructor;
|
|
|
+
|
|
|
+ x = new Ctor(x);
|
|
|
+ if (dp === void 0) return x;
|
|
|
+
|
|
|
+ checkInt32(dp, 0, MAX_DIGITS);
|
|
|
+
|
|
|
+ if (rm === void 0) rm = Ctor.rounding;
|
|
|
+ else checkInt32(rm, 0, 8);
|
|
|
+
|
|
|
+ return finalise(x, dp + x.e + 1, rm);
|
|
|
+};
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a string representing the value of this Decimal in exponential notation rounded to
|
|
|
+ * `dp` fixed decimal places using rounding mode `rounding`.
|
|
|
+ *
|
|
|
+ * [dp] {number} Decimal places. Integer, 0 to MAX_DIGITS inclusive.
|
|
|
+ * [rm] {number} Rounding mode. Integer, 0 to 8 inclusive.
|
|
|
+ *
|
|
|
+ */
|
|
|
+P.toExponential = function(dp, rm) {
|
|
|
+ var str,
|
|
|
+ x = this,
|
|
|
+ Ctor = x.constructor;
|
|
|
+
|
|
|
+ if (dp === void 0) {
|
|
|
+ str = finiteToString(x, true);
|
|
|
+ } else {
|
|
|
+ checkInt32(dp, 0, MAX_DIGITS);
|
|
|
+
|
|
|
+ if (rm === void 0) rm = Ctor.rounding;
|
|
|
+ else checkInt32(rm, 0, 8);
|
|
|
+
|
|
|
+ x = finalise(new Ctor(x), dp + 1, rm);
|
|
|
+ str = finiteToString(x, true, dp + 1);
|
|
|
+ }
|
|
|
+
|
|
|
+ return x.isNeg() && !x.isZero() ? '-' + str : str;
|
|
|
+};
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a string representing the value of this Decimal in normal (fixed-point) notation to
|
|
|
+ * `dp` fixed decimal places and rounded using rounding mode `rm` or `rounding` if `rm` is
|
|
|
+ * omitted.
|
|
|
+ *
|
|
|
+ * As with JavaScript numbers, (-0).toFixed(0) is '0', but e.g. (-0.00001).toFixed(0) is '-0'.
|
|
|
+ *
|
|
|
+ * [dp] {number} Decimal places. Integer, 0 to MAX_DIGITS inclusive.
|
|
|
+ * [rm] {number} Rounding mode. Integer, 0 to 8 inclusive.
|
|
|
+ *
|
|
|
+ * (-0).toFixed(0) is '0', but (-0.1).toFixed(0) is '-0'.
|
|
|
+ * (-0).toFixed(1) is '0.0', but (-0.01).toFixed(1) is '-0.0'.
|
|
|
+ * (-0).toFixed(3) is '0.000'.
|
|
|
+ * (-0.5).toFixed(0) is '-0'.
|
|
|
+ *
|
|
|
+ */
|
|
|
+P.toFixed = function(dp, rm) {
|
|
|
+ var str, y,
|
|
|
+ x = this,
|
|
|
+ Ctor = x.constructor;
|
|
|
+
|
|
|
+ if (dp === void 0) {
|
|
|
+ str = finiteToString(x);
|
|
|
+ } else {
|
|
|
+ checkInt32(dp, 0, MAX_DIGITS);
|
|
|
+
|
|
|
+ if (rm === void 0) rm = Ctor.rounding;
|
|
|
+ else checkInt32(rm, 0, 8);
|
|
|
+
|
|
|
+ y = finalise(new Ctor(x), dp + x.e + 1, rm);
|
|
|
+ str = finiteToString(y, false, dp + y.e + 1);
|
|
|
+ }
|
|
|
+
|
|
|
+ // To determine whether to add the minus sign look at the value before it was rounded,
|
|
|
+ // i.e. look at `x` rather than `y`.
|
|
|
+ return x.isNeg() && !x.isZero() ? '-' + str : str;
|
|
|
+};
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return an array representing the value of this Decimal as a simple fraction with an integer
|
|
|
+ * numerator and an integer denominator.
|
|
|
+ *
|
|
|
+ * The denominator will be a positive non-zero value less than or equal to the specified maximum
|
|
|
+ * denominator. If a maximum denominator is not specified, the denominator will be the lowest
|
|
|
+ * value necessary to represent the number exactly.
|
|
|
+ *
|
|
|
+ * [maxD] {number|string|Decimal} Maximum denominator. Integer >= 1 and < Infinity.
|
|
|
+ *
|
|
|
+ */
|
|
|
+P.toFraction = function(maxD) {
|
|
|
+ var d, d0, d1, d2, e, k, n, n0, n1, pr, q, r,
|
|
|
+ x = this,
|
|
|
+ xd = x.d,
|
|
|
+ Ctor = x.constructor;
|
|
|
+
|
|
|
+ if (!xd) return new Ctor(x);
|
|
|
+
|
|
|
+ n1 = d0 = new Ctor(1);
|
|
|
+ d1 = n0 = new Ctor(0);
|
|
|
+
|
|
|
+ d = new Ctor(d1);
|
|
|
+ e = d.e = getPrecision(xd) - x.e - 1;
|
|
|
+ k = e % LOG_BASE;
|
|
|
+ d.d[0] = mathpow(10, k < 0 ? LOG_BASE + k : k);
|
|
|
+
|
|
|
+ if (maxD == null) {
|
|
|
+
|
|
|
+ // d is 10**e, the minimum max-denominator needed.
|
|
|
+ maxD = e > 0 ? d : n1;
|
|
|
+ } else {
|
|
|
+ n = new Ctor(maxD);
|
|
|
+ if (!n.isInt() || n.lt(n1)) throw Error(invalidArgument + n);
|
|
|
+ maxD = n.gt(d) ? (e > 0 ? d : n1) : n;
|
|
|
+ }
|
|
|
+
|
|
|
+ external = false;
|
|
|
+ n = new Ctor(digitsToString(xd));
|
|
|
+ pr = Ctor.precision;
|
|
|
+ Ctor.precision = e = xd.length * LOG_BASE * 2;
|
|
|
+
|
|
|
+ for (;;) {
|
|
|
+ q = divide(n, d, 0, 1, 1);
|
|
|
+ d2 = d0.plus(q.times(d1));
|
|
|
+ if (d2.cmp(maxD) == 1) break;
|
|
|
+ d0 = d1;
|
|
|
+ d1 = d2;
|
|
|
+ d2 = n1;
|
|
|
+ n1 = n0.plus(q.times(d2));
|
|
|
+ n0 = d2;
|
|
|
+ d2 = d;
|
|
|
+ d = n.minus(q.times(d2));
|
|
|
+ n = d2;
|
|
|
+ }
|
|
|
+
|
|
|
+ d2 = divide(maxD.minus(d0), d1, 0, 1, 1);
|
|
|
+ n0 = n0.plus(d2.times(n1));
|
|
|
+ d0 = d0.plus(d2.times(d1));
|
|
|
+ n0.s = n1.s = x.s;
|
|
|
+
|
|
|
+ // Determine which fraction is closer to x, n0/d0 or n1/d1?
|
|
|
+ r = divide(n1, d1, e, 1).minus(x).abs().cmp(divide(n0, d0, e, 1).minus(x).abs()) < 1 ? [n1, d1] : [n0, d0];
|
|
|
+
|
|
|
+ Ctor.precision = pr;
|
|
|
+ external = true;
|
|
|
+
|
|
|
+ return r;
|
|
|
+};
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a string representing the value of this Decimal in base 16, round to `sd` significant
|
|
|
+ * digits using rounding mode `rm`.
|
|
|
+ *
|
|
|
+ * If the optional `sd` argument is present then return binary exponential notation.
|
|
|
+ *
|
|
|
+ * [sd] {number} Significant digits. Integer, 1 to MAX_DIGITS inclusive.
|
|
|
+ * [rm] {number} Rounding mode. Integer, 0 to 8 inclusive.
|
|
|
+ *
|
|
|
+ */
|
|
|
+P.toHexadecimal = P.toHex = function(sd, rm) {
|
|
|
+ return toStringBinary(this, 16, sd, rm);
|
|
|
+};
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Returns a new Decimal whose value is the nearest multiple of `y` in the direction of rounding
|
|
|
+ * mode `rm`, or `Decimal.rounding` if `rm` is omitted, to the value of this Decimal.
|
|
|
+ *
|
|
|
+ * The return value will always have the same sign as this Decimal, unless either this Decimal
|
|
|
+ * or `y` is NaN, in which case the return value will be also be NaN.
|
|
|
+ *
|
|
|
+ * The return value is not affected by the value of `precision`.
|
|
|
+ *
|
|
|
+ * y {number|string|Decimal} The magnitude to round to a multiple of.
|
|
|
+ * [rm] {number} Rounding mode. Integer, 0 to 8 inclusive.
|
|
|
+ *
|
|
|
+ * 'toNearest() rounding mode not an integer: {rm}'
|
|
|
+ * 'toNearest() rounding mode out of range: {rm}'
|
|
|
+ *
|
|
|
+ */
|
|
|
+P.toNearest = function(y, rm) {
|
|
|
+ var x = this,
|
|
|
+ Ctor = x.constructor;
|
|
|
+
|
|
|
+ x = new Ctor(x);
|
|
|
+
|
|
|
+ if (y == null) {
|
|
|
+
|
|
|
+ // If x is not finite, return x.
|
|
|
+ if (!x.d) return x;
|
|
|
+
|
|
|
+ y = new Ctor(1);
|
|
|
+ rm = Ctor.rounding;
|
|
|
+ } else {
|
|
|
+ y = new Ctor(y);
|
|
|
+ if (rm === void 0) {
|
|
|
+ rm = Ctor.rounding;
|
|
|
+ } else {
|
|
|
+ checkInt32(rm, 0, 8);
|
|
|
+ }
|
|
|
+
|
|
|
+ // If x is not finite, return x if y is not NaN, else NaN.
|
|
|
+ if (!x.d) return y.s ? x : y;
|
|
|
+
|
|
|
+ // If y is not finite, return Infinity with the sign of x if y is Infinity, else NaN.
|
|
|
+ if (!y.d) {
|
|
|
+ if (y.s) y.s = x.s;
|
|
|
+ return y;
|
|
|
+ }
|
|
|
+ }
|
|
|
+
|
|
|
+ // If y is not zero, calculate the nearest multiple of y to x.
|
|
|
+ if (y.d[0]) {
|
|
|
+ external = false;
|
|
|
+ x = divide(x, y, 0, rm, 1).times(y);
|
|
|
+ external = true;
|
|
|
+ finalise(x);
|
|
|
+
|
|
|
+ // If y is zero, return zero with the sign of x.
|
|
|
+ } else {
|
|
|
+ y.s = x.s;
|
|
|
+ x = y;
|
|
|
+ }
|
|
|
+
|
|
|
+ return x;
|
|
|
+};
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return the value of this Decimal converted to a number primitive.
|
|
|
+ * Zero keeps its sign.
|
|
|
+ *
|
|
|
+ */
|
|
|
+P.toNumber = function() {
|
|
|
+ return +this;
|
|
|
+};
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a string representing the value of this Decimal in base 8, round to `sd` significant
|
|
|
+ * digits using rounding mode `rm`.
|
|
|
+ *
|
|
|
+ * If the optional `sd` argument is present then return binary exponential notation.
|
|
|
+ *
|
|
|
+ * [sd] {number} Significant digits. Integer, 1 to MAX_DIGITS inclusive.
|
|
|
+ * [rm] {number} Rounding mode. Integer, 0 to 8 inclusive.
|
|
|
+ *
|
|
|
+ */
|
|
|
+P.toOctal = function(sd, rm) {
|
|
|
+ return toStringBinary(this, 8, sd, rm);
|
|
|
+};
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is the value of this Decimal raised to the power `y`, rounded
|
|
|
+ * to `precision` significant digits using rounding mode `rounding`.
|
|
|
+ *
|
|
|
+ * ECMAScript compliant.
|
|
|
+ *
|
|
|
+ * pow(x, NaN) = NaN
|
|
|
+ * pow(x, ±0) = 1
|
|
|
+
|
|
|
+ * pow(NaN, non-zero) = NaN
|
|
|
+ * pow(abs(x) > 1, +Infinity) = +Infinity
|
|
|
+ * pow(abs(x) > 1, -Infinity) = +0
|
|
|
+ * pow(abs(x) == 1, ±Infinity) = NaN
|
|
|
+ * pow(abs(x) < 1, +Infinity) = +0
|
|
|
+ * pow(abs(x) < 1, -Infinity) = +Infinity
|
|
|
+ * pow(+Infinity, y > 0) = +Infinity
|
|
|
+ * pow(+Infinity, y < 0) = +0
|
|
|
+ * pow(-Infinity, odd integer > 0) = -Infinity
|
|
|
+ * pow(-Infinity, even integer > 0) = +Infinity
|
|
|
+ * pow(-Infinity, odd integer < 0) = -0
|
|
|
+ * pow(-Infinity, even integer < 0) = +0
|
|
|
+ * pow(+0, y > 0) = +0
|
|
|
+ * pow(+0, y < 0) = +Infinity
|
|
|
+ * pow(-0, odd integer > 0) = -0
|
|
|
+ * pow(-0, even integer > 0) = +0
|
|
|
+ * pow(-0, odd integer < 0) = -Infinity
|
|
|
+ * pow(-0, even integer < 0) = +Infinity
|
|
|
+ * pow(finite x < 0, finite non-integer) = NaN
|
|
|
+ *
|
|
|
+ * For non-integer or very large exponents pow(x, y) is calculated using
|
|
|
+ *
|
|
|
+ * x^y = exp(y*ln(x))
|
|
|
+ *
|
|
|
+ * Assuming the first 15 rounding digits are each equally likely to be any digit 0-9, the
|
|
|
+ * probability of an incorrectly rounded result
|
|
|
+ * P([49]9{14} | [50]0{14}) = 2 * 0.2 * 10^-14 = 4e-15 = 1/2.5e+14
|
|
|
+ * i.e. 1 in 250,000,000,000,000
|
|
|
+ *
|
|
|
+ * If a result is incorrectly rounded the maximum error will be 1 ulp (unit in last place).
|
|
|
+ *
|
|
|
+ * y {number|string|Decimal} The power to which to raise this Decimal.
|
|
|
+ *
|
|
|
+ */
|
|
|
+P.toPower = P.pow = function(y) {
|
|
|
+ var e, k, pr, r, rm, s,
|
|
|
+ x = this,
|
|
|
+ Ctor = x.constructor,
|
|
|
+ yn = +(y = new Ctor(y));
|
|
|
+
|
|
|
+ // Either ±Infinity, NaN or ±0?
|
|
|
+ if (!x.d || !y.d || !x.d[0] || !y.d[0]) return new Ctor(mathpow(+x, yn));
|
|
|
+
|
|
|
+ x = new Ctor(x);
|
|
|
+
|
|
|
+ if (x.eq(1)) return x;
|
|
|
+
|
|
|
+ pr = Ctor.precision;
|
|
|
+ rm = Ctor.rounding;
|
|
|
+
|
|
|
+ if (y.eq(1)) return finalise(x, pr, rm);
|
|
|
+
|
|
|
+ // y exponent
|
|
|
+ e = mathfloor(y.e / LOG_BASE);
|
|
|
+
|
|
|
+ // If y is a small integer use the 'exponentiation by squaring' algorithm.
|
|
|
+ if (e >= y.d.length - 1 && (k = yn < 0 ? -yn : yn) <= MAX_SAFE_INTEGER) {
|
|
|
+ r = intPow(Ctor, x, k, pr);
|
|
|
+ return y.s < 0 ? new Ctor(1).div(r) : finalise(r, pr, rm);
|
|
|
+ }
|
|
|
+
|
|
|
+ s = x.s;
|
|
|
+
|
|
|
+ // if x is negative
|
|
|
+ if (s < 0) {
|
|
|
+
|
|
|
+ // if y is not an integer
|
|
|
+ if (e < y.d.length - 1) return new Ctor(NaN);
|
|
|
+
|
|
|
+ // Result is positive if x is negative and the last digit of integer y is even.
