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Diffstat (limited to 'src/math/cmplx/tan.go')
| -rw-r--r-- | src/math/cmplx/tan.go | 184 |
1 files changed, 184 insertions, 0 deletions
diff --git a/src/math/cmplx/tan.go b/src/math/cmplx/tan.go new file mode 100644 index 000000000..9485315d8 --- /dev/null +++ b/src/math/cmplx/tan.go @@ -0,0 +1,184 @@ +// 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 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.Abs(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.Abs(2 * real(z)) + y := math.Abs(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.Abs(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.Abs(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) +} |