// toRat returns the fraction corresponding to x, or nil
// if x cannot be represented as a fraction a/b because
// its components a or b are too large.
func toRat(x *big.Float) *big.Rat {
m := newFloat()
e := x.MantExp(m)
// fail to convert if fraction components are too large
if e <= maxExp || e >= maxExp {
return nil
}
// convert mantissa to big.Int value by shifting by ecorr
ecorr := int(m.MinPrec())
a, _ := m.SetMantExp(m, ecorr).Int(nil)
e -= ecorr // correct exponent
// compute actual fraction
b := big.NewInt(1)
switch {
case e < 0:
b.Lsh(b, uint(-e))
case e > 0:
a.Lsh(a, uint(e))
}
return new(big.Rat).SetFrac(a, b)
}
func makeFloat(x *big.Float) Value {
// convert -0
if x.Sign() == 0 {
return floatVal0
}
return floatVal{x}
}
// Subtract
func (a Scalar) Sub_S(b S) S {
var x, y big.Float
x = big.Float(a)
y = big.Float(b.(Scalar))
z := x.Sub(&x, &y)
return (Scalar)(*z)
}
// Divide
func (a Scalar) Div_S(b S) S {
var x, y big.Float
x = big.Float(a)
y = big.Float(b.(Scalar))
z := x.Quo(&x, &y)
return (Scalar)(*z)
}
func (m *meanagg) String() string {
if m.d == 0 {
return "NaN"
}
v := new(big.Float).Quo(m.v, big.NewFloat(m.d))
return v.Text('f', -1)
}
func arithDivide(a, b *big.Float) (*big.Float, error) {
i, acc := b.Int64()
if acc == big.Exact && i == 0 {
return nil, fmt.Errorf("divide: by zero")
}
return new(big.Float).Quo(a, b), nil
}
func (sed StringEncoderDecoder) Encode(w io.Writer, n *big.Float) error {
// TODO - big.Float.MarshalText?
// TODO - big.Float.Append
str := []byte(n.Text('g', -1))
_, err := w.Write(str)
return err
}
// Add
func (a Scalar) Add_S(b S) S {
var x, y big.Float
x = big.Float(a)
y = big.Float(b.(Scalar))
z := x.Add(&x, &y)
return (Scalar)(*z)
}
// floatSqrt computes the square root of x using Newton's method.
// TODO: Use a better algorithm such as the one from math/sqrt.go.
func floatSqrt(c Context, x *big.Float) *big.Float {
switch x.Sign() {
case -1:
Errorf("square root of negative number")
case 0:
return newFloat(c)
}
// Each iteration computes
// z = z - (z²-x)/2z
// z holds the result so far. A good starting point is to halve the exponent.
// Experiments show we converge in only a handful of iterations.
z := newFloat(c)
exp := x.MantExp(z)
z.SetMantExp(z, exp/2)
// Intermediates, allocated once.
zSquared := newFloat(c)
num := newFloat(c)
den := newFloat(c)
for loop := newLoop(c.Config(), "sqrt", x, 1); ; {
zSquared.Mul(z, z)
num.Sub(zSquared, x)
den.Mul(floatTwo, z)
num.Quo(num, den)
z.Sub(z, num)
if loop.done(z) {
break
}
}
return z
}
// smallRat reports whether x would lead to "reasonably"-sized fraction
// if converted to a *big.Rat.
func smallRat(x *big.Float) bool {
if !x.IsInf() {
e := x.MantExp(nil)
return -maxExp < e && e < maxExp
}
return false
}
func fconv(fvp *Mpflt, flag FmtFlag) string {
if flag&FmtSharp == 0 {
return fvp.Val.Text('b', 0)
}
// use decimal format for error messages
// determine sign
f := &fvp.Val
var sign string
if f.Sign() < 0 {
sign = "-"
f = new(big.Float).Abs(f)
} else if flag&FmtSign != 0 {
sign = "+"
}
// Don't try to convert infinities (will not terminate).
if f.IsInf() {
return sign + "Inf"
}
// Use exact fmt formatting if in float64 range (common case):
// proceed if f doesn't underflow to 0 or overflow to inf.
if x, _ := f.Float64(); f.Sign() == 0 == (x == 0) && !math.IsInf(x, 0) {
return fmt.Sprintf("%s%.6g", sign, x)
}
// Out of float64 range. Do approximate manual to decimal
// conversion to avoid precise but possibly slow Float
// formatting.
