MathFunctions.h

// Copyright (c) 2020 PaddlePaddle Authors. All Rights Reserved.
//
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//
//     http://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.

// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2006-2010 Benoit Jacob <jacob.benoit.1@gmail.com>
//
// 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_MATHFUNCTIONS_H
#define EIGEN_MATHFUNCTIONS_H

// source: http://www.geom.uiuc.edu/~huberty/math5337/groupe/digits.html
// TODO this should better be moved to NumTraits
#define EIGEN_PI \
  3.141592653589793238462643383279502884197169399375105820974944592307816406L

namespace Eigen {

// On WINCE, std::abs is defined for int only, so let's defined our own
// overloads:
// This issue has been confirmed with MSVC 2008 only, but the issue might exist
// for more recent versions too.
#if EIGEN_OS_WINCE && EIGEN_COMP_MSVC && EIGEN_COMP_MSVC <= 1500
long abs(long x) { return (labs(x)); }
double abs(double x) { return (fabs(x)); }
float abs(float x) { return (fabsf(x)); }
long double abs(long double x) { return (fabsl(x)); }
#endif

namespace internal {

/** \internal \class global_math_functions_filtering_base
  *
  * What it does:
  * Defines a typedef 'type' as follows:
  * - if type T has a member typedef
 * Eigen_BaseClassForSpecializationOfGlobalMathFuncImpl, then
  *   global_math_functions_filtering_base<T>::type is a typedef for it.
  * - otherwise, global_math_functions_filtering_base<T>::type is a typedef for
 * T.
  *
  * How it's used:
  * To allow to defined the global math functions (like sin...) in certain
 * cases, like the Array expressions.
  * When you do sin(array1+array2), the object array1+array2 has a complicated
 * expression type, all what you want to know
  * is that it inherits ArrayBase. So we implement a partial specialization of
 * sin_impl for ArrayBase<Derived>.
  * So we must make sure to use sin_impl<ArrayBase<Derived> > and not
 * sin_impl<Derived>, otherwise our partial specialization
  * won't be used. How does sin know that? That's exactly what
 * global_math_functions_filtering_base tells it.
  *
  * How it's implemented:
  * SFINAE in the style of enable_if. Highly susceptible of breaking compilers.
 * With GCC, it sure does work, but if you replace
  * the typename dummy by an integer template parameter, it doesn't work
 * anymore!
  */

template <typename T, typename dummy = void>
struct global_math_functions_filtering_base {
  typedef T type;
};

template <typename T>
struct always_void {
  typedef void type;
};

template <typename T>
struct global_math_functions_filtering_base<
    T,
    typename always_void<
        typename T::Eigen_BaseClassForSpecializationOfGlobalMathFuncImpl>::
        type> {
  typedef typename T::Eigen_BaseClassForSpecializationOfGlobalMathFuncImpl type;
};

#define EIGEN_MATHFUNC_IMPL(func, scalar)                             \
  Eigen::internal::func##_impl<                                       \
      typename Eigen::internal::global_math_functions_filtering_base< \
          scalar>::type>
#define EIGEN_MATHFUNC_RETVAL(func, scalar)                           \
  typename Eigen::internal::func##_retval<                            \
      typename Eigen::internal::global_math_functions_filtering_base< \
          scalar>::type>::type

/****************************************************************************
* Implementation of real                                                 *
****************************************************************************/

template <typename Scalar, bool IsComplex = NumTraits<Scalar>::IsComplex>
struct real_default_impl {
  typedef typename NumTraits<Scalar>::Real RealScalar;
  EIGEN_DEVICE_FUNC
  static inline RealScalar run(const Scalar& x) { return x; }
};

template <typename Scalar>
struct real_default_impl<Scalar, true> {
  typedef typename NumTraits<Scalar>::Real RealScalar;
  EIGEN_DEVICE_FUNC
  static inline RealScalar run(const Scalar& x) {
    using std::real;
    return real(x);
  }
};

template <typename Scalar>
struct real_impl : real_default_impl<Scalar> {};

#if defined(EIGEN_GPU_COMPILE_PHASE)
template <typename T>
struct real_impl<std::complex<T>> {
  typedef T RealScalar;
  EIGEN_DEVICE_FUNC
  static inline T run(const std::complex<T>& x) { return x.real(); }
};
#endif

template <typename Scalar>
struct real_retval {
  typedef typename NumTraits<Scalar>::Real type;
};

/****************************************************************************
* Implementation of imag                                                 *
****************************************************************************/

template <typename Scalar, bool IsComplex = NumTraits<Scalar>::IsComplex>
struct imag_default_impl {
  typedef typename NumTraits<Scalar>::Real RealScalar;
  EIGEN_DEVICE_FUNC
  static inline RealScalar run(const Scalar&) { return RealScalar(0); }
};

template <typename Scalar>
struct imag_default_impl<Scalar, true> {
  typedef typename NumTraits<Scalar>::Real RealScalar;
  EIGEN_DEVICE_FUNC
  static inline RealScalar run(const Scalar& x) {
    using std::imag;
    return imag(x);
  }
};

template <typename Scalar>
struct imag_impl : imag_default_impl<Scalar> {};

#if defined(EIGEN_GPU_COMPILE_PHASE)
template <typename T>
struct imag_impl<std::complex<T>> {
  typedef T RealScalar;
  EIGEN_DEVICE_FUNC
  static inline T run(const std::complex<T>& x) { return x.imag(); }
};
#endif

template <typename Scalar>
struct imag_retval {
  typedef typename NumTraits<Scalar>::Real type;
};

/****************************************************************************
* Implementation of real_ref                                             *
****************************************************************************/

template <typename Scalar>
struct real_ref_impl {
  typedef typename NumTraits<Scalar>::Real RealScalar;
  EIGEN_DEVICE_FUNC
  static inline RealScalar& run(Scalar& x) {
    return reinterpret_cast<RealScalar*>(&x)[0];
  }
  EIGEN_DEVICE_FUNC
  static inline const RealScalar& run(const Scalar& x) {
    return reinterpret_cast<const RealScalar*>(&x)[0];
  }
};

template <typename Scalar>
struct real_ref_retval {
  typedef typename NumTraits<Scalar>::Real& type;
};

/****************************************************************************
* Implementation of imag_ref                                             *
****************************************************************************/

template <typename Scalar, bool IsComplex>
struct imag_ref_default_impl {
  typedef typename NumTraits<Scalar>::Real RealScalar;
  EIGEN_DEVICE_FUNC
  static inline RealScalar& run(Scalar& x) {
    return reinterpret_cast<RealScalar*>(&x)[1];
  }
  EIGEN_DEVICE_FUNC
  static inline const RealScalar& run(const Scalar& x) {
    return reinterpret_cast<RealScalar*>(&x)[1];
  }
};

template <typename Scalar>
struct imag_ref_default_impl<Scalar, false> {
  EIGEN_DEVICE_FUNC
  static inline Scalar run(Scalar&) { return Scalar(0); }
  EIGEN_DEVICE_FUNC
  static inline const Scalar run(const Scalar&) { return Scalar(0); }
};

template <typename Scalar>
struct imag_ref_impl
    : imag_ref_default_impl<Scalar, NumTraits<Scalar>::IsComplex> {};

template <typename Scalar>
struct imag_ref_retval {
  typedef typename NumTraits<Scalar>::Real& type;
};

/****************************************************************************
* Implementation of conj                                                 *
****************************************************************************/

template <typename Scalar, bool IsComplex = NumTraits<Scalar>::IsComplex>
struct conj_default_impl {
  EIGEN_DEVICE_FUNC
  static inline Scalar run(const Scalar& x) { return x; }
};

template <typename Scalar>
struct conj_default_impl<Scalar, true> {
  EIGEN_DEVICE_FUNC
  static inline Scalar run(const Scalar& x) {
    using std::conj;
    return conj(x);
  }
};

template <typename Scalar>
struct conj_impl : conj_default_impl<Scalar> {};

