libgo/go/math/cmplx/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                 if real(x) == 0 {
  61                         return complex(0, 0)
  62                 }
  63                 if real(x) < 0 {
  64                         return complex(0, math.Sqrt(-real(x)))
  65                 }
  66                 return complex(math.Sqrt(real(x)), 0)
  67         }
  68         if real(x) == 0 {
  69                 if imag(x) < 0 {
  70                         r := math.Sqrt(-0.5 * imag(x))
  71                         return complex(r, -r)
  72                 }
  73                 r := math.Sqrt(0.5 * imag(x))
  74                 return complex(r, r)
  75         }
  76         a := real(x)
  77         b := imag(x)
  78         var scale float64
  79         // Rescale to avoid internal overflow or underflow.
  80         if math.Abs(a) > 4 || math.Abs(b) > 4 {
  81                 a *= 0.25
  82                 b *= 0.25
  83                 scale = 2
  84         } else {
  85                 a *= 1.8014398509481984e16 // 2**54
  86                 b *= 1.8014398509481984e16
  87                 scale = 7.450580596923828125e-9 // 2**-27
  88         }
  89         r := math.Hypot(a, b)
  90         var t float64
  91         if a > 0 {
  92                 t = math.Sqrt(0.5*r + 0.5*a)
  93                 r = scale * math.Abs((0.5*b)/t)
  94                 t *= scale
  95         } else {
  96                 r = math.Sqrt(0.5*r - 0.5*a)
  97                 t = scale * math.Abs((0.5*b)/r)
  98                 r *= scale
  99         }
 100         if b < 0 {
 101                 return complex(t, -r)
 102         }
 103         return complex(t, r)
 104 }