decimal.go

// Package decimal implements an arbitrary precision fixed-point decimal.
//
// The zero-value of a Decimal is 0, as you would expect.
//
// The best way to create a new Decimal is to use decimal.NewFromString, ex:
//
//     n, err := decimal.NewFromString("-123.4567")
//     n.String() // output: "-123.4567"
//
// To use Decimal as part of a struct:
//
//     type Struct struct {
//         Number Decimal
//     }
//
// Note: This can "only" represent numbers with a maximum of 2^31 digits after the decimal point.
package decimal

import (
	"database/sql/driver"
	"encoding/binary"
	"fmt"
	"math"
	"math/big"
	"regexp"
	"strconv"
	"strings"
)

// DivisionPrecision is the number of decimal places in the result when it
// doesn't divide exactly.
//
// Example:
//
//     d1 := decimal.NewFromFloat(2).Div(decimal.NewFromFloat(3))
//     d1.String() // output: "0.6666666666666667"
//     d2 := decimal.NewFromFloat(2).Div(decimal.NewFromFloat(30000))
//     d2.String() // output: "0.0000666666666667"
//     d3 := decimal.NewFromFloat(20000).Div(decimal.NewFromFloat(3))
//     d3.String() // output: "6666.6666666666666667"
//     decimal.DivisionPrecision = 3
//     d4 := decimal.NewFromFloat(2).Div(decimal.NewFromFloat(3))
//     d4.String() // output: "0.667"
//
var DivisionPrecision = 16

// MarshalJSONWithoutQuotes should be set to true if you want the decimal to
// be JSON marshaled as a number, instead of as a string.
// WARNING: this is dangerous for decimals with many digits, since many JSON
// unmarshallers (ex: Javascript's) will unmarshal JSON numbers to IEEE 754
// double-precision floating point numbers, which means you can potentially
// silently lose precision.
var MarshalJSONWithoutQuotes = false

// ExpMaxIterations specifies the maximum number of iterations needed to calculate
// precise natural exponent value using ExpHullAbrham method.
var ExpMaxIterations = 1000

// Zero constant, to make computations faster.
// Zero should never be compared with == or != directly, please use decimal.Equal or decimal.Cmp instead.
var Zero = New(0, 1)

var zeroInt = big.NewInt(0)
var oneInt = big.NewInt(1)
var twoInt = big.NewInt(2)
var fourInt = big.NewInt(4)
var fiveInt = big.NewInt(5)
var tenInt = big.NewInt(10)
var twentyInt = big.NewInt(20)

var factorials = []Decimal{New(1, 0)}

// Decimal represents a fixed-point decimal. It is immutable.
// number = value * 10 ^ exp
type Decimal struct {
	value *big.Int

	// NOTE(vadim): this must be an int32, because we cast it to float64 during
	// calculations. If exp is 64 bit, we might lose precision.
	// If we cared about being able to represent every possible decimal, we
	// could make exp a *big.Int but it would hurt performance and numbers
	// like that are unrealistic.
	exp int32
}

// New returns a new fixed-point decimal, value * 10 ^ exp.
func New(value int64, exp int32) Decimal {
	return Decimal{
		value: big.NewInt(value),
		exp:   exp,
	}
}

// NewFromInt converts a int64 to Decimal.
//
// Example:
//
//     NewFromInt(123).String() // output: "123"
//     NewFromInt(-10).String() // output: "-10"
func NewFromInt(value int64) Decimal {
	return Decimal{
		value: big.NewInt(value),
		exp:   0,
	}
}

// NewFromInt32 converts a int32 to Decimal.
//
// Example:
//
//     NewFromInt(123).String() // output: "123"
//     NewFromInt(-10).String() // output: "-10"
func NewFromInt32(value int32) Decimal {
	return Decimal{
		value: big.NewInt(int64(value)),
		exp:   0,
	}
}

// NewFromBigInt returns a new Decimal from a big.Int, value * 10 ^ exp
func NewFromBigInt(value *big.Int, exp int32) Decimal {
	return Decimal{
		value: new(big.Int).Set(value),
		exp:   exp,
	}
}

// NewFromString returns a new Decimal from a string representation.
// Trailing zeroes are not trimmed.
//
// Example:
//
//     d, err := NewFromString("-123.45")
//     d2, err := NewFromString(".0001")
//     d3, err := NewFromString("1.47000")
//
func NewFromString(value string) (Decimal, error) {
	originalInput := value
	var intString string
	var exp int64

	// Check if number is using scientific notation
	eIndex := strings.IndexAny(value, "Ee")
	if eIndex != -1 {
		expInt, err := strconv.ParseInt(value[eIndex+1:], 10, 32)
		if err != nil {
			if e, ok := err.(*strconv.NumError); ok && e.Err == strconv.ErrRange {
				return Decimal{}, fmt.Errorf("can't convert %s to decimal: fractional part too long", value)
			}
			return Decimal{}, fmt.Errorf("can't convert %s to decimal: exponent is not numeric", value)
		}
		value = value[:eIndex]
		exp = expInt
	}

	pIndex := -1
	vLen := len(value)
	for i := 0; i < vLen; i++ {
		if value[i] == '.' {
			if pIndex > -1 {
				return Decimal{}, fmt.Errorf("can't convert %s to decimal: too many .s", value)
			}
			pIndex = i
		}
	}

	if pIndex == -1 {
		// There is no decimal point, we can just parse the original string as
		// an int
		intString = value
	} else {
		if pIndex+1 < vLen {
			intString = value[:pIndex] + value[pIndex+1:]
		} else {
			intString = value[:pIndex]
		}
		expInt := -len(value[pIndex+1:])
		exp += int64(expInt)
	}

	var dValue *big.Int
	// strconv.ParseInt is faster than new(big.Int).SetString so this is just a shortcut for strings we know won't overflow
	if len(intString) <= 18 {
		parsed64, err := strconv.ParseInt(intString, 10, 64)
		if err != nil {
			return Decimal{}, fmt.Errorf("can't convert %s to decimal", value)
		}
		dValue = big.NewInt(parsed64)
	} else {
		dValue = new(big.Int)
		_, ok := dValue.SetString(intString, 10)
		if !ok {
			return Decimal{}, fmt.Errorf("can't convert %s to decimal", value)
		}
	}

	if exp < math.MinInt32 || exp > math.MaxInt32 {
		// NOTE(vadim): I doubt a string could realistically be this long
		return Decimal{}, fmt.Errorf("can't convert %s to decimal: fractional part too long", originalInput)
	}

	return Decimal{
		value: dValue,
		exp:   int32(exp),
	}, nil
}

// NewFromFormattedString returns a new Decimal from a formatted string representation.
// The second argument - replRegexp, is a regular expression that is used to find characters that should be
// removed from given decimal string representation. All matched characters will be replaced with an empty string.
//
// Example:
//
//     r := regexp.MustCompile("[$,]")
//     d1, err := NewFromFormattedString("$5,125.99", r)
//
//     r2 := regexp.MustCompile("[_]")
//     d2, err := NewFromFormattedString("1_000_000", r2)
//
//     r3 := regexp.MustCompile("[USD\\s]")
//     d3, err := NewFromFormattedString("5000 USD", r3)
//
func NewFromFormattedString(value string, replRegexp *regexp.Regexp) (Decimal, error) {
	parsedValue := replRegexp.ReplaceAllString(value, "")
	d, err := NewFromString(parsedValue)
	if err != nil {
		return Decimal{}, err
	}
	return d, nil
}

// RequireFromString returns a new Decimal from a string representation
// or panics if NewFromString would have returned an error.
//
// Example:
//
//     d := RequireFromString("-123.45")
//     d2 := RequireFromString(".0001")
//
func RequireFromString(value string) Decimal {
	dec, err := NewFromString(value)
	if err != nil {
		panic(err)
	}
	return dec
}