|
|
|
+ if ((y.d[e] & 1) == 0) s = 1;
|
|
|
+
|
|
|
+ // if x.eq(-1)
|
|
|
+ if (x.e == 0 && x.d[0] == 1 && x.d.length == 1) {
|
|
|
+ x.s = s;
|
|
|
+ return x;
|
|
|
+ }
|
|
|
+ }
|
|
|
+
|
|
|
+ // Estimate result exponent.
|
|
|
+ // x^y = 10^e, where e = y * log10(x)
|
|
|
+ // log10(x) = log10(x_significand) + x_exponent
|
|
|
+ // log10(x_significand) = ln(x_significand) / ln(10)
|
|
|
+ k = mathpow(+x, yn);
|
|
|
+ e = k == 0 || !isFinite(k) ?
|
|
|
+ mathfloor(yn * (Math.log('0.' + digitsToString(x.d)) / Math.LN10 + x.e + 1)) :
|
|
|
+ new Ctor(k + '').e;
|
|
|
+
|
|
|
+ // Exponent estimate may be incorrect e.g. x: 0.999999999999999999, y: 2.29, e: 0, r.e: -1.
|
|
|
+
|
|
|
+ // Overflow/underflow?
|
|
|
+ if (e > Ctor.maxE + 1 || e < Ctor.minE - 1) return new Ctor(e > 0 ? s / 0 : 0);
|
|
|
+
|
|
|
+ external = false;
|
|
|
+ Ctor.rounding = x.s = 1;
|
|
|
+
|
|
|
+ // Estimate the extra guard digits needed to ensure five correct rounding digits from
|
|
|
+ // naturalLogarithm(x). Example of failure without these extra digits (precision: 10):
|
|
|
+ // new Decimal(2.32456).pow('2087987436534566.46411')
|
|
|
+ // should be 1.162377823e+764914905173815, but is 1.162355823e+764914905173815
|
|
|
+ k = Math.min(12, (e + '').length);
|
|
|
+
|
|
|
+ // r = x^y = exp(y*ln(x))
|
|
|
+ r = naturalExponential(y.times(naturalLogarithm(x, pr + k)), pr);
|
|
|
+
|
|
|
+ // r may be Infinity, e.g. (0.9999999999999999).pow(-1e+40)
|
|
|
+ if (r.d) {
|
|
|
+
|
|
|
+ // Truncate to the required precision plus five rounding digits.
|
|
|
+ r = finalise(r, pr + 5, 1);
|
|
|
+
|
|
|
+ // If the rounding digits are [49]9999 or [50]0000 increase the precision by 10 and recalculate
|
|
|
+ // the result.
|
|
|
+ if (checkRoundingDigits(r.d, pr, rm)) {
|
|
|
+ e = pr + 10;
|
|
|
+
|
|
|
+ // Truncate to the increased precision plus five rounding digits.
|
|
|
+ r = finalise(naturalExponential(y.times(naturalLogarithm(x, e + k)), e), e + 5, 1);
|
|
|
+
|
|
|
+ // Check for 14 nines from the 2nd rounding digit (the first rounding digit may be 4 or 9).
|
|
|
+ if (+digitsToString(r.d).slice(pr + 1, pr + 15) + 1 == 1e14) {
|
|
|
+ r = finalise(r, pr + 1, 0);
|
|
|
+ }
|
|
|
+ }
|
|
|
+ }
|
|
|
+
|
|
|
+ r.s = s;
|
|
|
+ external = true;
|
|
|
+ Ctor.rounding = rm;
|
|
|
+
|
|
|
+ return finalise(r, pr, rm);
|
|
|
+};
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a string representing the value of this Decimal rounded to `sd` significant digits
|
|
|
+ * using rounding mode `rounding`.
|
|
|
+ *
|
|
|
+ * Return exponential notation if `sd` is less than the number of digits necessary to represent
|
|
|
+ * the integer part of the value in normal notation.
|
|
|
+ *
|
|
|
+ * [sd] {number} Significant digits. Integer, 1 to MAX_DIGITS inclusive.
|
|
|
+ * [rm] {number} Rounding mode. Integer, 0 to 8 inclusive.
|
|
|
+ *
|
|
|
+ */
|
|
|
+P.toPrecision = function(sd, rm) {
|
|
|
+ var str,
|
|
|
+ x = this,
|
|
|
+ Ctor = x.constructor;
|
|
|
+
|
|
|
+ if (sd === void 0) {
|
|
|
+ str = finiteToString(x, x.e <= Ctor.toExpNeg || x.e >= Ctor.toExpPos);
|
|
|
+ } else {
|
|
|
+ checkInt32(sd, 1, MAX_DIGITS);
|
|
|
+
|
|
|
+ if (rm === void 0) rm = Ctor.rounding;
|
|
|
+ else checkInt32(rm, 0, 8);
|
|
|
+
|
|
|
+ x = finalise(new Ctor(x), sd, rm);
|
|
|
+ str = finiteToString(x, sd <= x.e || x.e <= Ctor.toExpNeg, sd);
|
|
|
+ }
|
|
|
+
|
|
|
+ return x.isNeg() && !x.isZero() ? '-' + str : str;
|
|
|
+};
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is the value of this Decimal rounded to a maximum of `sd`
|
|
|
+ * significant digits using rounding mode `rm`, or to `precision` and `rounding` respectively if
|
|
|
+ * omitted.
|
|
|
+ *
|
|
|
+ * [sd] {number} Significant digits. Integer, 1 to MAX_DIGITS inclusive.
|
|
|
+ * [rm] {number} Rounding mode. Integer, 0 to 8 inclusive.
|
|
|
+ *
|
|
|
+ * 'toSD() digits out of range: {sd}'
|
|
|
+ * 'toSD() digits not an integer: {sd}'
|
|
|
+ * 'toSD() rounding mode not an integer: {rm}'
|
|
|
+ * 'toSD() rounding mode out of range: {rm}'
|
|
|
+ *
|
|
|
+ */
|
|
|
+P.toSignificantDigits = P.toSD = function(sd, rm) {
|
|
|
+ var x = this,
|
|
|
+ Ctor = x.constructor;
|
|
|
+
|
|
|
+ if (sd === void 0) {
|
|
|
+ sd = Ctor.precision;
|
|
|
+ rm = Ctor.rounding;
|
|
|
+ } else {
|
|
|
+ checkInt32(sd, 1, MAX_DIGITS);
|
|
|
+
|
|
|
+ if (rm === void 0) rm = Ctor.rounding;
|
|
|
+ else checkInt32(rm, 0, 8);
|
|
|
+ }
|
|
|
+
|
|
|
+ return finalise(new Ctor(x), sd, rm);
|
|
|
+};
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a string representing the value of this Decimal.
|
|
|
+ *
|
|
|
+ * Return exponential notation if this Decimal has a positive exponent equal to or greater than
|
|
|
+ * `toExpPos`, or a negative exponent equal to or less than `toExpNeg`.
|
|
|
+ *
|
|
|
+ */
|
|
|
+P.toString = function() {
|
|
|
+ var x = this,
|
|
|
+ Ctor = x.constructor,
|
|
|
+ str = finiteToString(x, x.e <= Ctor.toExpNeg || x.e >= Ctor.toExpPos);
|
|
|
+
|
|
|
+ return x.isNeg() && !x.isZero() ? '-' + str : str;
|
|
|
+};
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is the value of this Decimal truncated to a whole number.
|
|
|
+ *
|
|
|
+ */
|
|
|
+P.truncated = P.trunc = function() {
|
|
|
+ return finalise(new this.constructor(this), this.e + 1, 1);
|
|
|
+};
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a string representing the value of this Decimal.
|
|
|
+ * Unlike `toString`, negative zero will include the minus sign.
|
|
|
+ *
|
|
|
+ */
|
|
|
+P.valueOf = P.toJSON = function() {
|
|
|
+ var x = this,
|
|
|
+ Ctor = x.constructor,
|
|
|
+ str = finiteToString(x, x.e <= Ctor.toExpNeg || x.e >= Ctor.toExpPos);
|
|
|
+
|
|
|
+ return x.isNeg() ? '-' + str : str;
|
|
|
+};
|
|
|
+
|
|
|
+
|
|
|
+// Helper functions for Decimal.prototype (P) and/or Decimal methods, and their callers.
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * digitsToString P.cubeRoot, P.logarithm, P.squareRoot, P.toFraction, P.toPower,
|
|
|
+ * finiteToString, naturalExponential, naturalLogarithm
|
|
|
+ * checkInt32 P.toDecimalPlaces, P.toExponential, P.toFixed, P.toNearest,
|
|
|
+ * P.toPrecision, P.toSignificantDigits, toStringBinary, random
|
|
|
+ * checkRoundingDigits P.logarithm, P.toPower, naturalExponential, naturalLogarithm
|
|
|
+ * convertBase toStringBinary, parseOther
|
|
|
+ * cos P.cos
|
|
|
+ * divide P.atanh, P.cubeRoot, P.dividedBy, P.dividedToIntegerBy,
|
|
|
+ * P.logarithm, P.modulo, P.squareRoot, P.tan, P.tanh, P.toFraction,
|
|
|
+ * P.toNearest, toStringBinary, naturalExponential, naturalLogarithm,
|
|
|
+ * taylorSeries, atan2, parseOther
|
|
|
+ * finalise P.absoluteValue, P.atan, P.atanh, P.ceil, P.cos, P.cosh,
|
|
|
+ * P.cubeRoot, P.dividedToIntegerBy, P.floor, P.logarithm, P.minus,
|
|
|
+ * P.modulo, P.negated, P.plus, P.round, P.sin, P.sinh, P.squareRoot,
|
|
|
+ * P.tan, P.times, P.toDecimalPlaces, P.toExponential, P.toFixed,
|
|
|
+ * P.toNearest, P.toPower, P.toPrecision, P.toSignificantDigits,
|
|
|
+ * P.truncated, divide, getLn10, getPi, naturalExponential,
|
|
|
+ * naturalLogarithm, ceil, floor, round, trunc
|
|
|
+ * finiteToString P.toExponential, P.toFixed, P.toPrecision, P.toString, P.valueOf,
|
|
|
+ * toStringBinary
|
|
|
+ * getBase10Exponent P.minus, P.plus, P.times, parseOther
|
|
|
+ * getLn10 P.logarithm, naturalLogarithm
|
|
|
+ * getPi P.acos, P.asin, P.atan, toLessThanHalfPi, atan2
|
|
|
+ * getPrecision P.precision, P.toFraction
|
|
|
+ * getZeroString digitsToString, finiteToString
|
|
|
+ * intPow P.toPower, parseOther
|
|
|
+ * isOdd toLessThanHalfPi
|
|
|
+ * maxOrMin max, min
|
|
|
+ * naturalExponential P.naturalExponential, P.toPower
|
|
|
+ * naturalLogarithm P.acosh, P.asinh, P.atanh, P.logarithm, P.naturalLogarithm,
|
|
|
+ * P.toPower, naturalExponential
|
|
|
+ * nonFiniteToString finiteToString, toStringBinary
|
|
|
+ * parseDecimal Decimal
|
|
|
+ * parseOther Decimal
|
|
|
+ * sin P.sin
|
|
|
+ * taylorSeries P.cosh, P.sinh, cos, sin
|
|
|
+ * toLessThanHalfPi P.cos, P.sin
|
|
|
+ * toStringBinary P.toBinary, P.toHexadecimal, P.toOctal
|
|
|
+ * truncate intPow
|
|
|
+ *
|
|
|
+ * Throws: P.logarithm, P.precision, P.toFraction, checkInt32, getLn10, getPi,
|
|
|
+ * naturalLogarithm, config, parseOther, random, Decimal
|
|
|
+ */
|
|
|
+
|
|
|
+
|
|
|
+function digitsToString(d) {
|
|
|
+ var i, k, ws,
|
|
|
+ indexOfLastWord = d.length - 1,
|
|
|
+ str = '',
|
|
|
+ w = d[0];
|
|
|
+
|
|
|
+ if (indexOfLastWord > 0) {
|
|
|
+ str += w;
|
|
|
+ for (i = 1; i < indexOfLastWord; i++) {
|
|
|
+ ws = d[i] + '';
|
|
|
+ k = LOG_BASE - ws.length;
|
|
|
+ if (k) str += getZeroString(k);
|
|
|
+ str += ws;
|
|
|
+ }
|
|
|
+
|
|
|
+ w = d[i];
|
|
|
+ ws = w + '';
|
|
|
+ k = LOG_BASE - ws.length;
|
|
|
+ if (k) str += getZeroString(k);
|
|
|
+ } else if (w === 0) {
|
|
|
+ return '0';
|
|
|
+ }
|
|
|
+
|
|
|
+ // Remove trailing zeros of last w.
|
|
|
+ for (; w % 10 === 0;) w /= 10;
|
|
|
+
|
|
|
+ return str + w;
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+function checkInt32(i, min, max) {
|
|
|
+ if (i !== ~~i || i < min || i > max) {
|
|
|
+ throw Error(invalidArgument + i);
|
|
|
+ }
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Check 5 rounding digits if `repeating` is null, 4 otherwise.
|
|
|
+ * `repeating == null` if caller is `log` or `pow`,
|
|
|
+ * `repeating != null` if caller is `naturalLogarithm` or `naturalExponential`.
|
|
|
+ */
|
|
|
+function checkRoundingDigits(d, i, rm, repeating) {
|
|
|
+ var di, k, r, rd;
|
|
|
+
|
|
|
+ // Get the length of the first word of the array d.
|
|
|
+ for (k = d[0]; k >= 10; k /= 10) --i;
|
|
|
+
|
|
|
+ // Is the rounding digit in the first word of d?
|
|
|
+ if (--i < 0) {
|
|
|
+ i += LOG_BASE;
|
|
|
+ di = 0;
|
|
|
+ } else {
|
|
|
+ di = Math.ceil((i + 1) / LOG_BASE);
|
|
|
+ i %= LOG_BASE;
|
|
|
+ }
|
|
|
+
|
|
|
+ // i is the index (0 - 6) of the rounding digit.
|
|
|
+ // E.g. if within the word 3487563 the first rounding digit is 5,
|
|
|
+ // then i = 4, k = 1000, rd = 3487563 % 1000 = 563
|
|
|
+ k = mathpow(10, LOG_BASE - i);
|
|
|
+ rd = d[di] % k | 0;
|
|
|
+
|
|
|
+ if (repeating == null) {
|
|
|
+ if (i < 3) {
|
|
|
+ if (i == 0) rd = rd / 100 | 0;
|
|
|
+ else if (i == 1) rd = rd / 10 | 0;
|
|
|
+ r = rm < 4 && rd == 99999 || rm > 3 && rd == 49999 || rd == 50000 || rd == 0;
|
|
|
+ } else {
|
|
|
+ r = (rm < 4 && rd + 1 == k || rm > 3 && rd + 1 == k / 2) &&
|
|
|
+ (d[di + 1] / k / 100 | 0) == mathpow(10, i - 2) - 1 ||
|
|
|
+ (rd == k / 2 || rd == 0) && (d[di + 1] / k / 100 | 0) == 0;
|
|
|
+ }
|
|
|
+ } else {
|
|
|
+ if (i < 4) {
|
|
|
+ if (i == 0) rd = rd / 1000 | 0;
|
|
|
+ else if (i == 1) rd = rd / 100 | 0;
|
|
|
+ else if (i == 2) rd = rd / 10 | 0;
|
|
|
+ r = (repeating || rm < 4) && rd == 9999 || !repeating && rm > 3 && rd == 4999;
|
|
|
+ } else {
|
|
|
+ r = ((repeating || rm < 4) && rd + 1 == k ||
|
|
|
+ (!repeating && rm > 3) && rd + 1 == k / 2) &&
|
|
|
+ (d[di + 1] / k / 1000 | 0) == mathpow(10, i - 3) - 1;
|
|
|
+ }
|
|
|
+ }
|
|
|
+
|
|
|
+ return r;
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+// Convert string of `baseIn` to an array of numbers of `baseOut`.
