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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 circular tangent // // DESCRIPTION: // // If // z = x + iy, // // then // // sin 2x + i sinh 2y // w = --------------------. // cos 2x + cosh 2y // // On the real axis the denominator is zero at odd multiples // of PI/2. The denominator is evaluated by its Taylor // series near these points. // // ctan(z) = -i ctanh(iz). // // ACCURACY: // // Relative error: // arithmetic domain # trials peak rms // DEC -10,+10 5200 7.1e-17 1.6e-17 // IEEE -10,+10 30000 7.2e-16 1.2e-16 // Also tested by ctan * ccot = 1 and catan(ctan(z)) = z. // Tan returns the tangent of x. func Tan(x complex128) complex128 { d := math.Cos(2*real(x)) + math.Cosh(2*imag(x)) if math.Fabs(d) < 0.25 { d = tanSeries(x) } if d == 0 { return Inf() } return complex(math.Sin(2*real(x))/d, math.Sinh(2*imag(x))/d) } // Complex hyperbolic tangent // // DESCRIPTION: // // tanh z = (sinh 2x + i sin 2y) / (cosh 2x + cos 2y) . // // ACCURACY: // // Relative error: // arithmetic domain # trials peak rms // IEEE -10,+10 30000 1.7e-14 2.4e-16 // Tanh returns the hyperbolic tangent of x. func Tanh(x complex128) complex128 { d := math.Cosh(2*real(x)) + math.Cos(2*imag(x)) if d == 0 { return Inf() } return complex(math.Sinh(2*real(x))/d, math.Sin(2*imag(x))/d) } // Program to subtract nearest integer multiple of PI func reducePi(x float64) float64 { const ( // extended precision value of PI: DP1 = 3.14159265160560607910E0 // ?? 0x400921fb54000000 DP2 = 1.98418714791870343106E-9 // ?? 0x3e210b4610000000 DP3 = 1.14423774522196636802E-17 // ?? 0x3c6a62633145c06e ) t := x / math.Pi if t >= 0 { t += 0.5 } else { t -= 0.5 } t = float64(int64(t)) // int64(t) = the multiple return ((x - t*DP1) - t*DP2) - t*DP3 } // Taylor series expansion for cosh(2y) - cos(2x) func tanSeries(z complex128) float64 { const MACHEP = 1.0 / (1 << 53) x := math.Fabs(2 * real(z)) y := math.Fabs(2 * imag(z)) x = reducePi(x) x = x * x y = y * y x2 := 1.0 y2 := 1.0 f := 1.0 rn := 0.0 d := 0.0 for { rn += 1 f *= rn rn += 1 f *= rn x2 *= x y2 *= y t := y2 + x2 t /= f d += t rn += 1 f *= rn rn += 1 f *= rn x2 *= x y2 *= y t = y2 - x2 t /= f d += t if math.Fabs(t/d) <= MACHEP { break } } return d } // Complex circular cotangent // // DESCRIPTION: // // If // z = x + iy, // // then // // sin 2x - i sinh 2y // w = --------------------. // cosh 2y - cos 2x // // On the real axis, the denominator has zeros at even // multiples of PI/2. Near these points it is evaluated // by a Taylor series. // // ACCURACY: // // Relative error: // arithmetic domain # trials peak rms // DEC -10,+10 3000 6.5e-17 1.6e-17 // IEEE -10,+10 30000 9.2e-16 1.2e-16 // Also tested by ctan * ccot = 1 + i0. // Cot returns the cotangent of x. func Cot(x complex128) complex128 { d := math.Cosh(2*imag(x)) - math.Cos(2*real(x)) if math.Fabs(d) < 0.25 { d = tanSeries(x) } if d == 0 { return Inf() } return complex(math.Sin(2*real(x))/d, -math.Sinh(2*imag(x))/d) }