Eigen/src/Core/GenericPacketMath.h

// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2008 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2006-2008 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_GENERIC_PACKET_MATH_H
#define EIGEN_GENERIC_PACKET_MATH_H

// IWYU pragma: private
#include "./InternalHeaderCheck.h"

namespace Eigen {

namespace internal {

/** \internal
 * \file GenericPacketMath.h
 *
 * Default implementation for types not supported by the vectorization.
 * In practice these functions are provided to make easier the writing
 * of generic vectorized code.
 */

#ifndef EIGEN_DEBUG_ALIGNED_LOAD
#define EIGEN_DEBUG_ALIGNED_LOAD
#endif

#ifndef EIGEN_DEBUG_UNALIGNED_LOAD
#define EIGEN_DEBUG_UNALIGNED_LOAD
#endif

#ifndef EIGEN_DEBUG_ALIGNED_STORE
#define EIGEN_DEBUG_ALIGNED_STORE
#endif

#ifndef EIGEN_DEBUG_UNALIGNED_STORE
#define EIGEN_DEBUG_UNALIGNED_STORE
#endif

struct default_packet_traits {
  enum {
    // Ops that are implemented for most types.
    HasAdd = 1,
    HasSub = 1,
    HasShift = 1,
    HasMul = 1,
    HasNegate = 1,
    HasAbs = 1,
    HasAbs2 = 1,
    HasMin = 1,
    HasMax = 1,
    HasConj = 1,
    HasSetLinear = 1,
    HasSign = 1,

    HasArg = 0,
    HasAbsDiff = 0,
    HasBlend = 0,
    // This flag is used to indicate whether packet comparison is supported.
    // pcmp_eq, pcmp_lt and pcmp_le should be defined for it to be true.
    HasCmp = 0,
    HasRound = 0,
    HasRint = 0,
    HasFloor = 0,
    HasCeil = 0,

    HasDiv = 0,
    HasReciprocal = 0,
    HasSqrt = 0,
    HasRsqrt = 0,
    HasExp = 0,
    HasExpm1 = 0,
    HasLog = 0,
    HasLog1p = 0,
    HasLog10 = 0,
    HasPow = 0,
    HasSin = 0,
    HasCos = 0,
    HasTan = 0,
    HasASin = 0,
    HasACos = 0,
    HasATan = 0,
    HasATanh = 0,
    HasSinh = 0,
    HasCosh = 0,
    HasTanh = 0,
    HasLGamma = 0,
    HasDiGamma = 0,
    HasZeta = 0,
    HasPolygamma = 0,
    HasErf = 0,
    HasErfc = 0,
    HasNdtri = 0,
    HasBessel = 0,
    HasIGamma = 0,
    HasIGammaDerA = 0,
    HasGammaSampleDerAlpha = 0,
    HasIGammac = 0,
    HasBetaInc = 0
  };
};

template <typename T>
struct packet_traits : default_packet_traits {
  typedef T type;
  typedef T half;
  enum {
    Vectorizable = 0,
    size = 1,
    AlignedOnScalar = 0,
  };
  enum {
    HasAdd = 0,
    HasSub = 0,
    HasMul = 0,
    HasNegate = 0,
    HasAbs = 0,
    HasAbs2 = 0,
    HasMin = 0,
    HasMax = 0,
    HasConj = 0,
    HasSetLinear = 0
  };
};

template <typename T>
struct packet_traits<const T> : packet_traits<T> {};

template <typename T>
struct unpacket_traits {
  typedef T type;
  typedef T half;
  typedef typename numext::get_integer_by_size<sizeof(T)>::signed_type integer_packet;
  enum {
    size = 1,
    alignment = alignof(T),
    vectorizable = false,
    masked_load_available = false,
    masked_store_available = false
  };
};

template <typename T>
struct unpacket_traits<const T> : unpacket_traits<T> {};

/** \internal A convenience utility for determining if the type is a scalar.
 * This is used to enable some generic packet implementations.
 */
template <typename Packet>
struct is_scalar {
  using Scalar = typename unpacket_traits<Packet>::type;
  enum { value = internal::is_same<Packet, Scalar>::value };
};

