File size: 7,156 Bytes
c87d79d |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 |
// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2014 Pedro Gonnet (pedro.gonnet@gmail.com)
// Copyright (C) 2016 Gael Guennebaud <gael.guennebaud@inria.fr>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
#ifndef EIGEN_MATHFUNCTIONSIMPL_H
#define EIGEN_MATHFUNCTIONSIMPL_H
namespace Eigen {
namespace internal {
/** \internal \returns the hyperbolic tan of \a a (coeff-wise)
Doesn't do anything fancy, just a 13/6-degree rational interpolant which
is accurate up to a couple of ulps in the (approximate) range [-8, 8],
outside of which tanh(x) = +/-1 in single precision. The input is clamped
to the range [-c, c]. The value c is chosen as the smallest value where
the approximation evaluates to exactly 1. In the reange [-0.0004, 0.0004]
the approxmation tanh(x) ~= x is used for better accuracy as x tends to zero.
This implementation works on both scalars and packets.
*/
template<typename T>
T generic_fast_tanh_float(const T& a_x)
{
// Clamp the inputs to the range [-c, c]
#ifdef EIGEN_VECTORIZE_FMA
const T plus_clamp = pset1<T>(7.99881172180175781f);
const T minus_clamp = pset1<T>(-7.99881172180175781f);
#else
const T plus_clamp = pset1<T>(7.90531110763549805f);
const T minus_clamp = pset1<T>(-7.90531110763549805f);
#endif
const T tiny = pset1<T>(0.0004f);
const T x = pmax(pmin(a_x, plus_clamp), minus_clamp);
const T tiny_mask = pcmp_lt(pabs(a_x), tiny);
// The monomial coefficients of the numerator polynomial (odd).
const T alpha_1 = pset1<T>(4.89352455891786e-03f);
const T alpha_3 = pset1<T>(6.37261928875436e-04f);
const T alpha_5 = pset1<T>(1.48572235717979e-05f);
const T alpha_7 = pset1<T>(5.12229709037114e-08f);
const T alpha_9 = pset1<T>(-8.60467152213735e-11f);
const T alpha_11 = pset1<T>(2.00018790482477e-13f);
const T alpha_13 = pset1<T>(-2.76076847742355e-16f);
// The monomial coefficients of the denominator polynomial (even).
const T beta_0 = pset1<T>(4.89352518554385e-03f);
const T beta_2 = pset1<T>(2.26843463243900e-03f);
const T beta_4 = pset1<T>(1.18534705686654e-04f);
const T beta_6 = pset1<T>(1.19825839466702e-06f);
// Since the polynomials are odd/even, we need x^2.
const T x2 = pmul(x, x);
// Evaluate the numerator polynomial p.
T p = pmadd(x2, alpha_13, alpha_11);
p = pmadd(x2, p, alpha_9);
p = pmadd(x2, p, alpha_7);
p = pmadd(x2, p, alpha_5);
p = pmadd(x2, p, alpha_3);
p = pmadd(x2, p, alpha_1);
p = pmul(x, p);
// Evaluate the denominator polynomial q.
T q = pmadd(x2, beta_6, beta_4);
q = pmadd(x2, q, beta_2);
q = pmadd(x2, q, beta_0);
// Divide the numerator by the denominator.
return pselect(tiny_mask, x, pdiv(p, q));
}
template<typename RealScalar>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
RealScalar positive_real_hypot(const RealScalar& x, const RealScalar& y)
{
// IEEE IEC 6059 special cases.
if ((numext::isinf)(x) || (numext::isinf)(y))
return NumTraits<RealScalar>::infinity();
if ((numext::isnan)(x) || (numext::isnan)(y))
return NumTraits<RealScalar>::quiet_NaN();
EIGEN_USING_STD(sqrt);
RealScalar p, qp;
p = numext::maxi(x,y);
if(p==RealScalar(0)) return RealScalar(0);
qp = numext::mini(y,x) / p;
return p * sqrt(RealScalar(1) + qp*qp);
}
template<typename Scalar>
struct hypot_impl
{
typedef typename NumTraits<Scalar>::Real RealScalar;
static EIGEN_DEVICE_FUNC
inline RealScalar run(const Scalar& x, const Scalar& y)
{
EIGEN_USING_STD(abs);
return positive_real_hypot<RealScalar>(abs(x), abs(y));
}
};
// Generic complex sqrt implementation that correctly handles corner cases
// according to https://en.cppreference.com/w/cpp/numeric/complex/sqrt
template<typename T>
EIGEN_DEVICE_FUNC std::complex<T> complex_sqrt(const std::complex<T>& z) {
// Computes the principal sqrt of the input.
