Add implementation of natural logarithm (shopspring#339)

* Add initial implementation of natural logarithm
* Add constApproxmation struct to represent mathematical constants with their approximations

Author: mwoss

const.go

@@ -0,0 +1,63 @@
+package decimal
+
+import (
+	"strings"
+)
+
+const (
+	strLn10 = "2.302585092994045684017991454684364207601101488628772976033327900967572609677352480235997205089598298341967784042286248633409525465082806756666287369098781689482907208325554680843799894826233198528393505308965377732628846163366222287698219886746543667474404243274365155048934314939391479619404400222105101714174800368808401264708068556774321622835522011480466371565912137345074785694768346361679210180644507064800027750268491674655058685693567342067058113642922455440575892572420824131469568901675894025677631135691929203337658714166023010570308963457207544037084746994016826928280848118428931484852494864487192780967627127577539702766860595249671667418348570442250719796500471495105049221477656763693866297697952211071826454973477266242570942932258279850258550978526538320760672631716430950599508780752371033310119785754733154142180842754386359177811705430982748238504564801909561029929182431823752535770975053956518769751037497088869218020518933950723853920514463419726528728696511086257149219884997874887377134568620916705849807828059751193854445009978131146915934666241071846692310107598438319191292230792503747298650929009880391941702654416816335727555703151596113564846546190897042819763365836983716328982174407366009162177850541779276367731145041782137660111010731042397832521894898817597921798666394319523936855916447118246753245630912528778330963604262982153040874560927760726641354787576616262926568298704957954913954918049209069438580790032763017941503117866862092408537949861264933479354871737451675809537088281067452440105892444976479686075120275724181874989395971643105518848195288330746699317814634930000321200327765654130472621883970596794457943468343218395304414844803701305753674262153675579814770458031413637793236291560128185336498466942261465206459942072917119370602444929358037007718981097362533224548366988505528285966192805098447175198503666680874970496982273220244823343097169111136813588418696549323714996941979687803008850408979618598756579894836445212043698216415292987811742973332588607915912510967187510929248475023930572665446276200923068791518135803477701295593646298412366497023355174586195564772461857717369368404676577047874319780573853271810933883496338813069945569399346101090745616033312247949360455361849123333063704751724871276379140924398331810164737823379692265637682071706935846394531616949411701841938119405416449466111274712819705817783293841742231409930022911502362192186723337268385688273533371925103412930705632544426611429765388301822384091026198582888433587455960453004548370789052578473166283701953392231047527564998119228742789713715713228319641003422124210082180679525276689858180956119208391760721080919923461516952599099473782780648128058792731993893453415320185969711021407542282796298237068941764740642225757212455392526179373652434440560595336591539160312524480149313234572453879524389036839236450507881731359711238145323701508413491122324390927681724749607955799151363982881058285740538000653371655553014196332241918087621018204919492651483892"
+)
+
+var (
+	ln10 = newConstApproximation(strLn10)
+)
+
+type constApproximation struct {
+	exact          Decimal
+	approximations []Decimal
+}
+
+func newConstApproximation(value string) constApproximation {
+	parts := strings.Split(value, ".")
+	coeff, fractional := parts[0], parts[1]
+
+	coeffLen := len(coeff)
+	maxPrecision := len(fractional)
+
+	var approximations []Decimal
+	for p := 1; p < maxPrecision; p *= 2 {
+		r := RequireFromString(value[:coeffLen+p])
+		approximations = append(approximations, r)
+	}
+
+	return constApproximation{
+		RequireFromString(value),
+		approximations,
+	}
+}
+
+// Returns the smallest approximation available that's at least as precise
+// as the passed precision (places after decimal point), i.e. Floor[ log2(precision) ] + 1
+func (c constApproximation) withPrecision(precision int32) Decimal {
+	i := 0
+
+	if precision >= 1 {
+		i++
+	}
+
+	for precision >= 16 {
+		precision /= 16
+		i += 4
+	}
+
+	for precision >= 2 {
+		precision /= 2
+		i++
+	}
+
+	if i >= len(c.approximations) {
+		return c.exact
+	}
+
+	return c.approximations[i]
+}

