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Source file src/math/cmplx/sqrt.go

Documentation: math/cmplx

  // Copyright 2010 The Go Authors. All rights reserved.
  // Use of this source code is governed by a BSD-style
  // license that can be found in the LICENSE file.
  
  package cmplx
  
  import "math"
  
  // The original C code, the long comment, and the constants
  // below are from http://netlib.sandia.gov/cephes/c9x-complex/clog.c.
  // The go code is a simplified version of the original C.
  //
  // Cephes Math Library Release 2.8:  June, 2000
  // Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
  //
  // The readme file at http://netlib.sandia.gov/cephes/ says:
  //    Some software in this archive may be from the book _Methods and
  // Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster
  // International, 1989) or from the Cephes Mathematical Library, a
  // commercial product. In either event, it is copyrighted by the author.
  // What you see here may be used freely but it comes with no support or
  // guarantee.
  //
  //   The two known misprints in the book are repaired here in the
  // source listings for the gamma function and the incomplete beta
  // integral.
  //
  //   Stephen L. Moshier
  //   moshier@na-net.ornl.gov
  
  // Complex square root
  //
  // DESCRIPTION:
  //
  // If z = x + iy,  r = |z|, then
  //
  //                       1/2
  // Re w  =  [ (r + x)/2 ]   ,
  //
  //                       1/2
  // Im w  =  [ (r - x)/2 ]   .
  //
  // Cancelation error in r-x or r+x is avoided by using the
  // identity  2 Re w Im w  =  y.
  //
  // Note that -w is also a square root of z. The root chosen
  // is always in the right half plane and Im w has the same sign as y.
  //
  // ACCURACY:
  //
  //                      Relative error:
  // arithmetic   domain     # trials      peak         rms
  //    DEC       -10,+10     25000       3.2e-17     9.6e-18
  //    IEEE      -10,+10   1,000,000     2.9e-16     6.1e-17
  
  // Sqrt returns the square root of x.
  // The result r is chosen so that real(r) ≥ 0 and imag(r) has the same sign as imag(x).
  func Sqrt(x complex128) complex128 {
  	if imag(x) == 0 {
  		if real(x) == 0 {
  			return complex(0, 0)
  		}
  		if real(x) < 0 {
  			return complex(0, math.Sqrt(-real(x)))
  		}
  		return complex(math.Sqrt(real(x)), 0)
  	}
  	if real(x) == 0 {
  		if imag(x) < 0 {
  			r := math.Sqrt(-0.5 * imag(x))
  			return complex(r, -r)
  		}
  		r := math.Sqrt(0.5 * imag(x))
  		return complex(r, r)
  	}
  	a := real(x)
  	b := imag(x)
  	var scale float64
  	// Rescale to avoid internal overflow or underflow.
  	if math.Abs(a) > 4 || math.Abs(b) > 4 {
  		a *= 0.25
  		b *= 0.25
  		scale = 2
  	} else {
  		a *= 1.8014398509481984e16 // 2**54
  		b *= 1.8014398509481984e16
  		scale = 7.450580596923828125e-9 // 2**-27
  	}
  	r := math.Hypot(a, b)
  	var t float64
  	if a > 0 {
  		t = math.Sqrt(0.5*r + 0.5*a)
  		r = scale * math.Abs((0.5*b)/t)
  		t *= scale
  	} else {
  		r = math.Sqrt(0.5*r - 0.5*a)
  		t = scale * math.Abs((0.5*b)/r)
  		r *= scale
  	}
  	if b < 0 {
  		return complex(t, -r)
  	}
  	return complex(t, r)
  }
  

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