// NewFromFloat converts a float64 to Decimal.
//
// The converted number will contain the number of significant digits that can be
// represented in a float with reliable roundtrip.
// This is typically 15 digits, but may be more in some cases.
// See https://www.exploringbinary.com/decimal-precision-of-binary-floating-point-numbers/ for more information.
//
// For slightly faster conversion, use NewFromFloatWithExponent where you can specify the precision in absolute terms.
//
// NOTE: this will panic on NaN, +/-inf
func NewFromFloat(value float64) Decimal {
	if value == 0 {
		return New(0, 0)
	}
	return newFromFloat(value, math.Float64bits(value), &float64info)
}

// NewFromFloat32 converts a float32 to Decimal.
//
// The converted number will contain the number of significant digits that can be
// represented in a float with reliable roundtrip.
// This is typically 6-8 digits depending on the input.
// See https://www.exploringbinary.com/decimal-precision-of-binary-floating-point-numbers/ for more information.
//
// For slightly faster conversion, use NewFromFloatWithExponent where you can specify the precision in absolute terms.
//
// NOTE: this will panic on NaN, +/-inf
func NewFromFloat32(value float32) Decimal {
	if value == 0 {
		return New(0, 0)
	}
	// XOR is workaround for https://github.com/golang/go/issues/26285
	a := math.Float32bits(value) ^ 0x80808080
	return newFromFloat(float64(value), uint64(a)^0x80808080, &float32info)
}

func newFromFloat(val float64, bits uint64, flt *floatInfo) Decimal {
	if math.IsNaN(val) || math.IsInf(val, 0) {
		panic(fmt.Sprintf("Cannot create a Decimal from %v", val))
	}
	exp := int(bits>>flt.mantbits) & (1<<flt.expbits - 1)
	mant := bits & (uint64(1)<<flt.mantbits - 1)

	switch exp {
	case 0:
		// denormalized
		exp++

	default:
		// add implicit top bit
		mant |= uint64(1) << flt.mantbits
	}
	exp += flt.bias

	var d decimal
	d.Assign(mant)
	d.Shift(exp - int(flt.mantbits))
	d.neg = bits>>(flt.expbits+flt.mantbits) != 0

	roundShortest(&d, mant, exp, flt)
	// If less than 19 digits, we can do calculation in an int64.
	if d.nd < 19 {
		tmp := int64(0)
		m := int64(1)
		for i := d.nd - 1; i >= 0; i-- {
			tmp += m * int64(d.d[i]-'0')
			m *= 10
		}
		if d.neg {
			tmp *= -1
		}
		return Decimal{value: big.NewInt(tmp), exp: int32(d.dp) - int32(d.nd)}
	}
	dValue := new(big.Int)
	dValue, ok := dValue.SetString(string(d.d[:d.nd]), 10)
	if ok {
		return Decimal{value: dValue, exp: int32(d.dp) - int32(d.nd)}
	}

	return NewFromFloatWithExponent(val, int32(d.dp)-int32(d.nd))
}

// NewFromFloatWithExponent converts a float64 to Decimal, with an arbitrary
// number of fractional digits.
//
// Example:
//
//     NewFromFloatWithExponent(123.456, -2).String() // output: "123.46"
//
func NewFromFloatWithExponent(value float64, exp int32) Decimal {
	if math.IsNaN(value) || math.IsInf(value, 0) {
		panic(fmt.Sprintf("Cannot create a Decimal from %v", value))
	}

	bits := math.Float64bits(value)
	mant := bits & (1<<52 - 1)
	exp2 := int32((bits >> 52) & (1<<11 - 1))
	sign := bits >> 63

	if exp2 == 0 {
		// specials
		if mant == 0 {
			return Decimal{}
		}
		// subnormal
		exp2++
	} else {
		// normal
		mant |= 1 << 52
	}

	exp2 -= 1023 + 52

	// normalizing base-2 values
	for mant&1 == 0 {
		mant = mant >> 1
		exp2++
	}

	// maximum number of fractional base-10 digits to represent 2^N exactly cannot be more than -N if N<0
	if exp < 0 && exp < exp2 {
		if exp2 < 0 {
			exp = exp2
		} else {
			exp = 0
		}
	}

	// representing 10^M * 2^N as 5^M * 2^(M+N)
	exp2 -= exp

	temp := big.NewInt(1)
	dMant := big.NewInt(int64(mant))

	// applying 5^M
	if exp > 0 {
		temp = temp.SetInt64(int64(exp))
		temp = temp.Exp(fiveInt, temp, nil)
	} else if exp < 0 {
		temp = temp.SetInt64(-int64(exp))
		temp = temp.Exp(fiveInt, temp, nil)
		dMant = dMant.Mul(dMant, temp)
		temp = temp.SetUint64(1)
	}

	// applying 2^(M+N)
	if exp2 > 0 {
		dMant = dMant.Lsh(dMant, uint(exp2))
	} else if exp2 < 0 {
		temp = temp.Lsh(temp, uint(-exp2))
	}

	// rounding and downscaling
	if exp > 0 || exp2 < 0 {
		halfDown := new(big.Int).Rsh(temp, 1)
		dMant = dMant.Add(dMant, halfDown)
		dMant = dMant.Quo(dMant, temp)
	}

	if sign == 1 {
		dMant = dMant.Neg(dMant)
	}

	return Decimal{
		value: dMant,
		exp:   exp,
	}
}

// Copy returns a copy of decimal with the same value and exponent, but a different pointer to value.
func (d Decimal) Copy() Decimal {
	d.ensureInitialized()
	return Decimal{
		value: &(*d.value),
		exp:   d.exp,
	}
}

// rescale returns a rescaled version of the decimal. Returned
// decimal may be less precise if the given exponent is bigger
// than the initial exponent of the Decimal.
// NOTE: this will truncate, NOT round
//
// Example:
//
// 	d := New(12345, -4)
//	d2 := d.rescale(-1)
//	d3 := d2.rescale(-4)
//	println(d1)
//	println(d2)
//	println(d3)
//
// Output:
//
//	1.2345
//	1.2
//	1.2000
//
func (d Decimal) rescale(exp int32) Decimal {
	d.ensureInitialized()

	if d.exp == exp {
		return Decimal{
			new(big.Int).Set(d.value),
			d.exp,
		}
	}

	// NOTE(vadim): must convert exps to float64 before - to prevent overflow
	diff := math.Abs(float64(exp) - float64(d.exp))
	value := new(big.Int).Set(d.value)

	expScale := new(big.Int).Exp(tenInt, big.NewInt(int64(diff)), nil)
	if exp > d.exp {
		value = value.Quo(value, expScale)
	} else if exp < d.exp {
		value = value.Mul(value, expScale)
	}

	return Decimal{
		value: value,
		exp:   exp,
	}
}

// Abs returns the absolute value of the decimal.
func (d Decimal) Abs() Decimal {
	if !d.IsNegative() {
		return d
	}
	d.ensureInitialized()
	d2Value := new(big.Int).Abs(d.value)
	return Decimal{
		value: d2Value,
		exp:   d.exp,
	}
}

// Add returns d + d2.
func (d Decimal) Add(d2 Decimal) Decimal {
	rd, rd2 := RescalePair(d, d2)

	d3Value := new(big.Int).Add(rd.value, rd2.value)
	return Decimal{
		value: d3Value,
		exp:   rd.exp,
	}
}

// Sub returns d - d2.
func (d Decimal) Sub(d2 Decimal) Decimal {
	rd, rd2 := RescalePair(d, d2)

	d3Value := new(big.Int).Sub(rd.value, rd2.value)
	return Decimal{
		value: d3Value,
		exp:   rd.exp,
	}
}

// Neg returns -d.
func (d Decimal) Neg() Decimal {
	d.ensureInitialized()
	val := new(big.Int).Neg(d.value)
	return Decimal{
		value: val,
		exp:   d.exp,
	}
}