|
|
|
+// Eg. convertBase('255', 10, 16) returns [15, 15].
|
|
|
+// Eg. convertBase('ff', 16, 10) returns [2, 5, 5].
|
|
|
+function convertBase(str, baseIn, baseOut) {
|
|
|
+ var j,
|
|
|
+ arr = [0],
|
|
|
+ arrL,
|
|
|
+ i = 0,
|
|
|
+ strL = str.length;
|
|
|
+
|
|
|
+ for (; i < strL;) {
|
|
|
+ for (arrL = arr.length; arrL--;) arr[arrL] *= baseIn;
|
|
|
+ arr[0] += NUMERALS.indexOf(str.charAt(i++));
|
|
|
+ for (j = 0; j < arr.length; j++) {
|
|
|
+ if (arr[j] > baseOut - 1) {
|
|
|
+ if (arr[j + 1] === void 0) arr[j + 1] = 0;
|
|
|
+ arr[j + 1] += arr[j] / baseOut | 0;
|
|
|
+ arr[j] %= baseOut;
|
|
|
+ }
|
|
|
+ }
|
|
|
+ }
|
|
|
+
|
|
|
+ return arr.reverse();
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * cos(x) = 1 - x^2/2! + x^4/4! - ...
|
|
|
+ * |x| < pi/2
|
|
|
+ *
|
|
|
+ */
|
|
|
+function cosine(Ctor, x) {
|
|
|
+ var k, len, y;
|
|
|
+
|
|
|
+ if (x.isZero()) return x;
|
|
|
+
|
|
|
+ // Argument reduction: cos(4x) = 8*(cos^4(x) - cos^2(x)) + 1
|
|
|
+ // i.e. cos(x) = 8*(cos^4(x/4) - cos^2(x/4)) + 1
|
|
|
+
|
|
|
+ // Estimate the optimum number of times to use the argument reduction.
|
|
|
+ len = x.d.length;
|
|
|
+ if (len < 32) {
|
|
|
+ k = Math.ceil(len / 3);
|
|
|
+ y = (1 / tinyPow(4, k)).toString();
|
|
|
+ } else {
|
|
|
+ k = 16;
|
|
|
+ y = '2.3283064365386962890625e-10';
|
|
|
+ }
|
|
|
+
|
|
|
+ Ctor.precision += k;
|
|
|
+
|
|
|
+ x = taylorSeries(Ctor, 1, x.times(y), new Ctor(1));
|
|
|
+
|
|
|
+ // Reverse argument reduction
|
|
|
+ for (var i = k; i--;) {
|
|
|
+ var cos2x = x.times(x);
|
|
|
+ x = cos2x.times(cos2x).minus(cos2x).times(8).plus(1);
|
|
|
+ }
|
|
|
+
|
|
|
+ Ctor.precision -= k;
|
|
|
+
|
|
|
+ return x;
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Perform division in the specified base.
|
|
|
+ */
|
|
|
+var divide = (function() {
|
|
|
+
|
|
|
+ // Assumes non-zero x and k, and hence non-zero result.
|
|
|
+ function multiplyInteger(x, k, base) {
|
|
|
+ var temp,
|
|
|
+ carry = 0,
|
|
|
+ i = x.length;
|
|
|
+
|
|
|
+ for (x = x.slice(); i--;) {
|
|
|
+ temp = x[i] * k + carry;
|
|
|
+ x[i] = temp % base | 0;
|
|
|
+ carry = temp / base | 0;
|
|
|
+ }
|
|
|
+
|
|
|
+ if (carry) x.unshift(carry);
|
|
|
+
|
|
|
+ return x;
|
|
|
+ }
|
|
|
+
|
|
|
+ function compare(a, b, aL, bL) {
|
|
|
+ var i, r;
|
|
|
+
|
|
|
+ if (aL != bL) {
|
|
|
+ r = aL > bL ? 1 : -1;
|
|
|
+ } else {
|
|
|
+ for (i = r = 0; i < aL; i++) {
|
|
|
+ if (a[i] != b[i]) {
|
|
|
+ r = a[i] > b[i] ? 1 : -1;
|
|
|
+ break;
|
|
|
+ }
|
|
|
+ }
|
|
|
+ }
|
|
|
+
|
|
|
+ return r;
|
|
|
+ }
|
|
|
+
|
|
|
+ function subtract(a, b, aL, base) {
|
|
|
+ var i = 0;
|
|
|
+
|
|
|
+ // Subtract b from a.
|
|
|
+ for (; aL--;) {
|
|
|
+ a[aL] -= i;
|
|
|
+ i = a[aL] < b[aL] ? 1 : 0;
|
|
|
+ a[aL] = i * base + a[aL] - b[aL];
|
|
|
+ }
|
|
|
+
|
|
|
+ // Remove leading zeros.
|
|
|
+ for (; !a[0] && a.length > 1;) a.shift();
|
|
|
+ }
|
|
|
+
|
|
|
+ return function(x, y, pr, rm, dp, base) {
|
|
|
+ var cmp, e, i, k, logBase, more, prod, prodL, q, qd, rem, remL, rem0, sd, t, xi, xL, yd0,
|
|
|
+ yL, yz,
|
|
|
+ Ctor = x.constructor,
|
|
|
+ sign = x.s == y.s ? 1 : -1,
|
|
|
+ xd = x.d,
|
|
|
+ yd = y.d;
|
|
|
+
|
|
|
+ // Either NaN, Infinity or 0?
|
|
|
+ if (!xd || !xd[0] || !yd || !yd[0]) {
|
|
|
+
|
|
|
+ return new Ctor( // Return NaN if either NaN, or both Infinity or 0.
|
|
|
+ !x.s || !y.s || (xd ? yd && xd[0] == yd[0] : !yd) ? NaN :
|
|
|
+
|
|
|
+ // Return ±0 if x is 0 or y is ±Infinity, or return ±Infinity as y is 0.
|
|
|
+ xd && xd[0] == 0 || !yd ? sign * 0 : sign / 0);
|
|
|
+ }
|
|
|
+
|
|
|
+ if (base) {
|
|
|
+ logBase = 1;
|
|
|
+ e = x.e - y.e;
|
|
|
+ } else {
|
|
|
+ base = BASE;
|
|
|
+ logBase = LOG_BASE;
|
|
|
+ e = mathfloor(x.e / logBase) - mathfloor(y.e / logBase);
|
|
|
+ }
|
|
|
+
|
|
|
+ yL = yd.length;
|
|
|
+ xL = xd.length;
|
|
|
+ q = new Ctor(sign);
|
|
|
+ qd = q.d = [];
|
|
|
+
|
|
|
+ // Result exponent may be one less than e.
|
|
|
+ // The digit array of a Decimal from toStringBinary may have trailing zeros.
|
|
|
+ for (i = 0; yd[i] == (xd[i] || 0); i++);
|
|
|
+
|
|
|
+ if (yd[i] > (xd[i] || 0)) e--;
|
|
|
+
|
|
|
+ if (pr == null) {
|
|
|
+ sd = pr = Ctor.precision;
|
|
|
+ rm = Ctor.rounding;
|
|
|
+ } else if (dp) {
|
|
|
+ sd = pr + (x.e - y.e) + 1;
|
|
|
+ } else {
|
|
|
+ sd = pr;
|
|
|
+ }
|
|
|
+
|
|
|
+ if (sd < 0) {
|
|
|
+ qd.push(1);
|
|
|
+ more = true;
|
|
|
+ } else {
|
|
|
+
|
|
|
+ // Convert precision in number of base 10 digits to base 1e7 digits.
|
|
|
+ sd = sd / logBase + 2 | 0;
|
|
|
+ i = 0;
|
|
|
+
|
|
|
+ // divisor < 1e7
|
|
|
+ if (yL == 1) {
|
|
|
+ k = 0;
|
|
|
+ yd = yd[0];
|
|
|
+ sd++;
|
|
|
+
|
|
|
+ // k is the carry.
|
|
|
+ for (;
|
|
|
+ (i < xL || k) && sd--; i++) {
|
|
|
+ t = k * base + (xd[i] || 0);
|
|
|
+ qd[i] = t / yd | 0;
|
|
|
+ k = t % yd | 0;
|
|
|
+ }
|
|
|
+
|
|
|
+ more = k || i < xL;
|
|
|
+
|
|
|
+ // divisor >= 1e7
|
|
|
+ } else {
|
|
|
+
|
|
|
+ // Normalise xd and yd so highest order digit of yd is >= base/2
|
|
|
+ k = base / (yd[0] + 1) | 0;
|
|
|
+
|
|
|
+ if (k > 1) {
|
|
|
+ yd = multiplyInteger(yd, k, base);
|
|
|
+ xd = multiplyInteger(xd, k, base);
|
|
|
+ yL = yd.length;
|
|
|
+ xL = xd.length;
|
|
|
+ }
|
|
|
+
|
|
|
+ xi = yL;
|
|
|
+ rem = xd.slice(0, yL);
|
|
|
+ remL = rem.length;
|
|
|
+
|
|
|
+ // Add zeros to make remainder as long as divisor.
|
|
|
+ for (; remL < yL;) rem[remL++] = 0;
|
|
|
+
|
|
|
+ yz = yd.slice();
|
|
|
+ yz.unshift(0);
|
|
|
+ yd0 = yd[0];
|
|
|
+
|
|
|
+ if (yd[1] >= base / 2) ++yd0;
|
|
|
+
|
|
|
+ do {
|
|
|
+ k = 0;
|
|
|
+
|
|
|
+ // Compare divisor and remainder.
|
|
|
+ cmp = compare(yd, rem, yL, remL);
|
|
|
+
|
|
|
+ // If divisor < remainder.
|
|
|
+ if (cmp < 0) {
|
|
|
+
|
|
|
+ // Calculate trial digit, k.
|
|
|
+ rem0 = rem[0];
|
|
|
+ if (yL != remL) rem0 = rem0 * base + (rem[1] || 0);
|
|
|
+
|
|
|
+ // k will be how many times the divisor goes into the current remainder.
|
|
|
+ k = rem0 / yd0 | 0;
|
|
|
+
|
|
|
+ // Algorithm:
|
|
|
+ // 1. product = divisor * trial digit (k)
|
|
|
+ // 2. if product > remainder: product -= divisor, k--
|
|
|
+ // 3. remainder -= product
|
|
|
+ // 4. if product was < remainder at 2:
|
|
|
+ // 5. compare new remainder and divisor
|
|
|
+ // 6. If remainder > divisor: remainder -= divisor, k++
|
|
|
+
|
|
|
+ if (k > 1) {
|
|
|
+ if (k >= base) k = base - 1;
|
|
|
+
|
|
|
+ // product = divisor * trial digit.
|
|
|
+ prod = multiplyInteger(yd, k, base);
|
|
|
+ prodL = prod.length;
|
|
|
+ remL = rem.length;
|
|
|
+
|
|
|
+ // Compare product and remainder.
|
|
|
+ cmp = compare(prod, rem, prodL, remL);
|
|
|
+
|
|
|
+ // product > remainder.
|
|
|
+ if (cmp == 1) {
|
|
|
+ k--;
|
|
|
+
|
|
|
+ // Subtract divisor from product.
|
|
|
+ subtract(prod, yL < prodL ? yz : yd, prodL, base);
|
|
|
+ }
|
|
|
+ } else {
|
|
|
+
|
|
|
+ // cmp is -1.
|
|
|
+ // If k is 0, there is no need to compare yd and rem again below, so change cmp to 1
|
|
|
+ // to avoid it. If k is 1 there is a need to compare yd and rem again below.
|
|
|
+ if (k == 0) cmp = k = 1;
|
|
|
+ prod = yd.slice();
|
|
|
+ }
|
|
|
+
|
|
|
+ prodL = prod.length;
|
|
|
+ if (prodL < remL) prod.unshift(0);
|
|
|
+
|
|
|
+ // Subtract product from remainder.
|
|
|
+ subtract(rem, prod, remL, base);
|
|
|
+
|
|
|
+ // If product was < previous remainder.
|
|
|
+ if (cmp == -1) {
|
|
|
+ remL = rem.length;
|
|
|
+
|
|
|
+ // Compare divisor and new remainder.
|
|
|
+ cmp = compare(yd, rem, yL, remL);
|
|
|
+
|
|
|
+ // If divisor < new remainder, subtract divisor from remainder.
|
|
|
+ if (cmp < 1) {
|
|
|
+ k++;
|
|
|
+
|
|
|
+ // Subtract divisor from remainder.
|
|
|
+ subtract(rem, yL < remL ? yz : yd, remL, base);
|
|
|
+ }
|
|
|
+ }
|
|
|
+
|
|
|
+ remL = rem.length;
|
|
|
+ } else if (cmp === 0) {
|
|
|
+ k++;
|
|
|
+ rem = [0];
|
|
|
+ } // if cmp === 1, k will be 0
|
|
|
+
|
|
|
+ // Add the next digit, k, to the result array.
|
|
|
+ qd[i++] = k;
|
|
|
+
|
|
|
+ // Update the remainder.
|
|
|
+ if (cmp && rem[0]) {
|
|
|
+ rem[remL++] = xd[xi] || 0;
|
|
|
+ } else {
|
|
|
+ rem = [xd[xi]];
|
|
|
+ remL = 1;
|
|
|
+ }
|
|
|
+
|
|
|
+ } while ((xi++ < xL || rem[0] !== void 0) && sd--);
|
|
|
+
|
|
|
+ more = rem[0] !== void 0;
|
|
|
+ }
|
|
|
+
|
|
|
+ // Leading zero?
|
|
|
+ if (!qd[0]) qd.shift();
|
|
|
+ }
|
|
|
+
|
|
|
+ // logBase is 1 when divide is being used for base conversion.
|
|
|
+ if (logBase == 1) {
|
|
|
+ q.e = e;
|
|
|
+ inexact = more;
|
|
|
+ } else {
|
|
|
+
|
|
|
+ // To calculate q.e, first get the number of digits of qd[0].
|
|
|
+ for (i = 1, k = qd[0]; k >= 10; k /= 10) i++;
|
|
|
+ q.e = i + e * logBase - 1;
|
|
|
+
|
|
|
+ finalise(q, dp ? pr + q.e + 1 : pr, rm, more);
|
|
|
+ }
|
|
|
+
|
|
|
+ return q;
|
|
|
+ };
|
|
|
+})();
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Round `x` to `sd` significant digits using rounding mode `rm`.
|
|
|
+ * Check for over/under-flow.
|
|
|
+ */
|
|
|
+function finalise(x, sd, rm, isTruncated) {
|
|
|
+ var digits, i, j, k, rd, roundUp, w, xd, xdi,
|
|
|
+ Ctor = x.constructor;
|
|
|
+
|
|
|
+ // Don't round if sd is null or undefined.
|
|
|
+ out: if (sd != null) {
|
|
|
+ xd = x.d;
|
|
|
+
|
|
|
+ // Infinity/NaN.
|
|
|
+ if (!xd) return x;
|
|
|
+
|
|
|
+ // rd: the rounding digit, i.e. the digit after the digit that may be rounded up.
|
|
|
+ // w: the word of xd containing rd, a base 1e7 number.
|
|
|
+ // xdi: the index of w within xd.
|
|
|
+ // digits: the number of digits of w.
|
|
|
+ // i: what would be the index of rd within w if all the numbers were 7 digits long (i.e. if
|
|
|
+ // they had leading zeros)
|
|
|
+ // j: if > 0, the actual index of rd within w (if < 0, rd is a leading zero).
|
|
|
+
|
|
|
+ // Get the length of the first word of the digits array xd.
