sqrt.go

     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  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 square root
    32  //
    33  // DESCRIPTION:
    34  //
    35  // If z = x + iy,  r = |z|, then
    36  //
    37  //                       1/2
    38  // Re w  =  [ (r + x)/2 ]   ,
    39  //
    40  //                       1/2
    41  // Im w  =  [ (r - x)/2 ]   .
    42  //
    43  // Cancellation error in r-x or r+x is avoided by using the
    44  // identity  2 Re w Im w  =  y.
    45  //
    46  // Note that -w is also a square root of z. The root chosen
    47  // is always in the right half plane and Im w has the same sign as y.
    48  //
    49  // ACCURACY:
    50  //
    51  //                      Relative error:
    52  // arithmetic   domain     # trials      peak         rms
    53  //    DEC       -10,+10     25000       3.2e-17     9.6e-18
    54  //    IEEE      -10,+10   1,000,000     2.9e-16     6.1e-17
    55  
    56  // Sqrt returns the square root of x.
    57  // The result r is chosen so that real(r) ≥ 0 and imag(r) has the same sign as imag(x).
    58  func Sqrt(x complex128) complex128 {
    59  	if imag(x) == 0 {
    60  		// Ensure that imag(r) has the same sign as imag(x) for imag(x) == signed zero.
    61  		if real(x) == 0 {
    62  			return complex(0, imag(x))
    63  		}
    64  		if real(x) < 0 {
    65  			return complex(0, math.Copysign(math.Sqrt(-real(x)), imag(x)))
    66  		}
    67  		return complex(math.Sqrt(real(x)), imag(x))
    68  	} else if math.IsInf(imag(x), 0) {
    69  		return complex(math.Inf(1.0), imag(x))
    70  	}
    71  	if real(x) == 0 {
    72  		if imag(x) < 0 {
    73  			r := math.Sqrt(-0.5 * imag(x))
    74  			return complex(r, -r)
    75  		}
    76  		r := math.Sqrt(0.5 * imag(x))
    77  		return complex(r, r)
    78  	}
    79  	a := real(x)
    80  	b := imag(x)
    81  	var scale float64
    82  	// Rescale to avoid internal overflow or underflow.
    83  	if math.Abs(a) > 4 || math.Abs(b) > 4 {
    84  		a *= 0.25
    85  		b *= 0.25
    86  		scale = 2
    87  	} else {
    88  		a *= 1.8014398509481984e16 // 2**54
    89  		b *= 1.8014398509481984e16
    90  		scale = 7.450580596923828125e-9 // 2**-27
    91  	}
    92  	r := math.Hypot(a, b)
    93  	var t float64
    94  	if a > 0 {
    95  		t = math.Sqrt(0.5*r + 0.5*a)
    96  		r = scale * math.Abs((0.5*b)/t)
    97  		t *= scale
    98  	} else {
    99  		r = math.Sqrt(0.5*r - 0.5*a)
   100  		t = scale * math.Abs((0.5*b)/r)
   101  		r *= scale
   102  	}
   103  	if b < 0 {
   104  		return complex(t, -r)
   105  	}
   106  	return complex(t, r)
   107  }
   108
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