#if defined(EIGEN_GPU_COMPILE_PHASE)
template <typename T>
struct conj_impl<std::complex<T>> {
  EIGEN_DEVICE_FUNC
  static inline std::complex<T> run(const std::complex<T>& x) {
    return std::complex<T>(x.real(), -x.imag());
  }
};
#endif

template <typename Scalar>
struct conj_retval {
  typedef Scalar type;
};

/****************************************************************************
* Implementation of abs2                                                 *
****************************************************************************/

template <typename Scalar, bool IsComplex>
struct abs2_impl_default {
  typedef typename NumTraits<Scalar>::Real RealScalar;
  EIGEN_DEVICE_FUNC
  static inline RealScalar run(const Scalar& x) { return x * x; }
};

template <typename Scalar>
struct abs2_impl_default<Scalar, true>  // IsComplex
{
  typedef typename NumTraits<Scalar>::Real RealScalar;
  EIGEN_DEVICE_FUNC
  static inline RealScalar run(const Scalar& x) {
    return x.real() * x.real() + x.imag() * x.imag();
  }
};

template <typename Scalar>
struct abs2_impl {
  typedef typename NumTraits<Scalar>::Real RealScalar;
  EIGEN_DEVICE_FUNC
  static inline RealScalar run(const Scalar& x) {
    return abs2_impl_default<Scalar, NumTraits<Scalar>::IsComplex>::run(x);
  }
};

template <typename Scalar>
struct abs2_retval {
  typedef typename NumTraits<Scalar>::Real type;
};

/****************************************************************************
* Implementation of norm1                                                *
****************************************************************************/

template <typename Scalar, bool IsComplex>
struct norm1_default_impl;

template <typename Scalar>
struct norm1_default_impl<Scalar, true> {
  typedef typename NumTraits<Scalar>::Real RealScalar;
  EIGEN_DEVICE_FUNC
  static inline RealScalar run(const Scalar& x) {
    EIGEN_USING_STD_MATH(abs);
    return abs(x.real()) + abs(x.imag());
  }
};

template <typename Scalar>
struct norm1_default_impl<Scalar, false> {
  EIGEN_DEVICE_FUNC
  static inline Scalar run(const Scalar& x) {
    EIGEN_USING_STD_MATH(abs);
    return abs(x);
  }
};

template <typename Scalar>
struct norm1_impl : norm1_default_impl<Scalar, NumTraits<Scalar>::IsComplex> {};

template <typename Scalar>
struct norm1_retval {
  typedef typename NumTraits<Scalar>::Real type;
};

/****************************************************************************
* Implementation of hypot                                                *
****************************************************************************/

template <typename Scalar>
struct hypot_impl;

template <typename Scalar>
struct hypot_retval {
  typedef typename NumTraits<Scalar>::Real type;
};

/****************************************************************************
* Implementation of cast                                                 *
****************************************************************************/

template <typename OldType, typename NewType>
struct cast_impl {
  EIGEN_DEVICE_FUNC
  static inline NewType run(const OldType& x) {
    return static_cast<NewType>(x);
  }
};

// here, for once, we're plainly returning NewType: we don't want cast to do
// weird things.

template <typename OldType, typename NewType>
EIGEN_DEVICE_FUNC inline NewType cast(const OldType& x) {
  return cast_impl<OldType, NewType>::run(x);
}

/****************************************************************************
* Implementation of round                                                   *
****************************************************************************/

#if EIGEN_HAS_CXX11_MATH
template <typename Scalar>
struct round_impl {
  EIGEN_DEVICE_FUNC
  static inline Scalar run(const Scalar& x) {
    EIGEN_STATIC_ASSERT((!NumTraits<Scalar>::IsComplex),
                        NUMERIC_TYPE_MUST_BE_REAL)
    EIGEN_USING_STD_MATH(round);
    return round(x);
  }
};
#else
template <typename Scalar>
struct round_impl {
  EIGEN_DEVICE_FUNC
  static inline Scalar run(const Scalar& x) {
    EIGEN_STATIC_ASSERT((!NumTraits<Scalar>::IsComplex),
                        NUMERIC_TYPE_MUST_BE_REAL)
    EIGEN_USING_STD_MATH(floor);
    EIGEN_USING_STD_MATH(ceil);
    return (x > Scalar(0)) ? floor(x + Scalar(0.5)) : ceil(x - Scalar(0.5));
  }
};
#endif

template <typename Scalar>
struct round_retval {
  typedef Scalar type;
};

/****************************************************************************
* Implementation of rint                                                    *
****************************************************************************/

template <typename Scalar>
struct rint_impl {
  EIGEN_DEVICE_FUNC
  static inline Scalar run(const Scalar& x) {
    EIGEN_STATIC_ASSERT((!NumTraits<Scalar>::IsComplex),
                        NUMERIC_TYPE_MUST_BE_REAL)
#if EIGEN_HAS_CXX11_MATH
    EIGEN_USING_STD_MATH(rint);
#endif
    return rint(x);
  }
};

#if !EIGEN_HAS_CXX11_MATH
template <>
struct rint_impl<double> {
  EIGEN_DEVICE_FUNC
  static inline double run(const double& x) { return ::rint(x); }
};
template <>
struct rint_impl<float> {
  EIGEN_DEVICE_FUNC
  static inline float run(const float& x) { return ::rintf(x); }
};
#endif

template <typename Scalar>
struct rint_retval {
  typedef Scalar type;
};

/****************************************************************************
* Implementation of arg                                                     *
****************************************************************************/

#if EIGEN_HAS_CXX11_MATH
template <typename Scalar>
struct arg_impl {
  EIGEN_DEVICE_FUNC
  static inline Scalar run(const Scalar& x) {
#if defined(EIGEN_HIP_DEVICE_COMPILE)
    // HIP does not seem to have a native device side implementation for the
    // math routine "arg"
    using std::arg;
#else
    EIGEN_USING_STD_MATH(arg);
#endif
    return arg(x);
  }
};
#else
template <typename Scalar, bool IsComplex = NumTraits<Scalar>::IsComplex>
struct arg_default_impl {
  typedef typename NumTraits<Scalar>::Real RealScalar;
  EIGEN_DEVICE_FUNC
  static inline RealScalar run(const Scalar& x) {
    return (x < Scalar(0)) ? Scalar(EIGEN_PI) : Scalar(0);
  }
};

template <typename Scalar>
struct arg_default_impl<Scalar, true> {
  typedef typename NumTraits<Scalar>::Real RealScalar;
  EIGEN_DEVICE_FUNC
  static inline RealScalar run(const Scalar& x) {
    EIGEN_USING_STD_MATH(arg);
    return arg(x);
  }
};

template <typename Scalar>
struct arg_impl : arg_default_impl<Scalar> {};
#endif

template <typename Scalar>
struct arg_retval {
  typedef typename NumTraits<Scalar>::Real type;
};

/****************************************************************************
* Implementation of expm1                                                   *
****************************************************************************/

// This implementation is based on GSL Math's expm1.
namespace std_fallback {
// fallback expm1 implementation in case there is no expm1(Scalar) function in
// namespace of Scalar,
// or that there is no suitable std::expm1 function available. Implementation
// attributed to Kahan. See: http://www.plunk.org/~hatch/rightway.php.
template <typename Scalar>
EIGEN_DEVICE_FUNC inline Scalar expm1(const Scalar& x) {
  EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar)
  typedef typename NumTraits<Scalar>::Real RealScalar;

  EIGEN_USING_STD_MATH(exp);
  Scalar u = exp(x);
  if (numext::equal_strict(u, Scalar(1))) {
    return x;
  }
  Scalar um1 = u - RealScalar(1);
  if (numext::equal_strict(um1, Scalar(-1))) {
    return RealScalar(-1);
  }

  EIGEN_USING_STD_MATH(log);
  Scalar logu = log(u);
  return numext::equal_strict(u, logu) ? u : (u - RealScalar(1)) * x / logu;
}
}

template <typename Scalar>
struct expm1_impl {
  EIGEN_DEVICE_FUNC static inline Scalar run(const Scalar& x) {
    EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar)
#if EIGEN_HAS_CXX11_MATH
    using std::expm1;
#else
    using std_fallback::expm1;
#endif
    return expm1(x);
  }
};