// Mul returns d * d2.
func (d Decimal) Mul(d2 Decimal) Decimal {
	d.ensureInitialized()
	d2.ensureInitialized()

	expInt64 := int64(d.exp) + int64(d2.exp)
	if expInt64 > math.MaxInt32 || expInt64 < math.MinInt32 {
		// NOTE(vadim): better to panic than give incorrect results, as
		// Decimals are usually used for money
		panic(fmt.Sprintf("exponent %v overflows an int32!", expInt64))
	}

	d3Value := new(big.Int).Mul(d.value, d2.value)
	return Decimal{
		value: d3Value,
		exp:   int32(expInt64),
	}
}

// Shift shifts the decimal in base 10.
// It shifts left when shift is positive and right if shift is negative.
// In simpler terms, the given value for shift is added to the exponent
// of the decimal.
func (d Decimal) Shift(shift int32) Decimal {
	d.ensureInitialized()
	return Decimal{
		value: new(big.Int).Set(d.value),
		exp:   d.exp + shift,
	}
}

// Div returns d / d2. If it doesn't divide exactly, the result will have
// DivisionPrecision digits after the decimal point.
func (d Decimal) Div(d2 Decimal) Decimal {
	return d.DivRound(d2, int32(DivisionPrecision))
}

// QuoRem does division with remainder
// d.QuoRem(d2,precision) returns quotient q and remainder r such that
//   d = d2 * q + r, q an integer multiple of 10^(-precision)
//   0 <= r < abs(d2) * 10 ^(-precision) if d>=0
//   0 >= r > -abs(d2) * 10 ^(-precision) if d<0
// Note that precision<0 is allowed as input.
func (d Decimal) QuoRem(d2 Decimal, precision int32) (Decimal, Decimal) {
	d.ensureInitialized()
	d2.ensureInitialized()
	if d2.value.Sign() == 0 {
		panic("decimal division by 0")
	}
	scale := -precision
	e := int64(d.exp - d2.exp - scale)
	if e > math.MaxInt32 || e < math.MinInt32 {
		panic("overflow in decimal QuoRem")
	}
	var aa, bb, expo big.Int
	var scalerest int32
	// d = a 10^ea
	// d2 = b 10^eb
	if e < 0 {
		aa = *d.value
		expo.SetInt64(-e)
		bb.Exp(tenInt, &expo, nil)
		bb.Mul(d2.value, &bb)
		scalerest = d.exp
		// now aa = a
		//     bb = b 10^(scale + eb - ea)
	} else {
		expo.SetInt64(e)
		aa.Exp(tenInt, &expo, nil)
		aa.Mul(d.value, &aa)
		bb = *d2.value
		scalerest = scale + d2.exp
		// now aa = a ^ (ea - eb - scale)
		//     bb = b
	}
	var q, r big.Int
	q.QuoRem(&aa, &bb, &r)
	dq := Decimal{value: &q, exp: scale}
	dr := Decimal{value: &r, exp: scalerest}
	return dq, dr
}

// DivRound divides and rounds to a given precision
// i.e. to an integer multiple of 10^(-precision)
//   for a positive quotient digit 5 is rounded up, away from 0
//   if the quotient is negative then digit 5 is rounded down, away from 0
// Note that precision<0 is allowed as input.
func (d Decimal) DivRound(d2 Decimal, precision int32) Decimal {
	// QuoRem already checks initialization
	q, r := d.QuoRem(d2, precision)
	// the actual rounding decision is based on comparing r*10^precision and d2/2
	// instead compare 2 r 10 ^precision and d2
	var rv2 big.Int
	rv2.Abs(r.value)
	rv2.Lsh(&rv2, 1)
	// now rv2 = abs(r.value) * 2
	r2 := Decimal{value: &rv2, exp: r.exp + precision}
	// r2 is now 2 * r * 10 ^ precision
	var c = r2.Cmp(d2.Abs())

	if c < 0 {
		return q
	}

	if d.value.Sign()*d2.value.Sign() < 0 {
		return q.Sub(New(1, -precision))
	}

	return q.Add(New(1, -precision))
}

// Mod returns d % d2.
func (d Decimal) Mod(d2 Decimal) Decimal {
	quo := d.DivRound(d2, -d.exp+1).Truncate(0)
	return d.Sub(d2.Mul(quo))
}

// Pow returns d to the power d2
func (d Decimal) Pow(d2 Decimal) Decimal {
	var temp Decimal
	if d2.IntPart() == 0 {
		return NewFromFloat(1)
	}
	temp = d.Pow(d2.Div(NewFromFloat(2)))
	if d2.IntPart()%2 == 0 {
		return temp.Mul(temp)
	}
	if d2.IntPart() > 0 {
		return temp.Mul(temp).Mul(d)
	}
	return temp.Mul(temp).Div(d)
}

// ExpHullAbrham calculates the natural exponent of decimal (e to the power of d) using Hull-Abraham algorithm.
// OverallPrecision argument specifies the overall precision of the result (integer part + decimal part).
//
// ExpHullAbrham is faster than ExpTaylor for small precision values, but it is much slower for large precision values.
//
// Example:
//
//     NewFromFloat(26.1).ExpHullAbrham(2).String()    // output: "220000000000"
//     NewFromFloat(26.1).ExpHullAbrham(20).String()   // output: "216314672147.05767284"
//
func (d Decimal) ExpHullAbrham(overallPrecision uint32) (Decimal, error) {
	// Algorithm based on Variable precision exponential function.
	// ACM Transactions on Mathematical Software by T. E. Hull & A. Abrham.
	if d.IsZero() {
		return Decimal{oneInt, 0}, nil
	}

	currentPrecision := overallPrecision

	// Algorithm does not work if currentPrecision * 23 < |x|.
	// Precision is automatically increased in such cases, so the value can be calculated precisely.
	// If newly calculated precision is higher than ExpMaxIterations the currentPrecision will not be changed.
	f := d.Abs().InexactFloat64()
	if ncp := f / 23; ncp > float64(currentPrecision) && ncp < float64(ExpMaxIterations) {
		currentPrecision = uint32(math.Ceil(ncp))
	}

	// fail if abs(d) beyond an over/underflow threshold
	overflowThreshold := New(23*int64(currentPrecision), 0)
	if d.Abs().Cmp(overflowThreshold) > 0 {
		return Decimal{}, fmt.Errorf("over/underflow threshold, exp(x) cannot be calculated precisely")
	}

	// Return 1 if abs(d) small enough; this also avoids later over/underflow
	overflowThreshold2 := New(9, -int32(currentPrecision)-1)
	if d.Abs().Cmp(overflowThreshold2) <= 0 {
		return Decimal{oneInt, d.exp}, nil
	}

	// t is the smallest integer >= 0 such that the corresponding abs(d/k) < 1
	t := d.exp + int32(d.NumDigits()) // Add d.NumDigits because the paper assumes that d.value [0.1, 1)

	if t < 0 {
		t = 0
	}

	k := New(1, t)                                     // reduction factor
	r := Decimal{new(big.Int).Set(d.value), d.exp - t} // reduced argument
	p := int32(currentPrecision) + t + 2               // precision for calculating the sum