const_test.go

@@ -0,0 +1,34 @@
+package decimal
+
+import "testing"
+
+func TestConstApproximation(t *testing.T) {
+	for _, testCase := range []struct {
+		Const                 string
+		Precision             int32
+		ExpectedApproximation string
+	}{
+		{"2.3025850929940456840179914546", 0, "2"},
+		{"2.3025850929940456840179914546", 1, "2.3"},
+		{"2.3025850929940456840179914546", 3, "2.302"},
+		{"2.3025850929940456840179914546", 5, "2.302585"},
+		{"2.3025850929940456840179914546", 10, "2.302585092994045"},
+		{"2.3025850929940456840179914546", 100, "2.3025850929940456840179914546"},
+		{"2.3025850929940456840179914546", -1, "2"},
+		{"2.3025850929940456840179914546", -5, "2"},
+		{"3.14159265359", 0, "3"},
+		{"3.14159265359", 1, "3.1"},
+		{"3.14159265359", 2, "3.141"},
+		{"3.14159265359", 4, "3.1415926"},
+		{"3.14159265359", 13, "3.14159265359"},
+	} {
+		ca := newConstApproximation(testCase.Const)
+		expected, _ := NewFromString(testCase.ExpectedApproximation)
+
+		approximation := ca.withPrecision(testCase.Precision)
+
+		if approximation.Cmp(expected) != 0 {
+			t.Errorf("expected approximation %s, got %s - for const with %s precision %d", testCase.ExpectedApproximation, approximation.String(), testCase.Const, testCase.Precision)
+		}
+	}
+}

decimal.go

@@ -808,6 +808,123 @@ func (d Decimal) ExpTaylor(precision int32) (Decimal, error) {
 	return result, nil
 }
 
+// Ln calculates natural logarithm of d.
+// Precision argument specifies how precise the result must be (number of digits after decimal point).
+// Negative precision is allowed.
+//
+// Example:
+//
+//	d1, err := NewFromFloat(13.3).Ln(2)
+//	d1.String()  // output: "2.59"
+//
+//	d2, err := NewFromFloat(579.161).Ln(10)
+//	d2.String()  // output: "6.3615805046"
+func (d Decimal) Ln(precision int32) (Decimal, error) {
+	// Algorithm based on The Use of Iteration Methods for Approximating the Natural Logarithm,
+	// James F. Epperson, The American Mathematical Monthly, Vol. 96, No. 9, November 1989, pp. 831-835.
+	if d.IsNegative() {
+		return Decimal{}, fmt.Errorf("cannot calculate natural logarithm for negative decimals")
+	}
+
+	if d.IsZero() {
+		return Decimal{}, fmt.Errorf("cannot represent natural logarithm of 0, result: -infinity")
+	}
+
+	calcPrecision := precision + 2
+	z := d.Copy()
+
+	var comp1, comp3, comp2, comp4, reduceAdjust Decimal
+	comp1 = z.Sub(Decimal{oneInt, 0})
+	comp3 = Decimal{oneInt, -1}
+
+	// for decimal in range [0.9, 1.1] where ln(d) is close to 0
+	usePowerSeries := false
+
+	if comp1.Abs().Cmp(comp3) <= 0 {
+		usePowerSeries = true
+	} else {
+		// reduce input decimal to range [0.1, 1)
+		expDelta := int32(z.NumDigits()) + z.exp
+		z.exp -= expDelta
+
+		// Input decimal was reduced by factor of 10^expDelta, thus we will need to add
+		// ln(10^expDelta) = expDelta * ln(10)
+		// to the result to compensate that
+		ln10 := ln10.withPrecision(calcPrecision)
+		reduceAdjust = NewFromInt32(expDelta)
+		reduceAdjust = reduceAdjust.Mul(ln10)
+
+		comp1 = z.Sub(Decimal{oneInt, 0})
+
+		if comp1.Abs().Cmp(comp3) <= 0 {
+			usePowerSeries = true
+		} else {
+			// initial estimate using floats
+			zFloat := z.InexactFloat64()
+			comp1 = NewFromFloat(math.Log(zFloat))
+		}
+	}
+
+	epsilon := Decimal{oneInt, -calcPrecision}
+
+	if usePowerSeries {
+		// Power Series - https://en.wikipedia.org/wiki/Logarithm#Power_series
+		// Calculating n-th term of formula: ln(z+1) = 2 sum [ 1 / (2n+1) * (z / (z+2))^(2n+1) ]
+		// until the difference between current and next term is smaller than epsilon.
+		// Coverage quite fast for decimals close to 1.0
+
+		// z + 2
+		comp2 = comp1.Add(Decimal{twoInt, 0})
+		// z / (z + 2)
+		comp3 = comp1.DivRound(comp2, calcPrecision)
+		// 2 * (z / (z + 2))
+		comp1 = comp3.Add(comp3)
+		comp2 = comp1.Copy()
+
+		for n := 1; ; n++ {
+			// 2 * (z / (z+2))^(2n+1)
+			comp2 = comp2.Mul(comp3).Mul(comp3)
+
+			// 1 / (2n+1) * 2 * (z / (z+2))^(2n+1)
+			comp4 = NewFromInt(int64(2*n + 1))
+			comp4 = comp2.DivRound(comp4, calcPrecision)
+
+			// comp1 = 2 sum [ 1 / (2n+1) * (z / (z+2))^(2n+1) ]
+			comp1 = comp1.Add(comp4)
+
+			if comp4.Abs().Cmp(epsilon) <= 0 {
+				break
+			}
+		}
+	} else {
+		// Halley's Iteration.
+		// Calculating n-th term of formula: a_(n+1) = a_n - 2 * (exp(a_n) - z) / (exp(a_n) + z),
+		// until the difference between current and next term is smaller than epsilon
+		for {
+			// exp(a_n)
+			comp3, _ = comp1.ExpTaylor(calcPrecision)
+			// exp(a_n) - z
+			comp2 = comp3.Sub(z)
+			// 2 * (exp(a_n) - z)
+			comp2 = comp2.Add(comp2)
+			// exp(a_n) + z
+			comp4 = comp3.Add(z)
+			// 2 * (exp(a_n) - z) / (exp(a_n) + z)
+			comp3 = comp2.DivRound(comp4, calcPrecision)
+			// comp1 = a_(n+1) = a_n - 2 * (exp(a_n) - z) / (exp(a_n) + z)
+			comp1 = comp1.Sub(comp3)
+
+			if comp3.Abs().Cmp(epsilon) <= 0 {
+				break
+			}
+		}
+	}
+
+	comp1 = comp1.Add(reduceAdjust)
+
+	return comp1.Round(precision), 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 {