// automatically and succinctly define combinations of pcast<SrcPacket,TgtPacket> when
// 1) the packets are the same type, or
// 2) the packets differ only in sign.
// In both of these cases, preinterpret (bit_cast) is equivalent to pcast (static_cast)
template <typename SrcPacket, typename TgtPacket,
          bool Scalar = is_scalar<SrcPacket>::value && is_scalar<TgtPacket>::value>
struct is_degenerate_helper : is_same<SrcPacket, TgtPacket> {};
template <>
struct is_degenerate_helper<int8_t, uint8_t, true> : std::true_type {};
template <>
struct is_degenerate_helper<int16_t, uint16_t, true> : std::true_type {};
template <>
struct is_degenerate_helper<int32_t, uint32_t, true> : std::true_type {};
template <>
struct is_degenerate_helper<int64_t, uint64_t, true> : std::true_type {};

template <typename SrcPacket, typename TgtPacket>
struct is_degenerate_helper<SrcPacket, TgtPacket, false> {
  using SrcScalar = typename unpacket_traits<SrcPacket>::type;
  static constexpr int SrcSize = unpacket_traits<SrcPacket>::size;
  using TgtScalar = typename unpacket_traits<TgtPacket>::type;
  static constexpr int TgtSize = unpacket_traits<TgtPacket>::size;
  static constexpr bool value = is_degenerate_helper<SrcScalar, TgtScalar, true>::value && (SrcSize == TgtSize);
};

// is_degenerate<T1,T2>::value == is_degenerate<T2,T1>::value
template <typename SrcPacket, typename TgtPacket>
struct is_degenerate {
  static constexpr bool value =
      is_degenerate_helper<SrcPacket, TgtPacket>::value || is_degenerate_helper<TgtPacket, SrcPacket>::value;
};

template <typename Packet>
struct is_half {
  using Scalar = typename unpacket_traits<Packet>::type;
  static constexpr int Size = unpacket_traits<Packet>::size;
  using DefaultPacket = typename packet_traits<Scalar>::type;
  static constexpr int DefaultSize = unpacket_traits<DefaultPacket>::size;
  static constexpr bool value = Size < DefaultSize;
};

template <typename Src, typename Tgt>
struct type_casting_traits {
  enum {
    VectorizedCast =
        is_degenerate<Src, Tgt>::value && packet_traits<Src>::Vectorizable && packet_traits<Tgt>::Vectorizable,
    SrcCoeffRatio = 1,
    TgtCoeffRatio = 1
  };
};

// provides a succint template to define vectorized casting traits with respect to the largest accessible packet types
template <typename Src, typename Tgt>
struct vectorized_type_casting_traits {
  enum : int {
    DefaultSrcPacketSize = packet_traits<Src>::size,
    DefaultTgtPacketSize = packet_traits<Tgt>::size,
    VectorizedCast = 1,
    SrcCoeffRatio = plain_enum_max(DefaultTgtPacketSize / DefaultSrcPacketSize, 1),
    TgtCoeffRatio = plain_enum_max(DefaultSrcPacketSize / DefaultTgtPacketSize, 1)
  };
};

/** \internal Wrapper to ensure that multiple packet types can map to the same
    same underlying vector type. */
template <typename T, int unique_id = 0>
struct eigen_packet_wrapper {
  EIGEN_ALWAYS_INLINE operator T&() { return m_val; }
  EIGEN_ALWAYS_INLINE operator const T&() const { return m_val; }
  EIGEN_ALWAYS_INLINE eigen_packet_wrapper() = default;
  EIGEN_ALWAYS_INLINE eigen_packet_wrapper(const T& v) : m_val(v) {}
  EIGEN_ALWAYS_INLINE eigen_packet_wrapper& operator=(const T& v) {
    m_val = v;
    return *this;
  }

  T m_val;
};

template <typename Target, typename Packet, bool IsSame = is_same<Target, Packet>::value>
struct preinterpret_generic;

template <typename Target, typename Packet>
struct preinterpret_generic<Target, Packet, false> {
  // the packets are not the same, attempt scalar bit_cast
  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Target run(const Packet& a) {
    return numext::bit_cast<Target, Packet>(a);
  }
};

template <typename Packet>
struct preinterpret_generic<Packet, Packet, true> {
  // the packets are the same type: do nothing
  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Packet run(const Packet& a) { return a; }
};