	// Determine n, the number of therms for calculating sum
	// use first Newton step (1.435p - 1.182) / log10(p/abs(r))
	// for solving appropriate equation, along with directed
	// roundings and simple rational bound for log10(p/abs(r))
	rf := r.Abs().InexactFloat64()
	pf := float64(p)
	nf := math.Ceil((1.453*pf - 1.182) / math.Log10(pf/rf))
	if nf > float64(ExpMaxIterations) || math.IsNaN(nf) {
		return Decimal{}, fmt.Errorf("exact value cannot be calculated in <=ExpMaxIterations iterations")
	}
	n := int64(nf)

	tmp := New(0, 0)
	sum := New(1, 0)
	one := New(1, 0)
	for i := n - 1; i > 0; i-- {
		tmp.value.SetInt64(i)
		sum = sum.Mul(r.DivRound(tmp, p))
		sum = sum.Add(one)
	}

	ki := k.IntPart()
	res := New(1, 0)
	for i := ki; i > 0; i-- {
		res = res.Mul(sum)
	}

	resNumDigits := int32(res.NumDigits())

	var roundDigits int32
	if resNumDigits > abs(res.exp) {
		roundDigits = int32(currentPrecision) - resNumDigits - res.exp
	} else {
		roundDigits = int32(currentPrecision)
	}

	res = res.Round(roundDigits)

	return res, nil
}

// ExpTaylor calculates the natural exponent of decimal (e to the power of d) using Taylor series expansion.
// Precision argument specifies how precise the result must be (number of digits after decimal point).
// Negative precision is allowed.
//
// ExpTaylor is much faster for large precision values than ExpHullAbrham.
//
// Example:
//
//     d, err := NewFromFloat(26.1).ExpTaylor(2).String()
//     d.String()  // output: "216314672147.06"
//
//     NewFromFloat(26.1).ExpTaylor(20).String()
//     d.String()  // output: "216314672147.05767284062928674083"
//
//     NewFromFloat(26.1).ExpTaylor(-10).String()
//     d.String()  // output: "220000000000"
//
func (d Decimal) ExpTaylor(precision int32) (Decimal, error) {
	// Note(mwoss): Implementation can be optimized by exclusively using big.Int API only
	if d.IsZero() {
		return Decimal{oneInt, 0}.Round(precision), nil
	}

	var epsilon Decimal
	var divPrecision int32
	if precision < 0 {
		epsilon = New(1, -1)
		divPrecision = 8
	} else {
		epsilon = New(1, -precision-1)
		divPrecision = precision + 1
	}

	decAbs := d.Abs()
	pow := d.Abs()
	factorial := New(1, 0)

	result := New(1, 0)

	for i := int64(1); ; {
		step := pow.DivRound(factorial, divPrecision)
		result = result.Add(step)

		// Stop Taylor series when current step is smaller than epsilon
		if step.Cmp(epsilon) < 0 {
			break
		}

		pow = pow.Mul(decAbs)

		i++

		// Calculate next factorial number or retrieve cached value
		if len(factorials) >= int(i) && !factorials[i-1].IsZero() {
			factorial = factorials[i-1]
		} else {
			// To avoid any race conditions, firstly the zero value is appended to a slice to create
			// a spot for newly calculated factorial. After that, the zero value is replaced by calculated
			// factorial using the index notation.
			factorial = factorials[i-2].Mul(New(i, 0))
			factorials = append(factorials, Zero)
			factorials[i-1] = factorial
		}
	}

	if d.Sign() < 0 {
		result = New(1, 0).DivRound(result, precision+1)
	}

	result = result.Round(precision)
	return result, nil
}

// NumDigits returns the number of digits of the decimal coefficient (d.Value)
// Note: Current implementation is extremely slow for large decimals and/or decimals with large fractional part
func (d Decimal) NumDigits() int {
	// Note(mwoss): It can be optimized, unnecessary cast of big.Int to string
	if d.IsNegative() {
		return len(d.value.String()) - 1
	}
	return len(d.value.String())
}

// IsInteger returns true when decimal can be represented as an integer value, otherwise, it returns false.
func (d Decimal) IsInteger() bool {
	// The most typical case, all decimal with exponent higher or equal 0 can be represented as integer
	if d.exp >= 0 {
		return true
	}
	// When the exponent is negative we have to check every number after the decimal place
	// If all of them are zeroes, we are sure that given decimal can be represented as an integer
	var r big.Int
	q := new(big.Int).Set(d.value)
	for z := abs(d.exp); z > 0; z-- {
		q.QuoRem(q, tenInt, &r)
		if r.Cmp(zeroInt) != 0 {
			return false
		}
	}
	return true
}

// Abs calculates absolute value of any int32. Used for calculating absolute value of decimal's exponent.
func abs(n int32) int32 {
	if n < 0 {
		return -n
	}
	return n
}

// Cmp compares the numbers represented by d and d2 and returns:
//
//     -1 if d <  d2
//      0 if d == d2
//     +1 if d >  d2
//
func (d Decimal) Cmp(d2 Decimal) int {
	d.ensureInitialized()
	d2.ensureInitialized()

	if d.exp == d2.exp {
		return d.value.Cmp(d2.value)
	}

	rd, rd2 := RescalePair(d, d2)

	return rd.value.Cmp(rd2.value)
}

// Equal returns whether the numbers represented by d and d2 are equal.
func (d Decimal) Equal(d2 Decimal) bool {
	return d.Cmp(d2) == 0
}

// Equals is deprecated, please use Equal method instead
func (d Decimal) Equals(d2 Decimal) bool {
	return d.Equal(d2)
}

// GreaterThan (GT) returns true when d is greater than d2.
func (d Decimal) GreaterThan(d2 Decimal) bool {
	return d.Cmp(d2) == 1
}

// GreaterThanOrEqual (GTE) returns true when d is greater than or equal to d2.
func (d Decimal) GreaterThanOrEqual(d2 Decimal) bool {
	cmp := d.Cmp(d2)
	return cmp == 1 || cmp == 0
}

// LessThan (LT) returns true when d is less than d2.
func (d Decimal) LessThan(d2 Decimal) bool {
	return d.Cmp(d2) == -1
}

// LessThanOrEqual (LTE) returns true when d is less than or equal to d2.
func (d Decimal) LessThanOrEqual(d2 Decimal) bool {
	cmp := d.Cmp(d2)
	return cmp == -1 || cmp == 0
}

// Sign returns:
//
//	-1 if d <  0
//	 0 if d == 0
//	+1 if d >  0
//
func (d Decimal) Sign() int {
	if d.value == nil {
		return 0
	}
	return d.value.Sign()
}

// IsPositive return
//
//	true if d > 0
//	false if d == 0
//	false if d < 0
func (d Decimal) IsPositive() bool {
	return d.Sign() == 1
}

// IsNegative return
//
//	true if d < 0
//	false if d == 0
//	false if d > 0
func (d Decimal) IsNegative() bool {
	return d.Sign() == -1
}

// IsZero return
//
//	true if d == 0
//	false if d > 0
//	false if d < 0
func (d Decimal) IsZero() bool {
	return d.Sign() == 0
}

// Exponent returns the exponent, or scale component of the decimal.
func (d Decimal) Exponent() int32 {
	return d.exp
}

// Coefficient returns the coefficient of the decimal. It is scaled by 10^Exponent()
func (d Decimal) Coefficient() *big.Int {
	d.ensureInitialized()
	// we copy the coefficient so that mutating the result does not mutate the Decimal.
	return new(big.Int).Set(d.value)
}

// CoefficientInt64 returns the coefficient of the decimal as int64. It is scaled by 10^Exponent()
// If coefficient cannot be represented in an int64, the result will be undefined.
func (d Decimal) CoefficientInt64() int64 {
	d.ensureInitialized()
	return d.value.Int64()
}

// IntPart returns the integer component of the decimal.
func (d Decimal) IntPart() int64 {
	scaledD := d.rescale(0)
	return scaledD.value.Int64()
}

// BigInt returns integer component of the decimal as a BigInt.
func (d Decimal) BigInt() *big.Int {
	return new(big.Int).Set(d.value)
}

// BigFloat returns decimal as BigFloat.
// Be aware that casting decimal to BigFloat might cause a loss of precision.
func (d Decimal) BigFloat() *big.Float {
	f := &big.Float{}
	f.SetString(d.String())
	return f
}

// Rat returns a rational number representation of the decimal.
func (d Decimal) Rat() *big.Rat {
	d.ensureInitialized()
	if d.exp <= 0 {
		// NOTE(vadim): must negate after casting to prevent int32 overflow
		denom := new(big.Int).Exp(tenInt, big.NewInt(-int64(d.exp)), nil)
		return new(big.Rat).SetFrac(d.value, denom)
	}

	mul := new(big.Int).Exp(tenInt, big.NewInt(int64(d.exp)), nil)
	num := new(big.Int).Mul(d.value, mul)
	return new(big.Rat).SetFrac(num, oneInt)
}

