func quo(x, y *complexFloat) *complexFloat {
z := newComplexFloat()
denominator := new(big.Float).SetPrec(prec)
c2 := new(big.Float).SetPrec(prec)
d2 := new(big.Float).SetPrec(prec)
c2.Mul(y.r, y.r)
d2.Mul(y.i, y.i)
denominator.Add(c2, d2)
if denominator.Cmp(zero) == 0 || denominator.IsInf() {
return newComplexFloat()
}
ac := new(big.Float).SetPrec(prec)
bd := new(big.Float).SetPrec(prec)
ac.Mul(x.r, y.r)
bd.Mul(x.i, y.i)
bc := new(big.Float).SetPrec(prec)
ad := new(big.Float).SetPrec(prec)
bc.Mul(x.i, y.r)
ad.Mul(x.r, y.i)
z.r.Add(ac, bd)
z.r.Quo(z.r, denominator)
z.i.Add(bc, ad.Neg(ad))
z.i.Quo(z.i, denominator)
return z
}
// sincos iterates a sin or cos Taylor series.
func sincos(name string, c Context, index int, x *big.Float, z *big.Float, exp uint64, factorial *big.Float) *big.Float {
term := newFloat(c).Set(floatOne)
for j := 0; j < index; j++ {
term.Mul(term, x)
}
xN := newFloat(c).Set(term)
x2 := newFloat(c).Mul(x, x)
n := newFloat(c)
for loop := newLoop(c.Config(), name, x, 4); ; {
// Invariant: factorial holds -1ⁿ*exponent!.
factorial.Neg(factorial)
term.Quo(term, factorial)
z.Add(z, term)
if loop.done(z) {
break
}
// Advance x**index (multiply by x²).
term.Mul(xN, x2)
xN.Set(term)
// Advance factorial.
factorial.Mul(factorial, n.SetUint64(exp+1))
factorial.Mul(factorial, n.SetUint64(exp+2))
exp += 2
}
return z
}
func mul(x, y *complexFloat) *complexFloat {
t1 := new(big.Float).SetPrec(prec)
t2 := new(big.Float).SetPrec(prec)
z := newComplexFloat()
t1.Mul(x.r, y.r)
t2.Mul(x.i, y.i)
t2.Neg(t2)
z.r.Add(t1, t2)
t1.Mul(x.r, y.i)
t2.Mul(x.i, y.r)
z.i.Add(t1, t2)
return z
}
// floatSin computes sin(x) using argument reduction and a Taylor series.
func floatSin(c Context, x *big.Float) *big.Float {
negate := false
if x.Sign() < 0 {
x.Neg(x)
negate = true
}
twoPiReduce(c, x)
// sin(x) = x - x³/3! + x⁵/5! - ...
// First term to compute in loop will be -x³/3!
factorial := newFloat(c).SetInt64(6)
result := sincos("sin", c, 3, x, newFloat(c).Set(x), 3, factorial)
if negate {
result.Neg(result)
}
return result
}
// 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())
}
func (f BigFloat) String() string {
var mant big.Float
exp := f.Float.MantExp(&mant)
positive := 1
if exp < 0 {
positive = 0
exp = -exp
}
verb, prec := byte('g'), 12
format := conf.Format()
if format != "" {
v, p, ok := conf.FloatFormat()
if ok {
verb, prec = v, p
}
}
// Printing huge floats can be very slow using
// big.Float's native methods; see issue #11068.
// For example 1e5000000 takes a minute of CPU time just
// to print. The code below is instantaneous, by rescaling
// first. It is however less feature-complete.
// (Big ints are problematic too, but if you print 1e50000000
// as an integer you probably won't be surprised it's slow.)
if fastFloatPrint && exp > 10000 {
// We always use %g to print the fraction, and it will
// never have an exponent, but if the format is %E we
// need to use a capital E.
eChar := 'e'
if verb == 'E' || verb == 'G' {
eChar = 'E'
}
fexp := newF().SetInt64(int64(exp))
fexp.Mul(fexp, floatLog2)
fexp.Quo(fexp, floatLog10)
// We now have a floating-point base 10 exponent.
// Break into the integer part and the fractional part.
// The integer part is what we will show.
// The 10**(fractional part) will be multiplied back in.
iexp, _ := fexp.Int(nil)
fraction := fexp.Sub(fexp, newF().SetInt(iexp))
// Now compute 10**(fractional part).