// Specialization for complex types that are not supported by std::expm1.
template <typename RealScalar>
struct expm1_impl<std::complex<RealScalar>> {
  EIGEN_DEVICE_FUNC static inline std::complex<RealScalar> run(
      const std::complex<RealScalar>& x) {
    EIGEN_STATIC_ASSERT_NON_INTEGER(RealScalar)
    RealScalar xr = x.real();
    RealScalar xi = x.imag();
    // expm1(z) = exp(z) - 1
    //          = exp(x +  i * y) - 1
    //          = exp(x) * (cos(y) + i * sin(y)) - 1
    //          = exp(x) * cos(y) - 1 + i * exp(x) * sin(y)
    // Imag(expm1(z)) = exp(x) * sin(y)
    // Real(expm1(z)) = exp(x) * cos(y) - 1
    //          = exp(x) * cos(y) - 1.
    //          = expm1(x) + exp(x) * (cos(y) - 1)
    //          = expm1(x) + exp(x) * (2 * sin(y / 2) ** 2)

    // TODO better use numext::expm1 and numext::sin (but that would require
    // forward declarations or moving this specialization down).
    RealScalar erm1 = expm1_impl<RealScalar>::run(xr);
    RealScalar er = erm1 + RealScalar(1.);
    EIGEN_USING_STD_MATH(sin);
    RealScalar sin2 = sin(xi / RealScalar(2.));
    sin2 = sin2 * sin2;
    RealScalar s = sin(xi);
    RealScalar real_part = erm1 - RealScalar(2.) * er * sin2;
    return std::complex<RealScalar>(real_part, er * s);
  }
};

template <typename Scalar>
struct expm1_retval {
  typedef Scalar type;
};

/****************************************************************************
* Implementation of log1p                                                   *
****************************************************************************/

namespace std_fallback {
// fallback log1p implementation in case there is no log1p(Scalar) function in
// namespace of Scalar,
// or that there is no suitable std::log1p function available
template <typename Scalar>
EIGEN_DEVICE_FUNC inline Scalar log1p(const Scalar& x) {
  EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar)
  typedef typename NumTraits<Scalar>::Real RealScalar;
  EIGEN_USING_STD_MATH(log);
  Scalar x1p = RealScalar(1) + x;
  Scalar log_1p = log(x1p);
  const bool is_small = numext::equal_strict(x1p, Scalar(1));
  const bool is_inf = numext::equal_strict(x1p, log_1p);
  return (is_small || is_inf) ? x : x * (log_1p / (x1p - RealScalar(1)));
}
}

template <typename Scalar>
struct log1p_impl {
  EIGEN_DEVICE_FUNC static inline Scalar run(const Scalar& x) {
    EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar)
#if EIGEN_HAS_CXX11_MATH
    using std::log1p;
#else
    using std_fallback::log1p;
#endif
    return log1p(x);
  }
};

// Specialization for complex types that are not supported by std::log1p.
template <typename RealScalar>
struct log1p_impl<std::complex<RealScalar>> {
  EIGEN_DEVICE_FUNC static inline std::complex<RealScalar> run(
      const std::complex<RealScalar>& x) {
    EIGEN_STATIC_ASSERT_NON_INTEGER(RealScalar)
    return std_fallback::log1p(x);
  }
};

template <typename Scalar>
struct log1p_retval {
  typedef Scalar type;
};

/****************************************************************************
* Implementation of pow                                                  *
****************************************************************************/

template <typename ScalarX,
          typename ScalarY,
          bool IsInteger =
              NumTraits<ScalarX>::IsInteger&& NumTraits<ScalarY>::IsInteger>
struct pow_impl {
  // typedef Scalar retval;
  typedef typename ScalarBinaryOpTraits<
      ScalarX,
      ScalarY,
      internal::scalar_pow_op<ScalarX, ScalarY>>::ReturnType result_type;
  static EIGEN_DEVICE_FUNC inline result_type run(const ScalarX& x,
                                                  const ScalarY& y) {
    EIGEN_USING_STD_MATH(pow);
    return pow(x, y);
  }
};

template <typename ScalarX, typename ScalarY>
struct pow_impl<ScalarX, ScalarY, true> {
  typedef ScalarX result_type;
  static EIGEN_DEVICE_FUNC inline ScalarX run(ScalarX x, ScalarY y) {
    ScalarX res(1);
    eigen_assert(!NumTraits<ScalarY>::IsSigned || y >= 0);
    if (y & 1) res *= x;
    y >>= 1;
    while (y) {
      x *= x;
      if (y & 1) res *= x;
      y >>= 1;
    }
    return res;
  }
};

/****************************************************************************
* Implementation of random                                               *
****************************************************************************/

template <typename Scalar, bool IsComplex, bool IsInteger>
struct random_default_impl {};

template <typename Scalar>
struct random_impl : random_default_impl<Scalar,
                                         NumTraits<Scalar>::IsComplex,
                                         NumTraits<Scalar>::IsInteger> {};

template <typename Scalar>
struct random_retval {
  typedef Scalar type;
};

template <typename Scalar>
inline EIGEN_MATHFUNC_RETVAL(random, Scalar)
    random(const Scalar& x, const Scalar& y);
template <typename Scalar>
inline EIGEN_MATHFUNC_RETVAL(random, Scalar) random();

template <typename Scalar>
struct random_default_impl<Scalar, false, false> {
  static inline Scalar run(const Scalar& x, const Scalar& y) {
    return x + (y - x) * Scalar(std::rand()) / Scalar(RAND_MAX);
  }
  static inline Scalar run() {
    return run(Scalar(NumTraits<Scalar>::IsSigned ? -1 : 0), Scalar(1));
  }
};

enum {
  meta_floor_log2_terminate,
  meta_floor_log2_move_up,
  meta_floor_log2_move_down,
  meta_floor_log2_bogus
};

template <unsigned int n, int lower, int upper>
struct meta_floor_log2_selector {
  enum {
    middle = (lower + upper) / 2,
    value = (upper <= lower + 1)
                ? int(meta_floor_log2_terminate)
                : (n < (1 << middle)) ? int(meta_floor_log2_move_down)
                                      : (n == 0) ? int(meta_floor_log2_bogus)
                                                 : int(meta_floor_log2_move_up)
  };
};

template <unsigned int n,
          int lower = 0,
          int upper = sizeof(unsigned int) * CHAR_BIT - 1,
          int selector = meta_floor_log2_selector<n, lower, upper>::value>
struct meta_floor_log2 {};

template <unsigned int n, int lower, int upper>
struct meta_floor_log2<n, lower, upper, meta_floor_log2_move_down> {
  enum {
    value = meta_floor_log2<
        n,
        lower,
        meta_floor_log2_selector<n, lower, upper>::middle>::value
  };
};

template <unsigned int n, int lower, int upper>
struct meta_floor_log2<n, lower, upper, meta_floor_log2_move_up> {
  enum {
    value = meta_floor_log2<n,
                            meta_floor_log2_selector<n, lower, upper>::middle,
                            upper>::value
  };
};

template <unsigned int n, int lower, int upper>
struct meta_floor_log2<n, lower, upper, meta_floor_log2_terminate> {
  enum {
    value = (n >= ((unsigned int)(1) << (lower + 1))) ? lower + 1 : lower
  };
};

template <unsigned int n, int lower, int upper>
struct meta_floor_log2<n, lower, upper, meta_floor_log2_bogus> {
  // no value, error at compile time
};

