// Compute the square root of n using Newton's Method. We start with
// an initial estimate for sqrt(n), and then iterate
// x_{i+1} = 1/2 * ( x_i + (n / x_i) )
// Result is returned in x
func (e *Pslq) Sqrt(n, x *big.Float) {
if n == x {
panic("need distinct input and output")
}
if n.Sign() == 0 {
x.Set(n)
return
} else if n.Sign() < 0 {
panic("Sqrt of negative number")
}
prec := n.Prec()
// Use the floating point square root as initial estimate
nFloat64, _ := n.Float64()
x.SetPrec(prec).SetFloat64(math.Sqrt(nFloat64))
// We use t as a temporary variable. There's no need to set its precision
// since big.Float values with unset (== 0) precision automatically assume
// the largest precision of the arguments when used as the result (receiver)
// of a big.Float operation.
var t big.Float
// Iterate.
for {
t.Quo(n, x) // t = n / x_i
t.Add(x, &t) // t = x_i + (n / x_i)
t.Mul(&e.half, &t) // x_{i+1} = 0.5 * t
if x.Cmp(&t) == 0 {
// Exit loop if no change to result
break
}
x.Set(&t)
}
}
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 TestSqrt(t *testing.T) {
tests := []struct {
prec uint
in float64
}{
{16, 0},
{16, 1},
{16, 4},
{16, 10000},
{16, 2},
{64, 2},
{256, 2},
{1024, 1.5},
}
for _, test := range tests {
x := new(big.Float).SetPrec(test.prec)
x.SetFloat64(test.in)
var got, got2, diff big.Float
pslq := New(test.prec)
pslq.Sqrt(x, &got)
got2.SetPrec(test.prec).Mul(&got, &got)
diff.Sub(&got2, x)
if diff.MinPrec() > 1 {
t.Errorf("sqrt(%f) prec %d wrong got %.20f square %.20f expecting %f diff %g minprec %d", test.in, test.prec, &got, &got2, x, &diff, diff.MinPrec())
}
}
}
func (a *Mpint) SetFloat(b *Mpflt) int {
// avoid converting huge floating-point numbers to integers
// (2*Mpprec is large enough to permit all tests to pass)
if b.Val.MantExp(nil) > 2*Mpprec {
return -1
}
if _, acc := b.Val.Int(&a.Val); acc == big.Exact {
return 0
}
const delta = 16 // a reasonably small number of bits > 0
var t big.Float
t.SetPrec(Mpprec - delta)
// try rounding down a little
t.SetMode(big.ToZero)
t.Set(&b.Val)
if _, acc := t.Int(&a.Val); acc == big.Exact {
return 0
}
// try rounding up a little
t.SetMode(big.AwayFromZero)
t.Set(&b.Val)
if _, acc := t.Int(&a.Val); acc == big.Exact {
return 0
}
return -1
}
func (sed StringEncoderDecoder) Decode(r io.Reader, n *big.Float) error {
n.SetFloat64(0)
n.SetPrec(ENCODER_DECODER_PREC)
buf := make([]byte, 256)
if _, err := r.Read(buf); err != nil {
return err
}
_, _, err := n.Parse(string(buf), 10)
return err
}
// NearestInt set res to the nearest integer to x
func (e *Pslq) NearestInt(x *big.Float, res *big.Int) {
prec := x.Prec()
var tmp big.Float
tmp.SetPrec(prec)
if x.Sign() >= 0 {
tmp.Add(x, &e.half)
} else {
tmp.Sub(x, &e.half)
}
tmp.Int(res)
}
// ToInt converts x to an Int value if x is representable as an Int.
// Otherwise it returns an Unknown.
func ToInt(x Value) Value {
switch x := x.(type) {
case int64Val, intVal:
return x
case ratVal:
if x.val.IsInt() {
return makeInt(x.val.Num())
}
case floatVal:
// avoid creation of huge integers
// (Existing tests require permitting exponents of at least 1024;
// allow any value that would also be permissible as a fraction.)
if smallRat(x.val) {
i := newInt()
if _, acc := x.val.Int(i); acc == big.Exact {
return makeInt(i)
}
// If we can get an integer by rounding up or down,
// assume x is not an integer because of rounding
// errors in prior computations.
