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| -rw-r--r-- | src/math/cmplx/tan.go | 297 |
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diff --git a/src/math/cmplx/tan.go b/src/math/cmplx/tan.go new file mode 100644 index 0000000..67a1133 --- /dev/null +++ b/src/math/cmplx/tan.go @@ -0,0 +1,297 @@ +// 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" + "math/bits" +) + +// 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 { + switch re, im := real(x), imag(x); { + case math.IsInf(im, 0): + switch { + case math.IsInf(re, 0) || math.IsNaN(re): + return complex(math.Copysign(0, re), math.Copysign(1, im)) + } + return complex(math.Copysign(0, math.Sin(2*re)), math.Copysign(1, im)) + case re == 0 && math.IsNaN(im): + return x + } + 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 { + switch re, im := real(x), imag(x); { + case math.IsInf(re, 0): + switch { + case math.IsInf(im, 0) || math.IsNaN(im): + return complex(math.Copysign(1, re), math.Copysign(0, im)) + } + return complex(math.Copysign(1, re), math.Copysign(0, math.Sin(2*im))) + case im == 0 && math.IsNaN(re): + return x + } + 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) +} + +// 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(x 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 reduceThreshold float64 = 1 << 30 + if math.Abs(x) < reduceThreshold { + // 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. + PI1 = 3.141592502593994 // 0x400921fb40000000 + PI2 = 1.5099578831723193e-07 // 0x3e84442d00000000 + PI3 = 1.0780605716316238e-14 // 0x3d08469898cc5170 + ) + t := x / math.Pi + t += 0.5 + t = float64(int64(t)) // int64(t) = the multiple + return ((x - t*PI1) - t*PI2) - t*PI3 + } + // Must apply Payne-Hanek range reduction + const ( + mask = 0x7FF + shift = 64 - 11 - 1 + bias = 1023 + fracMask = 1<<shift - 1 + ) + // Extract out the integer and exponent such that, + // x = ix * 2 ** exp. + ix := math.Float64bits(x) + exp := int(ix>>shift&mask) - bias - shift + ix &= fracMask + ix |= 1 << shift + + // 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 mPi = [...]uint64{ + 0x0000000000000000, + 0x517cc1b727220a94, + 0xfe13abe8fa9a6ee0, + 0x6db14acc9e21c820, + 0xff28b1d5ef5de2b0, + 0xdb92371d2126e970, + 0x0324977504e8c90e, + 0x7f0ef58e5894d39f, + 0x74411afa975da242, + 0x74ce38135a2fbf20, + 0x9cc8eb1cc1a99cfa, + 0x4e422fc5defc941d, + 0x8ffc4bffef02cc07, + 0xf79788c5ad05368f, + 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. + digit, bitshift := uint(exp+64)/64, uint(exp+64)%64 + z0 := (mPi[digit] << bitshift) | (mPi[digit+1] >> (64 - bitshift)) + z1 := (mPi[digit+1] << bitshift) | (mPi[digit+2] >> (64 - bitshift)) + z2 := (mPi[digit+2] << bitshift) | (mPi[digit+3] >> (64 - bitshift)) + // Multiply mantissa by the digits and extract the upper two digits (hi, lo). + z2hi, _ := bits.Mul64(z2, ix) + z1hi, z1lo := bits.Mul64(z1, ix) + z0lo := z0 * ix + lo, c := bits.Add64(z1lo, z2hi, 0) + hi, _ := bits.Add64(z0lo, z1hi, c) + // Find the magnitude of the fraction. + lz := uint(bits.LeadingZeros64(hi)) + e := uint64(bias - (lz + 1)) + // Clear implicit mantissa bit and shift into place. + hi = (hi << (lz + 1)) | (lo >> (64 - (lz + 1))) + hi >>= 64 - shift + // Include the exponent and convert to a float. + hi |= e << shift + x = math.Float64frombits(hi) + // map to (-Pi/2, Pi/2] + if x > 0.5 { + x-- + } + return math.Pi * x +} + +// 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) +} |