src/math/cmplx/tan.go

// 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++
		f *= rn
		rn++
		f *= rn
		x2 *= x
		y2 *= y
		t := y2 + x2
		t /= f
		d += t

		rn++
		f *= rn
		rn++
		f *= rn
		x2 *= x
		y2 *= y
		t = y2 - x2
		t /= f
		d += t
		if !(math.Abs(t/d) > MACHEP) {
			// Caution: Use ! and > instead of <= for correct behavior if t/d is NaN.
			// See issue 17577.
			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)
}