298 lines
8.5 KiB
Go
298 lines
8.5 KiB
Go
// Copyright 2010 The Go Authors. All rights reserved.
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// Use of this source code is governed by a BSD-style
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// license that can be found in the LICENSE file.
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package cmplx
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import (
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"math"
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"math/bits"
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)
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// The original C code, the long comment, and the constants
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// below are from http://netlib.sandia.gov/cephes/c9x-complex/clog.c.
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// The go code is a simplified version of the original C.
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//
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// Cephes Math Library Release 2.8: June, 2000
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// Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
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//
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// The readme file at http://netlib.sandia.gov/cephes/ says:
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// Some software in this archive may be from the book _Methods and
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// Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster
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// International, 1989) or from the Cephes Mathematical Library, a
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// commercial product. In either event, it is copyrighted by the author.
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// What you see here may be used freely but it comes with no support or
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// guarantee.
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//
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// The two known misprints in the book are repaired here in the
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// source listings for the gamma function and the incomplete beta
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// integral.
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//
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// Stephen L. Moshier
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// moshier@na-net.ornl.gov
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// Complex circular tangent
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//
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// DESCRIPTION:
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//
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// If
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// z = x + iy,
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//
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// then
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//
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// sin 2x + i sinh 2y
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// w = --------------------.
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// cos 2x + cosh 2y
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//
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// On the real axis the denominator is zero at odd multiples
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// of PI/2. The denominator is evaluated by its Taylor
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// series near these points.
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//
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// ctan(z) = -i ctanh(iz).
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//
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// ACCURACY:
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//
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// Relative error:
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// arithmetic domain # trials peak rms
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// DEC -10,+10 5200 7.1e-17 1.6e-17
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// IEEE -10,+10 30000 7.2e-16 1.2e-16
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// Also tested by ctan * ccot = 1 and catan(ctan(z)) = z.
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// Tan returns the tangent of x.
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func Tan(x complex128) complex128 {
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switch re, im := real(x), imag(x); {
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case math.IsInf(im, 0):
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switch {
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case math.IsInf(re, 0) || math.IsNaN(re):
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return complex(math.Copysign(0, re), math.Copysign(1, im))
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}
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return complex(math.Copysign(0, math.Sin(2*re)), math.Copysign(1, im))
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case re == 0 && math.IsNaN(im):
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return x
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}
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d := math.Cos(2*real(x)) + math.Cosh(2*imag(x))
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if math.Abs(d) < 0.25 {
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d = tanSeries(x)
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}
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if d == 0 {
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return Inf()
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}
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return complex(math.Sin(2*real(x))/d, math.Sinh(2*imag(x))/d)
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}
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// Complex hyperbolic tangent
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//
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// DESCRIPTION:
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//
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// tanh z = (sinh 2x + i sin 2y) / (cosh 2x + cos 2y) .
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//
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// ACCURACY:
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//
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// Relative error:
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// arithmetic domain # trials peak rms
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// IEEE -10,+10 30000 1.7e-14 2.4e-16
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// Tanh returns the hyperbolic tangent of x.
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func Tanh(x complex128) complex128 {
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switch re, im := real(x), imag(x); {
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case math.IsInf(re, 0):
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switch {
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case math.IsInf(im, 0) || math.IsNaN(im):
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return complex(math.Copysign(1, re), math.Copysign(0, im))
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}
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return complex(math.Copysign(1, re), math.Copysign(0, math.Sin(2*im)))
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case im == 0 && math.IsNaN(re):
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return x
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}
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d := math.Cosh(2*real(x)) + math.Cos(2*imag(x))
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if d == 0 {
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return Inf()
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}
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return complex(math.Sinh(2*real(x))/d, math.Sin(2*imag(x))/d)
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}
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// reducePi reduces the input argument x to the range (-Pi/2, Pi/2].
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// x must be greater than or equal to 0. For small arguments it
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// uses Cody-Waite reduction in 3 float64 parts based on:
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// "Elementary Function Evaluation: Algorithms and Implementation"
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// Jean-Michel Muller, 1997.
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// For very large arguments it uses Payne-Hanek range reduction based on:
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// "ARGUMENT REDUCTION FOR HUGE ARGUMENTS: Good to the Last Bit"
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// K. C. Ng et al, March 24, 1992.
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func reducePi(x float64) float64 {
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// reduceThreshold is the maximum value of x where the reduction using
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// Cody-Waite reduction still gives accurate results. This threshold
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// is set by t*PIn being representable as a float64 without error
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// where t is given by t = floor(x * (1 / Pi)) and PIn are the leading partial
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// terms of Pi. Since the leading terms, PI1 and PI2 below, have 30 and 32
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// trailing zero bits respectively, t should have less than 30 significant bits.
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// t < 1<<30 -> floor(x*(1/Pi)+0.5) < 1<<30 -> x < (1<<30-1) * Pi - 0.5
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// So, conservatively we can take x < 1<<30.
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const reduceThreshold float64 = 1 << 30
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if math.Abs(x) < reduceThreshold {
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// Use Cody-Waite reduction in three parts.
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const (
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// PI1, PI2 and PI3 comprise an extended precision value of PI
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// such that PI ~= PI1 + PI2 + PI3. The parts are chosen so
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// that PI1 and PI2 have an approximately equal number of trailing
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// zero bits. This ensures that t*PI1 and t*PI2 are exact for
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// large integer values of t. The full precision PI3 ensures the
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// approximation of PI is accurate to 102 bits to handle cancellation
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// during subtraction.