/** \internal \returns reinterpret_cast<Target>(a) */
template <typename Target, typename Packet>
EIGEN_DEVICE_FUNC inline Target preinterpret(const Packet& a) {
  return preinterpret_generic<Target, Packet>::run(a);
}

template <typename SrcPacket, typename TgtPacket, bool Degenerate = is_degenerate<SrcPacket, TgtPacket>::value,
          bool TgtIsHalf = is_half<TgtPacket>::value>
struct pcast_generic;

template <typename SrcPacket, typename TgtPacket>
struct pcast_generic<SrcPacket, TgtPacket, false, false> {
  // the packets are not degenerate: attempt scalar static_cast
  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE TgtPacket run(const SrcPacket& a) {
    return cast_impl<SrcPacket, TgtPacket>::run(a);
  }
};

template <typename Packet>
struct pcast_generic<Packet, Packet, true, false> {
  // the packets are the same: do nothing
  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Packet run(const Packet& a) { return a; }
};

template <typename SrcPacket, typename TgtPacket, bool TgtIsHalf>
struct pcast_generic<SrcPacket, TgtPacket, true, TgtIsHalf> {
  // the packets are degenerate: preinterpret is equivalent to pcast
  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE TgtPacket run(const SrcPacket& a) { return preinterpret<TgtPacket>(a); }
};

/** \internal \returns static_cast<TgtType>(a) (coeff-wise) */
template <typename SrcPacket, typename TgtPacket>
EIGEN_DEVICE_FUNC inline TgtPacket pcast(const SrcPacket& a) {
  return pcast_generic<SrcPacket, TgtPacket>::run(a);
}
template <typename SrcPacket, typename TgtPacket>
EIGEN_DEVICE_FUNC inline TgtPacket pcast(const SrcPacket& a, const SrcPacket& b) {
  return pcast_generic<SrcPacket, TgtPacket>::run(a, b);
}
template <typename SrcPacket, typename TgtPacket>
EIGEN_DEVICE_FUNC inline TgtPacket pcast(const SrcPacket& a, const SrcPacket& b, const SrcPacket& c,
                                         const SrcPacket& d) {
  return pcast_generic<SrcPacket, TgtPacket>::run(a, b, c, d);
}
template <typename SrcPacket, typename TgtPacket>
EIGEN_DEVICE_FUNC inline TgtPacket pcast(const SrcPacket& a, const SrcPacket& b, const SrcPacket& c, const SrcPacket& d,
                                         const SrcPacket& e, const SrcPacket& f, const SrcPacket& g,
                                         const SrcPacket& h) {
  return pcast_generic<SrcPacket, TgtPacket>::run(a, b, c, d, e, f, g, h);
}

template <typename SrcPacket, typename TgtPacket>
struct pcast_generic<SrcPacket, TgtPacket, false, true> {
  // TgtPacket is a half packet of some other type
  // perform cast and truncate result
  using DefaultTgtPacket = typename is_half<TgtPacket>::DefaultPacket;
  static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE TgtPacket run(const SrcPacket& a) {
    return preinterpret<TgtPacket>(pcast<SrcPacket, DefaultTgtPacket>(a));
  }
};

/** \internal \returns a + b (coeff-wise) */
template <typename Packet>
EIGEN_DEVICE_FUNC inline Packet padd(const Packet& a, const Packet& b) {
  return a + b;
}
// Avoid compiler warning for boolean algebra.
template <>
EIGEN_DEVICE_FUNC inline bool padd(const bool& a, const bool& b) {
  return a || b;
}

/** \internal \returns a packet version of \a *from, (un-aligned masked add)
 * There is no generic implementation. We only have implementations for specialized
 * cases. Generic case should not be called.
 */
template <typename Packet>
EIGEN_DEVICE_FUNC inline std::enable_if_t<unpacket_traits<Packet>::masked_fpops_available, Packet> padd(
    const Packet& a, const Packet& b, typename unpacket_traits<Packet>::mask_t umask);

/** \internal \returns a - b (coeff-wise) */
template <typename Packet>
EIGEN_DEVICE_FUNC inline Packet psub(const Packet& a, const Packet& b) {
  return a - b;
}

/** \internal \returns -a (coeff-wise) */
template <typename Packet>
EIGEN_DEVICE_FUNC inline Packet pnegate(const Packet& a) {
  EIGEN_STATIC_ASSERT((!is_same<typename unpacket_traits<Packet>::type, bool>::value),
                      NEGATE IS NOT DEFINED FOR BOOLEAN TYPES)
  return numext::negate(a);
}

/** \internal \returns conj(a) (coeff-wise) */
template <typename Packet>
EIGEN_DEVICE_FUNC inline Packet pconj(const Packet& a) {
  return numext::conj(a);
}