template <typename Scalar>
struct random_default_impl<Scalar, false, true> {
  static inline Scalar run(const Scalar& x, const Scalar& y) {
    if (y <= x) return x;
    // ScalarU is the unsigned counterpart of Scalar, possibly Scalar itself.
    typedef typename make_unsigned<Scalar>::type ScalarU;
    // ScalarX is the widest of ScalarU and unsigned int.
    // We'll deal only with ScalarX and unsigned int below thus avoiding signed
    // types and arithmetic and signed overflows (which are undefined behavior).
    typedef typename conditional<(ScalarU(-1) > unsigned(-1)),
                                 ScalarU,
                                 unsigned>::type ScalarX;
    // The following difference doesn't overflow, provided our integer types are
    // two's
    // complement and have the same number of padding bits in signed and
    // unsigned variants.
    // This is the case in most modern implementations of C++.
    ScalarX range = ScalarX(y) - ScalarX(x);
    ScalarX offset = 0;
    ScalarX divisor = 1;
    ScalarX multiplier = 1;
    const unsigned rand_max = RAND_MAX;
    if (range <= rand_max)
      divisor = (rand_max + 1) / (range + 1);
    else
      multiplier = 1 + range / (rand_max + 1);
    // Rejection sampling.
    do {
      offset = (unsigned(std::rand()) * multiplier) / divisor;
    } while (offset > range);
    return Scalar(ScalarX(x) + offset);
  }

  static inline Scalar run() {
#ifdef EIGEN_MAKING_DOCS
    return run(Scalar(NumTraits<Scalar>::IsSigned ? -10 : 0), Scalar(10));
#else
    enum {
        rand_bits = meta_floor_log2<(unsigned int)(RAND_MAX) + 1>::value,
        scalar_bits = sizeof(Scalar) * CHAR_BIT,
        shift = EIGEN_PLAIN_ENUM_MAX(0, int(rand_bits) - int(scalar_bits)),
        offset = NumTraits<Scalar>::IsSigned
                     ? (1 << (EIGEN_PLAIN_ENUM_MIN(rand_bits, scalar_bits) - 1))
                     : 0};
    return Scalar((std::rand() >> shift) - offset);
#endif
  }
};

template <typename Scalar>
struct random_default_impl<Scalar, true, false> {
  static inline Scalar run(const Scalar& x, const Scalar& y) {
    return Scalar(random(x.real(), y.real()), random(x.imag(), y.imag()));
  }
  static inline Scalar run() {
    typedef typename NumTraits<Scalar>::Real RealScalar;
    return Scalar(random<RealScalar>(), random<RealScalar>());
  }
};

template <typename Scalar>
inline EIGEN_MATHFUNC_RETVAL(random, Scalar)
    random(const Scalar& x, const Scalar& y) {
  return EIGEN_MATHFUNC_IMPL(random, Scalar)::run(x, y);
}

template <typename Scalar>
inline EIGEN_MATHFUNC_RETVAL(random, Scalar) random() {
  return EIGEN_MATHFUNC_IMPL(random, Scalar)::run();
}

// Implementation of is* functions

// std::is* do not work with fast-math and gcc, std::is* are available on MSVC
// 2013 and newer, as well as in clang.
#if (EIGEN_HAS_CXX11_MATH &&                               \
     !(EIGEN_COMP_GNUC_STRICT && __FINITE_MATH_ONLY__)) || \
    (EIGEN_COMP_MSVC >= 1800) || (EIGEN_COMP_CLANG)
#define EIGEN_USE_STD_FPCLASSIFY 1
#else
#define EIGEN_USE_STD_FPCLASSIFY 0
#endif

template <typename T>
EIGEN_DEVICE_FUNC
    typename internal::enable_if<internal::is_integral<T>::value, bool>::type
    isnan_impl(const T&) {
  return false;
}

template <typename T>
EIGEN_DEVICE_FUNC
    typename internal::enable_if<internal::is_integral<T>::value, bool>::type
    isinf_impl(const T&) {
  return false;
}

template <typename T>
EIGEN_DEVICE_FUNC
    typename internal::enable_if<internal::is_integral<T>::value, bool>::type
    isfinite_impl(const T&) {
  return true;
}

template <typename T>
EIGEN_DEVICE_FUNC
    typename internal::enable_if<(!internal::is_integral<T>::value) &&
                                     (!NumTraits<T>::IsComplex),
                                 bool>::type
    isfinite_impl(const T& x) {
#if defined(EIGEN_GPU_COMPILE_PHASE)
  return (::isfinite)(x);
#elif EIGEN_USE_STD_FPCLASSIFY
  using std::isfinite;
  return isfinite EIGEN_NOT_A_MACRO(x);
#else
  return x <= NumTraits<T>::highest() && x >= NumTraits<T>::lowest();
#endif
}

template <typename T>
EIGEN_DEVICE_FUNC
    typename internal::enable_if<(!internal::is_integral<T>::value) &&
                                     (!NumTraits<T>::IsComplex),
                                 bool>::type
    isinf_impl(const T& x) {
#if defined(EIGEN_GPU_COMPILE_PHASE)
  return (::isinf)(x);
#elif EIGEN_USE_STD_FPCLASSIFY
  using std::isinf;
  return isinf EIGEN_NOT_A_MACRO(x);
#else
  return x > NumTraits<T>::highest() || x < NumTraits<T>::lowest();
#endif
}

template <typename T>
EIGEN_DEVICE_FUNC
    typename internal::enable_if<(!internal::is_integral<T>::value) &&
                                     (!NumTraits<T>::IsComplex),
                                 bool>::type
    isnan_impl(const T& x) {
#if defined(EIGEN_GPU_COMPILE_PHASE)
  return (::isnan)(x);
#elif EIGEN_USE_STD_FPCLASSIFY
  using std::isnan;
  return isnan EIGEN_NOT_A_MACRO(x);
#else
  return x != x;
#endif
}

#if (!EIGEN_USE_STD_FPCLASSIFY)

#if EIGEN_COMP_MSVC

template <typename T>
EIGEN_DEVICE_FUNC bool isinf_msvc_helper(T x) {
  return _fpclass(x) == _FPCLASS_NINF || _fpclass(x) == _FPCLASS_PINF;
}

// MSVC defines a _isnan builtin function, but for double only
EIGEN_DEVICE_FUNC inline bool isnan_impl(const long double& x) {
  return _isnan(x) != 0;
}
EIGEN_DEVICE_FUNC inline bool isnan_impl(const double& x) {
  return _isnan(x) != 0;
}
EIGEN_DEVICE_FUNC inline bool isnan_impl(const float& x) {
  return _isnan(x) != 0;
}

EIGEN_DEVICE_FUNC inline bool isinf_impl(const long double& x) {
  return isinf_msvc_helper(x);
}
EIGEN_DEVICE_FUNC inline bool isinf_impl(const double& x) {
  return isinf_msvc_helper(x);
}
EIGEN_DEVICE_FUNC inline bool isinf_impl(const float& x) {
  return isinf_msvc_helper(x);
}

#elif (defined __FINITE_MATH_ONLY__ && __FINITE_MATH_ONLY__ && EIGEN_COMP_GNUC)

#if EIGEN_GNUC_AT_LEAST(5, 0)
#define EIGEN_TMP_NOOPT_ATTRIB \
  EIGEN_DEVICE_FUNC inline __attribute__((optimize("no-finite-math-only")))
#else
// NOTE the inline qualifier and noinline attribute are both needed: the former
// is to avoid linking issue (duplicate symbol),
//      while the second prevent too aggressive optimizations in fast-math mode:
#define EIGEN_TMP_NOOPT_ATTRIB \
  EIGEN_DEVICE_FUNC inline     \
      __attribute__((noinline, optimize("no-finite-math-only")))
#endif

template <>
EIGEN_TMP_NOOPT_ATTRIB bool isnan_impl(const long double& x) {
  return __builtin_isnan(x);
}
template <>
EIGEN_TMP_NOOPT_ATTRIB bool isnan_impl(const double& x) {
  return __builtin_isnan(x);
}
template <>
EIGEN_TMP_NOOPT_ATTRIB bool isnan_impl(const float& x) {
  return __builtin_isnan(x);
}
template <>
EIGEN_TMP_NOOPT_ATTRIB bool isinf_impl(const double& x) {
  return __builtin_isinf(x);
}
template <>
EIGEN_TMP_NOOPT_ATTRIB bool isinf_impl(const float& x) {
  return __builtin_isinf(x);
}
template <>
EIGEN_TMP_NOOPT_ATTRIB bool isinf_impl(const long double& x) {
  return __builtin_isinf(x);
}