// Fraction is in base 10. Move it to base e.
fraction.Mul(fraction, floatLog10)
scale := exponential(fraction)
if positive > 0 {
mant.Mul(&mant, scale)
} else {
mant.Quo(&mant, scale)
}
ten := newF().SetInt64(10)
i64exp := iexp.Int64()
// For numbers not too far from one, print without the E notation.
// Shouldn't happen (exp must be large to get here) but just
// in case, we keep this around.
if -4 <= i64exp && i64exp <= 11 {
if i64exp > 0 {
for i := 0; i < int(i64exp); i++ {
mant.Mul(&mant, ten)
}
} else {
for i := 0; i < int(-i64exp); i++ {
mant.Quo(&mant, ten)
}
}
return fmt.Sprintf("%g\n", &mant)
} else {
sign := ""
if mant.Sign() < 0 {
sign = "-"
mant.Neg(&mant)
}
// If it has a leading zero, rescale.
digits := mant.Text('g', prec)
for digits[0] == '0' {
mant.Mul(&mant, ten)
if positive > 0 {
i64exp--
} else {
i64exp++
}
digits = mant.Text('g', prec)
}
return fmt.Sprintf("%s%s%c%c%d", sign, digits, eChar, "-+"[positive], i64exp)
}
}
return f.Float.Text(verb, prec)
}
// Given a vector of real numbers x = [x_0, x_1, ..., x_n], this
// uses the PSLQ algorithm to find a list of integers
// [c_0, c_1, ..., c_n] such that
//
// |c_1 * x_1 + c_2 * x_2 + ... + c_n * x_n| < tolerance
//
// and such that max |c_k| < maxcoeff. If no such vector exists, Pslq
// returns one of the errors in this package depending on whether it
// has run out of iterations, precision or explored up to the
// maxcoeff. The tolerance defaults to 3/4 of the precision.
//
// This is a fairly direct translation of the pseudocode given by
// David Bailey, "The PSLQ Integer Relation Algorithm":
// http://www.cecm.sfu.ca/organics/papers/bailey/paper/html/node3.html
//
// If a result is returned, the first non-zero element will be positive
func (e *Pslq) Run(x []big.Float) ([]big.Int, error) {
n := len(x)
if n <= 1 {
return nil, ErrorBadArguments
}
// At too low precision, the algorithm becomes meaningless
if e.prec < 64 {
return nil, ErrorPrecisionTooLow
}
if e.verbose && int(e.prec)/max(2, int(n)) < 5 {
log.Printf("Warning: precision for PSLQ may be too low")
}
if e.verbose {
log.Printf("PSLQ using prec %d and tol %g", e.prec, e.tol)
}
if e.tol.Sign() == 0 {
return nil, ErrorToleranceRoundsToZero
}
// Temporary variables
tmp0 := new(big.Float).SetPrec(e.prec)
tmp1 := new(big.Float).SetPrec(e.prec)
bigTmp := new(big.Int)
// Convert to use 1-based indexing to allow us to be
// consistent with Bailey's indexing.
xNew := make([]big.Float, len(x)+1)
minx := new(big.Float).SetPrec(e.prec)
minxFirst := true
for i, xk := range x {
p := &xNew[i+1]
p.Set(&xk)
tmp0.Abs(p)
if minxFirst || tmp0.Cmp(minx) < 0 {
minxFirst = false
minx.Set(tmp0)
}
}
x = xNew
if debug {
printVector("x", x)
}
// Sanity check on magnitudes
if minx.Sign() == 0 {
return nil, ErrorZeroArguments
}
tmp1.SetInt64(128)
tmp0.Quo(&e.tol, tmp1)
if minx.Cmp(tmp0) < 0 { // minx < tol/128
return nil, ErrorArgumentTooSmall
}
tmp0.SetInt64(4)
tmp1.SetInt64(3)
tmp0.Quo(tmp0, tmp1)
var γ big.Float
e.Sqrt(tmp0, &γ) // sqrt(4<<prec)/3)
if debug {
fmt.Printf("γ = %f\n", &γ)
}
A := newBigIntMatrix(n+1, n+1)
B := newBigIntMatrix(n+1, n+1)
H := newMatrix(n+1, n+1)
// Initialization Step 1
//
// Set the n×n matrices A and B to the identity.
for i := 1; i <= n; i++ {
for j := 1; j <= n; j++ {
if i == j {
A[i][j].SetInt64(1)
B[i][j].SetInt64(1)
} else {
A[i][j].SetInt64(0)
B[i][j].SetInt64(0)
}
H[i][j].SetInt64(0)
}
}
if debug {
printBigIntMatrix("A", A)
printBigIntMatrix("B", B)
printMatrix("H", H)
}
// Initialization Step 2
//
// For k := 1 to n
// compute s_k := sqrt( sum_j=k^n x_j^2 )
// endfor.
