Add implementation of natural logarithm (#339)

* Add initial implementation of natural logarithm
* Add constApproxmation struct to represent mathematical constants with their approximations
This commit is contained in:
Mateusz Woś 2023-12-29 01:59:48 +01:00 committed by GitHub
parent 88705b71ce
commit 12e9241da2
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4 changed files with 275 additions and 0 deletions

63
const.go Normal file
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@ -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]
}

34
const_test.go Normal file
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@ -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)
}
}
}

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@ -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 {

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@ -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