/** \internal \returns a * b (coeff-wise) */
template <typename Packet>
EIGEN_DEVICE_FUNC inline Packet pmul(const Packet& a, const Packet& b) {
  return a * b;
}
// Avoid compiler warning for boolean algebra.
template <>
EIGEN_DEVICE_FUNC inline bool pmul(const bool& a, const bool& b) {
  return a && b;
}

/** \internal \returns a / b (coeff-wise) */
template <typename Packet>
EIGEN_DEVICE_FUNC inline Packet pdiv(const Packet& a, const Packet& b) {
  return a / b;
}

// In the generic case, memset to all one bits.
template <typename Packet, typename EnableIf = void>
struct ptrue_impl {
  static EIGEN_DEVICE_FUNC inline Packet run(const Packet& /*a*/) {
    Packet b;
    memset(static_cast<void*>(&b), 0xff, sizeof(Packet));
    return b;
  }
};

// For booleans, we can only directly set a valid `bool` value to avoid UB.
template <>
struct ptrue_impl<bool, void> {
  static EIGEN_DEVICE_FUNC inline bool run(const bool& /*a*/) { return true; }
};

// For non-trivial scalars, set to Scalar(1) (i.e. a non-zero value).
// Although this is technically not a valid bitmask, the scalar path for pselect
// uses a comparison to zero, so this should still work in most cases. We don't
// have another option, since the scalar type requires initialization.
template <typename T>
struct ptrue_impl<T, std::enable_if_t<is_scalar<T>::value && NumTraits<T>::RequireInitialization>> {
  static EIGEN_DEVICE_FUNC inline T run(const T& /*a*/) { return T(1); }
};

/** \internal \returns one bits. */
template <typename Packet>
EIGEN_DEVICE_FUNC inline Packet ptrue(const Packet& a) {
  return ptrue_impl<Packet>::run(a);
}

// In the general case, memset to zero.
template <typename Packet, typename EnableIf = void>
struct pzero_impl {
  static EIGEN_DEVICE_FUNC inline Packet run(const Packet& /*a*/) {
    Packet b;
    memset(static_cast<void*>(&b), 0x00, sizeof(Packet));
    return b;
  }
};

// For scalars, explicitly set to Scalar(0), since the underlying representation
// for zero may not consist of all-zero bits.
template <typename T>
struct pzero_impl<T, std::enable_if_t<is_scalar<T>::value>> {
  static EIGEN_DEVICE_FUNC inline T run(const T& /*a*/) { return T(0); }
};

/** \internal \returns packet of zeros */
template <typename Packet>
EIGEN_DEVICE_FUNC inline Packet pzero(const Packet& a) {
  return pzero_impl<Packet>::run(a);
}

/** \internal \returns a <= b as a bit mask */
template <typename Packet>
EIGEN_DEVICE_FUNC inline Packet pcmp_le(const Packet& a, const Packet& b) {
  return a <= b ? ptrue(a) : pzero(a);
}

/** \internal \returns a < b as a bit mask */
template <typename Packet>
EIGEN_DEVICE_FUNC inline Packet pcmp_lt(const Packet& a, const Packet& b) {
  return a < b ? ptrue(a) : pzero(a);
}

/** \internal \returns a == b as a bit mask */
template <typename Packet>
EIGEN_DEVICE_FUNC inline Packet pcmp_eq(const Packet& a, const Packet& b) {
  return a == b ? ptrue(a) : pzero(a);
}

/** \internal \returns a < b or a==NaN or b==NaN as a bit mask */
template <typename Packet>
EIGEN_DEVICE_FUNC inline Packet pcmp_lt_or_nan(const Packet& a, const Packet& b) {
  return a >= b ? pzero(a) : ptrue(a);
}

template <typename T>
struct bit_and {
  EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR EIGEN_ALWAYS_INLINE T operator()(const T& a, const T& b) const { return a & b; }
};

template <typename T>
struct bit_or {
  EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR EIGEN_ALWAYS_INLINE T operator()(const T& a, const T& b) const { return a | b; }
};

template <typename T>
struct bit_xor {
  EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR EIGEN_ALWAYS_INLINE T operator()(const T& a, const T& b) const { return a ^ b; }
};

template <typename T>
struct bit_not {
  EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR EIGEN_ALWAYS_INLINE T operator()(const T& a) const { return ~a; }
};

template <>
struct bit_and<bool> {
  EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR EIGEN_ALWAYS_INLINE bool operator()(const bool& a, const bool& b) const {
    return a && b;
  }
};

template <>
struct bit_or<bool> {
  EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR EIGEN_ALWAYS_INLINE bool operator()(const bool& a, const bool& b) const {
    return a || b;
  }
};

template <>
struct bit_xor<bool> {
  EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR EIGEN_ALWAYS_INLINE bool operator()(const bool& a, const bool& b) const {
    return a != b;
  }
};

template <>
struct bit_not<bool> {
  EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR EIGEN_ALWAYS_INLINE bool operator()(const bool& a) const { return !a; }
};