// Set t = 1/s1.
// For k := 1 to n:
// y_k := t * x_k
// s_k := t * s_k
// endfor.
s := make([]big.Float, n+1)
for i := 1; i <= n; i++ {
s[i].SetInt64(0)
}
for k := 1; k <= n; k++ {
var t big.Float
t.SetInt64(0)
for j := k; j <= n; j++ {
tmp0.Mul(&x[j], &x[j])
t.Add(&t, tmp0)
}
e.Sqrt(&t, &s[k])
}
if debug {
fmt.Println("Init Step 2")
printVector("s", s)
}
var t big.Float
t.Set(&s[1])
y := make([]big.Float, len(x))
copy(y, x)
for k := 1; k <= n; k++ {
// y[k] = (x[k] << prec) / t
y[k].Quo(&x[k], &t)
// s[k] = (s[k] << prec) / t
s[k].Quo(&s[k], &t)
}
if debug {
printVector("y", y)
printVector("s", s)
}
// Init Step 3
//
// Compute the n×(n−1) matrix H as follows:
// For i := 1 to n:
// for j := i + 1 to n − 1:
// set Hij := 0
// endfor
// if i ≤ n − 1 then set Hii := s_(i+1)/s_i
// for j := 1 to i−1:
// set Hij := −y_i * y_j / (s_j * s_(j+1))
// endfor
// endfor
for i := 1; i <= n; i++ {
for j := i + 1; j < n; j++ {
H[i][j].SetInt64(0)
}
if i <= n-1 {
if s[i].Sign() == 0 {
// Precision probably exhausted
return nil, ErrorPrecisionExhausted
}
// H[i][i] = (s[i+1] << prec) / s[i]
H[i][i].Quo(&s[i+1], &s[i])
}
for j := 1; j < i; j++ {
var sjj1 big.Float
sjj1.Mul(&s[j], &s[j+1])
if debug {
fmt.Printf("sjj1 = %f\n", &sjj1)
}
if sjj1.Sign() == 0 {
// Precision probably exhausted
return nil, ErrorPrecisionExhausted
}
// H[i][j] = ((-y[i] * y[j]) << prec) / sjj1
tmp0.Mul(&y[i], &y[j])
tmp0.Neg(tmp0)
H[i][j].Quo(tmp0, &sjj1)
}
}
if debug {
fmt.Println("Init Step 3")
printMatrix("H", H)
}
// Init Step 4
//
// Perform full reduction on H, simultaneously updating y, A and B:
//
// For i := 2 to n:
// for j := i−1 to 1 step−1:
// t := nint(Hij/Hjj)
// y_j := y_j + t * y_i
// for k := 1 to j:
// Hik := Hik − t * Hjk
// endfor
// for k := 1 to n:
// Aik := Aik − t * Ajk
// Bkj := Bkj + t * Bki
// endfor
// endfor
// endfor
for i := 2; i <= n; i++ {
for j := i - 1; j > 0; j-- {
//t = floor(H[i][j]/H[j,j] + 0.5)
var t big.Int
var tFloat big.Float
if H[j][j].Sign() == 0 {
// Precision probably exhausted
return nil, ErrorPrecisionExhausted
}
tmp0.Quo(&H[i][j], &H[j][j])
e.NearestInt(tmp0, &t)
tFloat.SetInt(&t).SetPrec(e.prec)
if debug {
fmt.Printf("H[i][j]=%f\n", &H[i][j])
fmt.Printf("H[j][j]=%f\n", &H[j][j])
fmt.Printf("tmp=%f\n", tmp0)
fmt.Printf("t=%d\n", &t)
}
// y[j] = y[j] + (t * y[i] >> prec)
tmp0.Mul(&y[i], &tFloat)
y[j].Add(&y[j], tmp0)
for k := 1; k <= j; k++ {
// H[i][k] = H[i][k] - (t * H[j][k] >> prec)
tmp0.Mul(&H[j][k], &tFloat)
H[i][k].Sub(&H[i][k], tmp0)
}
for k := 1; k <= n; k++ {
bigTmp.Mul(&t, &A[j][k])
A[i][k].Sub(&A[i][k], bigTmp)
bigTmp.Mul(&t, &B[k][i])
B[k][j].Add(&B[k][j], bigTmp)
}
}
}
if debug {
fmt.Println("Init Step 4")
printBigIntMatrix("A", A)
printBigIntMatrix("B", B)
printMatrix("H", H)
}
// Main algorithm
var REP int
var norm big.Int
vec := make([]big.Int, n)
for REP = 0; REP < e.maxsteps; REP++ {
// Step 1
//
// Select m such that γ^i * |Hii| is maximal when i = m.