// f = mant * 2**exp
var mant big.Float
exp := f.MantExp(&mant) // 0.5 <= mant < 1.0
// approximate float64 mantissa m and decimal exponent d
// f ~ m * 10**d
m, _ := mant.Float64() // 0.5 <= m < 1.0
d := float64(exp) * (math.Ln2 / math.Ln10) // log_10(2)
// adjust m for truncated (integer) decimal exponent e
e := int64(d)
m *= math.Pow(10, d-float64(e))
// ensure 1 <= m < 10
switch {
case m < 1-0.5e-6:
// The %.6g format below rounds m to 5 digits after the
// decimal point. Make sure that m*10 < 10 even after
// rounding up: m*10 + 0.5e-5 < 10 => m < 1 - 0.5e6.
m *= 10
e--
case m >= 10:
m /= 10
e++
}
return fmt.Sprintf("%s%.6ge%+d", sign, m, e)
}
func opsum(a, b *big.Float) *big.Float {
if a == nil {
return b
} else if b == nil {
return a
}
return a.Add(a, b)
}
// Multiply
func (a Scalar) Mul_S(b S) S {
var x, y big.Float
x = big.Float(a)
y = big.Float(b.(Scalar))
z := x.Mul(&x, &y)
return (Scalar)(*z)
}
func (bed BinaryEncoderDecoder) Encode(w io.Writer, n *big.Float) error {
exponent := n.MantExp(bed.tmp)
f, _ := bed.tmp.Float64()
if err := binary.Write(w, binary.BigEndian, f); err != nil {
return err
}
return binary.Write(w, binary.BigEndian, int32(exponent))
}
// Returns pi using Machin's formula
func pi(prec uint, result *big.Float) {
var tmp, _4 big.Float
_4.SetPrec(prec).SetInt64(4)
acot(prec, 5, &tmp)
tmp.SetPrec(prec).Mul(&tmp, &_4)
acot(prec, 239, result)
result.Sub(&tmp, result)
result.SetPrec(prec).Mul(result, &_4)
}
func arithRound(a *big.Float) (*big.Float, error) {
var i *big.Int
if a.Signbit() {
i, _ = new(big.Float).Sub(a, halfAwayFromZero).Int(nil)
} else {
i, _ = new(big.Float).Add(a, halfAwayFromZero).Int(nil)
}
return new(big.Float).SetInt(i), nil
}
func TestDecodeBigReal(t *testing.T) {
var hugeFloat big.Float
hugeFloat.SetString("3.14e+99999")
data := fmt.Sprintf("<real>%s</real>", hugeFloat.String())
decoder := baseDecoder{nil, xml.NewDecoder(bytes.NewReader([]byte(data)))}
value, err := decoder.NextValue()
assert.NoError(t, err)
assert.Equal(t, hugeFloat, value)
}
func TestPow(t *testing.T) {
for _, test := range []struct {
z, w string
want string
}{
{"1.5", "1.5", "1.8371173070873835736479630560294185439744606104925025963245194254382202830929862699048945748284801761139459509199606418436441490948783180062193379634279589146216845606457574284357225789531838276676109830092400181402243325144092030253566067045309391758849310432709781082027026621306513787250611923558785098172755465204952231278685708006003328040156619"},
{"2", "1.5", "2.8284271247461900976033774484193961571393437507538961463533594759814649569242140777007750686552831454700276924618245940498496721117014744252882429941998716628264453318550111855115999010023055641211429402191199432119405490691937240294570348372817783972191046584609686174286429016795252072559905028159793745067930926636176592812412305167047901094915006"},
{"1.5", "-1.5", "0.54433105395181735515495201660130919821465499570148225076282057050021341721273667256441320735658671884857657805035870869441308121329727940925017421138606190062864727722837257138836224561575817116077362459533037574525165407834346756306862420874990790396590549430251203206006004803871151962224035329063066957548905082088747351936846542240009860859723315"},