const delta = 4 // a small number of bits > 0
var t big.Float
t.SetPrec(prec - delta)
// try rounding down a little
t.SetMode(big.ToZero)
t.Set(x.val)
if _, acc := t.Int(i); acc == big.Exact {
return makeInt(i)
}
// try rounding up a little
t.SetMode(big.AwayFromZero)
t.Set(x.val)
if _, acc := t.Int(i); acc == big.Exact {
return makeInt(i)
}
}
case complexVal:
if re := ToFloat(x); re.Kind() == Float {
return ToInt(re)
}
}
return unknownVal{}
}
// sqrt for big.Float
func sqrt(given *big.Float) *big.Float {
const prec = 200
steps := int(math.Log2(prec))
given.SetPrec(prec)
half := new(big.Float).SetPrec(prec).SetFloat64(0.5)
x := new(big.Float).SetPrec(prec).SetInt64(1)
t := new(big.Float)
for i := 0; i <= steps; i++ {
t.Quo(given, x)
t.Add(x, t)
t.Mul(half, t)
}
return x
}
func ExampleFloat_Add() {
// Operating on numbers of different precision.
var x, y, z big.Float
x.SetInt64(1000) // x is automatically set to 64bit precision
y.SetFloat64(2.718281828) // y is automatically set to 53bit precision
z.SetPrec(32)
z.Add(&x, &y)
fmt.Printf("x = %s (%s, prec = %d, acc = %s)\n", &x, x.Format('p', 0), x.Prec(), x.Acc())
fmt.Printf("y = %s (%s, prec = %d, acc = %s)\n", &y, y.Format('p', 0), y.Prec(), y.Acc())
fmt.Printf("z = %s (%s, prec = %d, acc = %s)\n", &z, z.Format('p', 0), z.Prec(), z.Acc())
// Output:
// x = 1000 (0x.fap10, prec = 64, acc = Exact)
// y = 2.718281828 (0x.adf85458248cd8p2, prec = 53, acc = Exact)
// z = 1002.718282 (0x.faadf854p10, prec = 32, acc = Below)
}
// Exp returns a big.Float representation of exp(z). Precision is
// the same as the one of the argument. The function returns +Inf
// when z = +Inf, and 0 when z = -Inf.
func Exp(z *big.Float) *big.Float {
// exp(0) == 1
if z.Sign() == 0 {
return big.NewFloat(1).SetPrec(z.Prec())
}
// Exp(+Inf) = +Inf
if z.IsInf() && z.Sign() > 0 {
return big.NewFloat(math.Inf(+1)).SetPrec(z.Prec())
}
// Exp(-Inf) = 0
if z.IsInf() && z.Sign() < 0 {
return big.NewFloat(0).SetPrec(z.Prec())
}
guess := new(big.Float)
// try to get initial estimate using IEEE-754 math
zf, _ := z.Float64()
if zfs := math.Exp(zf); zfs == math.Inf(+1) || zfs == 0 {
// too big or too small for IEEE-754 math,
// perform argument reduction using
// e^{2z} = (e^z)²
halfZ := new(big.Float).Mul(z, big.NewFloat(0.5))
halfExp := Exp(halfZ.SetPrec(z.Prec() + 64))
return new(big.Float).Mul(halfExp, halfExp).SetPrec(z.Prec())
} else {
// we got a nice IEEE-754 estimate
guess.SetFloat64(zfs)
}
// f(t)/f'(t) = t*(log(t) - z)
f := func(t *big.Float) *big.Float {
x := new(big.Float)
x.Sub(Log(t), z)
return x.Mul(x, t)
}
x := newton(f, guess, z.Prec())
return x
}
// Read lines from in as big.Float
func read(in io.Reader, xs []big.Float) []big.Float {
scanner := bufio.NewScanner(in)
for scanner.Scan() {
var x big.Float
x.SetPrec(*prec)
text := strings.TrimSpace(scanner.Text())
if len(text) == 0 || text[0] == '#' {
continue
}
_, ok := x.SetString(text)
if !ok {
log.Fatalf("Failed to parse line %q", text)
}
xs = append(xs, x)
}
if err := scanner.Err(); err != nil {
log.Fatalf("Error reading input: %v", err)
}
return xs
}
// Pow returns a big.Float representation of z**w. Precision is the same as the one
// of the first argument. The function panics when z is negative.