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PI1 = 3.141592502593994 // 0x400921fb40000000
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PI2 = 1.5099578831723193e-07 // 0x3e84442d00000000
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PI3 = 1.0780605716316238e-14 // 0x3d08469898cc5170
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)
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t := x / math.Pi
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t += 0.5
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t = float64(int64(t)) // int64(t) = the multiple
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return ((x - t*PI1) - t*PI2) - t*PI3
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}
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// Must apply Payne-Hanek range reduction
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const (
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mask = 0x7FF
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shift = 64 - 11 - 1
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bias = 1023
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fracMask = 1<<shift - 1
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)
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// Extract out the integer and exponent such that,
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// x = ix * 2 ** exp.
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ix := math.Float64bits(x)
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exp := int(ix>>shift&mask) - bias - shift
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ix &= fracMask
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ix |= 1 << shift
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// mPi is the binary digits of 1/Pi as a uint64 array,
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// that is, 1/Pi = Sum mPi[i]*2^(-64*i).
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// 19 64-bit digits give 1216 bits of precision
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// to handle the largest possible float64 exponent.
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var mPi = [...]uint64{
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0x0000000000000000,
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0x517cc1b727220a94,
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0xfe13abe8fa9a6ee0,
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0x6db14acc9e21c820,
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0xff28b1d5ef5de2b0,
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0xdb92371d2126e970,
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0x0324977504e8c90e,
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0x7f0ef58e5894d39f,
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0x74411afa975da242,
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0x74ce38135a2fbf20,
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0x9cc8eb1cc1a99cfa,
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0x4e422fc5defc941d,
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0x8ffc4bffef02cc07,
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0xf79788c5ad05368f,
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0xb69b3f6793e584db,
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0xa7a31fb34f2ff516,
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0xba93dd63f5f2f8bd,
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0x9e839cfbc5294975,
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0x35fdafd88fc6ae84,
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0x2b0198237e3db5d5,
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}
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// Use the exponent to extract the 3 appropriate uint64 digits from mPi,
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// B ~ (z0, z1, z2), such that the product leading digit has the exponent -64.
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// Note, exp >= 50 since x >= reduceThreshold and exp < 971 for maximum float64.
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digit, bitshift := uint(exp+64)/64, uint(exp+64)%64
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z0 := (mPi[digit] << bitshift) | (mPi[digit+1] >> (64 - bitshift))
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z1 := (mPi[digit+1] << bitshift) | (mPi[digit+2] >> (64 - bitshift))
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z2 := (mPi[digit+2] << bitshift) | (mPi[digit+3] >> (64 - bitshift))
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// Multiply mantissa by the digits and extract the upper two digits (hi, lo).
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z2hi, _ := bits.Mul64(z2, ix)
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z1hi, z1lo := bits.Mul64(z1, ix)
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z0lo := z0 * ix
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lo, c := bits.Add64(z1lo, z2hi, 0)
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hi, _ := bits.Add64(z0lo, z1hi, c)
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// Find the magnitude of the fraction.
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lz := uint(bits.LeadingZeros64(hi))
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e := uint64(bias - (lz + 1))
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// Clear implicit mantissa bit and shift into place.
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hi = (hi << (lz + 1)) | (lo >> (64 - (lz + 1)))
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hi >>= 64 - shift
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// Include the exponent and convert to a float.
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hi |= e << shift
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x = math.Float64frombits(hi)
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// map to (-Pi/2, Pi/2]
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if x > 0.5 {
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x--
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}
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return math.Pi * x
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}
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// Taylor series expansion for cosh(2y) - cos(2x)
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func tanSeries(z complex128) float64 {
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const MACHEP = 1.0 / (1 << 53)
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x := math.Abs(2 * real(z))
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y := math.Abs(2 * imag(z))
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x = reducePi(x)
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x = x * x
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y = y * y
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x2 := 1.0
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y2 := 1.0
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f := 1.0
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rn := 0.0
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d := 0.0
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for {
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rn++
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f *= rn
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rn++
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f *= rn
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x2 *= x
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y2 *= y
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t := y2 + x2
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t /= f
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d += t
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rn++
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f *= rn
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rn++
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f *= rn
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x2 *= x
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y2 *= y
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t = y2 - x2
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t /= f
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d += t
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if !(math.Abs(t/d) > MACHEP) {
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// Caution: Use ! and > instead of <= for correct behavior if t/d is NaN.
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// See issue 17577.
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break
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}
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}
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return d
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}
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// Complex circular cotangent
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//
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// DESCRIPTION:
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//
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// If
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// z = x + iy,
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//
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// then
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//
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// sin 2x - i sinh 2y
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// w = --------------------.
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// cosh 2y - cos 2x
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//
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// On the real axis, the denominator has zeros at even
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// multiples of PI/2. Near these points it is evaluated
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// by a Taylor series.
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//
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// ACCURACY:
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//
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// Relative error:
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// arithmetic domain # trials peak rms
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// DEC -10,+10 3000 6.5e-17 1.6e-17
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// IEEE -10,+10 30000 9.2e-16 1.2e-16
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// Also tested by ctan * ccot = 1 + i0.
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// Cot returns the cotangent of x.
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func Cot(x complex128) complex128 {
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d := math.Cosh(2*imag(x)) - math.Cos(2*real(x))
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if math.Abs(d) < 0.25 {
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d = tanSeries(x)
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}
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if d == 0 {
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return Inf()
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}
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return complex(math.Sin(2*real(x))/d, -math.Sinh(2*imag(x))/d)
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}
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