|
|
|
+ for (digits = 1, k = xd[0]; k >= 10; k /= 10) digits++;
|
|
|
+ i = sd - digits;
|
|
|
+
|
|
|
+ // Is the rounding digit in the first word of xd?
|
|
|
+ if (i < 0) {
|
|
|
+ i += LOG_BASE;
|
|
|
+ j = sd;
|
|
|
+ w = xd[xdi = 0];
|
|
|
+
|
|
|
+ // Get the rounding digit at index j of w.
|
|
|
+ rd = w / mathpow(10, digits - j - 1) % 10 | 0;
|
|
|
+ } else {
|
|
|
+ xdi = Math.ceil((i + 1) / LOG_BASE);
|
|
|
+ k = xd.length;
|
|
|
+ if (xdi >= k) {
|
|
|
+ if (isTruncated) {
|
|
|
+
|
|
|
+ // Needed by `naturalExponential`, `naturalLogarithm` and `squareRoot`.
|
|
|
+ for (; k++ <= xdi;) xd.push(0);
|
|
|
+ w = rd = 0;
|
|
|
+ digits = 1;
|
|
|
+ i %= LOG_BASE;
|
|
|
+ j = i - LOG_BASE + 1;
|
|
|
+ } else {
|
|
|
+ break out;
|
|
|
+ }
|
|
|
+ } else {
|
|
|
+ w = k = xd[xdi];
|
|
|
+
|
|
|
+ // Get the number of digits of w.
|
|
|
+ for (digits = 1; k >= 10; k /= 10) digits++;
|
|
|
+
|
|
|
+ // Get the index of rd within w.
|
|
|
+ i %= LOG_BASE;
|
|
|
+
|
|
|
+ // Get the index of rd within w, adjusted for leading zeros.
|
|
|
+ // The number of leading zeros of w is given by LOG_BASE - digits.
|
|
|
+ j = i - LOG_BASE + digits;
|
|
|
+
|
|
|
+ // Get the rounding digit at index j of w.
|
|
|
+ rd = j < 0 ? 0 : w / mathpow(10, digits - j - 1) % 10 | 0;
|
|
|
+ }
|
|
|
+ }
|
|
|
+
|
|
|
+ // Are there any non-zero digits after the rounding digit?
|
|
|
+ isTruncated = isTruncated || sd < 0 ||
|
|
|
+ xd[xdi + 1] !== void 0 || (j < 0 ? w : w % mathpow(10, digits - j - 1));
|
|
|
+
|
|
|
+ // The expression `w % mathpow(10, digits - j - 1)` returns all the digits of w to the right
|
|
|
+ // of the digit at (left-to-right) index j, e.g. if w is 908714 and j is 2, the expression
|
|
|
+ // will give 714.
|
|
|
+
|
|
|
+ roundUp = rm < 4 ?
|
|
|
+ (rd || isTruncated) && (rm == 0 || rm == (x.s < 0 ? 3 : 2)) :
|
|
|
+ rd > 5 || rd == 5 && (rm == 4 || isTruncated || rm == 6 &&
|
|
|
+
|
|
|
+ // Check whether the digit to the left of the rounding digit is odd.
|
|
|
+ ((i > 0 ? j > 0 ? w / mathpow(10, digits - j) : 0 : xd[xdi - 1]) % 10) & 1 ||
|
|
|
+ rm == (x.s < 0 ? 8 : 7));
|
|
|
+
|
|
|
+ if (sd < 1 || !xd[0]) {
|
|
|
+ xd.length = 0;
|
|
|
+ if (roundUp) {
|
|
|
+
|
|
|
+ // Convert sd to decimal places.
|
|
|
+ sd -= x.e + 1;
|
|
|
+
|
|
|
+ // 1, 0.1, 0.01, 0.001, 0.0001 etc.
|
|
|
+ xd[0] = mathpow(10, (LOG_BASE - sd % LOG_BASE) % LOG_BASE);
|
|
|
+ x.e = -sd || 0;
|
|
|
+ } else {
|
|
|
+
|
|
|
+ // Zero.
|
|
|
+ xd[0] = x.e = 0;
|
|
|
+ }
|
|
|
+
|
|
|
+ return x;
|
|
|
+ }
|
|
|
+
|
|
|
+ // Remove excess digits.
|
|
|
+ if (i == 0) {
|
|
|
+ xd.length = xdi;
|
|
|
+ k = 1;
|
|
|
+ xdi--;
|
|
|
+ } else {
|
|
|
+ xd.length = xdi + 1;
|
|
|
+ k = mathpow(10, LOG_BASE - i);
|
|
|
+
|
|
|
+ // E.g. 56700 becomes 56000 if 7 is the rounding digit.
|
|
|
+ // j > 0 means i > number of leading zeros of w.
|
|
|
+ xd[xdi] = j > 0 ? (w / mathpow(10, digits - j) % mathpow(10, j) | 0) * k : 0;
|
|
|
+ }
|
|
|
+
|
|
|
+ if (roundUp) {
|
|
|
+ for (;;) {
|
|
|
+
|
|
|
+ // Is the digit to be rounded up in the first word of xd?
|
|
|
+ if (xdi == 0) {
|
|
|
+
|
|
|
+ // i will be the length of xd[0] before k is added.
|
|
|
+ for (i = 1, j = xd[0]; j >= 10; j /= 10) i++;
|
|
|
+ j = xd[0] += k;
|
|
|
+ for (k = 1; j >= 10; j /= 10) k++;
|
|
|
+
|
|
|
+ // if i != k the length has increased.
|
|
|
+ if (i != k) {
|
|
|
+ x.e++;
|
|
|
+ if (xd[0] == BASE) xd[0] = 1;
|
|
|
+ }
|
|
|
+
|
|
|
+ break;
|
|
|
+ } else {
|
|
|
+ xd[xdi] += k;
|
|
|
+ if (xd[xdi] != BASE) break;
|
|
|
+ xd[xdi--] = 0;
|
|
|
+ k = 1;
|
|
|
+ }
|
|
|
+ }
|
|
|
+ }
|
|
|
+
|
|
|
+ // Remove trailing zeros.
|
|
|
+ for (i = xd.length; xd[--i] === 0;) xd.pop();
|
|
|
+ }
|
|
|
+
|
|
|
+ if (external) {
|
|
|
+
|
|
|
+ // Overflow?
|
|
|
+ if (x.e > Ctor.maxE) {
|
|
|
+
|
|
|
+ // Infinity.
|
|
|
+ x.d = null;
|
|
|
+ x.e = NaN;
|
|
|
+
|
|
|
+ // Underflow?
|
|
|
+ } else if (x.e < Ctor.minE) {
|
|
|
+
|
|
|
+ // Zero.
|
|
|
+ x.e = 0;
|
|
|
+ x.d = [0];
|
|
|
+ // Ctor.underflow = true;
|
|
|
+ } // else Ctor.underflow = false;
|
|
|
+ }
|
|
|
+
|
|
|
+ return x;
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+function finiteToString(x, isExp, sd) {
|
|
|
+ if (!x.isFinite()) return nonFiniteToString(x);
|
|
|
+ var k,
|
|
|
+ e = x.e,
|
|
|
+ str = digitsToString(x.d),
|
|
|
+ len = str.length;
|
|
|
+
|
|
|
+ if (isExp) {
|
|
|
+ if (sd && (k = sd - len) > 0) {
|
|
|
+ str = str.charAt(0) + '.' + str.slice(1) + getZeroString(k);
|
|
|
+ } else if (len > 1) {
|
|
|
+ str = str.charAt(0) + '.' + str.slice(1);
|
|
|
+ }
|
|
|
+
|
|
|
+ str = str + (x.e < 0 ? 'e' : 'e+') + x.e;
|
|
|
+ } else if (e < 0) {
|
|
|
+ str = '0.' + getZeroString(-e - 1) + str;
|
|
|
+ if (sd && (k = sd - len) > 0) str += getZeroString(k);
|
|
|
+ } else if (e >= len) {
|
|
|
+ str += getZeroString(e + 1 - len);
|
|
|
+ if (sd && (k = sd - e - 1) > 0) str = str + '.' + getZeroString(k);
|
|
|
+ } else {
|
|
|
+ if ((k = e + 1) < len) str = str.slice(0, k) + '.' + str.slice(k);
|
|
|
+ if (sd && (k = sd - len) > 0) {
|
|
|
+ if (e + 1 === len) str += '.';
|
|
|
+ str += getZeroString(k);
|
|
|
+ }
|
|
|
+ }
|
|
|
+
|
|
|
+ return str;
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+// Calculate the base 10 exponent from the base 1e7 exponent.
|
|
|
+function getBase10Exponent(digits, e) {
|
|
|
+ var w = digits[0];
|
|
|
+
|
|
|
+ // Add the number of digits of the first word of the digits array.
|
|
|
+ for (e *= LOG_BASE; w >= 10; w /= 10) e++;
|
|
|
+ return e;
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+function getLn10(Ctor, sd, pr) {
|
|
|
+ if (sd > LN10_PRECISION) {
|
|
|
+
|
|
|
+ // Reset global state in case the exception is caught.
|
|
|
+ external = true;
|
|
|
+ if (pr) Ctor.precision = pr;
|
|
|
+ throw Error(precisionLimitExceeded);
|
|
|
+ }
|
|
|
+ return finalise(new Ctor(LN10), sd, 1, true);
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+function getPi(Ctor, sd, rm) {
|
|
|
+ if (sd > PI_PRECISION) throw Error(precisionLimitExceeded);
|
|
|
+ return finalise(new Ctor(PI), sd, rm, true);
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+function getPrecision(digits) {
|
|
|
+ var w = digits.length - 1,
|
|
|
+ len = w * LOG_BASE + 1;
|
|
|
+
|
|
|
+ w = digits[w];
|
|
|
+
|
|
|
+ // If non-zero...
|
|
|
+ if (w) {
|
|
|
+
|
|
|
+ // Subtract the number of trailing zeros of the last word.
|
|
|
+ for (; w % 10 == 0; w /= 10) len--;
|
|
|
+
|
|
|
+ // Add the number of digits of the first word.
|
|
|
+ for (w = digits[0]; w >= 10; w /= 10) len++;
|
|
|
+ }
|
|
|
+
|
|
|
+ return len;
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+function getZeroString(k) {
|
|
|
+ var zs = '';
|
|
|
+ for (; k--;) zs += '0';
|
|
|
+ return zs;
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is the value of Decimal `x` to the power `n`, where `n` is an
|
|
|
+ * integer of type number.
|
|
|
+ *
|
|
|
+ * Implements 'exponentiation by squaring'. Called by `pow` and `parseOther`.
|
|
|
+ *
|
|
|
+ */
|
|
|
+function intPow(Ctor, x, n, pr) {
|
|
|
+ var isTruncated,
|
|
|
+ r = new Ctor(1),
|
|
|
+
|
|
|
+ // Max n of 9007199254740991 takes 53 loop iterations.
|
|
|
+ // Maximum digits array length; leaves [28, 34] guard digits.
|
|
|
+ k = Math.ceil(pr / LOG_BASE + 4);
|
|
|
+
|
|
|
+ external = false;
|
|
|
+
|
|
|
+ for (;;) {
|
|
|
+ if (n % 2) {
|
|
|
+ r = r.times(x);
|
|
|
+ if (truncate(r.d, k)) isTruncated = true;
|
|
|
+ }
|
|
|
+
|
|
|
+ n = mathfloor(n / 2);
|
|
|
+ if (n === 0) {
|
|
|
+
|
|
|
+ // To ensure correct rounding when r.d is truncated, increment the last word if it is zero.
|
|
|
+ n = r.d.length - 1;
|
|
|
+ if (isTruncated && r.d[n] === 0) ++r.d[n];
|
|
|
+ break;
|
|
|
+ }
|
|
|
+
|
|
|
+ x = x.times(x);
|
|
|
+ truncate(x.d, k);
|
|
|
+ }
|
|
|
+
|
|
|
+ external = true;
|
|
|
+
|
|
|
+ return r;
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+function isOdd(n) {
|
|
|
+ return n.d[n.d.length - 1] & 1;
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Handle `max` and `min`. `ltgt` is 'lt' or 'gt'.
|
|
|
+ */
|
|
|
+function maxOrMin(Ctor, args, ltgt) {
|
|
|
+ var y,
|
|
|
+ x = new Ctor(args[0]),
|
|
|
+ i = 0;
|
|
|
+
|
|
|
+ for (; ++i < args.length;) {
|
|
|
+ y = new Ctor(args[i]);
|
|
|
+ if (!y.s) {
|
|
|
+ x = y;
|
|
|
+ break;
|
|
|
+ } else if (x[ltgt](y)) {
|
|
|
+ x = y;
|
|
|
+ }
|
|
|
+ }
|
|
|
+
|
|
|
+ return x;
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is the natural exponential of `x` rounded to `sd` significant
|
|
|
+ * digits.
|
|
|
+ *
|
|
|
+ * Taylor/Maclaurin series.
|
|
|
+ *
|
|
|
+ * exp(x) = x^0/0! + x^1/1! + x^2/2! + x^3/3! + ...
|
|
|
+ *
|
|
|
+ * Argument reduction:
|
|
|
+ * Repeat x = x / 32, k += 5, until |x| < 0.1
|
|
|
+ * exp(x) = exp(x / 2^k)^(2^k)
|
|
|
+ *
|
|
|
+ * Previously, the argument was initially reduced by
|
|
|
+ * exp(x) = exp(r) * 10^k where r = x - k * ln10, k = floor(x / ln10)
|
|
|
+ * to first put r in the range [0, ln10], before dividing by 32 until |x| < 0.1, but this was
|
|
|
+ * found to be slower than just dividing repeatedly by 32 as above.
|
|
|
+ *
|
|
|
+ * Max integer argument: exp('20723265836946413') = 6.3e+9000000000000000
|
|
|
+ * Min integer argument: exp('-20723265836946411') = 1.2e-9000000000000000
|
|
|
+ * (Math object integer min/max: Math.exp(709) = 8.2e+307, Math.exp(-745) = 5e-324)
|
|
|
+ *
|
|
|
+ * exp(Infinity) = Infinity
|
|
|
+ * exp(-Infinity) = 0
|
|
|
+ * exp(NaN) = NaN
|
|
|
+ * exp(±0) = 1
|
|
|
+ *
|
|
|
+ * exp(x) is non-terminating for any finite, non-zero x.
|
|
|
+ *
|
|
|
+ * The result will always be correctly rounded.
|
|
|
+ *
|
|
|
+ */
|
|
|
+function naturalExponential(x, sd) {
|
|
|
+ var denominator, guard, j, pow, sum, t, wpr,
|
|
|
+ rep = 0,
|
|
|
+ i = 0,
|
|
|
+ k = 0,
|
|
|
+ Ctor = x.constructor,
|
|
|
+ rm = Ctor.rounding,
|
|
|
+ pr = Ctor.precision;
|
|
|
+
|
|
|
+ // 0/NaN/Infinity?
|
|
|
+ if (!x.d || !x.d[0] || x.e > 17) {
|
|
|
+
|
|
|
+ return new Ctor(x.d ?
|
|
|
+ !x.d[0] ? 1 : x.s < 0 ? 0 : 1 / 0 :
|
|
|
+ x.s ? x.s < 0 ? 0 : x : 0 / 0);
|
|
|
+ }
|
|
|
+
|
|
|
+ if (sd == null) {
|
|
|
+ external = false;
|
|
|
+ wpr = pr;
|
|
|
+ } else {
|
|
|
+ wpr = sd;
|
|
|
+ }
|
|
|
+
|
|
|
+ t = new Ctor(0.03125);
|
|
|
+
|
|
|
+ // while abs(x) >= 0.1
|
|
|
+ while (x.e > -2) {
|
|
|
+
|
|
|
+ // x = x / 2^5
|
|
|
+ x = x.times(t);
|
|
|
+ k += 5;
|
|
|
+ }
|
|
|
+
|
|
|
+ // Use 2 * log10(2^k) + 5 (empirically derived) to estimate the increase in precision
|
|
|
+ // necessary to ensure the first 4 rounding digits are correct.
