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- // Copyright 2010 The Go Authors. All rights reserved.
- // Use of this source code is governed by a BSD-style
- // license that can be found in the LICENSE file.
- package cmplx
- import (
- "math"
- "math/bits"
- )
- // The original C code, the long comment, and the constants
- // below are from http://netlib.sandia.gov/cephes/c9x-complex/clog.c.
- // The go code is a simplified version of the original C.
- //
- // Cephes Math Library Release 2.8: June, 2000
- // Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
- //
- // The readme file at http://netlib.sandia.gov/cephes/ says:
- // Some software in this archive may be from the book _Methods and
- // Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster
- // International, 1989) or from the Cephes Mathematical Library, a
- // commercial product. In either event, it is copyrighted by the author.
- // What you see here may be used freely but it comes with no support or
- // guarantee.
- //
- // The two known misprints in the book are repaired here in the
- // source listings for the gamma function and the incomplete beta
- // integral.
- //
- // Stephen L. Moshier
- // moshier@na-net.ornl.gov
- // Complex circular tangent
- //
- // DESCRIPTION:
- //
- // If
- // z = x + iy,
- //
- // then
- //
- // sin 2x + i sinh 2y
- // w = --------------------.
- // cos 2x + cosh 2y
- //
- // On the real axis the denominator is zero at odd multiples
- // of PI/2. The denominator is evaluated by its Taylor
- // series near these points.
- //
- // ctan(z) = -i ctanh(iz).
- //
- // ACCURACY:
- //
- // Relative error:
- // arithmetic domain # trials peak rms
- // DEC -10,+10 5200 7.1e-17 1.6e-17
- // IEEE -10,+10 30000 7.2e-16 1.2e-16
- // Also tested by ctan * ccot = 1 and catan(ctan(z)) = z.
- // Tan returns the tangent of x.
- func Tan(x complex128) complex128 {
- switch re, im := real(x), imag(x); {
- case math.IsInf(im, 0):
- switch {
- case math.IsInf(re, 0) || math.IsNaN(re):
- return complex(math.Copysign(0, re), math.Copysign(1, im))
- }
- return complex(math.Copysign(0, math.Sin(2*re)), math.Copysign(1, im))
- case re == 0 && math.IsNaN(im):
- return x
- }
- d := math.Cos(2*real(x)) + math.Cosh(2*imag(x))
- if math.Abs(d) < 0.25 {
- d = tanSeries(x)
- }
- if d == 0 {
- return Inf()
- }
- return complex(math.Sin(2*real(x))/d, math.Sinh(2*imag(x))/d)
- }
- // Complex hyperbolic tangent
- //
- // DESCRIPTION:
- //
- // tanh z = (sinh 2x + i sin 2y) / (cosh 2x + cos 2y) .
- //
- // ACCURACY:
- //
- // Relative error:
- // arithmetic domain # trials peak rms
- // IEEE -10,+10 30000 1.7e-14 2.4e-16
- // Tanh returns the hyperbolic tangent of x.
- func Tanh(x complex128) complex128 {
- switch re, im := real(x), imag(x); {
- case math.IsInf(re, 0):
- switch {
- case math.IsInf(im, 0) || math.IsNaN(im):
- return complex(math.Copysign(1, re), math.Copysign(0, im))
- }
- return complex(math.Copysign(1, re), math.Copysign(0, math.Sin(2*im)))
- case im == 0 && math.IsNaN(re):
- return x
- }
- d := math.Cosh(2*real(x)) + math.Cos(2*imag(x))
- if d == 0 {
- return Inf()
- }
- return complex(math.Sinh(2*real(x))/d, math.Sin(2*imag(x))/d)
- }
- // reducePi reduces the input argument x to the range (-Pi/2, Pi/2].
- // x must be greater than or equal to 0. For small arguments it
- // uses Cody-Waite reduction in 3 float64 parts based on:
- // "Elementary Function Evaluation: Algorithms and Implementation"
- // Jean-Michel Muller, 1997.
- // For very large arguments it uses Payne-Hanek range reduction based on:
- // "ARGUMENT REDUCTION FOR HUGE ARGUMENTS: Good to the Last Bit"
- // K. C. Ng et al, March 24, 1992.
- func reducePi(x float64) float64 {
- // reduceThreshold is the maximum value of x where the reduction using
- // Cody-Waite reduction still gives accurate results. This threshold
- // is set by t*PIn being representable as a float64 without error
- // where t is given by t = floor(x * (1 / Pi)) and PIn are the leading partial
- // terms of Pi. Since the leading terms, PI1 and PI2 below, have 30 and 32
- // trailing zero bits respectively, t should have less than 30 significant bits.
- // t < 1<<30 -> floor(x*(1/Pi)+0.5) < 1<<30 -> x < (1<<30-1) * Pi - 0.5
- // So, conservatively we can take x < 1<<30.
- const reduceThreshold float64 = 1 << 30
- if math.Abs(x) < reduceThreshold {
- // Use Cody-Waite reduction in three parts.
- const (
- // PI1, PI2 and PI3 comprise an extended precision value of PI
- // such that PI ~= PI1 + PI2 + PI3. The parts are chosen so
- // that PI1 and PI2 have an approximately equal number of trailing
- // zero bits. This ensures that t*PI1 and t*PI2 are exact for
- // large integer values of t. The full precision PI3 ensures the
- // approximation of PI is accurate to 102 bits to handle cancellation
- // during subtraction.
