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 // 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 cmath 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 ] . // // Cancellation 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. 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.Fabs(a) > 4 || math.Fabs(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.Fabs((0.5*b)/t) t *= scale } else { r = math.Sqrt(0.5*r - 0.5*a) t = scale * math.Fabs((0.5*b)/r) r *= scale } if b < 0 { return complex(t, -r) } return complex(t, r) }