|
|
|
+ guard = Math.log(mathpow(2, k)) / Math.LN10 * 2 + 5 | 0;
|
|
|
+ wpr += guard;
|
|
|
+ denominator = pow = sum = new Ctor(1);
|
|
|
+ Ctor.precision = wpr;
|
|
|
+
|
|
|
+ for (;;) {
|
|
|
+ pow = finalise(pow.times(x), wpr, 1);
|
|
|
+ denominator = denominator.times(++i);
|
|
|
+ t = sum.plus(divide(pow, denominator, wpr, 1));
|
|
|
+
|
|
|
+ if (digitsToString(t.d).slice(0, wpr) === digitsToString(sum.d).slice(0, wpr)) {
|
|
|
+ j = k;
|
|
|
+ while (j--) sum = finalise(sum.times(sum), wpr, 1);
|
|
|
+
|
|
|
+ // Check to see if the first 4 rounding digits are [49]999.
|
|
|
+ // If so, repeat the summation with a higher precision, otherwise
|
|
|
+ // e.g. with precision: 18, rounding: 1
|
|
|
+ // exp(18.404272462595034083567793919843761) = 98372560.1229999999 (should be 98372560.123)
|
|
|
+ // `wpr - guard` is the index of first rounding digit.
|
|
|
+ if (sd == null) {
|
|
|
+
|
|
|
+ if (rep < 3 && checkRoundingDigits(sum.d, wpr - guard, rm, rep)) {
|
|
|
+ Ctor.precision = wpr += 10;
|
|
|
+ denominator = pow = t = new Ctor(1);
|
|
|
+ i = 0;
|
|
|
+ rep++;
|
|
|
+ } else {
|
|
|
+ return finalise(sum, Ctor.precision = pr, rm, external = true);
|
|
|
+ }
|
|
|
+ } else {
|
|
|
+ Ctor.precision = pr;
|
|
|
+ return sum;
|
|
|
+ }
|
|
|
+ }
|
|
|
+
|
|
|
+ sum = t;
|
|
|
+ }
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is the natural logarithm of `x` rounded to `sd` significant
|
|
|
+ * digits.
|
|
|
+ *
|
|
|
+ * ln(-n) = NaN
|
|
|
+ * ln(0) = -Infinity
|
|
|
+ * ln(-0) = -Infinity
|
|
|
+ * ln(1) = 0
|
|
|
+ * ln(Infinity) = Infinity
|
|
|
+ * ln(-Infinity) = NaN
|
|
|
+ * ln(NaN) = NaN
|
|
|
+ *
|
|
|
+ * ln(n) (n != 1) is non-terminating.
|
|
|
+ *
|
|
|
+ */
|
|
|
+function naturalLogarithm(y, sd) {
|
|
|
+ var c, c0, denominator, e, numerator, rep, sum, t, wpr, x1, x2,
|
|
|
+ n = 1,
|
|
|
+ guard = 10,
|
|
|
+ x = y,
|
|
|
+ xd = x.d,
|
|
|
+ Ctor = x.constructor,
|
|
|
+ rm = Ctor.rounding,
|
|
|
+ pr = Ctor.precision;
|
|
|
+
|
|
|
+ // Is x negative or Infinity, NaN, 0 or 1?
|
|
|
+ if (x.s < 0 || !xd || !xd[0] || !x.e && xd[0] == 1 && xd.length == 1) {
|
|
|
+ return new Ctor(xd && !xd[0] ? -1 / 0 : x.s != 1 ? NaN : xd ? 0 : x);
|
|
|
+ }
|
|
|
+
|
|
|
+ if (sd == null) {
|
|
|
+ external = false;
|
|
|
+ wpr = pr;
|
|
|
+ } else {
|
|
|
+ wpr = sd;
|
|
|
+ }
|
|
|
+
|
|
|
+ Ctor.precision = wpr += guard;
|
|
|
+ c = digitsToString(xd);
|
|
|
+ c0 = c.charAt(0);
|
|
|
+
|
|
|
+ if (Math.abs(e = x.e) < 1.5e15) {
|
|
|
+
|
|
|
+ // Argument reduction.
|
|
|
+ // The series converges faster the closer the argument is to 1, so using
|
|
|
+ // ln(a^b) = b * ln(a), ln(a) = ln(a^b) / b
|
|
|
+ // multiply the argument by itself until the leading digits of the significand are 7, 8, 9,
|
|
|
+ // 10, 11, 12 or 13, recording the number of multiplications so the sum of the series can
|
|
|
+ // later be divided by this number, then separate out the power of 10 using
|
|
|
+ // ln(a*10^b) = ln(a) + b*ln(10).
|
|
|
+
|
|
|
+ // max n is 21 (gives 0.9, 1.0 or 1.1) (9e15 / 21 = 4.2e14).
|
|
|
+ //while (c0 < 9 && c0 != 1 || c0 == 1 && c.charAt(1) > 1) {
|
|
|
+ // max n is 6 (gives 0.7 - 1.3)
|
|
|
+ while (c0 < 7 && c0 != 1 || c0 == 1 && c.charAt(1) > 3) {
|
|
|
+ x = x.times(y);
|
|
|
+ c = digitsToString(x.d);
|
|
|
+ c0 = c.charAt(0);
|
|
|
+ n++;
|
|
|
+ }
|
|
|
+
|
|
|
+ e = x.e;
|
|
|
+
|
|
|
+ if (c0 > 1) {
|
|
|
+ x = new Ctor('0.' + c);
|
|
|
+ e++;
|
|
|
+ } else {
|
|
|
+ x = new Ctor(c0 + '.' + c.slice(1));
|
|
|
+ }
|
|
|
+ } else {
|
|
|
+
|
|
|
+ // The argument reduction method above may result in overflow if the argument y is a massive
|
|
|
+ // number with exponent >= 1500000000000000 (9e15 / 6 = 1.5e15), so instead recall this
|
|
|
+ // function using ln(x*10^e) = ln(x) + e*ln(10).
|
|
|
+ t = getLn10(Ctor, wpr + 2, pr).times(e + '');
|
|
|
+ x = naturalLogarithm(new Ctor(c0 + '.' + c.slice(1)), wpr - guard).plus(t);
|
|
|
+ Ctor.precision = pr;
|
|
|
+
|
|
|
+ return sd == null ? finalise(x, pr, rm, external = true) : x;
|
|
|
+ }
|
|
|
+
|
|
|
+ // x1 is x reduced to a value near 1.
|
|
|
+ x1 = x;
|
|
|
+
|
|
|
+ // Taylor series.
|
|
|
+ // ln(y) = ln((1 + x)/(1 - x)) = 2(x + x^3/3 + x^5/5 + x^7/7 + ...)
|
|
|
+ // where x = (y - 1)/(y + 1) (|x| < 1)
|
|
|
+ sum = numerator = x = divide(x.minus(1), x.plus(1), wpr, 1);
|
|
|
+ x2 = finalise(x.times(x), wpr, 1);
|
|
|
+ denominator = 3;
|
|
|
+
|
|
|
+ for (;;) {
|
|
|
+ numerator = finalise(numerator.times(x2), wpr, 1);
|
|
|
+ t = sum.plus(divide(numerator, new Ctor(denominator), wpr, 1));
|
|
|
+
|
|
|
+ if (digitsToString(t.d).slice(0, wpr) === digitsToString(sum.d).slice(0, wpr)) {
|
|
|
+ sum = sum.times(2);
|
|
|
+
|
|
|
+ // Reverse the argument reduction. Check that e is not 0 because, besides preventing an
|
|
|
+ // unnecessary calculation, -0 + 0 = +0 and to ensure correct rounding -0 needs to stay -0.
|
|
|
+ if (e !== 0) sum = sum.plus(getLn10(Ctor, wpr + 2, pr).times(e + ''));
|
|
|
+ sum = divide(sum, new Ctor(n), wpr, 1);
|
|
|
+
|
|
|
+ // Is rm > 3 and the first 4 rounding digits 4999, or rm < 4 (or the summation has
|
|
|
+ // been repeated previously) and the first 4 rounding digits 9999?
|
|
|
+ // If so, restart the summation with a higher precision, otherwise
|
|
|
+ // e.g. with precision: 12, rounding: 1
|
|
|
+ // ln(135520028.6126091714265381533) = 18.7246299999 when it should be 18.72463.
|
|
|
+ // `wpr - guard` is the index of first rounding digit.
|
|
|
+ if (sd == null) {
|
|
|
+ if (checkRoundingDigits(sum.d, wpr - guard, rm, rep)) {
|
|
|
+ Ctor.precision = wpr += guard;
|
|
|
+ t = numerator = x = divide(x1.minus(1), x1.plus(1), wpr, 1);
|
|
|
+ x2 = finalise(x.times(x), wpr, 1);
|
|
|
+ denominator = rep = 1;
|
|
|
+ } else {
|
|
|
+ return finalise(sum, Ctor.precision = pr, rm, external = true);
|
|
|
+ }
|
|
|
+ } else {
|
|
|
+ Ctor.precision = pr;
|
|
|
+ return sum;
|
|
|
+ }
|
|
|
+ }
|
|
|
+
|
|
|
+ sum = t;
|
|
|
+ denominator += 2;
|
|
|
+ }
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+// ±Infinity, NaN.
|
|
|
+function nonFiniteToString(x) {
|
|
|
+ // Unsigned.
|
|
|
+ return String(x.s * x.s / 0);
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Parse the value of a new Decimal `x` from string `str`.
|
|
|
+ */
|
|
|
+function parseDecimal(x, str) {
|
|
|
+ var e, i, len;
|
|
|
+
|
|
|
+ // Decimal point?
|
|
|
+ if ((e = str.indexOf('.')) > -1) str = str.replace('.', '');
|
|
|
+
|
|
|
+ // Exponential form?
|
|
|
+ if ((i = str.search(/e/i)) > 0) {
|
|
|
+
|
|
|
+ // Determine exponent.
|
|
|
+ if (e < 0) e = i;
|
|
|
+ e += +str.slice(i + 1);
|
|
|
+ str = str.substring(0, i);
|
|
|
+ } else if (e < 0) {
|
|
|
+
|
|
|
+ // Integer.
|
|
|
+ e = str.length;
|
|
|
+ }
|
|
|
+
|
|
|
+ // Determine leading zeros.
|
|
|
+ for (i = 0; str.charCodeAt(i) === 48; i++);
|
|
|
+
|
|
|
+ // Determine trailing zeros.
|
|
|
+ for (len = str.length; str.charCodeAt(len - 1) === 48; --len);
|
|
|
+ str = str.slice(i, len);
|
|
|
+
|
|
|
+ if (str) {
|
|
|
+ len -= i;
|
|
|
+ x.e = e = e - i - 1;
|
|
|
+ x.d = [];
|
|
|
+
|
|
|
+ // Transform base
|
|
|
+
|
|
|
+ // e is the base 10 exponent.
|
|
|
+ // i is where to slice str to get the first word of the digits array.
|
|
|
+ i = (e + 1) % LOG_BASE;
|
|
|
+ if (e < 0) i += LOG_BASE;
|
|
|
+
|
|
|
+ if (i < len) {
|
|
|
+ if (i) x.d.push(+str.slice(0, i));
|
|
|
+ for (len -= LOG_BASE; i < len;) x.d.push(+str.slice(i, i += LOG_BASE));
|
|
|
+ str = str.slice(i);
|
|
|
+ i = LOG_BASE - str.length;
|
|
|
+ } else {
|
|
|
+ i -= len;
|
|
|
+ }
|
|
|
+
|
|
|
+ for (; i--;) str += '0';
|
|
|
+ x.d.push(+str);
|
|
|
+
|
|
|
+ if (external) {
|
|
|
+
|
|
|
+ // Overflow?
|
|
|
+ if (x.e > x.constructor.maxE) {
|
|
|
+
|
|
|
+ // Infinity.
|
|
|
+ x.d = null;
|
|
|
+ x.e = NaN;
|
|
|
+
|
|
|
+ // Underflow?
|
|
|
+ } else if (x.e < x.constructor.minE) {
|
|
|
+
|
|
|
+ // Zero.
|
|
|
+ x.e = 0;
|
|
|
+ x.d = [0];
|
|
|
+ // x.constructor.underflow = true;
|
|
|
+ } // else x.constructor.underflow = false;
|
|
|
+ }
|
|
|
+ } else {
|
|
|
+
|
|
|
+ // Zero.
|
|
|
+ x.e = 0;
|
|
|
+ x.d = [0];
|
|
|
+ }
|
|
|
+
|
|
|
+ return x;
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Parse the value of a new Decimal `x` from a string `str`, which is not a decimal value.
|
|
|
+ */
|
|
|
+function parseOther(x, str) {
|
|
|
+ var base, Ctor, divisor, i, isFloat, len, p, xd, xe;
|
|
|
+
|
|
|
+ if (str.indexOf('_') > -1) {
|
|
|
+ str = str.replace(/(\d)_(?=\d)/g, '$1');
|
|
|
+ if (isDecimal.test(str)) return parseDecimal(x, str);
|
|
|
+ } else if (str === 'Infinity' || str === 'NaN') {
|
|
|
+ if (!+str) x.s = NaN;
|
|
|
+ x.e = NaN;
|
|
|
+ x.d = null;
|
|
|
+ return x;
|
|
|
+ }
|
|
|
+
|
|
|
+ if (isHex.test(str)) {
|
|
|
+ base = 16;
|
|
|
+ str = str.toLowerCase();
|
|
|
+ } else if (isBinary.test(str)) {
|
|
|
+ base = 2;
|
|
|
+ } else if (isOctal.test(str)) {
|
|
|
+ base = 8;
|
|
|
+ } else {
|
|
|
+ throw Error(invalidArgument + str);
|
|
|
+ }
|
|
|
+
|
|
|
+ // Is there a binary exponent part?
|
|
|
+ i = str.search(/p/i);
|
|
|
+
|
|
|
+ if (i > 0) {
|
|
|
+ p = +str.slice(i + 1);
|
|
|
+ str = str.substring(2, i);
|
|
|
+ } else {
|
|
|
+ str = str.slice(2);
|
|
|
+ }
|
|
|
+
|
|
|
+ // Convert `str` as an integer then divide the result by `base` raised to a power such that the
|
|
|
+ // fraction part will be restored.
|
|
|
+ i = str.indexOf('.');
|
|
|
+ isFloat = i >= 0;
|
|
|
+ Ctor = x.constructor;
|
|
|
+
|
|
|
+ if (isFloat) {
|
|
|
+ str = str.replace('.', '');
|
|
|
+ len = str.length;
|
|
|
+ i = len - i;
|
|
|
+
|
|
|
+ // log[10](16) = 1.2041... , log[10](88) = 1.9444....
|
|
|
+ divisor = intPow(Ctor, new Ctor(base), i, i * 2);
|
|
|
+ }
|
|
|
+
|
|
|
+ xd = convertBase(str, base, BASE);
|
|
|
+ xe = xd.length - 1;
|
|
|
+
|
|
|
+ // Remove trailing zeros.
|
|
|
+ for (i = xe; xd[i] === 0; --i) xd.pop();
|
|
|
+ if (i < 0) return new Ctor(x.s * 0);
|
|
|
+ x.e = getBase10Exponent(xd, xe);
|
|
|
+ x.d = xd;
|
|
|
+ external = false;
|
|
|
+
|
|
|
+ // At what precision to perform the division to ensure exact conversion?
|
|
|
+ // maxDecimalIntegerPartDigitCount = ceil(log[10](b) * otherBaseIntegerPartDigitCount)
|
|
|
+ // log[10](2) = 0.30103, log[10](8) = 0.90309, log[10](16) = 1.20412
|
|
|
+ // E.g. ceil(1.2 * 3) = 4, so up to 4 decimal digits are needed to represent 3 hex int digits.
