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Source file src/math/cmplx/tan.go

Documentation: math/cmplx

  // 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"
  
  // 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 {
  	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 {
  	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)
  }
  
  // Program to subtract nearest integer multiple of PI
  func reducePi(x float64) float64 {
  	const (
  		// extended precision value of PI:
  		DP1 = 3.14159265160560607910E0   // ?? 0x400921fb54000000
  		DP2 = 1.98418714791870343106E-9  // ?? 0x3e210b4610000000
  		DP3 = 1.14423774522196636802E-17 // ?? 0x3c6a62633145c06e
  	)
  	t := x / math.Pi
  	if t >= 0 {
  		t += 0.5
  	} else {
  		t -= 0.5
  	}
  	t = float64(int64(t)) // int64(t) = the multiple
  	return ((x - t*DP1) - t*DP2) - t*DP3
  }
  
  // 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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