108 lines
3 KiB
Go
108 lines
3 KiB
Go
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// Copyright 2010 The Go Authors. All rights reserved.
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// Use of this source code is governed by a BSD-style
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// license that can be found in the LICENSE file.
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package cmplx
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import "math"
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// The original C code, the long comment, and the constants
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// below are from http://netlib.sandia.gov/cephes/c9x-complex/clog.c.
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// The go code is a simplified version of the original C.
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//
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// Cephes Math Library Release 2.8: June, 2000
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// Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
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//
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// The readme file at http://netlib.sandia.gov/cephes/ says:
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// Some software in this archive may be from the book _Methods and
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// Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster
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// International, 1989) or from the Cephes Mathematical Library, a
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// commercial product. In either event, it is copyrighted by the author.
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// What you see here may be used freely but it comes with no support or
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// guarantee.
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//
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// The two known misprints in the book are repaired here in the
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// source listings for the gamma function and the incomplete beta
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// integral.
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//
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// Stephen L. Moshier
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// moshier@na-net.ornl.gov
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// Complex square root
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//
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// DESCRIPTION:
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//
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// If z = x + iy, r = |z|, then
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//
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// 1/2
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// Re w = [ (r + x)/2 ] ,
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//
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// 1/2
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// Im w = [ (r - x)/2 ] .
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//
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// Cancellation error in r-x or r+x is avoided by using the
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// identity 2 Re w Im w = y.
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//
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// Note that -w is also a square root of z. The root chosen
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// is always in the right half plane and Im w has the same sign as y.
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//
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// ACCURACY:
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//
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// Relative error:
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// arithmetic domain # trials peak rms
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// DEC -10,+10 25000 3.2e-17 9.6e-18
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// IEEE -10,+10 1,000,000 2.9e-16 6.1e-17
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// Sqrt returns the square root of x.
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// The result r is chosen so that real(r) ≥ 0 and imag(r) has the same sign as imag(x).
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func Sqrt(x complex128) complex128 {
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if imag(x) == 0 {
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// Ensure that imag(r) has the same sign as imag(x) for imag(x) == signed zero.
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if real(x) == 0 {
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return complex(0, imag(x))
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}
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if real(x) < 0 {
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return complex(0, math.Copysign(math.Sqrt(-real(x)), imag(x)))
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}
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return complex(math.Sqrt(real(x)), imag(x))
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} else if math.IsInf(imag(x), 0) {
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return complex(math.Inf(1.0), imag(x))
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}
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if real(x) == 0 {
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if imag(x) < 0 {
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r := math.Sqrt(-0.5 * imag(x))
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return complex(r, -r)
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}
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r := math.Sqrt(0.5 * imag(x))
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return complex(r, r)
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}
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a := real(x)
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b := imag(x)
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var scale float64
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// Rescale to avoid internal overflow or underflow.
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if math.Abs(a) > 4 || math.Abs(b) > 4 {
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a *= 0.25
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b *= 0.25
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scale = 2
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} else {
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a *= 1.8014398509481984e16 // 2**54
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b *= 1.8014398509481984e16
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scale = 7.450580596923828125e-9 // 2**-27
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}
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r := math.Hypot(a, b)
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var t float64
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if a > 0 {
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t = math.Sqrt(0.5*r + 0.5*a)
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r = scale * math.Abs((0.5*b)/t)
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t *= scale
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} else {
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r = math.Sqrt(0.5*r - 0.5*a)
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t = scale * math.Abs((0.5*b)/r)
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r *= scale
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}
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if b < 0 {
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return complex(t, -r)
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}
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return complex(t, r)
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}
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