|
|
|
+ // maxDecimalFractionPartDigitCount = {Hex:4|Oct:3|Bin:1} * otherBaseFractionPartDigitCount
|
|
|
+ // Therefore using 4 * the number of digits of str will always be enough.
|
|
|
+ if (isFloat) x = divide(x, divisor, len * 4);
|
|
|
+
|
|
|
+ // Multiply by the binary exponent part if present.
|
|
|
+ if (p) x = x.times(Math.abs(p) < 54 ? mathpow(2, p) : Decimal.pow(2, p));
|
|
|
+ external = true;
|
|
|
+
|
|
|
+ return x;
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * sin(x) = x - x^3/3! + x^5/5! - ...
|
|
|
+ * |x| < pi/2
|
|
|
+ *
|
|
|
+ */
|
|
|
+function sine(Ctor, x) {
|
|
|
+ var k,
|
|
|
+ len = x.d.length;
|
|
|
+
|
|
|
+ if (len < 3) {
|
|
|
+ return x.isZero() ? x : taylorSeries(Ctor, 2, x, x);
|
|
|
+ }
|
|
|
+
|
|
|
+ // Argument reduction: sin(5x) = 16*sin^5(x) - 20*sin^3(x) + 5*sin(x)
|
|
|
+ // i.e. sin(x) = 16*sin^5(x/5) - 20*sin^3(x/5) + 5*sin(x/5)
|
|
|
+ // and sin(x) = sin(x/5)(5 + sin^2(x/5)(16sin^2(x/5) - 20))
|
|
|
+
|
|
|
+ // Estimate the optimum number of times to use the argument reduction.
|
|
|
+ k = 1.4 * Math.sqrt(len);
|
|
|
+ k = k > 16 ? 16 : k | 0;
|
|
|
+
|
|
|
+ x = x.times(1 / tinyPow(5, k));
|
|
|
+ x = taylorSeries(Ctor, 2, x, x);
|
|
|
+
|
|
|
+ // Reverse argument reduction
|
|
|
+ var sin2_x,
|
|
|
+ d5 = new Ctor(5),
|
|
|
+ d16 = new Ctor(16),
|
|
|
+ d20 = new Ctor(20);
|
|
|
+ for (; k--;) {
|
|
|
+ sin2_x = x.times(x);
|
|
|
+ x = x.times(d5.plus(sin2_x.times(d16.times(sin2_x).minus(d20))));
|
|
|
+ }
|
|
|
+
|
|
|
+ return x;
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+// Calculate Taylor series for `cos`, `cosh`, `sin` and `sinh`.
|
|
|
+function taylorSeries(Ctor, n, x, y, isHyperbolic) {
|
|
|
+ var j, t, u, x2,
|
|
|
+ i = 1,
|
|
|
+ pr = Ctor.precision,
|
|
|
+ k = Math.ceil(pr / LOG_BASE);
|
|
|
+
|
|
|
+ external = false;
|
|
|
+ x2 = x.times(x);
|
|
|
+ u = new Ctor(y);
|
|
|
+
|
|
|
+ for (;;) {
|
|
|
+ t = divide(u.times(x2), new Ctor(n++ * n++), pr, 1);
|
|
|
+ u = isHyperbolic ? y.plus(t) : y.minus(t);
|
|
|
+ y = divide(t.times(x2), new Ctor(n++ * n++), pr, 1);
|
|
|
+ t = u.plus(y);
|
|
|
+
|
|
|
+ if (t.d[k] !== void 0) {
|
|
|
+ for (j = k; t.d[j] === u.d[j] && j--;);
|
|
|
+ if (j == -1) break;
|
|
|
+ }
|
|
|
+
|
|
|
+ j = u;
|
|
|
+ u = y;
|
|
|
+ y = t;
|
|
|
+ t = j;
|
|
|
+ i++;
|
|
|
+ }
|
|
|
+
|
|
|
+ external = true;
|
|
|
+ t.d.length = k + 1;
|
|
|
+
|
|
|
+ return t;
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+// Exponent e must be positive and non-zero.
|
|
|
+function tinyPow(b, e) {
|
|
|
+ var n = b;
|
|
|
+ while (--e) n *= b;
|
|
|
+ return n;
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+// Return the absolute value of `x` reduced to less than or equal to half pi.
|
|
|
+function toLessThanHalfPi(Ctor, x) {
|
|
|
+ var t,
|
|
|
+ isNeg = x.s < 0,
|
|
|
+ pi = getPi(Ctor, Ctor.precision, 1),
|
|
|
+ halfPi = pi.times(0.5);
|
|
|
+
|
|
|
+ x = x.abs();
|
|
|
+
|
|
|
+ if (x.lte(halfPi)) {
|
|
|
+ quadrant = isNeg ? 4 : 1;
|
|
|
+ return x;
|
|
|
+ }
|
|
|
+
|
|
|
+ t = x.divToInt(pi);
|
|
|
+
|
|
|
+ if (t.isZero()) {
|
|
|
+ quadrant = isNeg ? 3 : 2;
|
|
|
+ } else {
|
|
|
+ x = x.minus(t.times(pi));
|
|
|
+
|
|
|
+ // 0 <= x < pi
|
|
|
+ if (x.lte(halfPi)) {
|
|
|
+ quadrant = isOdd(t) ? (isNeg ? 2 : 3) : (isNeg ? 4 : 1);
|
|
|
+ return x;
|
|
|
+ }
|
|
|
+
|
|
|
+ quadrant = isOdd(t) ? (isNeg ? 1 : 4) : (isNeg ? 3 : 2);
|
|
|
+ }
|
|
|
+
|
|
|
+ return x.minus(pi).abs();
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return the value of Decimal `x` as a string in base `baseOut`.
|
|
|
+ *
|
|
|
+ * If the optional `sd` argument is present include a binary exponent suffix.
|
|
|
+ */
|
|
|
+function toStringBinary(x, baseOut, sd, rm) {
|
|
|
+ var base, e, i, k, len, roundUp, str, xd, y,
|
|
|
+ Ctor = x.constructor,
|
|
|
+ isExp = sd !== void 0;
|
|
|
+
|
|
|
+ if (isExp) {
|
|
|
+ checkInt32(sd, 1, MAX_DIGITS);
|
|
|
+ if (rm === void 0) rm = Ctor.rounding;
|
|
|
+ else checkInt32(rm, 0, 8);
|
|
|
+ } else {
|
|
|
+ sd = Ctor.precision;
|
|
|
+ rm = Ctor.rounding;
|
|
|
+ }
|
|
|
+
|
|
|
+ if (!x.isFinite()) {
|
|
|
+ str = nonFiniteToString(x);
|
|
|
+ } else {
|
|
|
+ str = finiteToString(x);
|
|
|
+ i = str.indexOf('.');
|
|
|
+
|
|
|
+ // Use exponential notation according to `toExpPos` and `toExpNeg`? No, but if required:
|
|
|
+ // maxBinaryExponent = floor((decimalExponent + 1) * log[2](10))
|
|
|
+ // minBinaryExponent = floor(decimalExponent * log[2](10))
|
|
|
+ // log[2](10) = 3.321928094887362347870319429489390175864
|
|
|
+
|
|
|
+ if (isExp) {
|
|
|
+ base = 2;
|
|
|
+ if (baseOut == 16) {
|
|
|
+ sd = sd * 4 - 3;
|
|
|
+ } else if (baseOut == 8) {
|
|
|
+ sd = sd * 3 - 2;
|
|
|
+ }
|
|
|
+ } else {
|
|
|
+ base = baseOut;
|
|
|
+ }
|
|
|
+
|
|
|
+ // Convert the number as an integer then divide the result by its base raised to a power such
|
|
|
+ // that the fraction part will be restored.
|
|
|
+
|
|
|
+ // Non-integer.
|
|
|
+ if (i >= 0) {
|
|
|
+ str = str.replace('.', '');
|
|
|
+ y = new Ctor(1);
|
|
|
+ y.e = str.length - i;
|
|
|
+ y.d = convertBase(finiteToString(y), 10, base);
|
|
|
+ y.e = y.d.length;
|
|
|
+ }
|
|
|
+
|
|
|
+ xd = convertBase(str, 10, base);
|
|
|
+ e = len = xd.length;
|
|
|
+
|
|
|
+ // Remove trailing zeros.
|
|
|
+ for (; xd[--len] == 0;) xd.pop();
|
|
|
+
|
|
|
+ if (!xd[0]) {
|
|
|
+ str = isExp ? '0p+0' : '0';
|
|
|
+ } else {
|
|
|
+ if (i < 0) {
|
|
|
+ e--;
|
|
|
+ } else {
|
|
|
+ x = new Ctor(x);
|
|
|
+ x.d = xd;
|
|
|
+ x.e = e;
|
|
|
+ x = divide(x, y, sd, rm, 0, base);
|
|
|
+ xd = x.d;
|
|
|
+ e = x.e;
|
|
|
+ roundUp = inexact;
|
|
|
+ }
|
|
|
+
|
|
|
+ // The rounding digit, i.e. the digit after the digit that may be rounded up.
|
|
|
+ i = xd[sd];
|
|
|
+ k = base / 2;
|
|
|
+ roundUp = roundUp || xd[sd + 1] !== void 0;
|
|
|
+
|
|
|
+ roundUp = rm < 4 ?
|
|
|
+ (i !== void 0 || roundUp) && (rm === 0 || rm === (x.s < 0 ? 3 : 2)) :
|
|
|
+ i > k || i === k && (rm === 4 || roundUp || rm === 6 && xd[sd - 1] & 1 ||
|
|
|
+ rm === (x.s < 0 ? 8 : 7));
|
|
|
+
|
|
|
+ xd.length = sd;
|
|
|
+
|
|
|
+ if (roundUp) {
|
|
|
+
|
|
|
+ // Rounding up may mean the previous digit has to be rounded up and so on.
|
|
|
+ for (; ++xd[--sd] > base - 1;) {
|
|
|
+ xd[sd] = 0;
|
|
|
+ if (!sd) {
|
|
|
+ ++e;
|
|
|
+ xd.unshift(1);
|
|
|
+ }
|
|
|
+ }
|
|
|
+ }
|
|
|
+
|
|
|
+ // Determine trailing zeros.
|
|
|
+ for (len = xd.length; !xd[len - 1]; --len);
|
|
|
+
|
|
|
+ // E.g. [4, 11, 15] becomes 4bf.
|
|
|
+ for (i = 0, str = ''; i < len; i++) str += NUMERALS.charAt(xd[i]);
|
|
|
+
|
|
|
+ // Add binary exponent suffix?
|
|
|
+ if (isExp) {
|
|
|
+ if (len > 1) {
|
|
|
+ if (baseOut == 16 || baseOut == 8) {
|
|
|
+ i = baseOut == 16 ? 4 : 3;
|
|
|
+ for (--len; len % i; len++) str += '0';
|
|
|
+ xd = convertBase(str, base, baseOut);
|
|
|
+ for (len = xd.length; !xd[len - 1]; --len);
|
|
|
+
|
|
|
+ // xd[0] will always be be 1
|
|
|
+ for (i = 1, str = '1.'; i < len; i++) str += NUMERALS.charAt(xd[i]);
|
|
|
+ } else {
|
|
|
+ str = str.charAt(0) + '.' + str.slice(1);
|
|
|
+ }
|
|
|
+ }
|
|
|
+
|
|
|
+ str = str + (e < 0 ? 'p' : 'p+') + e;
|
|
|
+ } else if (e < 0) {
|
|
|
+ for (; ++e;) str = '0' + str;
|
|
|
+ str = '0.' + str;
|
|
|
+ } else {
|
|
|
+ if (++e > len)
|
|
|
+ for (e -= len; e--;) str += '0';
|
|
|
+ else if (e < len) str = str.slice(0, e) + '.' + str.slice(e);
|
|
|
+ }
|
|
|
+ }
|
|
|
+
|
|
|
+ str = (baseOut == 16 ? '0x' : baseOut == 2 ? '0b' : baseOut == 8 ? '0o' : '') + str;
|
|
|
+ }
|
|
|
+
|
|
|
+ return x.s < 0 ? '-' + str : str;
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+// Does not strip trailing zeros.
|
|
|
+function truncate(arr, len) {
|
|
|
+ if (arr.length > len) {
|
|
|
+ arr.length = len;
|
|
|
+ return true;
|
|
|
+ }
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+// Decimal methods
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * abs
|
|
|
+ * acos
|
|
|
+ * acosh
|
|
|
+ * add
|
|
|
+ * asin
|
|
|
+ * asinh
|
|
|
+ * atan
|
|
|
+ * atanh
|
|
|
+ * atan2
|
|
|
+ * cbrt
|
|
|
+ * ceil
|
|
|
+ * clamp
|
|
|
+ * clone
|
|
|
+ * config
|
|
|
+ * cos
|
|
|
+ * cosh
|
|
|
+ * div
|
|
|
+ * exp
|
|
|
+ * floor
|
|
|
+ * hypot
|
|
|
+ * ln
|
|
|
+ * log
|
|
|
+ * log2
|
|
|
+ * log10
|
|
|
+ * max
|
|
|
+ * min
|
|
|
+ * mod
|
|
|
+ * mul
|
|
|
+ * pow
|
|
|
+ * random
|
|
|
+ * round
|
|
|
+ * set
|
|
|
+ * sign
|
|
|
+ * sin
|
|
|
+ * sinh
|
|
|
+ * sqrt
|
|
|
+ * sub
|
|
|
+ * sum
|
|
|
+ * tan
|
|
|
+ * tanh
|
|
|
+ * trunc
|
|
|
+ */
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is the absolute value of `x`.
|
|
|
+ *
|
|
|
+ * x {number|string|Decimal}
|
|
|
+ *
|
|
|
+ */
|
|
|
+function abs(x) {
|
|
|
+ return new this(x).abs();
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is the arccosine in radians of `x`.
|
|
|
+ *
|
|
|
+ * x {number|string|Decimal}
|
|
|
+ *
|
|
|
+ */
|
|
|
+function acos(x) {
|
|
|
+ return new this(x).acos();
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is the inverse of the hyperbolic cosine of `x`, rounded to
|
|
|
+ * `precision` significant digits using rounding mode `rounding`.
|
|
|
+ *
|
|
|
+ * x {number|string|Decimal} A value in radians.
|
|
|
+ *
|
|
|
+ */
|
|
|
+function acosh(x) {
|
|
|
+ return new this(x).acosh();
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is the sum of `x` and `y`, rounded to `precision` significant
|
|
|
+ * digits using rounding mode `rounding`.
|
|
|
+ *
|
|
|
+ * x {number|string|Decimal}
|
|
|
+ * y {number|string|Decimal}
|
|
|
+ *
|
|
|
+ */
|
|
|
+function add(x, y) {
|
|
|
+ return new this(x).plus(y);
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is the arcsine in radians of `x`, rounded to `precision`
|
|
|
+ * significant digits using rounding mode `rounding`.
|
|
|
+ *
|
|
|
+ * x {number|string|Decimal}
|
|
|
+ *
|
|
|
+ */
|
|
|
+function asin(x) {
|
|
|
+ return new this(x).asin();
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is the inverse of the hyperbolic sine of `x`, rounded to
|
|
|
+ * `precision` significant digits using rounding mode `rounding`.
|
|
|
+ *
|
|
|
+ * x {number|string|Decimal} A value in radians.
|
|
|
+ *
|
|
|
+ */
|
|
|
+function asinh(x) {
|
|
|
+ return new this(x).asinh();
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is the arctangent in radians of `x`, rounded to `precision`
|
|
|
+ * significant digits using rounding mode `rounding`.