// Use operators &, |, ^, ~.
template <typename T>
struct operator_bitwise_helper {
  EIGEN_DEVICE_FUNC static inline T bitwise_and(const T& a, const T& b) { return bit_and<T>()(a, b); }
  EIGEN_DEVICE_FUNC static inline T bitwise_or(const T& a, const T& b) { return bit_or<T>()(a, b); }
  EIGEN_DEVICE_FUNC static inline T bitwise_xor(const T& a, const T& b) { return bit_xor<T>()(a, b); }
  EIGEN_DEVICE_FUNC static inline T bitwise_not(const T& a) { return bit_not<T>()(a); }
};

// Apply binary operations byte-by-byte
template <typename T>
struct bytewise_bitwise_helper {
  EIGEN_DEVICE_FUNC static inline T bitwise_and(const T& a, const T& b) {
    return binary(a, b, bit_and<unsigned char>());
  }
  EIGEN_DEVICE_FUNC static inline T bitwise_or(const T& a, const T& b) { return binary(a, b, bit_or<unsigned char>()); }
  EIGEN_DEVICE_FUNC static inline T bitwise_xor(const T& a, const T& b) {
    return binary(a, b, bit_xor<unsigned char>());
  }
  EIGEN_DEVICE_FUNC static inline T bitwise_not(const T& a) { return unary(a, bit_not<unsigned char>()); }

 private:
  template <typename Op>
  EIGEN_DEVICE_FUNC static inline T unary(const T& a, Op op) {
    const unsigned char* a_ptr = reinterpret_cast<const unsigned char*>(&a);
    T c;
    unsigned char* c_ptr = reinterpret_cast<unsigned char*>(&c);
    for (size_t i = 0; i < sizeof(T); ++i) {
      *c_ptr++ = op(*a_ptr++);
    }
    return c;
  }

  template <typename Op>
  EIGEN_DEVICE_FUNC static inline T binary(const T& a, const T& b, Op op) {
    const unsigned char* a_ptr = reinterpret_cast<const unsigned char*>(&a);
    const unsigned char* b_ptr = reinterpret_cast<const unsigned char*>(&b);
    T c;
    unsigned char* c_ptr = reinterpret_cast<unsigned char*>(&c);
    for (size_t i = 0; i < sizeof(T); ++i) {
      *c_ptr++ = op(*a_ptr++, *b_ptr++);
    }
    return c;
  }
};

// In the general case, use byte-by-byte manipulation.
template <typename T, typename EnableIf = void>
struct bitwise_helper : public bytewise_bitwise_helper<T> {};

// For integers or non-trivial scalars, use binary operators.
template <typename T>
struct bitwise_helper<T, typename std::enable_if_t<is_scalar<T>::value &&
                                                   (NumTraits<T>::IsInteger || NumTraits<T>::RequireInitialization)>>
    : public operator_bitwise_helper<T> {};

/** \internal \returns the bitwise and of \a a and \a b */
template <typename Packet>
EIGEN_DEVICE_FUNC inline Packet pand(const Packet& a, const Packet& b) {
  return bitwise_helper<Packet>::bitwise_and(a, b);
}

/** \internal \returns the bitwise or of \a a and \a b */
template <typename Packet>
EIGEN_DEVICE_FUNC inline Packet por(const Packet& a, const Packet& b) {
  return bitwise_helper<Packet>::bitwise_or(a, b);
}

/** \internal \returns the bitwise xor of \a a and \a b */
template <typename Packet>
EIGEN_DEVICE_FUNC inline Packet pxor(const Packet& a, const Packet& b) {
  return bitwise_helper<Packet>::bitwise_xor(a, b);
}

/** \internal \returns the bitwise not of \a a */
template <typename Packet>
EIGEN_DEVICE_FUNC inline Packet pnot(const Packet& a) {
  return bitwise_helper<Packet>::bitwise_not(a);
}

/** \internal \returns the bitwise and of \a a and not \a b */
template <typename Packet>
EIGEN_DEVICE_FUNC inline Packet pandnot(const Packet& a, const Packet& b) {
  return pand(a, pnot(b));
}

