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Source file src/math/cmplx/asin.go

  // 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 arc sine
  //
  // DESCRIPTION:
  //
  // Inverse complex sine:
  //                               2
  // w = -i clog( iz + csqrt( 1 - z ) ).
  //
  // casin(z) = -i casinh(iz)
  //
  // ACCURACY:
  //
  //                      Relative error:
  // arithmetic   domain     # trials      peak         rms
  //    DEC       -10,+10     10100       2.1e-15     3.4e-16
  //    IEEE      -10,+10     30000       2.2e-14     2.7e-15
  // Larger relative error can be observed for z near zero.
  // Also tested by csin(casin(z)) = z.
  
  // Asin returns the inverse sine of x.
  func Asin(x complex128) complex128 {
  	if imag(x) == 0 {
  		if math.Abs(real(x)) > 1 {
  			return complex(math.Pi/2, 0) // DOMAIN error
  		}
  		return complex(math.Asin(real(x)), 0)
  	}
  	ct := complex(-imag(x), real(x)) // i * x
  	xx := x * x
  	x1 := complex(1-real(xx), -imag(xx)) // 1 - x*x
  	x2 := Sqrt(x1)                       // x2 = sqrt(1 - x*x)
  	w := Log(ct + x2)
  	return complex(imag(w), -real(w)) // -i * w
  }
  
  // Asinh returns the inverse hyperbolic sine of x.
  func Asinh(x complex128) complex128 {
  	// TODO check range
  	if imag(x) == 0 {
  		if math.Abs(real(x)) > 1 {
  			return complex(math.Pi/2, 0) // DOMAIN error
  		}
  		return complex(math.Asinh(real(x)), 0)
  	}
  	xx := x * x
  	x1 := complex(1+real(xx), imag(xx)) // 1 + x*x
  	return Log(x + Sqrt(x1))            // log(x + sqrt(1 + x*x))
  }
  
  // Complex circular arc cosine
  //
  // DESCRIPTION:
  //
  // w = arccos z  =  PI/2 - arcsin z.
  //
  // ACCURACY:
  //
  //                      Relative error:
  // arithmetic   domain     # trials      peak         rms
  //    DEC       -10,+10      5200      1.6e-15      2.8e-16
  //    IEEE      -10,+10     30000      1.8e-14      2.2e-15
  
  // Acos returns the inverse cosine of x.
  func Acos(x complex128) complex128 {
  	w := Asin(x)
  	return complex(math.Pi/2-real(w), -imag(w))
  }
  
  // Acosh returns the inverse hyperbolic cosine of x.
  func Acosh(x complex128) complex128 {
  	w := Acos(x)
  	if imag(w) <= 0 {
  		return complex(-imag(w), real(w)) // i * w
  	}
  	return complex(imag(w), -real(w)) // -i * w
  }
  
  // Complex circular arc tangent
  //
  // DESCRIPTION:
  //
  // If
  //     z = x + iy,
  //
  // then
  //          1       (    2x     )
  // Re w  =  - arctan(-----------)  +  k PI
  //          2       (     2    2)
  //                  (1 - x  - y )
  //
  //               ( 2         2)
  //          1    (x  +  (y+1) )
  // Im w  =  - log(------------)
  //          4    ( 2         2)
  //               (x  +  (y-1) )
  //
  // Where k is an arbitrary integer.
  //
  // catan(z) = -i catanh(iz).
  //
  // ACCURACY:
  //
  //                      Relative error:
  // arithmetic   domain     # trials      peak         rms
  //    DEC       -10,+10      5900       1.3e-16     7.8e-18
  //    IEEE      -10,+10     30000       2.3e-15     8.5e-17
  // The check catan( ctan(z) )  =  z, with |x| and |y| < PI/2,
  // had peak relative error 1.5e-16, rms relative error
  // 2.9e-17.  See also clog().
  
  // Atan returns the inverse tangent of x.
  func Atan(x complex128) complex128 {
  	if real(x) == 0 && imag(x) > 1 {
  		return NaN()
  	}
  
  	x2 := real(x) * real(x)
  	a := 1 - x2 - imag(x)*imag(x)
  	if a == 0 {
  		return NaN()
  	}
  	t := 0.5 * math.Atan2(2*real(x), a)
  	w := reducePi(t)
  
  	t = imag(x) - 1
  	b := x2 + t*t
  	if b == 0 {
  		return NaN()
  	}
  	t = imag(x) + 1
  	c := (x2 + t*t) / b
  	return complex(w, 0.25*math.Log(c))
  }
  
  // Atanh returns the inverse hyperbolic tangent of x.
  func Atanh(x complex128) complex128 {
  	z := complex(-imag(x), real(x)) // z = i * x
  	z = Atan(z)
  	return complex(imag(z), -real(z)) // z = -i * z
  }
  

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