#undef EIGEN_TMP_NOOPT_ATTRIB

#endif

#endif

// The following overload are defined at the end of this file
template <typename T>
EIGEN_DEVICE_FUNC bool isfinite_impl(const std::complex<T>& x);
template <typename T>
EIGEN_DEVICE_FUNC bool isnan_impl(const std::complex<T>& x);
template <typename T>
EIGEN_DEVICE_FUNC bool isinf_impl(const std::complex<T>& x);

template <typename T>
T generic_fast_tanh_float(const T& a_x);
}  // end namespace internal

/****************************************************************************
* Generic math functions                                                    *
****************************************************************************/

namespace numext {

#if (!defined(EIGEN_GPUCC))
template <typename T>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE T mini(const T& x, const T& y) {
  EIGEN_USING_STD_MATH(min);
  return min EIGEN_NOT_A_MACRO(x, y);
}

template <typename T>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE T maxi(const T& x, const T& y) {
  EIGEN_USING_STD_MATH(max);
  return max EIGEN_NOT_A_MACRO(x, y);
}
#else
template <typename T>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE T mini(const T& x, const T& y) {
  return y < x ? y : x;
}
template <>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE float mini(const float& x,
                                                 const float& y) {
  return fminf(x, y);
}
template <>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE double mini(const double& x,
                                                  const double& y) {
  return fmin(x, y);
}
template <>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE long double mini(const long double& x,
                                                       const long double& y) {
#if defined(EIGEN_HIPCC)
  // no "fminl" on HIP yet
  return (x < y) ? x : y;
#else
  return fminl(x, y);
#endif
}

template <typename T>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE T maxi(const T& x, const T& y) {
  return x < y ? y : x;
}
template <>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE float maxi(const float& x,
                                                 const float& y) {
  return fmaxf(x, y);
}
template <>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE double maxi(const double& x,
                                                  const double& y) {
  return fmax(x, y);
}
template <>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE long double maxi(const long double& x,
                                                       const long double& y) {
#if defined(EIGEN_HIPCC)
  // no "fmaxl" on HIP yet
  return (x > y) ? x : y;
#else
  return fmaxl(x, y);
#endif
}
#endif

#if defined(SYCL_DEVICE_ONLY)

#define SYCL_SPECIALIZE_SIGNED_INTEGER_TYPES_BINARY(NAME, FUNC) \
  SYCL_SPECIALIZE_BINARY_FUNC(NAME, FUNC, cl::sycl::cl_char)    \
  SYCL_SPECIALIZE_BINARY_FUNC(NAME, FUNC, cl::sycl::cl_short)   \
  SYCL_SPECIALIZE_BINARY_FUNC(NAME, FUNC, cl::sycl::cl_int)     \
  SYCL_SPECIALIZE_BINARY_FUNC(NAME, FUNC, cl::sycl::cl_long)
#define SYCL_SPECIALIZE_SIGNED_INTEGER_TYPES_UNARY(NAME, FUNC) \
  SYCL_SPECIALIZE_UNARY_FUNC(NAME, FUNC, cl::sycl::cl_char)    \
  SYCL_SPECIALIZE_UNARY_FUNC(NAME, FUNC, cl::sycl::cl_short)   \
  SYCL_SPECIALIZE_UNARY_FUNC(NAME, FUNC, cl::sycl::cl_int)     \
  SYCL_SPECIALIZE_UNARY_FUNC(NAME, FUNC, cl::sycl::cl_long)
#define SYCL_SPECIALIZE_UNSIGNED_INTEGER_TYPES_BINARY(NAME, FUNC) \
  SYCL_SPECIALIZE_BINARY_FUNC(NAME, FUNC, cl::sycl::cl_uchar)     \
  SYCL_SPECIALIZE_BINARY_FUNC(NAME, FUNC, cl::sycl::cl_ushort)    \
  SYCL_SPECIALIZE_BINARY_FUNC(NAME, FUNC, cl::sycl::cl_uint)      \
  SYCL_SPECIALIZE_BINARY_FUNC(NAME, FUNC, cl::sycl::cl_ulong)
#define SYCL_SPECIALIZE_UNSIGNED_INTEGER_TYPES_UNARY(NAME, FUNC) \
  SYCL_SPECIALIZE_UNARY_FUNC(NAME, FUNC, cl::sycl::cl_uchar)     \
  SYCL_SPECIALIZE_UNARY_FUNC(NAME, FUNC, cl::sycl::cl_ushort)    \
  SYCL_SPECIALIZE_UNARY_FUNC(NAME, FUNC, cl::sycl::cl_uint)      \
  SYCL_SPECIALIZE_UNARY_FUNC(NAME, FUNC, cl::sycl::cl_ulong)
#define SYCL_SPECIALIZE_INTEGER_TYPES_BINARY(NAME, FUNC)  \
  SYCL_SPECIALIZE_SIGNED_INTEGER_TYPES_BINARY(NAME, FUNC) \
  SYCL_SPECIALIZE_UNSIGNED_INTEGER_TYPES_BINARY(NAME, FUNC)
#define SYCL_SPECIALIZE_INTEGER_TYPES_UNARY(NAME, FUNC)  \
  SYCL_SPECIALIZE_SIGNED_INTEGER_TYPES_UNARY(NAME, FUNC) \
  SYCL_SPECIALIZE_UNSIGNED_INTEGER_TYPES_UNARY(NAME, FUNC)
#define SYCL_SPECIALIZE_FLOATING_TYPES_BINARY(NAME, FUNC)     \
  SYCL_SPECIALIZE_BINARY_FUNC(NAME, FUNC, cl::sycl::cl_float) \
  SYCL_SPECIALIZE_BINARY_FUNC(NAME, FUNC, cl::sycl::cl_double)
#define SYCL_SPECIALIZE_FLOATING_TYPES_UNARY(NAME, FUNC)     \
  SYCL_SPECIALIZE_UNARY_FUNC(NAME, FUNC, cl::sycl::cl_float) \
  SYCL_SPECIALIZE_UNARY_FUNC(NAME, FUNC, cl::sycl::cl_double)
#define SYCL_SPECIALIZE_FLOATING_TYPES_UNARY_FUNC_RET_TYPE(                \
    NAME, FUNC, RET_TYPE)                                                  \
  SYCL_SPECIALIZE_GEN_UNARY_FUNC(NAME, FUNC, RET_TYPE, cl::sycl::cl_float) \
  SYCL_SPECIALIZE_GEN_UNARY_FUNC(NAME, FUNC, RET_TYPE, cl::sycl::cl_double)

#define SYCL_SPECIALIZE_GEN_UNARY_FUNC(NAME, FUNC, RET_TYPE, ARG_TYPE)     \
  template <>                                                              \
  EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE RET_TYPE NAME(const ARG_TYPE& x) { \
    return cl::sycl::FUNC(x);                                              \
  }

#define SYCL_SPECIALIZE_UNARY_FUNC(NAME, FUNC, TYPE) \
  SYCL_SPECIALIZE_GEN_UNARY_FUNC(NAME, FUNC, TYPE, TYPE)

#define SYCL_SPECIALIZE_GEN1_BINARY_FUNC(                                   \
    NAME, FUNC, RET_TYPE, ARG_TYPE1, ARG_TYPE2)                             \
  template <>                                                               \
  EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE RET_TYPE NAME(const ARG_TYPE1& x,   \
                                                      const ARG_TYPE2& y) { \
    return cl::sycl::FUNC(x, y);                                            \
  }

#define SYCL_SPECIALIZE_GEN2_BINARY_FUNC(NAME, FUNC, RET_TYPE, ARG_TYPE) \
  SYCL_SPECIALIZE_GEN1_BINARY_FUNC(NAME, FUNC, RET_TYPE, ARG_TYPE, ARG_TYPE)

#define SYCL_SPECIALIZE_BINARY_FUNC(NAME, FUNC, TYPE) \
  SYCL_SPECIALIZE_GEN2_BINARY_FUNC(NAME, FUNC, TYPE, TYPE)

SYCL_SPECIALIZE_INTEGER_TYPES_BINARY(mini, min)
SYCL_SPECIALIZE_FLOATING_TYPES_BINARY(mini, fmin)
SYCL_SPECIALIZE_INTEGER_TYPES_BINARY(maxi, max)
SYCL_SPECIALIZE_FLOATING_TYPES_BINARY(maxi, fmax)