// Float64 returns the nearest float64 value for d and a bool indicating
// whether f represents d exactly.
// For more details, see the documentation for big.Rat.Float64
func (d Decimal) Float64() (f float64, exact bool) {
	return d.Rat().Float64()
}

// InexactFloat64 returns the nearest float64 value for d.
// It doesn't indicate if the returned value represents d exactly.
func (d Decimal) InexactFloat64() float64 {
	f, _ := d.Float64()
	return f
}

// String returns the string representation of the decimal
// with the fixed point.
//
// Example:
//
//     d := New(-12345, -3)
//     println(d.String())
//
// Output:
//
//     -12.345
//
func (d Decimal) String() string {
	return d.string(true)
}

// StringFixed returns a rounded fixed-point string with places digits after
// the decimal point.
//
// Example:
//
// 	   NewFromFloat(0).StringFixed(2) // output: "0.00"
// 	   NewFromFloat(0).StringFixed(0) // output: "0"
// 	   NewFromFloat(5.45).StringFixed(0) // output: "5"
// 	   NewFromFloat(5.45).StringFixed(1) // output: "5.5"
// 	   NewFromFloat(5.45).StringFixed(2) // output: "5.45"
// 	   NewFromFloat(5.45).StringFixed(3) // output: "5.450"
// 	   NewFromFloat(545).StringFixed(-1) // output: "550"
//
func (d Decimal) StringFixed(places int32) string {
	rounded := d.Round(places)
	return rounded.string(false)
}

// StringFixedBank returns a banker rounded fixed-point string with places digits
// after the decimal point.
//
// Example:
//
// 	   NewFromFloat(0).StringFixedBank(2) // output: "0.00"
// 	   NewFromFloat(0).StringFixedBank(0) // output: "0"
// 	   NewFromFloat(5.45).StringFixedBank(0) // output: "5"
// 	   NewFromFloat(5.45).StringFixedBank(1) // output: "5.4"
// 	   NewFromFloat(5.45).StringFixedBank(2) // output: "5.45"
// 	   NewFromFloat(5.45).StringFixedBank(3) // output: "5.450"
// 	   NewFromFloat(545).StringFixedBank(-1) // output: "540"
//
func (d Decimal) StringFixedBank(places int32) string {
	rounded := d.RoundBank(places)
	return rounded.string(false)
}

// StringFixedCash returns a Swedish/Cash rounded fixed-point string. For
// more details see the documentation at function RoundCash.
func (d Decimal) StringFixedCash(interval uint8) string {
	rounded := d.RoundCash(interval)
	return rounded.string(false)
}

// Round rounds the decimal to places decimal places.
// If places < 0, it will round the integer part to the nearest 10^(-places).
//
// Example:
//
// 	   NewFromFloat(5.45).Round(1).String() // output: "5.5"
// 	   NewFromFloat(545).Round(-1).String() // output: "550"
//
func (d Decimal) Round(places int32) Decimal {
	if d.exp == -places {
		return d
	}
	// truncate to places + 1
	ret := d.rescale(-places - 1)

	// add sign(d) * 0.5
	if ret.value.Sign() < 0 {
		ret.value.Sub(ret.value, fiveInt)
	} else {
		ret.value.Add(ret.value, fiveInt)
	}

	// floor for positive numbers, ceil for negative numbers
	_, m := ret.value.DivMod(ret.value, tenInt, new(big.Int))
	ret.exp++
	if ret.value.Sign() < 0 && m.Cmp(zeroInt) != 0 {
		ret.value.Add(ret.value, oneInt)
	}

	return ret
}

// RoundCeil rounds the decimal towards +infinity.
//
// Example:
//
//     NewFromFloat(545).RoundCeil(-2).String()   // output: "600"
//     NewFromFloat(500).RoundCeil(-2).String()   // output: "500"
//     NewFromFloat(1.1001).RoundCeil(2).String() // output: "1.11"
//     NewFromFloat(-1.454).RoundCeil(1).String() // output: "-1.5"
//
func (d Decimal) RoundCeil(places int32) Decimal {
	if d.exp >= -places {
		return d
	}

	rescaled := d.rescale(-places)
	if d.Equal(rescaled) {
		return d
	}

	if d.value.Sign() > 0 {
		rescaled.value.Add(rescaled.value, oneInt)
	}

	return rescaled
}

// RoundFloor rounds the decimal towards -infinity.
//
// Example:
//
//     NewFromFloat(545).RoundFloor(-2).String()   // output: "500"
//     NewFromFloat(-500).RoundFloor(-2).String()   // output: "-500"
//     NewFromFloat(1.1001).RoundFloor(2).String() // output: "1.1"
//     NewFromFloat(-1.454).RoundFloor(1).String() // output: "-1.4"
//
func (d Decimal) RoundFloor(places int32) Decimal {
	if d.exp >= -places {
		return d
	}

	rescaled := d.rescale(-places)
	if d.Equal(rescaled) {
		return d
	}

	if d.value.Sign() < 0 {
		rescaled.value.Sub(rescaled.value, oneInt)
	}

	return rescaled
}

// RoundUp rounds the decimal away from zero.
//
// Example:
//
//     NewFromFloat(545).RoundUp(-2).String()   // output: "600"
//     NewFromFloat(500).RoundUp(-2).String()   // output: "500"
//     NewFromFloat(1.1001).RoundUp(2).String() // output: "1.11"
//     NewFromFloat(-1.454).RoundUp(1).String() // output: "-1.4"
//
func (d Decimal) RoundUp(places int32) Decimal {
	if d.exp >= -places {
		return d
	}

	rescaled := d.rescale(-places)
	if d.Equal(rescaled) {
		return d
	}

	if d.value.Sign() > 0 {
		rescaled.value.Add(rescaled.value, oneInt)
	} else if d.value.Sign() < 0 {
		rescaled.value.Sub(rescaled.value, oneInt)
	}

	return rescaled
}

// RoundDown rounds the decimal towards zero.
//
// Example:
//
//     NewFromFloat(545).RoundDown(-2).String()   // output: "500"
//     NewFromFloat(-500).RoundDown(-2).String()   // output: "-500"
//     NewFromFloat(1.1001).RoundDown(2).String() // output: "1.1"
//     NewFromFloat(-1.454).RoundDown(1).String() // output: "-1.5"
//
func (d Decimal) RoundDown(places int32) Decimal {
	if d.exp >= -places {
		return d
	}

	rescaled := d.rescale(-places)
	if d.Equal(rescaled) {
		return d
	}
	return rescaled
}