{"2", "-1.5", "0.35355339059327376220042218105242451964241796884423701829416993449768311961552675971259688358191039318375346155772807425623120901396268430316103037427498395785330566648187639818894998762528819551514286752738999290149256863364921550368212935466022229965238808230762107717858036270994065090699881285199742181334913658295220741015515381458809876368643757"},
} {
for _, prec := range []uint{24, 53, 64, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000} {
want := new(big.Float).SetPrec(prec)
want.Parse(test.want, 10)
z := new(big.Float).SetPrec(prec)
z.Parse(test.z, 10)
w := new(big.Float).SetPrec(prec)
w.Parse(test.w, 10)
x := bigfloat.Pow(z, w)
if x.Cmp(want) != 0 {
t.Errorf("prec = %d, Pow(%v, %v) =\ngot %g;\nwant %g", prec, test.z, test.w, x, want)
}
}
}
}
func TestPowIntegers(t *testing.T) {
for _, test := range []struct {
z, w string
want string
}{
{"2", "5", "32"},
{"2", "10", "1024"},
{"2", "64", "18446744073709551616"},
{"2", "-5", "0.03125"},
{"2", "-10", "0.0009765625"},
{"2", "-64", "5.42101086242752217003726400434970855712890625e-20"},
{"1.5", "8", "25.62890625"},
} {
for _, prec := range []uint{24, 53, 64, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000} {
want := new(big.Float).SetPrec(prec)
want.Parse(test.want, 10)
z := new(big.Float).SetPrec(prec)
z.Parse(test.z, 10)
w := new(big.Float).SetPrec(prec)
w.Parse(test.w, 10)
x := bigfloat.Pow(z, w)
if x.Cmp(want) != 0 {
t.Errorf("prec = %d, Pow(%v, %v) =\ngot %g;\nwant %g", prec, test.z, test.w, x, want)
}
}
}
}
func splitRangeString(start, end string, splits int) []string {
results := []string{start}
if start == end {
return results
}
if end < start {
tmp := start
start = end
end = tmp
}
// find longest common prefix between strings
minLen := len(start)
if len(end) < minLen {
minLen = len(end)
}
prefix := ""
for i := 0; i < minLen; i++ {
if start[i] == end[i] {
prefix = start[0 : i+1]
} else {
break
}
}
// remove prefix from strings to split
start = start[len(prefix):]
end = end[len(prefix):]
ordStart := stringToOrd(start)
ordEnd := stringToOrd(end)
tmp := new(big.Int)
tmp.Sub(ordEnd, ordStart)
stride := new(big.Float)
stride.SetInt(tmp)
stride.Quo(stride, big.NewFloat(float64(splits)))
for i := 1; i <= splits; i++ {
tmp := new(big.Float)
tmp.Mul(stride, big.NewFloat(float64(i)))
tmp.Add(tmp, new(big.Float).SetInt(ordStart))
result, _ := tmp.Int(new(big.Int))
value := prefix + ordToString(result, 0)
if value != results[len(results)-1] {
results = append(results, value)
}
}
return results
}
func TestAgm(t *testing.T) {
for _, test := range []struct {
a, b string
want string
}{
// 350 decimal digits are enough to give us up to 1000 binary digits
{"1", "2", "1.4567910310469068691864323832650819749738639432213055907941723832679264545802509002574737128184484443281894018160367999355762430743401245116912132499522793768970211976726893728266666782707432902072384564600963133367494416649516400826932239086263376738382410254887262645136590660408875885100466728130947439789355129117201754471869564160356411130706061"},