decimal_test.go

@@ -2749,6 +2749,67 @@ func TestDecimal_ExpTaylor(t *testing.T) {
 	}
 }
 
+func TestDecimal_Ln(t *testing.T) {
+	for _, testCase := range []struct {
+		Dec       string
+		Precision int32
+		Expected  string
+	}{
+		{"0.1", 25, "-2.3025850929940456840179915"},
+		{"0.01", 25, "-4.6051701859880913680359829"},
+		{"0.001", 25, "-6.9077552789821370520539744"},
+		{"0.00000001", 25, "-18.4206807439523654721439316"},
+		{"1.0", 10, "0.0"},
+		{"1.01", 25, "0.0099503308531680828482154"},
+		{"1.001", 25, "0.0009995003330835331668094"},
+		{"1.0001", 25, "0.0000999950003333083353332"},
+		{"1.1", 25, "0.0953101798043248600439521"},
+		{"1.13", 25, "0.1222176327242492005461486"},
+		{"3.13", 10, "1.1410330046"},
+		{"3.13", 25, "1.1410330045520618486427824"},
+		{"3.13", 50, "1.14103300455206184864278239988848193837089629107972"},
+		{"3.13", 100, "1.1410330045520618486427823998884819383708962910797239760817078430268177216960996098918971117211892839"},
+		{"5.71", 25, "1.7422190236679188486939833"},
+		{"5.7185108151957193571930205", 50, "1.74370842450178929149992165925283704012576949094645"},
+		{"839101.0351", 25, "13.6400864014410013994397240"},
+		{"839101.0351094726488848490572028502", 50, "13.64008640145229044389152437468283605382056561604272"},
+		{"5023583755703750094849.03519358513093500275017501750602739169823", 25, "49.9684305274348922267409953"},
+		{"5023583755703750094849.03519358513093500275017501750602739169823", -1, "50.0"},
+	} {
+		d, _ := NewFromString(testCase.Dec)
+		expected, _ := NewFromString(testCase.Expected)
+
+		ln, err := d.Ln(testCase.Precision)
+		if err != nil {
+			t.Fatal(err)
+		}
+
+		if ln.Cmp(expected) != 0 {
+			t.Errorf("expected %s, got %s, for decimal %s", testCase.Expected, ln.String(), testCase.Dec)
+		}
+	}
+}
+
+func TestDecimal_LnZero(t *testing.T) {
+	d := New(0, 0)
+
+	_, err := d.Ln(5)
+
+	if err == nil {
+		t.Errorf("expected error, natural logarithm of 0 cannot be represented (-infinity)")
+	}
+}
+
+func TestDecimal_LnNegative(t *testing.T) {
+	d := New(-20, 2)
+
+	_, err := d.Ln(5)
+
+	if err == nil {
+		t.Errorf("expected error, natural logarithm cannot be calculated for nagative decimals")
+	}
+}
+
 func TestDecimal_NumDigits(t *testing.T) {
 	for _, testCase := range []struct {
 		Dec               string