// RoundBank rounds the decimal to places decimal places.
// If the final digit to round is equidistant from the nearest two integers the
// rounded value is taken as the even number
//
// If places < 0, it will round the integer part to the nearest 10^(-places).
//
// Examples:
//
// 	   NewFromFloat(5.45).RoundBank(1).String() // output: "5.4"
// 	   NewFromFloat(545).RoundBank(-1).String() // output: "540"
// 	   NewFromFloat(5.46).RoundBank(1).String() // output: "5.5"
// 	   NewFromFloat(546).RoundBank(-1).String() // output: "550"
// 	   NewFromFloat(5.55).RoundBank(1).String() // output: "5.6"
// 	   NewFromFloat(555).RoundBank(-1).String() // output: "560"
//
func (d Decimal) RoundBank(places int32) Decimal {

	round := d.Round(places)
	remainder := d.Sub(round).Abs()

	half := New(5, -places-1)
	if remainder.Cmp(half) == 0 && round.value.Bit(0) != 0 {
		if round.value.Sign() < 0 {
			round.value.Add(round.value, oneInt)
		} else {
			round.value.Sub(round.value, oneInt)
		}
	}

	return round
}

// RoundCash aka Cash/Penny/öre rounding rounds decimal to a specific
// interval. The amount payable for a cash transaction is rounded to the nearest
// multiple of the minimum currency unit available. The following intervals are
// available: 5, 10, 25, 50 and 100; any other number throws a panic.
//	    5:   5 cent rounding 3.43 => 3.45
// 	   10:  10 cent rounding 3.45 => 3.50 (5 gets rounded up)
// 	   25:  25 cent rounding 3.41 => 3.50
// 	   50:  50 cent rounding 3.75 => 4.00
// 	  100: 100 cent rounding 3.50 => 4.00
// For more details: https://en.wikipedia.org/wiki/Cash_rounding
func (d Decimal) RoundCash(interval uint8) Decimal {
	var iVal *big.Int
	switch interval {
	case 5:
		iVal = twentyInt
	case 10:
		iVal = tenInt
	case 25:
		iVal = fourInt
	case 50:
		iVal = twoInt
	case 100:
		iVal = oneInt
	default:
		panic(fmt.Sprintf("Decimal does not support this Cash rounding interval `%d`. Supported: 5, 10, 25, 50, 100", interval))
	}
	dVal := Decimal{
		value: iVal,
	}

	// TODO: optimize those calculations to reduce the high allocations (~29 allocs).
	return d.Mul(dVal).Round(0).Div(dVal).Truncate(2)
}

// Floor returns the nearest integer value less than or equal to d.
func (d Decimal) Floor() Decimal {
	d.ensureInitialized()

	if d.exp >= 0 {
		return d
	}

	exp := big.NewInt(10)

	// NOTE(vadim): must negate after casting to prevent int32 overflow
	exp.Exp(exp, big.NewInt(-int64(d.exp)), nil)

	z := new(big.Int).Div(d.value, exp)
	return Decimal{value: z, exp: 0}
}

// Ceil returns the nearest integer value greater than or equal to d.
func (d Decimal) Ceil() Decimal {
	d.ensureInitialized()

	if d.exp >= 0 {
		return d
	}

	exp := big.NewInt(10)

	// NOTE(vadim): must negate after casting to prevent int32 overflow
	exp.Exp(exp, big.NewInt(-int64(d.exp)), nil)

	z, m := new(big.Int).DivMod(d.value, exp, new(big.Int))
	if m.Cmp(zeroInt) != 0 {
		z.Add(z, oneInt)
	}
	return Decimal{value: z, exp: 0}
}

// Truncate truncates off digits from the number, without rounding.
//
// NOTE: precision is the last digit that will not be truncated (must be >= 0).
//
// Example:
//
//     decimal.NewFromString("123.456").Truncate(2).String() // "123.45"
//
func (d Decimal) Truncate(precision int32) Decimal {
	d.ensureInitialized()
	if precision >= 0 && -precision > d.exp {
		return d.rescale(-precision)
	}
	return d
}

// UnmarshalJSON implements the json.Unmarshaler interface.
func (d *Decimal) UnmarshalJSON(decimalBytes []byte) error {
	if string(decimalBytes) == "null" {
		return nil
	}

	str, err := unquoteIfQuoted(decimalBytes)
	if err != nil {
		return fmt.Errorf("error decoding string '%s': %s", decimalBytes, err)
	}

	decimal, err := NewFromString(str)
	*d = decimal
	if err != nil {
		return fmt.Errorf("error decoding string '%s': %s", str, err)
	}
	return nil
}

// MarshalJSON implements the json.Marshaler interface.
func (d Decimal) MarshalJSON() ([]byte, error) {
	var str string
	if MarshalJSONWithoutQuotes {
		str = d.String()
	} else {
		str = "\"" + d.String() + "\""
	}
	return []byte(str), nil
}

// UnmarshalBinary implements the encoding.BinaryUnmarshaler interface. As a string representation
// is already used when encoding to text, this method stores that string as []byte
func (d *Decimal) UnmarshalBinary(data []byte) error {
	// Verify we have at least 4 bytes for the exponent. The GOB encoded value
	// may be empty.
	if len(data) < 4 {
		return fmt.Errorf("error decoding binary %v: expected at least 4 bytes, got %d", data, len(data))
	}

	// Extract the exponent
	d.exp = int32(binary.BigEndian.Uint32(data[:4]))

	// Extract the value
	d.value = new(big.Int)
	if err := d.value.GobDecode(data[4:]); err != nil {
		return fmt.Errorf("error decoding binary %v: %s", data, err)
	}

	return nil
}

// MarshalBinary implements the encoding.BinaryMarshaler interface.
func (d Decimal) MarshalBinary() (data []byte, err error) {
	// Write the exponent first since it's a fixed size
	v1 := make([]byte, 4)
	binary.BigEndian.PutUint32(v1, uint32(d.exp))

	// Add the value
	var v2 []byte
	if v2, err = d.value.GobEncode(); err != nil {
		return
	}

	// Return the byte array
	data = append(v1, v2...)
	return
}

// Scan implements the sql.Scanner interface for database deserialization.
func (d *Decimal) Scan(value interface{}) error {
	// first try to see if the data is stored in database as a Numeric datatype
	switch v := value.(type) {

	case float32:
		*d = NewFromFloat(float64(v))
		return nil

	case float64:
		// numeric in sqlite3 sends us float64
		*d = NewFromFloat(v)
		return nil

	case int64:
		// at least in sqlite3 when the value is 0 in db, the data is sent
		// to us as an int64 instead of a float64 ...
		*d = New(v, 0)
		return nil

	default:
		// default is trying to interpret value stored as string
		str, err := unquoteIfQuoted(v)
		if err != nil {
			return err
		}
		*d, err = NewFromString(str)
		return err
	}
}

// Value implements the driver.Valuer interface for database serialization.
func (d Decimal) Value() (driver.Value, error) {
	return d.String(), nil
}

// UnmarshalText implements the encoding.TextUnmarshaler interface for XML
// deserialization.
func (d *Decimal) UnmarshalText(text []byte) error {
	str := string(text)

	dec, err := NewFromString(str)
	*d = dec
	if err != nil {
		return fmt.Errorf("error decoding string '%s': %s", str, err)
	}

	return nil
}

// MarshalText implements the encoding.TextMarshaler interface for XML
// serialization.
func (d Decimal) MarshalText() (text []byte, err error) {
	return []byte(d.String()), nil
}

// GobEncode implements the gob.GobEncoder interface for gob serialization.
func (d Decimal) GobEncode() ([]byte, error) {
	return d.MarshalBinary()
}

// GobDecode implements the gob.GobDecoder interface for gob serialization.
func (d *Decimal) GobDecode(data []byte) error {
	return d.UnmarshalBinary(data)
}

// StringScaled first scales the decimal then calls .String() on it.
// NOTE: buggy, unintuitive, and DEPRECATED! Use StringFixed instead.
func (d Decimal) StringScaled(exp int32) string {
	return d.rescale(exp).String()
}

func (d Decimal) string(trimTrailingZeros bool) string {
	if d.exp >= 0 {
		return d.rescale(0).value.String()
	}

	abs := new(big.Int).Abs(d.value)
	str := abs.String()

	var intPart, fractionalPart string

	// NOTE(vadim): this cast to int will cause bugs if d.exp == INT_MIN
	// and you are on a 32-bit machine. Won't fix this super-edge case.
	dExpInt := int(d.exp)
	if len(str) > -dExpInt {
		intPart = str[:len(str)+dExpInt]
		fractionalPart = str[len(str)+dExpInt:]
	} else {
		intPart = "0"

		num0s := -dExpInt - len(str)
		fractionalPart = strings.Repeat("0", num0s) + str
	}

	if trimTrailingZeros {
		i := len(fractionalPart) - 1
		for ; i >= 0; i-- {
			if fractionalPart[i] != '0' {
				break
			}
		}
		fractionalPart = fractionalPart[:i+1]
	}

	number := intPart
	if len(fractionalPart) > 0 {
		number += "." + fractionalPart
	}

	if d.value.Sign() < 0 {
		return "-" + number
	}

	return number
}

func (d *Decimal) ensureInitialized() {
	if d.value == nil {
		d.value = new(big.Int)
	}
}