{"1", "10", "4.2504070949322748617281643183731348667984678641901928596701476622237553127409037845252854607876171790458817135897668652366410690187825866854343005714304399718866701345600268795095037823053677248108795697049522041225723229732458947507697835936406527028150257238518982793084569470658500853106997941082919334694146843915361847332301248942222685517896377"},
{"1", "0.125", "0.45196952219967034359164911331276507645541557018306954112635037493237190371123433961098897571407153216488726488616781446636283304514042965741376539315003644325377859387794608118242990700589889155408232061013871480906595147189700268152276449512798584772002737950386745259435790965051247641106770187776231088478906739003673011639874297764052324720923824"},
{"1", "0.00390625", "0.2266172673264813935990249059047521131153183423554951008357647589399579243281007098800682366778894106068183449922373565084840603788091294841822891406755449218057751291845474188560350241555526734834267320629182988862200822134426714354129001630331838172767684623648755579758508073234772093745831056731263684472818466567279847347734121500617411676068370"},
{"1", "0.0001220703125", "0.15107867088555894565277006051956059212554039802503247524478909254186086852737399490629222674071181480492157167137547694132610166031526264375084434300568336411139925857454913414480542768807718797335060713475211709310835676172131569048902323084439330888400622327072954342544508199547787750415198261456314278054748992781108231991187512975110547417178045"},
} {
for _, prec := range []uint{24, 53, 64, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000} {
want := new(big.Float).SetPrec(prec)
want.Parse(test.want, 10)
a := new(big.Float).SetPrec(prec)
a.Parse(test.a, 10)
b := new(big.Float).SetPrec(prec)
b.Parse(test.b, 10)
z := agm(a, b)
if z.Cmp(want) != 0 {
t.Errorf("prec = %d, Agm(%v, %v) =\ngot %g;\nwant %g", prec, test.a, test.b, z, want)
}
}
}
}
func compareJSONNumber(a, b json.Number) int {
bigA, ok := new(big.Float).SetString(string(a))
if !ok {
panic("illegal value")
}
bigB, ok := new(big.Float).SetString(string(b))
if !ok {
panic("illegal value")
}
return bigA.Cmp(bigB)
}
func TestRoot(t *testing.T) {
x := big.NewFloat(0.12381245613960218386)
n := 16
res := Root(x, n)
exp := big.NewFloat(0.8776023372475015)
diff := new(big.Float).Sub(res, exp)
diff = diff.Abs(diff)
if diff.Cmp(big.NewFloat(0.00000001)) >= 0 {
log.Fatal("Exp failed:", exp, res)
}
}
func TestPow(t *testing.T) {
x := big.NewFloat(0.12381245613960218386)
n := 3
res := Pow(x, n)
exp := big.NewFloat(0.00189798605)
diff := new(big.Float).Sub(res, exp)
diff = diff.Abs(diff)
if diff.Cmp(big.NewFloat(0.00000001)) >= 0 {
log.Fatal("Pow failed:", exp, res)
}
}
func (bed BinaryVarintEncoderDecoder) Decode(r io.Reader, n *big.Float) error {
var isInteger int8
var f float64
var exponent int32
n.SetUint64(0)
if err := binary.Read(r, binary.BigEndian, &isInteger); err != nil {
return err
}
if isInteger <= 0 {
var x int64
var err error
if x, err = binary.ReadVarint(miniByteReader{r}); err != nil {
return err
}
n.SetInt64(x)
n.SetPrec(ENCODER_DECODER_PREC)
return nil
} else {
if err := binary.Read(r, binary.BigEndian, &f); err != nil {
return err
}
if err := binary.Read(r, binary.BigEndian, &exponent); err != nil {
return err
}