|
|
|
+ *
|
|
|
+ * x {number|string|Decimal}
|
|
|
+ *
|
|
|
+ */
|
|
|
+function atan(x) {
|
|
|
+ return new this(x).atan();
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is the inverse of the hyperbolic tangent of `x`, rounded to
|
|
|
+ * `precision` significant digits using rounding mode `rounding`.
|
|
|
+ *
|
|
|
+ * x {number|string|Decimal} A value in radians.
|
|
|
+ *
|
|
|
+ */
|
|
|
+function atanh(x) {
|
|
|
+ return new this(x).atanh();
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is the arctangent in radians of `y/x` in the range -pi to pi
|
|
|
+ * (inclusive), rounded to `precision` significant digits using rounding mode `rounding`.
|
|
|
+ *
|
|
|
+ * Domain: [-Infinity, Infinity]
|
|
|
+ * Range: [-pi, pi]
|
|
|
+ *
|
|
|
+ * y {number|string|Decimal} The y-coordinate.
|
|
|
+ * x {number|string|Decimal} The x-coordinate.
|
|
|
+ *
|
|
|
+ * atan2(±0, -0) = ±pi
|
|
|
+ * atan2(±0, +0) = ±0
|
|
|
+ * atan2(±0, -x) = ±pi for x > 0
|
|
|
+ * atan2(±0, x) = ±0 for x > 0
|
|
|
+ * atan2(-y, ±0) = -pi/2 for y > 0
|
|
|
+ * atan2(y, ±0) = pi/2 for y > 0
|
|
|
+ * atan2(±y, -Infinity) = ±pi for finite y > 0
|
|
|
+ * atan2(±y, +Infinity) = ±0 for finite y > 0
|
|
|
+ * atan2(±Infinity, x) = ±pi/2 for finite x
|
|
|
+ * atan2(±Infinity, -Infinity) = ±3*pi/4
|
|
|
+ * atan2(±Infinity, +Infinity) = ±pi/4
|
|
|
+ * atan2(NaN, x) = NaN
|
|
|
+ * atan2(y, NaN) = NaN
|
|
|
+ *
|
|
|
+ */
|
|
|
+function atan2(y, x) {
|
|
|
+ y = new this(y);
|
|
|
+ x = new this(x);
|
|
|
+ var r,
|
|
|
+ pr = this.precision,
|
|
|
+ rm = this.rounding,
|
|
|
+ wpr = pr + 4;
|
|
|
+
|
|
|
+ // Either NaN
|
|
|
+ if (!y.s || !x.s) {
|
|
|
+ r = new this(NaN);
|
|
|
+
|
|
|
+ // Both ±Infinity
|
|
|
+ } else if (!y.d && !x.d) {
|
|
|
+ r = getPi(this, wpr, 1).times(x.s > 0 ? 0.25 : 0.75);
|
|
|
+ r.s = y.s;
|
|
|
+
|
|
|
+ // x is ±Infinity or y is ±0
|
|
|
+ } else if (!x.d || y.isZero()) {
|
|
|
+ r = x.s < 0 ? getPi(this, pr, rm) : new this(0);
|
|
|
+ r.s = y.s;
|
|
|
+
|
|
|
+ // y is ±Infinity or x is ±0
|
|
|
+ } else if (!y.d || x.isZero()) {
|
|
|
+ r = getPi(this, wpr, 1).times(0.5);
|
|
|
+ r.s = y.s;
|
|
|
+
|
|
|
+ // Both non-zero and finite
|
|
|
+ } else if (x.s < 0) {
|
|
|
+ this.precision = wpr;
|
|
|
+ this.rounding = 1;
|
|
|
+ r = this.atan(divide(y, x, wpr, 1));
|
|
|
+ x = getPi(this, wpr, 1);
|
|
|
+ this.precision = pr;
|
|
|
+ this.rounding = rm;
|
|
|
+ r = y.s < 0 ? r.minus(x) : r.plus(x);
|
|
|
+ } else {
|
|
|
+ r = this.atan(divide(y, x, wpr, 1));
|
|
|
+ }
|
|
|
+
|
|
|
+ return r;
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is the cube root of `x`, rounded to `precision` significant
|
|
|
+ * digits using rounding mode `rounding`.
|
|
|
+ *
|
|
|
+ * x {number|string|Decimal}
|
|
|
+ *
|
|
|
+ */
|
|
|
+function cbrt(x) {
|
|
|
+ return new this(x).cbrt();
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is `x` rounded to an integer using `ROUND_CEIL`.
|
|
|
+ *
|
|
|
+ * x {number|string|Decimal}
|
|
|
+ *
|
|
|
+ */
|
|
|
+function ceil(x) {
|
|
|
+ return finalise(x = new this(x), x.e + 1, 2);
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is `x` clamped to the range delineated by `min` and `max`.
|
|
|
+ *
|
|
|
+ * x {number|string|Decimal}
|
|
|
+ * min {number|string|Decimal}
|
|
|
+ * max {number|string|Decimal}
|
|
|
+ *
|
|
|
+ */
|
|
|
+function clamp(x, min, max) {
|
|
|
+ return new this(x).clamp(min, max);
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Configure global settings for a Decimal constructor.
|
|
|
+ *
|
|
|
+ * `obj` is an object with one or more of the following properties,
|
|
|
+ *
|
|
|
+ * precision {number}
|
|
|
+ * rounding {number}
|
|
|
+ * toExpNeg {number}
|
|
|
+ * toExpPos {number}
|
|
|
+ * maxE {number}
|
|
|
+ * minE {number}
|
|
|
+ * modulo {number}
|
|
|
+ * crypto {boolean|number}
|
|
|
+ * defaults {true}
|
|
|
+ *
|
|
|
+ * E.g. Decimal.config({ precision: 20, rounding: 4 })
|
|
|
+ *
|
|
|
+ */
|
|
|
+function config(obj) {
|
|
|
+ if (!obj || typeof obj !== 'object') throw Error(decimalError + 'Object expected');
|
|
|
+ var i, p, v,
|
|
|
+ useDefaults = obj.defaults === true,
|
|
|
+ ps = [
|
|
|
+ 'precision', 1, MAX_DIGITS,
|
|
|
+ 'rounding', 0, 8,
|
|
|
+ 'toExpNeg', -EXP_LIMIT, 0,
|
|
|
+ 'toExpPos', 0, EXP_LIMIT,
|
|
|
+ 'maxE', 0, EXP_LIMIT,
|
|
|
+ 'minE', -EXP_LIMIT, 0,
|
|
|
+ 'modulo', 0, 9
|
|
|
+ ];
|
|
|
+
|
|
|
+ for (i = 0; i < ps.length; i += 3) {
|
|
|
+ if (p = ps[i], useDefaults) this[p] = DEFAULTS[p];
|
|
|
+ if ((v = obj[p]) !== void 0) {
|
|
|
+ if (mathfloor(v) === v && v >= ps[i + 1] && v <= ps[i + 2]) this[p] = v;
|
|
|
+ else throw Error(invalidArgument + p + ': ' + v);
|
|
|
+ }
|
|
|
+ }
|
|
|
+
|
|
|
+ if (p = 'crypto', useDefaults) this[p] = DEFAULTS[p];
|
|
|
+ if ((v = obj[p]) !== void 0) {
|
|
|
+ if (v === true || v === false || v === 0 || v === 1) {
|
|
|
+ if (v) {
|
|
|
+ if (typeof crypto != 'undefined' && crypto &&
|
|
|
+ (crypto.getRandomValues || crypto.randomBytes)) {
|
|
|
+ this[p] = true;
|
|
|
+ } else {
|
|
|
+ throw Error(cryptoUnavailable);
|
|
|
+ }
|
|
|
+ } else {
|
|
|
+ this[p] = false;
|
|
|
+ }
|
|
|
+ } else {
|
|
|
+ throw Error(invalidArgument + p + ': ' + v);
|
|
|
+ }
|
|
|
+ }
|
|
|
+
|
|
|
+ return this;
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is the cosine of `x`, rounded to `precision` significant
|
|
|
+ * digits using rounding mode `rounding`.
|
|
|
+ *
|
|
|
+ * x {number|string|Decimal} A value in radians.
|
|
|
+ *
|
|
|
+ */
|
|
|
+function cos(x) {
|
|
|
+ return new this(x).cos();
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is the hyperbolic cosine of `x`, rounded to precision
|
|
|
+ * significant digits using rounding mode `rounding`.
|
|
|
+ *
|
|
|
+ * x {number|string|Decimal} A value in radians.
|
|
|
+ *
|
|
|
+ */
|
|
|
+function cosh(x) {
|
|
|
+ return new this(x).cosh();
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Create and return a Decimal constructor with the same configuration properties as this Decimal
|
|
|
+ * constructor.
|
|
|
+ *
|
|
|
+ */
|
|
|
+function clone(obj) {
|
|
|
+ var i, p, ps;
|
|
|
+
|
|
|
+ /*
|
|
|
+ * The Decimal constructor and exported function.
|
|
|
+ * Return a new Decimal instance.
|
|
|
+ *
|
|
|
+ * v {number|string|Decimal} A numeric value.
|
|
|
+ *
|
|
|
+ */
|
|
|
+ function Decimal(v) {
|
|
|
+ var e, i, t,
|
|
|
+ x = this;
|
|
|
+
|
|
|
+ // Decimal called without new.
|
|
|
+ if (!(x instanceof Decimal)) return new Decimal(v);
|
|
|
+
|
|
|
+ // Retain a reference to this Decimal constructor, and shadow Decimal.prototype.constructor
|
|
|
+ // which points to Object.
|
|
|
+ x.constructor = Decimal;
|
|
|
+
|
|
|
+ // Duplicate.
|
|
|
+ if (isDecimalInstance(v)) {
|
|
|
+ x.s = v.s;
|
|
|
+
|
|
|
+ if (external) {
|
|
|
+ if (!v.d || v.e > Decimal.maxE) {
|
|
|
+
|
|
|
+ // Infinity.
|
|
|
+ x.e = NaN;
|
|
|
+ x.d = null;
|
|
|
+ } else if (v.e < Decimal.minE) {
|
|
|
+
|
|
|
+ // Zero.
|
|
|
+ x.e = 0;
|
|
|
+ x.d = [0];
|
|
|
+ } else {
|
|
|
+ x.e = v.e;
|
|
|
+ x.d = v.d.slice();
|
|
|
+ }
|
|
|
+ } else {
|
|
|
+ x.e = v.e;
|
|
|
+ x.d = v.d ? v.d.slice() : v.d;
|
|
|
+ }
|
|
|
+
|
|
|
+ return;
|
|
|
+ }
|
|
|
+
|
|
|
+ t = typeof v;
|
|
|
+
|
|
|
+ if (t === 'number') {
|
|
|
+ if (v === 0) {
|
|
|
+ x.s = 1 / v < 0 ? -1 : 1;
|
|
|
+ x.e = 0;
|
|
|
+ x.d = [0];
|
|
|
+ return;
|
|
|
+ }
|
|
|
+
|
|
|
+ if (v < 0) {
|
|
|
+ v = -v;
|
|
|
+ x.s = -1;
|
|
|
+ } else {
|
|
|
+ x.s = 1;
|
|
|
+ }
|
|
|
+
|
|
|
+ // Fast path for small integers.
|
|
|
+ if (v === ~~v && v < 1e7) {
|
|
|
+ for (e = 0, i = v; i >= 10; i /= 10) e++;
|
|
|
+
|
|
|
+ if (external) {
|
|
|
+ if (e > Decimal.maxE) {
|
|
|
+ x.e = NaN;
|
|
|
+ x.d = null;
|
|
|
+ } else if (e < Decimal.minE) {
|
|
|
+ x.e = 0;
|
|
|
+ x.d = [0];
|
|
|
+ } else {
|
|
|
+ x.e = e;
|
|
|
+ x.d = [v];
|
|
|
+ }
|
|
|
+ } else {
|
|
|
+ x.e = e;
|
|
|
+ x.d = [v];
|
|
|
+ }
|
|
|
+
|
|
|
+ return;
|
|
|
+
|
|
|
+ // Infinity, NaN.
|
|
|
+ } else if (v * 0 !== 0) {
|
|
|
+ if (!v) x.s = NaN;
|
|
|
+ x.e = NaN;
|
|
|
+ x.d = null;
|
|
|
+ return;
|
|
|
+ }
|
|
|
+
|
|
|
+ return parseDecimal(x, v.toString());
|
|
|
+
|
|
|
+ } else if (t !== 'string') {
|
|
|
+ throw Error(invalidArgument + v);
|
|
|
+ }
|
|
|
+
|
|
|
+ // Minus sign?
|
|
|
+ if ((i = v.charCodeAt(0)) === 45) {
|
|
|
+ v = v.slice(1);
|
|
|
+ x.s = -1;
|
|
|
+ } else {
|
|
|
+ // Plus sign?
|
|
|
+ if (i === 43) v = v.slice(1);
|
|
|
+ x.s = 1;
|
|
|
+ }
|
|
|
+
|
|
|
+ return isDecimal.test(v) ? parseDecimal(x, v) : parseOther(x, v);
|
|
|
+ }
|
|
|
+
|
|
|
+ Decimal.prototype = P;
|
|
|
+
|
|
|
+ Decimal.ROUND_UP = 0;
|
|
|
+ Decimal.ROUND_DOWN = 1;
|
|
|
+ Decimal.ROUND_CEIL = 2;
|
|
|
+ Decimal.ROUND_FLOOR = 3;
|
|
|
+ Decimal.ROUND_HALF_UP = 4;
|
|
|
+ Decimal.ROUND_HALF_DOWN = 5;
|
|
|
+ Decimal.ROUND_HALF_EVEN = 6;
|
|
|
+ Decimal.ROUND_HALF_CEIL = 7;
|
|
|
+ Decimal.ROUND_HALF_FLOOR = 8;
|
|
|
+ Decimal.EUCLID = 9;
|
|
|
+
|
|
|
+ Decimal.config = Decimal.set = config;
|
|
|
+ Decimal.clone = clone;
|
|
|
+ Decimal.isDecimal = isDecimalInstance;
|
|
|
+
|
|
|
+ Decimal.abs = abs;
|
|
|
+ Decimal.acos = acos;
|
|
|
+ Decimal.acosh = acosh; // ES6
|
|
|
+ Decimal.add = add;
|
|
|
+ Decimal.asin = asin;
|
|
|
+ Decimal.asinh = asinh; // ES6
|
|
|
+ Decimal.atan = atan;
|
|
|
+ Decimal.atanh = atanh; // ES6
|
|
|
+ Decimal.atan2 = atan2;
|
|
|
+ Decimal.cbrt = cbrt; // ES6
|
|
|
+ Decimal.ceil = ceil;
|
|
|
+ Decimal.clamp = clamp;
|
|
|
+ Decimal.cos = cos;
|
|
|
+ Decimal.cosh = cosh; // ES6
|
|
|
+ Decimal.div = div;
|
|
|
+ Decimal.exp = exp;
|
|
|
+ Decimal.floor = floor;
|
|
|
+ Decimal.hypot = hypot; // ES6
|
|
|
+ Decimal.ln = ln;
|
|
|
+ Decimal.log = log;
|
|
|
+ Decimal.log10 = log10; // ES6
|
|
|
+ Decimal.log2 = log2; // ES6
|
|
|
+ Decimal.max = max;
|
|
|
+ Decimal.min = min;
|
|
|
+ Decimal.mod = mod;
|
|
|
+ Decimal.mul = mul;
|
|
|
+ Decimal.pow = pow;
|
|
|
+ Decimal.random = random;
|
|
|
+ Decimal.round = round;
|
|
|
+ Decimal.sign = sign; // ES6
|
|
|
+ Decimal.sin = sin;
|
|
|
+ Decimal.sinh = sinh; // ES6
|
|
|
+ Decimal.sqrt = sqrt;
|
|
|
+ Decimal.sub = sub;
|
|
|
+ Decimal.sum = sum;
|
|
|
+ Decimal.tan = tan;
|
|
|
+ Decimal.tanh = tanh; // ES6
|
|
|
+ Decimal.trunc = trunc; // ES6
|
|
|
+
|
|
|
+ if (obj === void 0) obj = {};
|
|
|
+ if (obj) {
|
|
|
+ if (obj.defaults !== true) {
|
|
|
+ ps = ['precision', 'rounding', 'toExpNeg', 'toExpPos', 'maxE', 'minE', 'modulo', 'crypto'];
|
|
|
+ for (i = 0; i < ps.length;)
|
|
|
+ if (!obj.hasOwnProperty(p = ps[i++])) obj[p] = this[p];
|
|
|
+ }
|
|
|
+ }
|
|
|
+
|
|
|
+ Decimal.config(obj);
|
|
|
+
|
|
|
+ return Decimal;
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is `x` divided by `y`, rounded to `precision` significant
|
|
|
+ * digits using rounding mode `rounding`.
