// Use of this source code is governed by a BSD-style

package cmplx

import (

)

// 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 ( complex128) complex128 {
switch ,  := real(), imag(); {
case math.IsInf(, 0):
switch {
case math.IsInf(, 0) || math.IsNaN():
return complex(math.Copysign(0, ), math.Copysign(1, ))
}
return complex(math.Copysign(0, math.Sin(2*)), math.Copysign(1, ))
case  == 0 && math.IsNaN():
return
}
:= math.Cos(2*real()) + math.Cosh(2*imag())
if math.Abs() < 0.25 {
= tanSeries()
}
if  == 0 {
return Inf()
}
return complex(math.Sin(2*real())/, math.Sinh(2*imag())/)
}

// 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 ( complex128) complex128 {
switch ,  := real(), imag(); {
case math.IsInf(, 0):
switch {
case math.IsInf(, 0) || math.IsNaN():
return complex(math.Copysign(1, ), math.Copysign(0, ))
}
return complex(math.Copysign(1, ), math.Copysign(0, math.Sin(2*)))
case  == 0 && math.IsNaN():
return
}
:= math.Cosh(2*real()) + math.Cos(2*imag())
if  == 0 {
return Inf()
}
return complex(math.Sinh(2*real())/, math.Sin(2*imag())/)
}

// reducePi reduces the input argument x to the range (-Pi/2, Pi/2].
// x must be greater than or equal to 0. For small arguments it
// uses Cody-Waite reduction in 3 float64 parts based on:
// "Elementary Function Evaluation:  Algorithms and Implementation"
// Jean-Michel Muller, 1997.
// For very large arguments it uses Payne-Hanek range reduction based on:
// "ARGUMENT REDUCTION FOR HUGE ARGUMENTS: Good to the Last Bit"
// K. C. Ng et al, March 24, 1992.
func reducePi( float64) float64 {
// reduceThreshold is the maximum value of x where the reduction using
// Cody-Waite reduction still gives accurate results. This threshold
// is set by t*PIn being representable as a float64 without error
// where t is given by t = floor(x * (1 / Pi)) and PIn are the leading partial
// terms of Pi. Since the leading terms, PI1 and PI2 below, have 30 and 32
// trailing zero bits respectively, t should have less than 30 significant bits.
//	t < 1<<30  -> floor(x*(1/Pi)+0.5) < 1<<30 -> x < (1<<30-1) * Pi - 0.5
// So, conservatively we can take x < 1<<30.
const  float64 = 1 << 30
if math.Abs() <  {
// Use Cody-Waite reduction in three parts.
const (
// PI1, PI2 and PI3 comprise an extended precision value of PI
// such that PI ~= PI1 + PI2 + PI3. The parts are chosen so
// that PI1 and PI2 have an approximately equal number of trailing
// zero bits. This ensures that t*PI1 and t*PI2 are exact for
// large integer values of t. The full precision PI3 ensures the
// approximation of PI is accurate to 102 bits to handle cancellation
// during subtraction.
= 3.141592502593994      // 0x400921fb40000000
= 1.5099578831723193e-07 // 0x3e84442d00000000
= 1.0780605716316238e-14 // 0x3d08469898cc5170
)
:=  / math.Pi
+= 0.5
= float64(int64()) // int64(t) = the multiple
return (( - *) - *) - *
}
// Must apply Payne-Hanek range reduction
const (
= 0x7FF
= 64 - 11 - 1
= 1023
= 1<< - 1
)
// Extract out the integer and exponent such that,
// x = ix * 2 ** exp.
:= math.Float64bits()
:= int(>>&) -  -
&=
|= 1 <<

// mPi is the binary digits of 1/Pi as a uint64 array,
// that is, 1/Pi = Sum mPi[i]*2^(-64*i).
// 19 64-bit digits give 1216 bits of precision
// to handle the largest possible float64 exponent.
var  = [...]uint64{
0x0000000000000000,
0x517cc1b727220a94,
0xfe13abe8fa9a6ee0,
0x6db14acc9e21c820,
0xff28b1d5ef5de2b0,
0xdb92371d2126e970,
0x0324977504e8c90e,
0x7f0ef58e5894d39f,
0x74411afa975da242,
0x74ce38135a2fbf20,
0x9cc8eb1cc1a99cfa,
0x4e422fc5defc941d,
0x8ffc4bffef02cc07,
0xb69b3f6793e584db,
0xa7a31fb34f2ff516,
0xba93dd63f5f2f8bd,
0x9e839cfbc5294975,
0x35fdafd88fc6ae84,
0x2b0198237e3db5d5,
}
// Use the exponent to extract the 3 appropriate uint64 digits from mPi,
// B ~ (z0, z1, z2), such that the product leading digit has the exponent -64.
// Note, exp >= 50 since x >= reduceThreshold and exp < 971 for maximum float64.
,  := uint(+64)/64, uint(+64)%64
:= ([] << ) | ([+1] >> (64 - ))
:= ([+1] << ) | ([+2] >> (64 - ))
:= ([+2] << ) | ([+3] >> (64 - ))
// Multiply mantissa by the digits and extract the upper two digits (hi, lo).
,  := bits.Mul64(, )
,  := bits.Mul64(, )
:=  *
// Find the magnitude of the fraction.
:= uint64( - ( + 1))
// Clear implicit mantissa bit and shift into place.
= ( << ( + 1)) | ( >> (64 - ( + 1)))
>>= 64 -
// Include the exponent and convert to a float.
|=  <<
= math.Float64frombits()
// map to (-Pi/2, Pi/2]
if  > 0.5 {
--
}
return math.Pi *
}

// Taylor series expansion for cosh(2y) - cos(2x)
func tanSeries( complex128) float64 {
const  = 1.0 / (1 << 53)
:= math.Abs(2 * real())
:= math.Abs(2 * imag())
= reducePi()
=  *
=  *
:= 1.0
:= 1.0
:= 1.0
:= 0.0
:= 0.0
for {
++
*=
++
*=
*=
*=
:=  +
/=
+=

++
*=
++
*=
*=
*=
=  -
/=
+=
if !(math.Abs(/) > ) {
// Caution: Use ! and > instead of <= for correct behavior if t/d is NaN.
// See issue 17577.
break
}
}
return
}

// 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 ( complex128) complex128 {
:= math.Cosh(2*imag()) - math.Cos(2*real())
if math.Abs() < 0.25 {
= tanSeries()
}
if  == 0 {
return Inf()
}
return complex(math.Sin(2*real())/, -math.Sinh(2*imag())/)
}