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# Source file src/math/cmplx/tan.go

## Documentation: math/cmplx

```     1  // Copyright 2010 The Go Authors. All rights reserved.
2  // Use of this source code is governed by a BSD-style
4
5  package cmplx
6
7  import "math"
8
9  // The original C code, the long comment, and the constants
10  // below are from http://netlib.sandia.gov/cephes/c9x-complex/clog.c.
11  // The go code is a simplified version of the original C.
12  //
13  // Cephes Math Library Release 2.8:  June, 2000
14  // Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
15  //
16  // The readme file at http://netlib.sandia.gov/cephes/ says:
17  //    Some software in this archive may be from the book _Methods and
18  // Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster
19  // International, 1989) or from the Cephes Mathematical Library, a
20  // commercial product. In either event, it is copyrighted by the author.
21  // What you see here may be used freely but it comes with no support or
22  // guarantee.
23  //
24  //   The two known misprints in the book are repaired here in the
25  // source listings for the gamma function and the incomplete beta
26  // integral.
27  //
28  //   Stephen L. Moshier
29  //   moshier@na-net.ornl.gov
30
31  // Complex circular tangent
32  //
33  // DESCRIPTION:
34  //
35  // If
36  //     z = x + iy,
37  //
38  // then
39  //
40  //           sin 2x  +  i sinh 2y
41  //     w  =  --------------------.
42  //            cos 2x  +  cosh 2y
43  //
44  // On the real axis the denominator is zero at odd multiples
45  // of PI/2.  The denominator is evaluated by its Taylor
46  // series near these points.
47  //
48  // ctan(z) = -i ctanh(iz).
49  //
50  // ACCURACY:
51  //
52  //                      Relative error:
53  // arithmetic   domain     # trials      peak         rms
54  //    DEC       -10,+10      5200       7.1e-17     1.6e-17
55  //    IEEE      -10,+10     30000       7.2e-16     1.2e-16
56  // Also tested by ctan * ccot = 1 and catan(ctan(z))  =  z.
57
58  // Tan returns the tangent of x.
59  func Tan(x complex128) complex128 {
60  	d := math.Cos(2*real(x)) + math.Cosh(2*imag(x))
61  	if math.Abs(d) < 0.25 {
62  		d = tanSeries(x)
63  	}
64  	if d == 0 {
65  		return Inf()
66  	}
67  	return complex(math.Sin(2*real(x))/d, math.Sinh(2*imag(x))/d)
68  }
69
70  // Complex hyperbolic tangent
71  //
72  // DESCRIPTION:
73  //
74  // tanh z = (sinh 2x  +  i sin 2y) / (cosh 2x + cos 2y) .
75  //
76  // ACCURACY:
77  //
78  //                      Relative error:
79  // arithmetic   domain     # trials      peak         rms
80  //    IEEE      -10,+10     30000       1.7e-14     2.4e-16
81
82  // Tanh returns the hyperbolic tangent of x.
83  func Tanh(x complex128) complex128 {
84  	d := math.Cosh(2*real(x)) + math.Cos(2*imag(x))
85  	if d == 0 {
86  		return Inf()
87  	}
88  	return complex(math.Sinh(2*real(x))/d, math.Sin(2*imag(x))/d)
89  }
90
91  // Program to subtract nearest integer multiple of PI
92  func reducePi(x float64) float64 {
93  	const (
94  		// extended precision value of PI:
95  		DP1 = 3.14159265160560607910e0   // ?? 0x400921fb54000000
96  		DP2 = 1.98418714791870343106e-9  // ?? 0x3e210b4610000000
97  		DP3 = 1.14423774522196636802e-17 // ?? 0x3c6a62633145c06e
98  	)
99  	t := x / math.Pi
100  	if t >= 0 {
101  		t += 0.5
102  	} else {
103  		t -= 0.5
104  	}
105  	t = float64(int64(t)) // int64(t) = the multiple
106  	return ((x - t*DP1) - t*DP2) - t*DP3
107  }
108
109  // Taylor series expansion for cosh(2y) - cos(2x)
110  func tanSeries(z complex128) float64 {
111  	const MACHEP = 1.0 / (1 << 53)
112  	x := math.Abs(2 * real(z))
113  	y := math.Abs(2 * imag(z))
114  	x = reducePi(x)
115  	x = x * x
116  	y = y * y
117  	x2 := 1.0
118  	y2 := 1.0
119  	f := 1.0
120  	rn := 0.0
121  	d := 0.0
122  	for {
123  		rn++
124  		f *= rn
125  		rn++
126  		f *= rn
127  		x2 *= x
128  		y2 *= y
129  		t := y2 + x2
130  		t /= f
131  		d += t
132
133  		rn++
134  		f *= rn
135  		rn++
136  		f *= rn
137  		x2 *= x
138  		y2 *= y
139  		t = y2 - x2
140  		t /= f
141  		d += t
142  		if !(math.Abs(t/d) > MACHEP) {
143  			// Caution: Use ! and > instead of <= for correct behavior if t/d is NaN.
144  			// See issue 17577.
145  			break
146  		}
147  	}
148  	return d
149  }
150
151  // Complex circular cotangent
152  //
153  // DESCRIPTION:
154  //
155  // If
156  //     z = x + iy,
157  //
158  // then
159  //
160  //           sin 2x  -  i sinh 2y
161  //     w  =  --------------------.
162  //            cosh 2y  -  cos 2x
163  //
164  // On the real axis, the denominator has zeros at even
165  // multiples of PI/2.  Near these points it is evaluated
166  // by a Taylor series.
167  //
168  // ACCURACY:
169  //
170  //                      Relative error:
171  // arithmetic   domain     # trials      peak         rms
172  //    DEC       -10,+10      3000       6.5e-17     1.6e-17
173  //    IEEE      -10,+10     30000       9.2e-16     1.2e-16
174  // Also tested by ctan * ccot = 1 + i0.
175
176  // Cot returns the cotangent of x.
177  func Cot(x complex128) complex128 {
178  	d := math.Cosh(2*imag(x)) - math.Cos(2*real(x))
179  	if math.Abs(d) < 0.25 {
180  		d = tanSeries(x)
181  	}
182  	if d == 0 {
183  		return Inf()
184  	}
185  	return complex(math.Sin(2*real(x))/d, -math.Sinh(2*imag(x))/d)
186  }
187
```

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