#endif

template <typename Scalar>
EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(real, Scalar)
    real(const Scalar& x) {
  return EIGEN_MATHFUNC_IMPL(real, Scalar)::run(x);
}

template <typename Scalar>
EIGEN_DEVICE_FUNC inline typename internal::add_const_on_value_type<
    EIGEN_MATHFUNC_RETVAL(real_ref, Scalar)>::type
real_ref(const Scalar& x) {
  return internal::real_ref_impl<Scalar>::run(x);
}

template <typename Scalar>
EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(real_ref, Scalar)
    real_ref(Scalar& x) {
  return EIGEN_MATHFUNC_IMPL(real_ref, Scalar)::run(x);
}

template <typename Scalar>
EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(imag, Scalar)
    imag(const Scalar& x) {
  return EIGEN_MATHFUNC_IMPL(imag, Scalar)::run(x);
}

template <typename Scalar>
EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(arg, Scalar)
    arg(const Scalar& x) {
  return EIGEN_MATHFUNC_IMPL(arg, Scalar)::run(x);
}

template <typename Scalar>
EIGEN_DEVICE_FUNC inline typename internal::add_const_on_value_type<
    EIGEN_MATHFUNC_RETVAL(imag_ref, Scalar)>::type
imag_ref(const Scalar& x) {
  return internal::imag_ref_impl<Scalar>::run(x);
}

template <typename Scalar>
EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(imag_ref, Scalar)
    imag_ref(Scalar& x) {
  return EIGEN_MATHFUNC_IMPL(imag_ref, Scalar)::run(x);
}

template <typename Scalar>
EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(conj, Scalar)
    conj(const Scalar& x) {
  return EIGEN_MATHFUNC_IMPL(conj, Scalar)::run(x);
}

template <typename Scalar>
EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(abs2, Scalar)
    abs2(const Scalar& x) {
  return EIGEN_MATHFUNC_IMPL(abs2, Scalar)::run(x);
}

EIGEN_DEVICE_FUNC
inline bool abs2(bool x) { return x; }

template <typename T>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE T absdiff(const T& x, const T& y) {
  return x > y ? x - y : y - x;
}
template <>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE float absdiff(const float& x,
                                                    const float& y) {
  return fabsf(x - y);
}
template <>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE double absdiff(const double& x,
                                                     const double& y) {
  return fabs(x - y);
}
template <>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE long double absdiff(
    const long double& x, const long double& y) {
#if defined(EIGEN_HIPCC)
  // no "fabsl" on HIP yet
  return (x > y) ? x : y;
#else
  return fabsl(x - y);
#endif
}

template <typename Scalar>
EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(norm1, Scalar)
    norm1(const Scalar& x) {
  return EIGEN_MATHFUNC_IMPL(norm1, Scalar)::run(x);
}

template <typename Scalar>
EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(hypot, Scalar)
    hypot(const Scalar& x, const Scalar& y) {
  return EIGEN_MATHFUNC_IMPL(hypot, Scalar)::run(x, y);
}

#if defined(SYCL_DEVICE_ONLY)
SYCL_SPECIALIZE_FLOATING_TYPES_BINARY(hypot, hypot)
#endif

template <typename Scalar>
EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(log1p, Scalar)
    log1p(const Scalar& x) {
  return EIGEN_MATHFUNC_IMPL(log1p, Scalar)::run(x);
}

#if defined(SYCL_DEVICE_ONLY)
SYCL_SPECIALIZE_FLOATING_TYPES_UNARY(log1p, log1p)
#endif

#if defined(EIGEN_GPUCC)
template <>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE float log1p(const float& x) {
  return ::log1pf(x);
}

template <>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE double log1p(const double& x) {
  return ::log1p(x);
}
#endif

template <typename ScalarX, typename ScalarY>
EIGEN_DEVICE_FUNC inline
    typename internal::pow_impl<ScalarX, ScalarY>::result_type
    pow(const ScalarX& x, const ScalarY& y) {
  return internal::pow_impl<ScalarX, ScalarY>::run(x, y);
}

#if defined(SYCL_DEVICE_ONLY)
SYCL_SPECIALIZE_FLOATING_TYPES_BINARY(pow, pow)
#endif

template <typename T>
EIGEN_DEVICE_FUNC bool(isnan)(const T& x) {
  return internal::isnan_impl(x);
}
template <typename T>
EIGEN_DEVICE_FUNC bool(isinf)(const T& x) {
  return internal::isinf_impl(x);
}
template <typename T>
EIGEN_DEVICE_FUNC bool(isfinite)(const T& x) {
  return internal::isfinite_impl(x);
}

#if defined(SYCL_DEVICE_ONLY)
SYCL_SPECIALIZE_FLOATING_TYPES_UNARY_FUNC_RET_TYPE(isnan, isnan, bool)
SYCL_SPECIALIZE_FLOATING_TYPES_UNARY_FUNC_RET_TYPE(isinf, isinf, bool)
SYCL_SPECIALIZE_FLOATING_TYPES_UNARY_FUNC_RET_TYPE(isfinite, isfinite, bool)
#endif

template <typename Scalar>
EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(rint, Scalar)
    rint(const Scalar& x) {
  return EIGEN_MATHFUNC_IMPL(rint, Scalar)::run(x);
}

template <typename Scalar>
EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(round, Scalar)
    round(const Scalar& x) {
  return EIGEN_MATHFUNC_IMPL(round, Scalar)::run(x);
}

#if defined(SYCL_DEVICE_ONLY)
SYCL_SPECIALIZE_FLOATING_TYPES_UNARY(round, round)
#endif

template <typename T>
EIGEN_DEVICE_FUNC T(floor)(const T& x) {
  EIGEN_USING_STD_MATH(floor);
  return floor(x);
}

#if defined(SYCL_DEVICE_ONLY)
SYCL_SPECIALIZE_FLOATING_TYPES_UNARY(floor, floor)
#endif

#if defined(EIGEN_GPUCC)
template <>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE float floor(const float& x) {
  return ::floorf(x);
}

template <>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE double floor(const double& x) {
  return ::floor(x);
}
#endif

template <typename T>
EIGEN_DEVICE_FUNC T(ceil)(const T& x) {
  EIGEN_USING_STD_MATH(ceil);
  return ceil(x);
}

#if defined(SYCL_DEVICE_ONLY)
SYCL_SPECIALIZE_FLOATING_TYPES_UNARY(ceil, ceil)
#endif

#if defined(EIGEN_GPUCC)
template <>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE float ceil(const float& x) {
  return ::ceilf(x);
}

template <>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE double ceil(const double& x) {
  return ::ceil(x);
}
#endif

/** Log base 2 for 32 bits positive integers.
  * Conveniently returns 0 for x==0. */
inline int log2(int x) {
  eigen_assert(x >= 0);
  unsigned int v(x);
  static const int table[32] = {0,  9,  1,  10, 13, 21, 2,  29, 11, 14, 16,
                                18, 22, 25, 3,  30, 8,  12, 20, 28, 15, 17,
                                24, 7,  19, 27, 23, 6,  26, 5,  4,  31};
  v |= v >> 1;
  v |= v >> 2;
  v |= v >> 4;
  v |= v >> 8;
  v |= v >> 16;
  return table[(v * 0x07C4ACDDU) >> 27];
}

/** \returns the square root of \a x.
  *
  * It is essentially equivalent to
  * \code using std::sqrt; return sqrt(x); \endcode
  * but slightly faster for float/double and some compilers (e.g., gcc), thanks
 * to
  * specializations when SSE is enabled.
  *
  * It's usage is justified in performance critical functions, like
 * norm/normalize.
  */
template <typename T>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE T sqrt(const T& x) {
  EIGEN_USING_STD_MATH(sqrt);
  return sqrt(x);
}

#if defined(SYCL_DEVICE_ONLY)
SYCL_SPECIALIZE_FLOATING_TYPES_UNARY(sqrt, sqrt)
#endif

template <typename T>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE T log(const T& x) {
  EIGEN_USING_STD_MATH(log);
  return log(x);
}

