Source file src/pkg/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	//   [email protected]
    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	func Sqrt(x complex128) complex128 {
    58		if imag(x) == 0 {
    59			if real(x) == 0 {
    60				return complex(0, 0)
    61			}
    62			if real(x) < 0 {
    63				return complex(0, math.Sqrt(-real(x)))
    64			}
    65			return complex(math.Sqrt(real(x)), 0)
    66		}
    67		if real(x) == 0 {
    68			if imag(x) < 0 {
    69				r := math.Sqrt(-0.5 * imag(x))
    70				return complex(r, -r)
    71			}
    72			r := math.Sqrt(0.5 * imag(x))
    73			return complex(r, r)
    74		}
    75		a := real(x)
    76		b := imag(x)
    77		var scale float64
    78		// Rescale to avoid internal overflow or underflow.
    79		if math.Abs(a) > 4 || math.Abs(b) > 4 {
    80			a *= 0.25
    81			b *= 0.25
    82			scale = 2
    83		} else {
    84			a *= 1.8014398509481984e16 // 2**54
    85			b *= 1.8014398509481984e16
    86			scale = 7.450580596923828125e-9 // 2**-27
    87		}
    88		r := math.Hypot(a, b)
    89		var t float64
    90		if a > 0 {
    91			t = math.Sqrt(0.5*r + 0.5*a)
    92			r = scale * math.Abs((0.5*b)/t)
    93			t *= scale
    94		} else {
    95			r = math.Sqrt(0.5*r - 0.5*a)
    96			t = scale * math.Abs((0.5*b)/r)
    97			r *= scale
    98		}
    99		if b < 0 {
   100			return complex(t, -r)
   101		}
   102		return complex(t, r)
   103	}