m := -1
var szmax big.Float
szmax.SetInt64(-1)
var γPower big.Float
γPower.Set(&γ)
for i := 1; i < n; i++ {
var absH big.Float
absH.Abs(&H[i][i])
var sz big.Float
sz.Mul(&γPower, &absH)
// sz := (g**i * abs(h)) >> (prec * (i - 1))
if sz.Cmp(&szmax) > 0 {
m = i
szmax.Set(&sz)
}
γPower.Mul(&γPower, &γ)
}
if debug {
fmt.Println("Step 1")
fmt.Printf("szmax=%f\n", &szmax)
fmt.Printf("m=%d\n", m)
}
// Step 2
//
// Exchange entries m and m+1 of y, corresponding rows
// of A and H, and corresponding columns of B.
y[m], y[m+1] = y[m+1], y[m]
for i := 1; i < n+1; i++ {
H[m][i], H[m+1][i] = H[m+1][i], H[m][i]
}
for i := 1; i < n+1; i++ {
A[m][i], A[m+1][i] = A[m+1][i], A[m][i]
}
for i := 1; i < n+1; i++ {
B[i][m], B[i][m+1] = B[i][m+1], B[i][m]
}
if debug {
fmt.Println("Step 2")
printVector("y", y)
printBigIntMatrix("A", A)
printBigIntMatrix("B", B)
printMatrix("H", H)
}
// Step 3
//
// If m ≤ n−2 then update H as follows:
//
// t0 := sqrt( Hmm^2 + H(m,m+1)^2 )
// t1 := Hmm/t0
// t2 := H(m,m+1)/t0.
// for i := m to n:
// t3 := Him
// t4 := Hi,m+1
// Him := t1t3 +t2t4
// Hi,m+1 := −t2t3 +t1t4
// endfor.
if m <= n-2 {
tmp0.Mul(&H[m][m], &H[m][m])
tmp1.Mul(&H[m][m+1], &H[m][m+1])
tmp0.Add(tmp0, tmp1)
var t0 big.Float
e.Sqrt(tmp0, &t0)
// Precision probably exhausted
if t0.Sign() == 0 {
return nil, ErrorPrecisionExhausted
}
var t1, t2 big.Float
t1.Quo(&H[m][m], &t0)
t2.Quo(&H[m][m+1], &t0)
for i := m; i <= n; i++ {
var t3, t4 big.Float
t3.Set(&H[i][m])
t4.Set(&H[i][m+1])
// H[i][m] = (t1*t3 + t2*t4) >> prec
tmp0.Mul(&t1, &t3)
tmp1.Mul(&t2, &t4)
H[i][m].Add(tmp0, tmp1)
// H[i][m+1] = (-t2*t3 + t1*t4) >> prec
tmp0.Mul(&t2, &t3)
tmp1.Mul(&t1, &t4)
H[i][m+1].Sub(tmp1, tmp0)
}
}
if debug {
fmt.Println("Step 3")
printMatrix("H", H)
}
// Step 4
// Perform block reduction on H, simultaneously updating y, A and B:
//
// For i := m+1 to n:
// for j := min(i−1, m+1) to 1 step −1:
// t := nint(Hij/Hjj)
// yj := yj + t * yi
// for k := 1 to j:
// Hik := Hik − tHjk
// endfor
// for k := 1 to n:
// Aik := Aik −tAjk
// Bkj := Bkj +tBki
// endfor
// endfor
// endfor.
for i := m + 1; i <= n; i++ {
var t big.Int
var tFloat big.Float
for j := min(i-1, m+1); j > 0; j-- {
if H[j][j].Sign() == 0 {
// Precision probably exhausted
return nil, ErrorPrecisionExhausted
}
tmp0.Quo(&H[i][j], &H[j][j])
e.NearestInt(tmp0, &t)
tFloat.SetInt(&t).SetPrec(e.prec)
// y[j] = y[j] + ((t * y[i]) >> prec)
tmp0.Mul(&y[i], &tFloat)
y[j].Add(&y[j], tmp0)
for k := 1; k <= j; k++ {
// H[i][k] = H[i][k] - (t * H[j][k] >> prec)
tmp0.Mul(&H[j][k], &tFloat)
H[i][k].Sub(&H[i][k], tmp0)
}
for k := 1; k <= n; k++ {
bigTmp.Mul(&t, &A[j][k])
A[i][k].Sub(&A[i][k], bigTmp)
bigTmp.Mul(&t, &B[k][i])
B[k][j].Add(&B[k][j], bigTmp)
}
}
}
if debug {
fmt.Println("Step 4")
printBigIntMatrix("A", A)
printBigIntMatrix("B", B)
printMatrix("H", H)
}
// Step 6
//
// Termination test: If the largest entry of A exceeds
// the level of numeric precision used, then precision
// is exhausted. If the smallest entry of the y vector
// is less than the detection threshold, a relation
// has been detected and is given in the corresponding
// column of B.