#if defined(SYCL_DEVICE_ONLY)
SYCL_SPECIALIZE_FLOATING_TYPES_UNARY(log, log)
#endif

#if defined(EIGEN_GPUCC)
template <>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE float log(const float& x) {
  return ::logf(x);
}

template <>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE double log(const double& x) {
  return ::log(x);
}
#endif

template <typename T>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE
    typename internal::enable_if<NumTraits<T>::IsSigned ||
                                     NumTraits<T>::IsComplex,
                                 typename NumTraits<T>::Real>::type
    abs(const T& x) {
  EIGEN_USING_STD_MATH(abs);
  return abs(x);
}

template <typename T>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE
    typename internal::enable_if<!(NumTraits<T>::IsSigned ||
                                   NumTraits<T>::IsComplex),
                                 typename NumTraits<T>::Real>::type
    abs(const T& x) {
  return x;
}

#if defined(SYCL_DEVICE_ONLY)
SYCL_SPECIALIZE_INTEGER_TYPES_UNARY(abs, abs)
SYCL_SPECIALIZE_FLOATING_TYPES_UNARY(abs, fabs)
#endif

#if defined(EIGEN_GPUCC)
template <>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE float abs(const float& x) {
  return ::fabsf(x);
}

template <>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE double abs(const double& x) {
  return ::fabs(x);
}

template <>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE float abs(const std::complex<float>& x) {
  return ::hypotf(x.real(), x.imag());
}

template <>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE double abs(
    const std::complex<double>& x) {
  return ::hypot(x.real(), x.imag());
}
#endif

template <typename T>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE T exp(const T& x) {
  EIGEN_USING_STD_MATH(exp);
  return exp(x);
}

#if defined(SYCL_DEVICE_ONLY)
SYCL_SPECIALIZE_FLOATING_TYPES_UNARY(exp, exp)
#endif

#if defined(EIGEN_GPUCC)
template <>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE float exp(const float& x) {
  return ::expf(x);
}

template <>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE double exp(const double& x) {
  return ::exp(x);
}

template <>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE std::complex<float> exp(
    const std::complex<float>& x) {
  float com = ::expf(x.real());
  float res_real = com * ::cosf(x.imag());
  float res_imag = com * ::sinf(x.imag());
  return std::complex<float>(res_real, res_imag);
}

template <>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE std::complex<double> exp(
    const std::complex<double>& x) {
  double com = ::exp(x.real());
  double res_real = com * ::cos(x.imag());
  double res_imag = com * ::sin(x.imag());
  return std::complex<double>(res_real, res_imag);
}
#endif

template <typename Scalar>
EIGEN_DEVICE_FUNC inline EIGEN_MATHFUNC_RETVAL(expm1, Scalar)
    expm1(const Scalar& x) {
  return EIGEN_MATHFUNC_IMPL(expm1, Scalar)::run(x);
}

#if defined(SYCL_DEVICE_ONLY)
SYCL_SPECIALIZE_FLOATING_TYPES_UNARY(expm1, expm1)
#endif

#if defined(EIGEN_GPUCC)
template <>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE float expm1(const float& x) {
  return ::expm1f(x);
}

template <>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE double expm1(const double& x) {
  return ::expm1(x);
}
#endif

template <typename T>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE T cos(const T& x) {
  EIGEN_USING_STD_MATH(cos);
  return cos(x);
}

#if defined(SYCL_DEVICE_ONLY)
SYCL_SPECIALIZE_FLOATING_TYPES_UNARY(cos, cos)
#endif

#if defined(EIGEN_GPUCC)
template <>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE float cos(const float& x) {
  return ::cosf(x);
}

template <>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE double cos(const double& x) {
  return ::cos(x);
}
#endif

template <typename T>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE T sin(const T& x) {
  EIGEN_USING_STD_MATH(sin);
  return sin(x);
}

#if defined(SYCL_DEVICE_ONLY)
SYCL_SPECIALIZE_FLOATING_TYPES_UNARY(sin, sin)
#endif

#if defined(EIGEN_GPUCC)
template <>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE float sin(const float& x) {
  return ::sinf(x);
}

template <>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE double sin(const double& x) {
  return ::sin(x);
}
#endif

template <typename T>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE T tan(const T& x) {
  EIGEN_USING_STD_MATH(tan);
  return tan(x);
}

#if defined(SYCL_DEVICE_ONLY)
SYCL_SPECIALIZE_FLOATING_TYPES_UNARY(tan, tan)
#endif

#if defined(EIGEN_GPUCC)
template <>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE float tan(const float& x) {
  return ::tanf(x);
}

template <>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE double tan(const double& x) {
  return ::tan(x);
}
#endif

template <typename T>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE T acos(const T& x) {
  EIGEN_USING_STD_MATH(acos);
  return acos(x);
}

#if EIGEN_HAS_CXX11_MATH
template <typename T>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE T acosh(const T& x) {
  EIGEN_USING_STD_MATH(acosh);
  return acosh(x);
}
#endif

#if defined(SYCL_DEVICE_ONLY)
SYCL_SPECIALIZE_FLOATING_TYPES_UNARY(acos, acos)
SYCL_SPECIALIZE_FLOATING_TYPES_UNARY(acosh, acosh)
#endif

#if defined(EIGEN_GPUCC)
template <>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE float acos(const float& x) {
  return ::acosf(x);
}

template <>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE double acos(const double& x) {
  return ::acos(x);
}
#endif

template <typename T>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE T asin(const T& x) {
  EIGEN_USING_STD_MATH(asin);
  return asin(x);
}

#if EIGEN_HAS_CXX11_MATH
template <typename T>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE T asinh(const T& x) {
  EIGEN_USING_STD_MATH(asinh);
  return asinh(x);
}
#endif

#if defined(SYCL_DEVICE_ONLY)
SYCL_SPECIALIZE_FLOATING_TYPES_UNARY(asin, asin)
SYCL_SPECIALIZE_FLOATING_TYPES_UNARY(asinh, asinh)
#endif

#if defined(EIGEN_GPUCC)
template <>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE float asin(const float& x) {
  return ::asinf(x);
}

template <>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE double asin(const double& x) {
  return ::asin(x);
}
#endif

template <typename T>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE T atan(const T& x) {
  EIGEN_USING_STD_MATH(atan);
  return atan(x);
}

#if EIGEN_HAS_CXX11_MATH
template <typename T>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE T atanh(const T& x) {
  EIGEN_USING_STD_MATH(atanh);
  return atanh(x);
}
#endif

#if defined(SYCL_DEVICE_ONLY)
SYCL_SPECIALIZE_FLOATING_TYPES_UNARY(atan, atan)
SYCL_SPECIALIZE_FLOATING_TYPES_UNARY(atanh, atanh)
#endif

#if defined(EIGEN_GPUCC)
template <>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE float atan(const float& x) {
  return ::atanf(x);
}

template <>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE double atan(const double& x) {
  return ::atan(x);
}
#endif

template <typename T>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE T cosh(const T& x) {
  EIGEN_USING_STD_MATH(cosh);
  return cosh(x);
}

#if defined(SYCL_DEVICE_ONLY)
SYCL_SPECIALIZE_FLOATING_TYPES_UNARY(cosh, cosh)
#endif

#if defined(EIGEN_GPUCC)
template <>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE float cosh(const float& x) {
  return ::coshf(x);
}

template <>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE double cosh(const double& x) {
  return ::cosh(x);
}
#endif

template <typename T>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE T sinh(const T& x) {
  EIGEN_USING_STD_MATH(sinh);
  return sinh(x);
}

#if defined(SYCL_DEVICE_ONLY)
SYCL_SPECIALIZE_FLOATING_TYPES_UNARY(sinh, sinh)
#endif

#if defined(EIGEN_GPUCC)
template <>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE float sinh(const float& x) {
  return ::sinhf(x);
}

template <>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE double sinh(const double& x) {
  return ::sinh(x);
}
#endif

template <typename T>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE T tanh(const T& x) {
  EIGEN_USING_STD_MATH(tanh);
  return tanh(x);
}