//
// For a complex square root of the number x + i*y. We want to find real
// numbers u and v such that
// (u + i*v)^2 = x + i*y <=>
// u^2 - v^2 + i*2*u*v = x + i*v.
// By equating the real and imaginary parts we get:
// u^2 - v^2 = x
// 2*u*v = y.
//
// For x >= 0, this has the numerically stable solution
// u = sqrt(0.5 * (x + sqrt(x^2 + y^2)))
// v = y / (2 * u)
// and for x < 0,
// v = sign(y) * sqrt(0.5 * (-x + sqrt(x^2 + y^2)))
// u = y / (2 * v)
//
// Letting w = sqrt(0.5 * (|x| + |z|)),
// if x == 0: u = w, v = sign(y) * w
// if x > 0: u = w, v = y / (2 * w)
// if x < 0: u = |y| / (2 * w), v = sign(y) * w
const T x = numext::real(z);
const T y = numext::imag(z);
const T zero = T(0);
const T w = numext::sqrt(T(0.5) * (numext::abs(x) + numext::hypot(x, y)));
return
(numext::isinf)(y) ? std::complex<T>(NumTraits<T>::infinity(), y)
: x == zero ? std::complex<T>(w, y < zero ? -w : w)
: x > zero ? std::complex<T>(w, y / (2 * w))
: std::complex<T>(numext::abs(y) / (2 * w), y < zero ? -w : w );
}
// Generic complex rsqrt implementation.
template<typename T>
EIGEN_DEVICE_FUNC std::complex<T> complex_rsqrt(const std::complex<T>& z) {
// Computes the principal reciprocal sqrt of the input.
//
// For a complex reciprocal square root of the number z = x + i*y. We want to
// find real numbers u and v such that
// (u + i*v)^2 = 1 / (x + i*y) <=>
// u^2 - v^2 + i*2*u*v = x/|z|^2 - i*v/|z|^2.
// By equating the real and imaginary parts we get:
// u^2 - v^2 = x/|z|^2
// 2*u*v = y/|z|^2.
//
// For x >= 0, this has the numerically stable solution
// u = sqrt(0.5 * (x + |z|)) / |z|
// v = -y / (2 * u * |z|)
// and for x < 0,
// v = -sign(y) * sqrt(0.5 * (-x + |z|)) / |z|
// u = -y / (2 * v * |z|)
//
// Letting w = sqrt(0.5 * (|x| + |z|)),
// if x == 0: u = w / |z|, v = -sign(y) * w / |z|
// if x > 0: u = w / |z|, v = -y / (2 * w * |z|)
// if x < 0: u = |y| / (2 * w * |z|), v = -sign(y) * w / |z|
const T x = numext::real(z);
const T y = numext::imag(z);
const T zero = T(0);
const T abs_z = numext::hypot(x, y);
const T w = numext::sqrt(T(0.5) * (numext::abs(x) + abs_z));
const T woz = w / abs_z;
// Corner cases consistent with 1/sqrt(z) on gcc/clang.
return
abs_z == zero ? std::complex<T>(NumTraits<T>::infinity(), NumTraits<T>::quiet_NaN())
: ((numext::isinf)(x) || (numext::isinf)(y)) ? std::complex<T>(zero, zero)
: x == zero ? std::complex<T>(woz, y < zero ? woz : -woz)
: x > zero ? std::complex<T>(woz, -y / (2 * w * abs_z))
: std::complex<T>(numext::abs(y) / (2 * w * abs_z), y < zero ? woz : -woz );
}
template<typename T>
EIGEN_DEVICE_FUNC std::complex<T> complex_log(const std::complex<T>& z) {
// Computes complex log.
T a = numext::abs(z);
EIGEN_USING_STD(atan2);
T b = atan2(z.imag(), z.real());
return std::complex<T>(numext::log(a), b);
}
} // end namespace internal
} // end namespace Eigen
#endif // EIGEN_MATHFUNCTIONSIMPL_H
|