// Min returns the smallest Decimal that was passed in the arguments.
//
// To call this function with an array, you must do:
//
//     Min(arr[0], arr[1:]...)
//
// This makes it harder to accidentally call Min with 0 arguments.
func Min(first Decimal, rest ...Decimal) Decimal {
	ans := first
	for _, item := range rest {
		if item.Cmp(ans) < 0 {
			ans = item
		}
	}
	return ans
}

// Max returns the largest Decimal that was passed in the arguments.
//
// To call this function with an array, you must do:
//
//     Max(arr[0], arr[1:]...)
//
// This makes it harder to accidentally call Max with 0 arguments.
func Max(first Decimal, rest ...Decimal) Decimal {
	ans := first
	for _, item := range rest {
		if item.Cmp(ans) > 0 {
			ans = item
		}
	}
	return ans
}

// Sum returns the combined total of the provided first and rest Decimals
func Sum(first Decimal, rest ...Decimal) Decimal {
	total := first
	for _, item := range rest {
		total = total.Add(item)
	}

	return total
}

// Avg returns the average value of the provided first and rest Decimals
func Avg(first Decimal, rest ...Decimal) Decimal {
	count := New(int64(len(rest)+1), 0)
	sum := Sum(first, rest...)
	return sum.Div(count)
}

// RescalePair rescales two decimals to common exponential value (minimal exp of both decimals)
func RescalePair(d1 Decimal, d2 Decimal) (Decimal, Decimal) {
	d1.ensureInitialized()
	d2.ensureInitialized()

	if d1.exp == d2.exp {
		return d1, d2
	}

	baseScale := min(d1.exp, d2.exp)
	if baseScale != d1.exp {
		return d1.rescale(baseScale), d2
	}
	return d1, d2.rescale(baseScale)
}

func min(x, y int32) int32 {
	if x >= y {
		return y
	}
	return x
}

func unquoteIfQuoted(value interface{}) (string, error) {
	var bytes []byte

	switch v := value.(type) {
	case string:
		bytes = []byte(v)
	case []byte:
		bytes = v
	default:
		return "", fmt.Errorf("could not convert value '%+v' to byte array of type '%T'",
			value, value)
	}

	// If the amount is quoted, strip the quotes
	if len(bytes) > 2 && bytes[0] == '"' && bytes[len(bytes)-1] == '"' {
		bytes = bytes[1 : len(bytes)-1]
	}
	return string(bytes), nil
}

// NullDecimal represents a nullable decimal with compatibility for
// scanning null values from the database.
type NullDecimal struct {
	Decimal Decimal
	Valid   bool
}

func NewNullDecimal(d Decimal) NullDecimal {
	return NullDecimal{
		Decimal: d,
		Valid:   true,
	}
}

// Scan implements the sql.Scanner interface for database deserialization.
func (d *NullDecimal) Scan(value interface{}) error {
	if value == nil {
		d.Valid = false
		return nil
	}
	d.Valid = true
	return d.Decimal.Scan(value)
}

// Value implements the driver.Valuer interface for database serialization.
func (d NullDecimal) Value() (driver.Value, error) {
	if !d.Valid {
		return nil, nil
	}
	return d.Decimal.Value()
}

// UnmarshalJSON implements the json.Unmarshaler interface.
func (d *NullDecimal) UnmarshalJSON(decimalBytes []byte) error {
	if string(decimalBytes) == "null" {
		d.Valid = false
		return nil
	}
	d.Valid = true
	return d.Decimal.UnmarshalJSON(decimalBytes)
}

// MarshalJSON implements the json.Marshaler interface.
func (d NullDecimal) MarshalJSON() ([]byte, error) {
	if !d.Valid {
		return []byte("null"), nil
	}
	return d.Decimal.MarshalJSON()
}

// UnmarshalText implements the encoding.TextUnmarshaler interface for XML
// deserialization
func (d *NullDecimal) UnmarshalText(text []byte) error {
	str := string(text)

	// check for empty XML or XML without body e.g., <tag></tag>
	if str == "" {
		d.Valid = false
		return nil
	}
	if err := d.Decimal.UnmarshalText(text); err != nil {
		d.Valid = false
		return err
	}
	d.Valid = true
	return nil
}

// MarshalText implements the encoding.TextMarshaler interface for XML
// serialization.
func (d NullDecimal) MarshalText() (text []byte, err error) {
	if !d.Valid {
		return []byte{}, nil
	}
	return d.Decimal.MarshalText()
}

// Trig functions

// Atan returns the arctangent, in radians, of x.
func (d Decimal) Atan() Decimal {
	if d.Equal(NewFromFloat(0.0)) {
		return d
	}
	if d.GreaterThan(NewFromFloat(0.0)) {
		return d.satan()
	}
	return d.Neg().satan().Neg()
}

func (d Decimal) xatan() Decimal {
	P0 := NewFromFloat(-8.750608600031904122785e-01)
	P1 := NewFromFloat(-1.615753718733365076637e+01)
	P2 := NewFromFloat(-7.500855792314704667340e+01)
	P3 := NewFromFloat(-1.228866684490136173410e+02)
	P4 := NewFromFloat(-6.485021904942025371773e+01)
	Q0 := NewFromFloat(2.485846490142306297962e+01)
	Q1 := NewFromFloat(1.650270098316988542046e+02)
	Q2 := NewFromFloat(4.328810604912902668951e+02)
	Q3 := NewFromFloat(4.853903996359136964868e+02)
	Q4 := NewFromFloat(1.945506571482613964425e+02)
	z := d.Mul(d)
	b1 := P0.Mul(z).Add(P1).Mul(z).Add(P2).Mul(z).Add(P3).Mul(z).Add(P4).Mul(z)
	b2 := z.Add(Q0).Mul(z).Add(Q1).Mul(z).Add(Q2).Mul(z).Add(Q3).Mul(z).Add(Q4)
	z = b1.Div(b2)
	z = d.Mul(z).Add(d)
	return z
}

// satan reduces its argument (known to be positive)
// to the range [0, 0.66] and calls xatan.
func (d Decimal) satan() Decimal {
	Morebits := NewFromFloat(6.123233995736765886130e-17) // pi/2 = PIO2 + Morebits
	Tan3pio8 := NewFromFloat(2.41421356237309504880)      // tan(3*pi/8)
	pi := NewFromFloat(3.14159265358979323846264338327950288419716939937510582097494459)

	if d.LessThanOrEqual(NewFromFloat(0.66)) {
		return d.xatan()
	}
	if d.GreaterThan(Tan3pio8) {
		return pi.Div(NewFromFloat(2.0)).Sub(NewFromFloat(1.0).Div(d).xatan()).Add(Morebits)
	}
	return pi.Div(NewFromFloat(4.0)).Add((d.Sub(NewFromFloat(1.0)).Div(d.Add(NewFromFloat(1.0)))).xatan()).Add(NewFromFloat(0.5).Mul(Morebits))
}

// sin coefficients
var _sin = [...]Decimal{
	NewFromFloat(1.58962301576546568060e-10), // 0x3de5d8fd1fd19ccd
	NewFromFloat(-2.50507477628578072866e-8), // 0xbe5ae5e5a9291f5d
	NewFromFloat(2.75573136213857245213e-6),  // 0x3ec71de3567d48a1
	NewFromFloat(-1.98412698295895385996e-4), // 0xbf2a01a019bfdf03
	NewFromFloat(8.33333333332211858878e-3),  // 0x3f8111111110f7d0
	NewFromFloat(-1.66666666666666307295e-1), // 0xbfc5555555555548
}