//
// Until a relation is found, the error typically decreases
// slowly (e.g. a factor 1-10) with each step TODO: we could
// compare err from two successive iterations. If there is a
// large drop (several orders of magnitude), that indicates a
// "high quality" relation was detected. Reporting this to
// the user somehow might be useful.
maxAPrecision := 0
for i := 1; i <= n; i++ {
for j := 1; j <= n; j++ {
precision := A[i][j].BitLen()
if precision > maxAPrecision {
maxAPrecision = precision
}
}
}
if debug {
log.Printf("Max A precision = %d, precision = %d, tolerance %d, ratio = %.3f\n", maxAPrecision, e.prec, e.target, float64(maxAPrecision)/float64(e.target))
}
if float64(maxAPrecision)/float64(e.target) > 0.85 {
if e.verbose {
log.Printf("CANCELLING after step %d/%d.", REP, e.maxsteps)
}
return nil, ErrorPrecisionExhausted
}
var best_err big.Float
best_err.Set(&e.maxcoeff_fp)
for i := 1; i <= n; i++ {
var err big.Float
err.Abs(&y[i])
// Maybe we are done?
if err.Cmp(&e.tol) < 0 {
// We are done if the coefficients are acceptable
var maxc big.Int
for j := 1; j <= n; j++ {
if debug {
fmt.Printf("vec[%d]=%d\n", j-1, &B[j][i])
}
t := B[j][i]
if debug {
fmt.Printf("vec[%d]=%d\n", j-1, t)
}
vec[j-1] = t
if t.Sign() < 0 {
t.Neg(&t)
}
if t.Cmp(&maxc) > 0 {
maxc.Set(&t)
}
}
if debug {
fmt.Printf("maxc = %d, maxcoeff = %d\n", maxc, e.maxcoeff)
}
if maxc.Cmp(&e.maxcoeff) < 0 {
if e.verbose {
log.Printf("FOUND relation at iter %d/%d, error: %g", REP, e.maxsteps, &err)
}
// Find sign of first non zero item
sign := 0
for i := range vec {
sign = vec[i].Sign()
if sign != 0 {
break
}
}
// Normalise vec making first non-zero argument positive
if sign < 0 {
for i := range vec {
vec[i].Neg(&vec[i])
}
}
return vec, nil
}
}
if err.Cmp(&best_err) < 0 {
best_err = err
}
}
// Step 5
//
// Norm bound: Compute M := 1/maxj |Hj|, where Hj
// denotes the j-th row of H.
//
// Then there can exist no relation vector whose
// Euclidean norm is less than M.
//
// Calculate a lower bound for the norm. We could do this
// more exactly (using the Euclidean norm) but there is probably
// no practical benefit.
var recnorm big.Float
recnorm.SetInt64(0)
for i := 1; i <= n; i++ {
for j := 1; j <= n; j++ {
tmp0.Abs(&H[i][j])
if tmp0.Cmp(&recnorm) > 0 {
recnorm.Set(tmp0)
}
}
}
norm.Set(&e.maxcoeff)
if recnorm.Sign() != 0 {
// norm = ((1 << (2 * prec)) / recnorm) >> prec
tmp0.Quo(&e.one, &recnorm)
tmp0.Int(&norm)
}
if e.verbose {
log.Printf("%2d/%2d: Error: %g Norm: %d", REP, e.maxsteps, &best_err, &norm)
}
if norm.Cmp(&e.maxcoeff) >= 0 {
if e.verbose {
log.Printf("CANCELLING after step %d/%d.", REP, e.maxsteps)
log.Printf("Could not find an integer relation. Norm bound: %d", &norm)
}
return nil, ErrorNoRelationFound
}
}
if e.verbose {
log.Printf("CANCELLING after step %d/%d.", REP, e.maxsteps)
log.Printf("Could not find an integer relation. Norm bound: %d", &norm)
}
return nil, ErrorIterationsExceeded
}