github.com/gopherd/gonum@v0.0.4/internal/cmplx64/sqrt.go (about)

     1  // Copyright 2010 The Go Authors. All rights reserved.
     2  // Use of this source code is governed by a BSD-style
     3  // license that can be found in the LICENSE file.
     4  
     5  // Copyright ©2017 The Gonum Authors. All rights reserved.
     6  // Use of this source code is governed by a BSD-style
     7  // license that can be found in the LICENSE file.
     8  
     9  package cmplx64
    10  
    11  import math "github.com/gopherd/gonum/internal/math32"
    12  
    13  // The original C code, the long comment, and the constants
    14  // below are from http://netlib.sandia.gov/cephes/c9x-complex/clog.c.
    15  // The go code is a simplified version of the original C.
    16  //
    17  // Cephes Math Library Release 2.8:  June, 2000
    18  // Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
    19  //
    20  // The readme file at http://netlib.sandia.gov/cephes/ says:
    21  //    Some software in this archive may be from the book _Methods and
    22  // Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster
    23  // International, 1989) or from the Cephes Mathematical Library, a
    24  // commercial product. In either event, it is copyrighted by the author.
    25  // What you see here may be used freely but it comes with no support or
    26  // guarantee.
    27  //
    28  //   The two known misprints in the book are repaired here in the
    29  // source listings for the gamma function and the incomplete beta
    30  // integral.
    31  //
    32  //   Stephen L. Moshier
    33  //   moshier@na-net.ornl.gov
    34  
    35  // Complex square root
    36  //
    37  // DESCRIPTION:
    38  //
    39  // If z = x + iy,  r = |z|, then
    40  //
    41  //                       1/2
    42  // Re w  =  [ (r + x)/2 ]   ,
    43  //
    44  //                       1/2
    45  // Im w  =  [ (r - x)/2 ]   .
    46  //
    47  // Cancelation error in r-x or r+x is avoided by using the
    48  // identity  2 Re w Im w  =  y.
    49  //
    50  // Note that -w is also a square root of z. The root chosen
    51  // is always in the right half plane and Im w has the same sign as y.
    52  //
    53  // ACCURACY:
    54  //
    55  //                      Relative error:
    56  // arithmetic   domain     # trials      peak         rms
    57  //    DEC       -10,+10     25000       3.2e-17     9.6e-18
    58  //    IEEE      -10,+10   1,000,000     2.9e-16     6.1e-17
    59  
    60  // Sqrt returns the square root of x.
    61  // The result r is chosen so that real(r) ≥ 0 and imag(r) has the same sign as imag(x).
    62  func Sqrt(x complex64) complex64 {
    63  	if imag(x) == 0 {
    64  		if real(x) == 0 {
    65  			return complex(0, 0)
    66  		}
    67  		if real(x) < 0 {
    68  			return complex(0, math.Sqrt(-real(x)))
    69  		}
    70  		return complex(math.Sqrt(real(x)), 0)
    71  	}
    72  	if real(x) == 0 {
    73  		if imag(x) < 0 {
    74  			r := math.Sqrt(-0.5 * imag(x))
    75  			return complex(r, -r)
    76  		}
    77  		r := math.Sqrt(0.5 * imag(x))
    78  		return complex(r, r)
    79  	}
    80  	a := real(x)
    81  	b := imag(x)
    82  	var scale float32
    83  	// Rescale to avoid internal overflow or underflow.
    84  	if math.Abs(a) > 4 || math.Abs(b) > 4 {
    85  		a *= 0.25
    86  		b *= 0.25
    87  		scale = 2
    88  	} else {
    89  		a *= 1.8014398509481984e16 // 2**54
    90  		b *= 1.8014398509481984e16
    91  		scale = 7.450580596923828125e-9 // 2**-27
    92  	}
    93  	r := math.Hypot(a, b)
    94  	var t float32
    95  	if a > 0 {
    96  		t = math.Sqrt(0.5*r + 0.5*a)
    97  		r = scale * math.Abs((0.5*b)/t)
    98  		t *= scale
    99  	} else {
   100  		r = math.Sqrt(0.5*r - 0.5*a)
   101  		t = scale * math.Abs((0.5*b)/r)
   102  		r *= scale
   103  	}
   104  	if b < 0 {
   105  		return complex(t, -r)
   106  	}
   107  	return complex(t, r)
   108  }