- PI1 = 3.141592502593994 // 0x400921fb40000000
- PI2 = 1.5099578831723193e-07 // 0x3e84442d00000000
- PI3 = 1.0780605716316238e-14 // 0x3d08469898cc5170
- )
- t := x / math.Pi
- t += 0.5
- t = float64(int64(t)) // int64(t) = the multiple
- return ((x - t*PI1) - t*PI2) - t*PI3
- }
- // Must apply Payne-Hanek range reduction
- const (
- mask = 0x7FF
- shift = 64 - 11 - 1
- bias = 1023
- fracMask = 1<<shift - 1
- )
- // Extract out the integer and exponent such that,
- // x = ix * 2 ** exp.
- ix := math.Float64bits(x)
- exp := int(ix>>shift&mask) - bias - shift
- ix &= fracMask
- ix |= 1 << shift
- // mPi is the binary digits of 1/Pi as a uint64 array,
- // that is, 1/Pi = Sum mPi[i]*2^(-64*i).
- // 19 64-bit digits give 1216 bits of precision
- // to handle the largest possible float64 exponent.
- var mPi = [...]uint64{
- 0x0000000000000000,
- 0x517cc1b727220a94,
- 0xfe13abe8fa9a6ee0,
- 0x6db14acc9e21c820,
- 0xff28b1d5ef5de2b0,
- 0xdb92371d2126e970,
- 0x0324977504e8c90e,
- 0x7f0ef58e5894d39f,
- 0x74411afa975da242,
- 0x74ce38135a2fbf20,
- 0x9cc8eb1cc1a99cfa,
- 0x4e422fc5defc941d,
- 0x8ffc4bffef02cc07,
- 0xf79788c5ad05368f,
- 0xb69b3f6793e584db,
- 0xa7a31fb34f2ff516,
- 0xba93dd63f5f2f8bd,
- 0x9e839cfbc5294975,
- 0x35fdafd88fc6ae84,
- 0x2b0198237e3db5d5,
- }
- // Use the exponent to extract the 3 appropriate uint64 digits from mPi,
- // B ~ (z0, z1, z2), such that the product leading digit has the exponent -64.
- // Note, exp >= 50 since x >= reduceThreshold and exp < 971 for maximum float64.
- digit, bitshift := uint(exp+64)/64, uint(exp+64)%64
- z0 := (mPi[digit] << bitshift) | (mPi[digit+1] >> (64 - bitshift))
- z1 := (mPi[digit+1] << bitshift) | (mPi[digit+2] >> (64 - bitshift))
- z2 := (mPi[digit+2] << bitshift) | (mPi[digit+3] >> (64 - bitshift))
- // Multiply mantissa by the digits and extract the upper two digits (hi, lo).
- z2hi, _ := bits.Mul64(z2, ix)
- z1hi, z1lo := bits.Mul64(z1, ix)
- z0lo := z0 * ix
- lo, c := bits.Add64(z1lo, z2hi, 0)
- hi, _ := bits.Add64(z0lo, z1hi, c)
- // Find the magnitude of the fraction.
- lz := uint(bits.LeadingZeros64(hi))
- e := uint64(bias - (lz + 1))
- // Clear implicit mantissa bit and shift into place.
- hi = (hi << (lz + 1)) | (lo >> (64 - (lz + 1)))
- hi >>= 64 - shift
- // Include the exponent and convert to a float.
- hi |= e << shift
- x = math.Float64frombits(hi)
- // map to (-Pi/2, Pi/2]
- if x > 0.5 {
- x--
- }
- return math.Pi * x
- }
- // Taylor series expansion for cosh(2y) - cos(2x)
- func tanSeries(z complex128) float64 {
- const MACHEP = 1.0 / (1 << 53)
- x := math.Abs(2 * real(z))
- y := math.Abs(2 * imag(z))
- x = reducePi(x)
- x = x * x
- y = y * y
- x2 := 1.0
- y2 := 1.0
- f := 1.0
- rn := 0.0
- d := 0.0
- for {
- rn++
- f *= rn
- rn++
- f *= rn
- x2 *= x
- y2 *= y
- t := y2 + x2
- t /= f
- d += t
- rn++
- f *= rn
- rn++
- f *= rn
- x2 *= x
- y2 *= y
- t = y2 - x2
- t /= f
- d += t
- if !(math.Abs(t/d) > MACHEP) {
- // Caution: Use ! and > instead of <= for correct behavior if t/d is NaN.
- // See issue 17577.
- break
- }
- }
- return d
- }
- // Complex circular cotangent
- //
- // DESCRIPTION:
- //
- // If
- // z = x + iy,
- //
- // then
- //
- // sin 2x - i sinh 2y
- // w = --------------------.
- // cosh 2y - cos 2x
- //
- // On the real axis, the denominator has zeros at even
- // multiples of PI/2. Near these points it is evaluated
- // by a Taylor series.
- //
- // ACCURACY:
- //
- // Relative error:
- // arithmetic domain # trials peak rms
- // DEC -10,+10 3000 6.5e-17 1.6e-17
- // IEEE -10,+10 30000 9.2e-16 1.2e-16
- // Also tested by ctan * ccot = 1 + i0.
- // Cot returns the cotangent of x.
- func Cot(x complex128) complex128 {
- d := math.Cosh(2*imag(x)) - math.Cos(2*real(x))
- if math.Abs(d) < 0.25 {
- d = tanSeries(x)
- }
- if d == 0 {
- return Inf()
- }
- return complex(math.Sin(2*real(x))/d, -math.Sinh(2*imag(x))/d)
- }
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