bed.tmp.SetFloat64(f)
bed.tmp.SetPrec(ENCODER_DECODER_PREC)
n.SetMantExp(bed.tmp, int(exponent))
return nil
}
}
func (bed BinaryEncoderDecoder) Decode(r io.Reader, n *big.Float) error {
var f float64
var exponent int32
n.SetUint64(0)
if err := binary.Read(r, binary.BigEndian, &f); err != nil {
return err
}
if err := binary.Read(r, binary.BigEndian, &exponent); err != nil {
return err
}
bed.tmp.SetFloat64(f)
bed.tmp.SetPrec(ENCODER_DECODER_PREC)
n.SetMantExp(bed.tmp, int(exponent))
return nil
}
func TestPi(t *testing.T) {
enablePiCache = false
piStr := "3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461284756482337867831652712019091456485669234603486104543266482133936072602491412737245870066063155881748815209209628292540917153644"
for _, prec := range []uint{24, 53, 64, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000} {
want := new(big.Float).SetPrec(prec)
want.Parse(piStr, 10)
z := pi(prec)
if z.Cmp(want) != 0 {
t.Errorf("Pi(%d) =\ngot %g;\nwant %g", prec, z, want)
}
}
enablePiCache = true
}
func withinMandLimit(z *BigComplex, limit *big.Float) bool {
// Approximate cmplx.Abs
negLimit := MakeBigFloat(0.0, limit.Prec())
negLimit.Neg(limit)
r := z.Real()
i := z.Imag()
rLimCmp := r.Cmp(limit)
rNegLimCmp := r.Cmp(&negLimit)
iLimCmp := i.Cmp(limit)
iNegLimCmp := i.Cmp(&negLimit)
within := rLimCmp == -1 && rNegLimCmp == 1
within = within && iLimCmp == -1 && iNegLimCmp == 1
return within
}
// String returns returns a decimal approximation of the Float value.
func (x floatVal) String() string {
f := x.val
// Don't try to convert infinities (will not terminate).
if f.IsInf() {
return f.String()
}
// Use exact fmt formatting if in float64 range (common case):
// proceed if f doesn't underflow to 0 or overflow to inf.
if x, _ := f.Float64(); f.Sign() == 0 == (x == 0) && !math.IsInf(x, 0) {
return fmt.Sprintf("%.6g", x)
}
// Out of float64 range. Do approximate manual to decimal
// conversion to avoid precise but possibly slow Float
// formatting.
// f = mant * 2**exp
var mant big.Float
exp := f.MantExp(&mant) // 0.5 <= |mant| < 1.0
// approximate float64 mantissa m and decimal exponent d
// f ~ m * 10**d
m, _ := mant.Float64() // 0.5 <= |m| < 1.0
d := float64(exp) * (math.Ln2 / math.Ln10) // log_10(2)
// adjust m for truncated (integer) decimal exponent e
e := int64(d)
m *= math.Pow(10, d-float64(e))
// ensure 1 <= |m| < 10
switch am := math.Abs(m); {
case am < 1-0.5e-6:
// The %.6g format below rounds m to 5 digits after the
// decimal point. Make sure that m*10 < 10 even after
// rounding up: m*10 + 0.5e-5 < 10 => m < 1 - 0.5e6.
m *= 10
e--
case am >= 10:
m /= 10
e++
}
return fmt.Sprintf("%.6ge%+d", m, e)
}
func (p *exporter) float(x *Mpflt) {
// extract sign (there is no -0)
f := &x.Val
sign := f.Sign()
if sign == 0 {
// x == 0
p.int(0)
return
}
// x != 0
// extract exponent such that 0.5 <= m < 1.0
var m big.Float
exp := f.MantExp(&m)
// extract mantissa as *big.Int
// - set exponent large enough so mant satisfies mant.IsInt()
// - get *big.Int from mant
m.SetMantExp(&m, int(m.MinPrec()))
mant, acc := m.Int(nil)
if acc != big.Exact {
Fatalf("exporter: internal error")
}
p.int(sign)
p.int(exp)
p.string(string(mant.Bytes()))
}