func Pow(z *big.Float, w *big.Float) *big.Float {
if z.Sign() < 0 {
panic("Pow: negative base")
}
// Pow(z, 0) = 1.0
if w.Sign() == 0 {
return big.NewFloat(1).SetPrec(z.Prec())
}
// Pow(z, 1) = z
// Pow(+Inf, n) = +Inf
if w.Cmp(big.NewFloat(1)) == 0 || z.IsInf() {
return new(big.Float).Copy(z)
}
// Pow(z, -w) = 1 / Pow(z, w)
if w.Sign() < 0 {
x := new(big.Float)
zExt := new(big.Float).Copy(z).SetPrec(z.Prec() + 64)
wNeg := new(big.Float).Neg(w)
return x.Quo(big.NewFloat(1), Pow(zExt, wNeg)).SetPrec(z.Prec())
}
// w integer fast path
if w.IsInt() {
wi, _ := w.Int64()
return powInt(z, int(wi))
}
// compute w**z as exp(z log(w))
x := new(big.Float).SetPrec(z.Prec() + 64)
logZ := Log(new(big.Float).Copy(z).SetPrec(z.Prec() + 64))
x.Mul(w, logZ)
x = Exp(x)
return x.SetPrec(z.Prec())
}
// 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)
}
// Returns acot(x) in result
func acot(prec uint, x int64, result *big.Float) {
var term, power, _x, _kp, x2, oldresult big.Float
_x.SetPrec(prec).SetInt64(x)
power.SetPrec(prec).SetInt64(1)
power.Quo(&power, &_x) // 1/x
x2.Mul(&_x, &_x)
result.SetPrec(prec).SetInt64(0)
positive := true
for k := int64(1); ; k += 2 {
oldresult.Set(result)
kp := k
if !positive {
kp = -k
}
positive = !positive
_kp.SetPrec(prec).SetInt64(kp)
term.Quo(&power, &_kp)
result.Add(result, &term)
if oldresult.Cmp(result) == 0 {
break
}
power.Quo(&power, &x2)
}
}
// Log returns a big.Float representation of the natural logarithm of
// z. Precision is the same as the one of the argument. The function
// panics if z is negative, returns -Inf when z = 0, and +Inf when z =
// +Inf
func Log(z *big.Float) *big.Float {
// panic on negative z
if z.Sign() == -1 {
panic("Log: argument is negative")
}
// Log(0) = -Inf
if z.Sign() == 0 {
return big.NewFloat(math.Inf(-1)).SetPrec(z.Prec())
}
prec := z.Prec() + 64 // guard digits
one := big.NewFloat(1).SetPrec(prec)
two := big.NewFloat(2).SetPrec(prec)
four := big.NewFloat(4).SetPrec(prec)
// Log(1) = 0
if z.Cmp(one) == 0 {
return big.NewFloat(0).SetPrec(z.Prec())
}
// Log(+Inf) = +Inf
if z.IsInf() {
return big.NewFloat(math.Inf(+1)).SetPrec(z.Prec())
}
x := new(big.Float).SetPrec(prec)
// if 0 < z < 1 we compute log(z) as -log(1/z)
var neg bool
if z.Cmp(one) < 0 {
x.Quo(one, z)
neg = true
} else {
x.Set(z)
}
// We scale up x until x >= 2**(prec/2), and then we'll be allowed
// to use the AGM formula for Log(x).
//
// Double x until the condition is met, and keep track of the
// number of doubling we did (needed to scale back later).
lim := new(big.Float)
lim.SetMantExp(two, int(prec/2))
k := 0
for x.Cmp(lim) < 0 {
x.Mul(x, x)
k++
}
// Compute the natural log of x using the fact that
// log(x) = π / (2 * AGM(1, 4/x))
// if
// x >= 2**(prec/2),
// where prec is the desired precision (in bits)
pi := pi(prec)
agm := agm(one, x.Quo(four, x)) // agm = AGM(1, 4/x)
x.Quo(pi, x.Mul(two, agm)) // reuse x, we don't need it
if neg {
x.Neg(x)
}
// scale the result back multiplying by 2**-k
// reuse lim to reduce allocations.
x.Mul(x, lim.SetMantExp(one, -k))
return x.SetPrec(z.Prec())
}
// used to ensure all of the big.Floats end up with the same precision
func newBigFloat(n uint64) *big.Float {
tmp := new(big.Float).SetUint64(n)
tmp.SetPrec(ENCODER_DECODER_PREC)
return tmp
}
// Evaluates a BBP term
//
// sum(k=0->inf)(1/base**k * (1/a*k + b))
func bbp(prec uint, base, a, b int64, result *big.Float) {
var term, power, aFp, bFp, _1, k, _base, oldresult big.Float
power.SetPrec(prec).SetInt64(1)
result.SetPrec(prec).SetInt64(0)
aFp.SetPrec(prec).SetInt64(a)
bFp.SetPrec(prec).SetInt64(b)
_1.SetPrec(prec).SetInt64(1)
k.SetPrec(prec).SetInt64(0)
_base.SetPrec(prec).SetInt64(base)
for {
oldresult.Set(result)
term.Mul(&aFp, &k)
term.Add(&term, &bFp)
term.Quo(&_1, &term)
term.Mul(&term, &power)
result.Add(result, &term)
if oldresult.Cmp(result) == 0 {
break
}
power.Quo(&power, &_base)
k.Add(&k, &_1)
}
}
