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Source file src/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	// Cancelation 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	}
   105	

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