// Sin returns the sine of the radian argument x.
func (d Decimal) Sin() Decimal {
	PI4A := NewFromFloat(7.85398125648498535156e-1)                             // 0x3fe921fb40000000, Pi/4 split into three parts
	PI4B := NewFromFloat(3.77489470793079817668e-8)                             // 0x3e64442d00000000,
	PI4C := NewFromFloat(2.69515142907905952645e-15)                            // 0x3ce8469898cc5170,
	M4PI := NewFromFloat(1.273239544735162542821171882678754627704620361328125) // 4/pi

	if d.Equal(NewFromFloat(0.0)) {
		return d
	}
	// make argument positive but save the sign
	sign := false
	if d.LessThan(NewFromFloat(0.0)) {
		d = d.Neg()
		sign = true
	}

	j := d.Mul(M4PI).IntPart()    // integer part of x/(Pi/4), as integer for tests on the phase angle
	y := NewFromFloat(float64(j)) // integer part of x/(Pi/4), as float

	// map zeros to origin
	if j&1 == 1 {
		j++
		y = y.Add(NewFromFloat(1.0))
	}
	j &= 7 // octant modulo 2Pi radians (360 degrees)
	// reflect in x axis
	if j > 3 {
		sign = !sign
		j -= 4
	}
	z := d.Sub(y.Mul(PI4A)).Sub(y.Mul(PI4B)).Sub(y.Mul(PI4C)) // Extended precision modular arithmetic
	zz := z.Mul(z)

	if j == 1 || j == 2 {
		w := zz.Mul(zz).Mul(_cos[0].Mul(zz).Add(_cos[1]).Mul(zz).Add(_cos[2]).Mul(zz).Add(_cos[3]).Mul(zz).Add(_cos[4]).Mul(zz).Add(_cos[5]))
		y = NewFromFloat(1.0).Sub(NewFromFloat(0.5).Mul(zz)).Add(w)
	} else {
		y = z.Add(z.Mul(zz).Mul(_sin[0].Mul(zz).Add(_sin[1]).Mul(zz).Add(_sin[2]).Mul(zz).Add(_sin[3]).Mul(zz).Add(_sin[4]).Mul(zz).Add(_sin[5])))
	}
	if sign {
		y = y.Neg()
	}
	return y
}

// cos coefficients
var _cos = [...]Decimal{
	NewFromFloat(-1.13585365213876817300e-11), // 0xbda8fa49a0861a9b
	NewFromFloat(2.08757008419747316778e-9),   // 0x3e21ee9d7b4e3f05
	NewFromFloat(-2.75573141792967388112e-7),  // 0xbe927e4f7eac4bc6
	NewFromFloat(2.48015872888517045348e-5),   // 0x3efa01a019c844f5
	NewFromFloat(-1.38888888888730564116e-3),  // 0xbf56c16c16c14f91
	NewFromFloat(4.16666666666665929218e-2),   // 0x3fa555555555554b
}

// Cos returns the cosine of the radian argument x.
func (d Decimal) Cos() Decimal {

	PI4A := NewFromFloat(7.85398125648498535156e-1)                             // 0x3fe921fb40000000, Pi/4 split into three parts
	PI4B := NewFromFloat(3.77489470793079817668e-8)                             // 0x3e64442d00000000,
	PI4C := NewFromFloat(2.69515142907905952645e-15)                            // 0x3ce8469898cc5170,
	M4PI := NewFromFloat(1.273239544735162542821171882678754627704620361328125) // 4/pi

	// make argument positive
	sign := false
	if d.LessThan(NewFromFloat(0.0)) {
		d = d.Neg()
	}

	j := d.Mul(M4PI).IntPart()    // integer part of x/(Pi/4), as integer for tests on the phase angle
	y := NewFromFloat(float64(j)) // integer part of x/(Pi/4), as float

	// map zeros to origin
	if j&1 == 1 {
		j++
		y = y.Add(NewFromFloat(1.0))
	}
	j &= 7 // octant modulo 2Pi radians (360 degrees)
	// reflect in x axis
	if j > 3 {
		sign = !sign
		j -= 4
	}
	if j > 1 {
		sign = !sign
	}

	z := d.Sub(y.Mul(PI4A)).Sub(y.Mul(PI4B)).Sub(y.Mul(PI4C)) // Extended precision modular arithmetic
	zz := z.Mul(z)

	if j == 1 || j == 2 {
		y = z.Add(z.Mul(zz).Mul(_sin[0].Mul(zz).Add(_sin[1]).Mul(zz).Add(_sin[2]).Mul(zz).Add(_sin[3]).Mul(zz).Add(_sin[4]).Mul(zz).Add(_sin[5])))
	} else {
		w := zz.Mul(zz).Mul(_cos[0].Mul(zz).Add(_cos[1]).Mul(zz).Add(_cos[2]).Mul(zz).Add(_cos[3]).Mul(zz).Add(_cos[4]).Mul(zz).Add(_cos[5]))
		y = NewFromFloat(1.0).Sub(NewFromFloat(0.5).Mul(zz)).Add(w)
	}
	if sign {
		y = y.Neg()
	}
	return y
}

var _tanP = [...]Decimal{
	NewFromFloat(-1.30936939181383777646e+4), // 0xc0c992d8d24f3f38
	NewFromFloat(1.15351664838587416140e+6),  // 0x413199eca5fc9ddd
	NewFromFloat(-1.79565251976484877988e+7), // 0xc1711fead3299176
}
var _tanQ = [...]Decimal{
	NewFromFloat(1.00000000000000000000e+0),
	NewFromFloat(1.36812963470692954678e+4),  //0x40cab8a5eeb36572
	NewFromFloat(-1.32089234440210967447e+6), //0xc13427bc582abc96
	NewFromFloat(2.50083801823357915839e+7),  //0x4177d98fc2ead8ef
	NewFromFloat(-5.38695755929454629881e+7), //0xc189afe03cbe5a31
}

// Tan returns the tangent of the radian argument x.
func (d Decimal) Tan() Decimal {

	PI4A := NewFromFloat(7.85398125648498535156e-1)                             // 0x3fe921fb40000000, Pi/4 split into three parts
	PI4B := NewFromFloat(3.77489470793079817668e-8)                             // 0x3e64442d00000000,
	PI4C := NewFromFloat(2.69515142907905952645e-15)                            // 0x3ce8469898cc5170,
	M4PI := NewFromFloat(1.273239544735162542821171882678754627704620361328125) // 4/pi

	if d.Equal(NewFromFloat(0.0)) {
		return d
	}

	// make argument positive but save the sign
	sign := false
	if d.LessThan(NewFromFloat(0.0)) {
		d = d.Neg()
		sign = true
	}

	j := d.Mul(M4PI).IntPart()    // integer part of x/(Pi/4), as integer for tests on the phase angle
	y := NewFromFloat(float64(j)) // integer part of x/(Pi/4), as float

	// map zeros to origin
	if j&1 == 1 {
		j++
		y = y.Add(NewFromFloat(1.0))
	}

	z := d.Sub(y.Mul(PI4A)).Sub(y.Mul(PI4B)).Sub(y.Mul(PI4C)) // Extended precision modular arithmetic
	zz := z.Mul(z)

	if zz.GreaterThan(NewFromFloat(1e-14)) {
		w := zz.Mul(_tanP[0].Mul(zz).Add(_tanP[1]).Mul(zz).Add(_tanP[2]))
		x := zz.Add(_tanQ[1]).Mul(zz).Add(_tanQ[2]).Mul(zz).Add(_tanQ[3]).Mul(zz).Add(_tanQ[4])
		y = z.Add(z.Mul(w.Div(x)))
	} else {
		y = z
	}
	if j&2 == 2 {
		y = NewFromFloat(-1.0).Div(y)
	}
	if sign {
		y = y.Neg()
	}
	return y
}