/** \internal \returns isnan(a) */
template <typename Packet>
EIGEN_DEVICE_FUNC inline Packet pisnan(const Packet& a) {
  return pandnot(ptrue(a), pcmp_eq(a, a));
}

// In the general case, use bitwise select.
template <typename Packet, typename EnableIf = void>
struct pselect_impl {
  static EIGEN_DEVICE_FUNC inline Packet run(const Packet& mask, const Packet& a, const Packet& b) {
    return por(pand(a, mask), pandnot(b, mask));
  }
};

// For scalars, use ternary select.
template <typename Packet>
struct pselect_impl<Packet, std::enable_if_t<is_scalar<Packet>::value>> {
  static EIGEN_DEVICE_FUNC inline Packet run(const Packet& mask, const Packet& a, const Packet& b) {
    return numext::equal_strict(mask, Packet(0)) ? b : a;
  }
};

/** \internal \returns \a or \b for each field in packet according to \mask */
template <typename Packet>
EIGEN_DEVICE_FUNC inline Packet pselect(const Packet& mask, const Packet& a, const Packet& b) {
  return pselect_impl<Packet>::run(mask, a, b);
}

template <>
EIGEN_DEVICE_FUNC inline bool pselect<bool>(const bool& cond, const bool& a, const bool& b) {
  return cond ? a : b;
}

/** \internal \returns the min or of \a a and \a b (coeff-wise)
    If either \a a or \a b are NaN, the result is implementation defined. */
template <int NaNPropagation>
struct pminmax_impl {
  template <typename Packet, typename Op>
  static EIGEN_DEVICE_FUNC inline Packet run(const Packet& a, const Packet& b, Op op) {
    return op(a, b);
  }
};

/** \internal \returns the min or max of \a a and \a b (coeff-wise)
    If either \a a or \a b are NaN, NaN is returned. */
template <>
struct pminmax_impl<PropagateNaN> {
  template <typename Packet, typename Op>
  static EIGEN_DEVICE_FUNC inline Packet run(const Packet& a, const Packet& b, Op op) {
    Packet not_nan_mask_a = pcmp_eq(a, a);
    Packet not_nan_mask_b = pcmp_eq(b, b);
    return pselect(not_nan_mask_a, pselect(not_nan_mask_b, op(a, b), b), a);
  }
};

/** \internal \returns the min or max of \a a and \a b (coeff-wise)
    If both \a a and \a b are NaN, NaN is returned.
    Equivalent to std::fmin(a, b).  */
template <>
struct pminmax_impl<PropagateNumbers> {
  template <typename Packet, typename Op>
  static EIGEN_DEVICE_FUNC inline Packet run(const Packet& a, const Packet& b, Op op) {
    Packet not_nan_mask_a = pcmp_eq(a, a);
    Packet not_nan_mask_b = pcmp_eq(b, b);
    return pselect(not_nan_mask_a, pselect(not_nan_mask_b, op(a, b), a), b);
  }
};

#define EIGEN_BINARY_OP_NAN_PROPAGATION(Type, Func) [](const Type& a, const Type& b) { return Func(a, b); }

/** \internal \returns the min of \a a and \a b  (coeff-wise).
    If \a a or \b b is NaN, the return value is implementation defined. */
template <typename Packet>
EIGEN_DEVICE_FUNC inline Packet pmin(const Packet& a, const Packet& b) {
  return numext::mini(a, b);
}

/** \internal \returns the min of \a a and \a b  (coeff-wise).
    NaNPropagation determines the NaN propagation semantics. */
template <int NaNPropagation, typename Packet>
EIGEN_DEVICE_FUNC inline Packet pmin(const Packet& a, const Packet& b) {
  return pminmax_impl<NaNPropagation>::run(a, b, EIGEN_BINARY_OP_NAN_PROPAGATION(Packet, (pmin<Packet>)));
}

/** \internal \returns the max of \a a and \a b  (coeff-wise)
    If \a a or \b b is NaN, the return value is implementation defined. */
template <typename Packet>
EIGEN_DEVICE_FUNC inline Packet pmax(const Packet& a, const Packet& b) {
  return numext::maxi(a, b);
}