|
|
|
+ *
|
|
|
+ * x {number|string|Decimal}
|
|
|
+ * y {number|string|Decimal}
|
|
|
+ *
|
|
|
+ */
|
|
|
+function div(x, y) {
|
|
|
+ return new this(x).div(y);
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is the natural exponential of `x`, rounded to `precision`
|
|
|
+ * significant digits using rounding mode `rounding`.
|
|
|
+ *
|
|
|
+ * x {number|string|Decimal} The power to which to raise the base of the natural log.
|
|
|
+ *
|
|
|
+ */
|
|
|
+function exp(x) {
|
|
|
+ return new this(x).exp();
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is `x` round to an integer using `ROUND_FLOOR`.
|
|
|
+ *
|
|
|
+ * x {number|string|Decimal}
|
|
|
+ *
|
|
|
+ */
|
|
|
+function floor(x) {
|
|
|
+ return finalise(x = new this(x), x.e + 1, 3);
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is the square root of the sum of the squares of the arguments,
|
|
|
+ * rounded to `precision` significant digits using rounding mode `rounding`.
|
|
|
+ *
|
|
|
+ * hypot(a, b, ...) = sqrt(a^2 + b^2 + ...)
|
|
|
+ *
|
|
|
+ * arguments {number|string|Decimal}
|
|
|
+ *
|
|
|
+ */
|
|
|
+function hypot() {
|
|
|
+ var i, n,
|
|
|
+ t = new this(0);
|
|
|
+
|
|
|
+ external = false;
|
|
|
+
|
|
|
+ for (i = 0; i < arguments.length;) {
|
|
|
+ n = new this(arguments[i++]);
|
|
|
+ if (!n.d) {
|
|
|
+ if (n.s) {
|
|
|
+ external = true;
|
|
|
+ return new this(1 / 0);
|
|
|
+ }
|
|
|
+ t = n;
|
|
|
+ } else if (t.d) {
|
|
|
+ t = t.plus(n.times(n));
|
|
|
+ }
|
|
|
+ }
|
|
|
+
|
|
|
+ external = true;
|
|
|
+
|
|
|
+ return t.sqrt();
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return true if object is a Decimal instance (where Decimal is any Decimal constructor),
|
|
|
+ * otherwise return false.
|
|
|
+ *
|
|
|
+ */
|
|
|
+function isDecimalInstance(obj) {
|
|
|
+ return obj instanceof Decimal || obj && obj.toStringTag === tag || false;
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is the natural logarithm of `x`, rounded to `precision`
|
|
|
+ * significant digits using rounding mode `rounding`.
|
|
|
+ *
|
|
|
+ * x {number|string|Decimal}
|
|
|
+ *
|
|
|
+ */
|
|
|
+function ln(x) {
|
|
|
+ return new this(x).ln();
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is the log of `x` to the base `y`, or to base 10 if no base
|
|
|
+ * is specified, rounded to `precision` significant digits using rounding mode `rounding`.
|
|
|
+ *
|
|
|
+ * log[y](x)
|
|
|
+ *
|
|
|
+ * x {number|string|Decimal} The argument of the logarithm.
|
|
|
+ * y {number|string|Decimal} The base of the logarithm.
|
|
|
+ *
|
|
|
+ */
|
|
|
+function log(x, y) {
|
|
|
+ return new this(x).log(y);
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is the base 2 logarithm of `x`, rounded to `precision`
|
|
|
+ * significant digits using rounding mode `rounding`.
|
|
|
+ *
|
|
|
+ * x {number|string|Decimal}
|
|
|
+ *
|
|
|
+ */
|
|
|
+function log2(x) {
|
|
|
+ return new this(x).log(2);
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is the base 10 logarithm of `x`, rounded to `precision`
|
|
|
+ * significant digits using rounding mode `rounding`.
|
|
|
+ *
|
|
|
+ * x {number|string|Decimal}
|
|
|
+ *
|
|
|
+ */
|
|
|
+function log10(x) {
|
|
|
+ return new this(x).log(10);
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is the maximum of the arguments.
|
|
|
+ *
|
|
|
+ * arguments {number|string|Decimal}
|
|
|
+ *
|
|
|
+ */
|
|
|
+function max() {
|
|
|
+ return maxOrMin(this, arguments, 'lt');
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is the minimum of the arguments.
|
|
|
+ *
|
|
|
+ * arguments {number|string|Decimal}
|
|
|
+ *
|
|
|
+ */
|
|
|
+function min() {
|
|
|
+ return maxOrMin(this, arguments, 'gt');
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is `x` modulo `y`, rounded to `precision` significant digits
|
|
|
+ * using rounding mode `rounding`.
|
|
|
+ *
|
|
|
+ * x {number|string|Decimal}
|
|
|
+ * y {number|string|Decimal}
|
|
|
+ *
|
|
|
+ */
|
|
|
+function mod(x, y) {
|
|
|
+ return new this(x).mod(y);
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is `x` multiplied by `y`, rounded to `precision` significant
|
|
|
+ * digits using rounding mode `rounding`.
|
|
|
+ *
|
|
|
+ * x {number|string|Decimal}
|
|
|
+ * y {number|string|Decimal}
|
|
|
+ *
|
|
|
+ */
|
|
|
+function mul(x, y) {
|
|
|
+ return new this(x).mul(y);
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is `x` raised to the power `y`, rounded to precision
|
|
|
+ * significant digits using rounding mode `rounding`.
|
|
|
+ *
|
|
|
+ * x {number|string|Decimal} The base.
|
|
|
+ * y {number|string|Decimal} The exponent.
|
|
|
+ *
|
|
|
+ */
|
|
|
+function pow(x, y) {
|
|
|
+ return new this(x).pow(y);
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Returns a new Decimal with a random value equal to or greater than 0 and less than 1, and with
|
|
|
+ * `sd`, or `Decimal.precision` if `sd` is omitted, significant digits (or less if trailing zeros
|
|
|
+ * are produced).
|
|
|
+ *
|
|
|
+ * [sd] {number} Significant digits. Integer, 0 to MAX_DIGITS inclusive.
|
|
|
+ *
|
|
|
+ */
|
|
|
+function random(sd) {
|
|
|
+ var d, e, k, n,
|
|
|
+ i = 0,
|
|
|
+ r = new this(1),
|
|
|
+ rd = [];
|
|
|
+
|
|
|
+ if (sd === void 0) sd = this.precision;
|
|
|
+ else checkInt32(sd, 1, MAX_DIGITS);
|
|
|
+
|
|
|
+ k = Math.ceil(sd / LOG_BASE);
|
|
|
+
|
|
|
+ if (!this.crypto) {
|
|
|
+ for (; i < k;) rd[i++] = Math.random() * 1e7 | 0;
|
|
|
+
|
|
|
+ // Browsers supporting crypto.getRandomValues.
|
|
|
+ } else if (crypto.getRandomValues) {
|
|
|
+ d = crypto.getRandomValues(new Uint32Array(k));
|
|
|
+
|
|
|
+ for (; i < k;) {
|
|
|
+ n = d[i];
|
|
|
+
|
|
|
+ // 0 <= n < 4294967296
|
|
|
+ // Probability n >= 4.29e9, is 4967296 / 4294967296 = 0.00116 (1 in 865).
|
|
|
+ if (n >= 4.29e9) {
|
|
|
+ d[i] = crypto.getRandomValues(new Uint32Array(1))[0];
|
|
|
+ } else {
|
|
|
+
|
|
|
+ // 0 <= n <= 4289999999
|
|
|
+ // 0 <= (n % 1e7) <= 9999999
|
|
|
+ rd[i++] = n % 1e7;
|
|
|
+ }
|
|
|
+ }
|
|
|
+
|
|
|
+ // Node.js supporting crypto.randomBytes.
|
|
|
+ } else if (crypto.randomBytes) {
|
|
|
+
|
|
|
+ // buffer
|
|
|
+ d = crypto.randomBytes(k *= 4);
|
|
|
+
|
|
|
+ for (; i < k;) {
|
|
|
+
|
|
|
+ // 0 <= n < 2147483648
|
|
|
+ n = d[i] + (d[i + 1] << 8) + (d[i + 2] << 16) + ((d[i + 3] & 0x7f) << 24);
|
|
|
+
|
|
|
+ // Probability n >= 2.14e9, is 7483648 / 2147483648 = 0.0035 (1 in 286).
|
|
|
+ if (n >= 2.14e9) {
|
|
|
+ crypto.randomBytes(4).copy(d, i);
|
|
|
+ } else {
|
|
|
+
|
|
|
+ // 0 <= n <= 2139999999
|
|
|
+ // 0 <= (n % 1e7) <= 9999999
|
|
|
+ rd.push(n % 1e7);
|
|
|
+ i += 4;
|
|
|
+ }
|
|
|
+ }
|
|
|
+
|
|
|
+ i = k / 4;
|
|
|
+ } else {
|
|
|
+ throw Error(cryptoUnavailable);
|
|
|
+ }
|
|
|
+
|
|
|
+ k = rd[--i];
|
|
|
+ sd %= LOG_BASE;
|
|
|
+
|
|
|
+ // Convert trailing digits to zeros according to sd.
|
|
|
+ if (k && sd) {
|
|
|
+ n = mathpow(10, LOG_BASE - sd);
|
|
|
+ rd[i] = (k / n | 0) * n;
|
|
|
+ }
|
|
|
+
|
|
|
+ // Remove trailing words which are zero.
|
|
|
+ for (; rd[i] === 0; i--) rd.pop();
|
|
|
+
|
|
|
+ // Zero?
|
|
|
+ if (i < 0) {
|
|
|
+ e = 0;
|
|
|
+ rd = [0];
|
|
|
+ } else {
|
|
|
+ e = -1;
|
|
|
+
|
|
|
+ // Remove leading words which are zero and adjust exponent accordingly.
|
|
|
+ for (; rd[0] === 0; e -= LOG_BASE) rd.shift();
|
|
|
+
|
|
|
+ // Count the digits of the first word of rd to determine leading zeros.
|
|
|
+ for (k = 1, n = rd[0]; n >= 10; n /= 10) k++;
|
|
|
+
|
|
|
+ // Adjust the exponent for leading zeros of the first word of rd.
|
|
|
+ if (k < LOG_BASE) e -= LOG_BASE - k;
|
|
|
+ }
|
|
|
+
|
|
|
+ r.e = e;
|
|
|
+ r.d = rd;
|
|
|
+
|
|
|
+ return r;
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is `x` rounded to an integer using rounding mode `rounding`.
|
|
|
+ *
|
|
|
+ * To emulate `Math.round`, set rounding to 7 (ROUND_HALF_CEIL).
|
|
|
+ *
|
|
|
+ * x {number|string|Decimal}
|
|
|
+ *
|
|
|
+ */
|
|
|
+function round(x) {
|
|
|
+ return finalise(x = new this(x), x.e + 1, this.rounding);
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return
|
|
|
+ * 1 if x > 0,
|
|
|
+ * -1 if x < 0,
|
|
|
+ * 0 if x is 0,
|
|
|
+ * -0 if x is -0,
|
|
|
+ * NaN otherwise
|
|
|
+ *
|
|
|
+ * x {number|string|Decimal}
|
|
|
+ *
|
|
|
+ */
|
|
|
+function sign(x) {
|
|
|
+ x = new this(x);
|
|
|
+ return x.d ? (x.d[0] ? x.s : 0 * x.s) : x.s || NaN;
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is the sine of `x`, rounded to `precision` significant digits
|
|
|
+ * using rounding mode `rounding`.
|
|
|
+ *
|
|
|
+ * x {number|string|Decimal} A value in radians.
|
|
|
+ *
|
|
|
+ */
|
|
|
+function sin(x) {
|
|
|
+ return new this(x).sin();
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is the hyperbolic sine of `x`, rounded to `precision`
|
|
|
+ * significant digits using rounding mode `rounding`.
|
|
|
+ *
|
|
|
+ * x {number|string|Decimal} A value in radians.
|
|
|
+ *
|
|
|
+ */
|
|
|
+function sinh(x) {
|
|
|
+ return new this(x).sinh();
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is the square root of `x`, rounded to `precision` significant
|
|
|
+ * digits using rounding mode `rounding`.
|
|
|
+ *
|
|
|
+ * x {number|string|Decimal}
|
|
|
+ *
|
|
|
+ */
|
|
|
+function sqrt(x) {
|
|
|
+ return new this(x).sqrt();
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is `x` minus `y`, rounded to `precision` significant digits
|
|
|
+ * using rounding mode `rounding`.
|
|
|
+ *
|
|
|
+ * x {number|string|Decimal}
|
|
|
+ * y {number|string|Decimal}
|
|
|
+ *
|
|
|
+ */
|
|
|
+function sub(x, y) {
|
|
|
+ return new this(x).sub(y);
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is the sum of the arguments, rounded to `precision`
|
|
|
+ * significant digits using rounding mode `rounding`.
|
|
|
+ *
|
|
|
+ * Only the result is rounded, not the intermediate calculations.
|
|
|
+ *
|
|
|
+ * arguments {number|string|Decimal}
|
|
|
+ *
|
|
|
+ */
|
|
|
+function sum() {
|
|
|
+ var i = 0,
|
|
|
+ args = arguments,
|
|
|
+ x = new this(args[i]);
|
|
|
+
|
|
|
+ external = false;
|
|
|
+ for (; x.s && ++i < args.length;) x = x.plus(args[i]);
|
|
|
+ external = true;
|
|
|
+
|
|
|
+ return finalise(x, this.precision, this.rounding);
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is the tangent of `x`, rounded to `precision` significant
|
|
|
+ * digits using rounding mode `rounding`.
|
|
|
+ *
|
|
|
+ * x {number|string|Decimal} A value in radians.
|
|
|
+ *
|
|
|
+ */
|
|
|
+function tan(x) {
|
|
|
+ return new this(x).tan();
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is the hyperbolic tangent of `x`, rounded to `precision`
|
|
|
+ * significant digits using rounding mode `rounding`.
|
|
|
+ *
|
|
|
+ * x {number|string|Decimal} A value in radians.
|
|
|
+ *
|
|
|
+ */
|
|
|
+function tanh(x) {
|
|
|
+ return new this(x).tanh();
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+/*
|
|
|
+ * Return a new Decimal whose value is `x` truncated to an integer.
|
|
|
+ *
|
|
|
+ * x {number|string|Decimal}
|
|
|
+ *
|
|
|
+ */
|
|
|
+function trunc(x) {
|
|
|
+ return finalise(x = new this(x), x.e + 1, 1);
|
|
|
+}
|
|
|
+
|
|
|
+
|
|
|
+// Create and configure initial Decimal constructor.
|
|
|
+Decimal = clone(DEFAULTS);
|
|
|
+Decimal.prototype.constructor = Decimal;
|
|
|
+Decimal['default'] = Decimal.Decimal = Decimal;
|
|
|
+
|
|
|
+// Create the internal constants from their string values.
|
|
|
+LN10 = new Decimal(LN10);
|
|
|
+PI = new Decimal(PI);
|
|
|
+
|
|
|
+export default Decimal
|
