4899 lines
124 KiB
JavaScript
4899 lines
124 KiB
JavaScript
/*!
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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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// ----------------------------------- EDITABLE DEFAULTS ------------------------------------ //
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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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// 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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// Base conversion alphabet.
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NUMERALS = '0123456789abcdef',
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// The natural logarithm of 10 (1025 digits).
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LN10 = '2.3025850929940456840179914546843642076011014886287729760333279009675726096773524802359972050895982983419677840422862486334095254650828067566662873690987816894829072083255546808437998948262331985283935053089653777326288461633662222876982198867465436674744042432743651550489343149393914796194044002221051017141748003688084012647080685567743216228355220114804663715659121373450747856947683463616792101806445070648000277502684916746550586856935673420670581136429224554405758925724208241314695689016758940256776311356919292033376587141660230105703089634572075440370847469940168269282808481184289314848524948644871927809676271275775397027668605952496716674183485704422507197965004714951050492214776567636938662976979522110718264549734772662425709429322582798502585509785265383207606726317164309505995087807523710333101197857547331541421808427543863591778117054309827482385045648019095610299291824318237525357709750539565187697510374970888692180205189339507238539205144634197265287286965110862571492198849978748873771345686209167058',
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// Pi (1025 digits).
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PI = '3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461284756482337867831652712019091456485669234603486104543266482133936072602491412737245870066063155881748815209209628292540917153643678925903600113305305488204665213841469519415116094330572703657595919530921861173819326117931051185480744623799627495673518857527248912279381830119491298336733624406566430860213949463952247371907021798609437027705392171762931767523846748184676694051320005681271452635608277857713427577896091736371787214684409012249534301465495853710507922796892589235420199561121290219608640344181598136297747713099605187072113499999983729780499510597317328160963185950244594553469083026425223082533446850352619311881710100031378387528865875332083814206171776691473035982534904287554687311595628638823537875937519577818577805321712268066130019278766111959092164201989380952572010654858632789',
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// The initial configuration properties of the Decimal constructor.
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DEFAULTS = {
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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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// 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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// 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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// 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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// 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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// 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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// 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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// 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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// 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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// ----------------------------------- END OF EDITABLE DEFAULTS ------------------------------- //
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inexact, quadrant,
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external = true,
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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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mathfloor = Math.floor,
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mathpow = Math.pow,
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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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BASE = 1e7,
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LOG_BASE = 7,
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MAX_SAFE_INTEGER = 9007199254740991,
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LN10_PRECISION = LN10.length - 1,
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PI_PRECISION = PI.length - 1,
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// Decimal.prototype object
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P = { toStringTag: tag };
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// Decimal prototype methods
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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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* 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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* 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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* 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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* 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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// 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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// Either zero?
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if (!xd[0] || !yd[0]) return xd[0] ? xs : yd[0] ? -ys : 0;
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// Signs differ?
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if (xs !== ys) return xs;
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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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xdL = xd.length;
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ydL = yd.length;
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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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// 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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* 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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if (!x.d) return new Ctor(NaN);
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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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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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x = cosine(Ctor, toLessThanHalfPi(Ctor, x));
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Ctor.precision = pr;
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Ctor.rounding = rm;
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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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* 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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if (!x.isFinite() || x.isZero()) return new Ctor(x);
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external = false;
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// Initial estimate.
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s = x.s * mathpow(x.s * x, 1 / 3);
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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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// 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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// 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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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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}
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r = new Ctor(n);
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r.s = x.s;
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} else {
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r = new Ctor(s.toString());
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}
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sd = (e = Ctor.precision) + 3;
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// Halley's method.
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// TODO? Compare Newton's method.
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for (;;) {
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t = r;
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t3 = t.times(t).times(t);
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t3plusx = t3.plus(x);
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r = divide(t3plusx.plus(x).times(t), t3plusx.plus(t3), sd + 2, 1);
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// TODO? Replace with for-loop and checkRoundingDigits.
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if (digitsToString(t.d).slice(0, sd) === (n = digitsToString(r.d)).slice(0, sd)) {
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n = n.slice(sd - 3, sd + 1);
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// The 4th rounding digit may be in error by -1 so if the 4 rounding digits are 9999 or 4999
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// , i.e. approaching a rounding boundary, continue the iteration.
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if (n == '9999' || !rep && n == '4999') {
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// On the first iteration only, check to see if rounding up gives the exact result as the
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// nines may infinitely repeat.
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if (!rep) {
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finalise(t, e + 1, 0);
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if (t.times(t).times(t).eq(x)) {
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r = t;
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break;
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}
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}
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sd += 4;
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rep = 1;
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} else {
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// If the rounding digits are null, 0{0,4} or 50{0,3}, check for an exact result.
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// If not, then there are further digits and m will be truthy.
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if (!+n || !+n.slice(1) && n.charAt(0) == '5') {
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// Truncate to the first rounding digit.
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finalise(r, e + 1, 1);
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m = !r.times(r).times(r).eq(x);
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}
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break;
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}
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}
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}
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external = true;
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return finalise(r, e, Ctor.rounding, m);
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};
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/*
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* Return the number of decimal places of the value of this Decimal.
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*
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*/
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P.decimalPlaces = P.dp = function () {
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var w,
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d = this.d,
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n = NaN;
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if (d) {
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w = d.length - 1;
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n = (w - mathfloor(this.e / LOG_BASE)) * LOG_BASE;
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// Subtract the number of trailing zeros of the last word.
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w = d[w];
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if (w) for (; w % 10 == 0; w /= 10) n--;
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if (n < 0) n = 0;
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}
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return n;
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};
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/*
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* n / 0 = I
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* n / N = N
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* n / I = 0
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* 0 / n = 0
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* 0 / 0 = N
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* 0 / N = N
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* 0 / I = 0
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* N / n = N
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* N / 0 = N
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* N / N = N
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* N / I = N
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* I / n = I
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* I / 0 = I
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* I / N = N
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* I / I = N
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*
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* Return a new Decimal whose value is the value of this Decimal divided by `y`, rounded to
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* `precision` significant digits using rounding mode `rounding`.
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*
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*/
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P.dividedBy = P.div = function (y) {
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return divide(this, new this.constructor(y));
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};
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/*
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* 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);
|
|
}
|
|
|
|
|
|
P[Symbol.for('nodejs.util.inspect.custom')] = P.toString;
|
|
P[Symbol.toStringTag] = 'Decimal';
|
|
|
|
// Create and configure initial Decimal constructor.
|
|
export var Decimal = P.constructor = clone(DEFAULTS);
|
|
|
|
// Create the internal constants from their string values.
|
|
LN10 = new Decimal(LN10);
|
|
PI = new Decimal(PI);
|
|
|
|
export default Decimal;
|