/** \internal \returns the max of \a a and \a b  (coeff-wise).
    NaNPropagation determines the NaN propagation semantics. */
template <int NaNPropagation, typename Packet>
EIGEN_DEVICE_FUNC inline Packet pmax(const Packet& a, const Packet& b) {
  return pminmax_impl<NaNPropagation>::run(a, b, EIGEN_BINARY_OP_NAN_PROPAGATION(Packet, (pmax<Packet>)));
}

/** \internal \returns the absolute value of \a a */
template <typename Packet>
EIGEN_DEVICE_FUNC inline Packet pabs(const Packet& a) {
  return numext::abs(a);
}
template <>
EIGEN_DEVICE_FUNC inline unsigned int pabs(const unsigned int& a) {
  return a;
}
template <>
EIGEN_DEVICE_FUNC inline unsigned long pabs(const unsigned long& a) {
  return a;
}
template <>
EIGEN_DEVICE_FUNC inline unsigned long long pabs(const unsigned long long& a) {
  return a;
}

/** \internal \returns the addsub value of \a a,b */
template <typename Packet>
EIGEN_DEVICE_FUNC inline Packet paddsub(const Packet& a, const Packet& b) {
  return pselect(peven_mask(a), padd(a, b), psub(a, b));
}

/** \internal \returns the phase angle of \a a */
template <typename Packet>
EIGEN_DEVICE_FUNC inline Packet parg(const Packet& a) {
  using numext::arg;
  return arg(a);
}

/** \internal \returns \a a arithmetically shifted by N bits to the right */
template <int N>
EIGEN_DEVICE_FUNC inline int parithmetic_shift_right(const int& a) {
  return a >> N;
}
template <int N>
EIGEN_DEVICE_FUNC inline long int parithmetic_shift_right(const long int& a) {
  return a >> N;
}

/** \internal \returns \a a logically shifted by N bits to the right */
template <int N>
EIGEN_DEVICE_FUNC inline int plogical_shift_right(const int& a) {
  return static_cast<int>(static_cast<unsigned int>(a) >> N);
}
template <int N>
EIGEN_DEVICE_FUNC inline long int plogical_shift_right(const long int& a) {
  return static_cast<long>(static_cast<unsigned long>(a) >> N);
}

/** \internal \returns \a a shifted by N bits to the left */
template <int N>
EIGEN_DEVICE_FUNC inline int plogical_shift_left(const int& a) {
  return a << N;
}
template <int N>
EIGEN_DEVICE_FUNC inline long int plogical_shift_left(const long int& a) {
  return a << N;
}

/** \internal \returns the significant and exponent of the underlying floating point numbers
 * See https://en.cppreference.com/w/cpp/numeric/math/frexp
 */
template <typename Packet>
EIGEN_DEVICE_FUNC inline Packet pfrexp(const Packet& a, Packet& exponent) {
  int exp;
  EIGEN_USING_STD(frexp);
  Packet result = static_cast<Packet>(frexp(a, &exp));
  exponent = static_cast<Packet>(exp);
  return result;
}

/** \internal \returns a * 2^((int)exponent)
 * See https://en.cppreference.com/w/cpp/numeric/math/ldexp
 */
template <typename Packet>
EIGEN_DEVICE_FUNC inline Packet pldexp(const Packet& a, const Packet& exponent) {
  EIGEN_USING_STD(ldexp)
  return static_cast<Packet>(ldexp(a, static_cast<int>(exponent)));
}

/** \internal \returns the min of \a a and \a b  (coeff-wise) */
template <typename Packet>
EIGEN_DEVICE_FUNC inline Packet pabsdiff(const Packet& a, const Packet& b) {
  return pselect(pcmp_lt(a, b), psub(b, a), psub(a, b));
}

/** \internal \returns a packet version of \a *from, from must be properly aligned */
template <typename Packet>
EIGEN_DEVICE_FUNC inline Packet pload(const typename unpacket_traits<Packet>::type* from) {
  return *from;
}

/** \internal \returns n elements of a packet version of \a *from, from must be properly aligned
 * offset indicates the starting element in which to load and
 * offset + n <= unpacket_traits::size
 * All elements before offset and after the last element loaded will initialized with zero */
template <typename Packet>
EIGEN_DEVICE_FUNC inline Packet pload_partial(const typename unpacket_traits<Packet>::type* from, const Index n,
                                              const Index offset = 0) {
  const Index packet_size = unpacket_traits<Packet>::size;
  eigen_assert(n + offset <= packet_size && "number of elements plus offset will read past end of packet");
  typedef typename unpacket_traits<Packet>::type Scalar;
  EIGEN_ALIGN_MAX Scalar elements[packet_size] = {Scalar(0)};
  for (Index i = offset; i < numext::mini(n + offset, packet_size); i++) {
    elements[i] = from[i - offset];
  }
  return pload<Packet>(elements);
}