#if (!defined(EIGEN_GPUCC)) && EIGEN_FAST_MATH && !defined(SYCL_DEVICE_ONLY)
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE float tanh(float x) {
  return internal::generic_fast_tanh_float(x);
}
#endif

#if defined(SYCL_DEVICE_ONLY)
SYCL_SPECIALIZE_FLOATING_TYPES_UNARY(tanh, tanh)
#endif

#if defined(EIGEN_GPUCC)
template <>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE float tanh(const float& x) {
  return ::tanhf(x);
}

template <>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE double tanh(const double& x) {
  return ::tanh(x);
}
#endif

template <typename T>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE T fmod(const T& a, const T& b) {
  EIGEN_USING_STD_MATH(fmod);
  return fmod(a, b);
}

#if defined(SYCL_DEVICE_ONLY)
SYCL_SPECIALIZE_FLOATING_TYPES_BINARY(fmod, fmod)
#endif

#if defined(EIGEN_GPUCC)
template <>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE float fmod(const float& a,
                                                 const float& b) {
  return ::fmodf(a, b);
}

template <>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE double fmod(const double& a,
                                                  const double& b) {
  return ::fmod(a, b);
}
#endif

#if defined(SYCL_DEVICE_ONLY)
#undef SYCL_SPECIALIZE_SIGNED_INTEGER_TYPES_BINARY
#undef SYCL_SPECIALIZE_SIGNED_INTEGER_TYPES_UNARY
#undef SYCL_SPECIALIZE_UNSIGNED_INTEGER_TYPES_BINARY
#undef SYCL_SPECIALIZE_UNSIGNED_INTEGER_TYPES_UNARY
#undef SYCL_SPECIALIZE_INTEGER_TYPES_BINARY
#undef SYCL_SPECIALIZE_UNSIGNED_INTEGER_TYPES_UNARY
#undef SYCL_SPECIALIZE_FLOATING_TYPES_BINARY
#undef SYCL_SPECIALIZE_FLOATING_TYPES_UNARY
#undef SYCL_SPECIALIZE_FLOATING_TYPES_UNARY_FUNC_RET_TYPE
#undef SYCL_SPECIALIZE_GEN_UNARY_FUNC
#undef SYCL_SPECIALIZE_UNARY_FUNC
#undef SYCL_SPECIALIZE_GEN1_BINARY_FUNC
#undef SYCL_SPECIALIZE_GEN2_BINARY_FUNC
#undef SYCL_SPECIALIZE_BINARY_FUNC
#endif

}  // end namespace numext

namespace internal {

template <typename T>
EIGEN_DEVICE_FUNC bool isfinite_impl(const std::complex<T>& x) {
  return (numext::isfinite)(numext::real(x)) &&
         (numext::isfinite)(numext::imag(x));
}

template <typename T>
EIGEN_DEVICE_FUNC bool isnan_impl(const std::complex<T>& x) {
  return (numext::isnan)(numext::real(x)) || (numext::isnan)(numext::imag(x));
}

template <typename T>
EIGEN_DEVICE_FUNC bool isinf_impl(const std::complex<T>& x) {
  return ((numext::isinf)(numext::real(x)) ||
          (numext::isinf)(numext::imag(x))) &&
         (!(numext::isnan)(x));
}

/****************************************************************************
* Implementation of fuzzy comparisons                                       *
****************************************************************************/

template <typename Scalar, bool IsComplex, bool IsInteger>
struct scalar_fuzzy_default_impl {};

template <typename Scalar>
struct scalar_fuzzy_default_impl<Scalar, false, false> {
  typedef typename NumTraits<Scalar>::Real RealScalar;
  template <typename OtherScalar>
  EIGEN_DEVICE_FUNC static inline bool isMuchSmallerThan(
      const Scalar& x, const OtherScalar& y, const RealScalar& prec) {
    return numext::abs(x) <= numext::abs(y) * prec;
  }
  EIGEN_DEVICE_FUNC
  static inline bool isApprox(const Scalar& x,
                              const Scalar& y,
                              const RealScalar& prec) {
    return numext::abs(x - y) <=
           numext::mini(numext::abs(x), numext::abs(y)) * prec;
  }
  EIGEN_DEVICE_FUNC
  static inline bool isApproxOrLessThan(const Scalar& x,
                                        const Scalar& y,
                                        const RealScalar& prec) {
    return x <= y || isApprox(x, y, prec);
  }
};

template <typename Scalar>
struct scalar_fuzzy_default_impl<Scalar, false, true> {
  typedef typename NumTraits<Scalar>::Real RealScalar;
  template <typename OtherScalar>
  EIGEN_DEVICE_FUNC static inline bool isMuchSmallerThan(const Scalar& x,
                                                         const Scalar&,
                                                         const RealScalar&) {
    return x == Scalar(0);
  }
  EIGEN_DEVICE_FUNC
  static inline bool isApprox(const Scalar& x,
                              const Scalar& y,
                              const RealScalar&) {
    return x == y;
  }
  EIGEN_DEVICE_FUNC
  static inline bool isApproxOrLessThan(const Scalar& x,
                                        const Scalar& y,
                                        const RealScalar&) {
    return x <= y;
  }
};

template <typename Scalar>
struct scalar_fuzzy_default_impl<Scalar, true, false> {
  typedef typename NumTraits<Scalar>::Real RealScalar;
  template <typename OtherScalar>
  EIGEN_DEVICE_FUNC static inline bool isMuchSmallerThan(
      const Scalar& x, const OtherScalar& y, const RealScalar& prec) {
    return numext::abs2(x) <= numext::abs2(y) * prec * prec;
  }
  EIGEN_DEVICE_FUNC
  static inline bool isApprox(const Scalar& x,
                              const Scalar& y,
                              const RealScalar& prec) {
    return numext::abs2(x - y) <=
           numext::mini(numext::abs2(x), numext::abs2(y)) * prec * prec;
  }
};

template <typename Scalar>
struct scalar_fuzzy_impl
    : scalar_fuzzy_default_impl<Scalar,
                                NumTraits<Scalar>::IsComplex,
                                NumTraits<Scalar>::IsInteger> {};

template <typename Scalar, typename OtherScalar>
EIGEN_DEVICE_FUNC inline bool isMuchSmallerThan(
    const Scalar& x,
    const OtherScalar& y,
    const typename NumTraits<Scalar>::Real& precision =
        NumTraits<Scalar>::dummy_precision()) {
  return scalar_fuzzy_impl<Scalar>::template isMuchSmallerThan<OtherScalar>(
      x, y, precision);
}

template <typename Scalar>
EIGEN_DEVICE_FUNC inline bool isApprox(
    const Scalar& x,
    const Scalar& y,
    const typename NumTraits<Scalar>::Real& precision =
        NumTraits<Scalar>::dummy_precision()) {
  return scalar_fuzzy_impl<Scalar>::isApprox(x, y, precision);
}

template <typename Scalar>
EIGEN_DEVICE_FUNC inline bool isApproxOrLessThan(
    const Scalar& x,
    const Scalar& y,
    const typename NumTraits<Scalar>::Real& precision =
        NumTraits<Scalar>::dummy_precision()) {
  return scalar_fuzzy_impl<Scalar>::isApproxOrLessThan(x, y, precision);
}

/******************************************
***  The special case of the  bool type ***
******************************************/

template <>
struct random_impl<bool> {
  static inline bool run() { return random<int>(0, 1) == 0 ? false : true; }
};

template <>
struct scalar_fuzzy_impl<bool> {
  typedef bool RealScalar;

  template <typename OtherScalar>
  EIGEN_DEVICE_FUNC static inline bool isMuchSmallerThan(const bool& x,
                                                         const bool&,
                                                         const bool&) {
    return !x;
  }

  EIGEN_DEVICE_FUNC
  static inline bool isApprox(bool x, bool y, bool) { return x == y; }

  EIGEN_DEVICE_FUNC
  static inline bool isApproxOrLessThan(const bool& x,
                                        const bool& y,
                                        const bool&) {
    return (!x) || y;
  }
};

}  // end namespace internal

}  // end namespace Eigen

#endif  // EIGEN_MATHFUNCTIONS_H