/** \internal \returns a packet version of \a *from, (un-aligned load) */
template <typename Packet>
EIGEN_DEVICE_FUNC inline Packet ploadu(const typename unpacket_traits<Packet>::type* from) {
  return *from;
}

/** \internal \returns n elements of a packet version of \a *from, (un-aligned load)
 * All elements after the last element loaded will initialized with zero */
template <typename Packet>
EIGEN_DEVICE_FUNC inline Packet ploadu_partial(const typename unpacket_traits<Packet>::type* from, const Index n,
                                               const Index offset = 0) {
  const Index packet_size = unpacket_traits<Packet>::size;
  eigen_assert(n + offset <= packet_size && "number of elements plus offset will read past end of packet");
  typedef typename unpacket_traits<Packet>::type Scalar;
  EIGEN_ALIGN_MAX Scalar elements[packet_size] = {Scalar(0)};
  for (Index i = offset; i < numext::mini(n + offset, packet_size); i++) {
    elements[i] = from[i - offset];
  }
  return pload<Packet>(elements);
}

/** \internal \returns a packet version of \a *from, (un-aligned masked load)
 * There is no generic implementation. We only have implementations for specialized
 * cases. Generic case should not be called.
 */
template <typename Packet>
EIGEN_DEVICE_FUNC inline std::enable_if_t<unpacket_traits<Packet>::masked_load_available, Packet> ploadu(
    const typename unpacket_traits<Packet>::type* from, typename unpacket_traits<Packet>::mask_t umask);

/** \internal \returns a packet with constant coefficients \a a, e.g.: (a,a,a,a) */
template <typename Packet>
EIGEN_DEVICE_FUNC inline Packet pset1(const typename unpacket_traits<Packet>::type& a) {
  return a;
}

/** \internal \returns a packet with constant coefficients set from bits */
template <typename Packet, typename BitsType>
EIGEN_DEVICE_FUNC inline Packet pset1frombits(BitsType a);

/** \internal \returns a packet with constant coefficients \a a[0], e.g.: (a[0],a[0],a[0],a[0]) */
template <typename Packet>
EIGEN_DEVICE_FUNC inline Packet pload1(const typename unpacket_traits<Packet>::type* a) {
  return pset1<Packet>(*a);
}

/** \internal \returns a packet with elements of \a *from duplicated.
 * For instance, for a packet of 8 elements, 4 scalars will be read from \a *from and
 * duplicated to form: {from[0],from[0],from[1],from[1],from[2],from[2],from[3],from[3]}
 * Currently, this function is only used for scalar * complex products.
 */
template <typename Packet>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Packet ploaddup(const typename unpacket_traits<Packet>::type* from) {
  return *from;
}

/** \internal \returns a packet with elements of \a *from quadrupled.
 * For instance, for a packet of 8 elements, 2 scalars will be read from \a *from and
 * replicated to form: {from[0],from[0],from[0],from[0],from[1],from[1],from[1],from[1]}
 * Currently, this function is only used in matrix products.
 * For packet-size smaller or equal to 4, this function is equivalent to pload1
 */
template <typename Packet>
EIGEN_DEVICE_FUNC inline Packet ploadquad(const typename unpacket_traits<Packet>::type* from) {
  return pload1<Packet>(from);
}

/** \internal equivalent to
 * \code
 * a0 = pload1(a+0);
 * a1 = pload1(a+1);
 * a2 = pload1(a+2);
 * a3 = pload1(a+3);
 * \endcode
 * \sa pset1, pload1, ploaddup, pbroadcast2
 */
template <typename Packet>
EIGEN_DEVICE_FUNC inline void pbroadcast4(const typename unpacket_traits<Packet>::type* a, Packet& a0, Packet& a1,
                                          Packet& a2, Packet& a3) {
  a0 = pload1<Packet>(a + 0);
  a1 = pload1<Packet>(a + 1);
  a2 = pload1<Packet>(a + 2);
  a3 = pload1<Packet>(a + 3);
}

/** \internal equivalent to
 * \code
 * a0 = pload1(a+0);
 * a1 = pload1(a+1);
 * \endcode
 * \sa pset1, pload1, ploaddup, pbroadcast4
 */
template <typename Packet>
EIGEN_DEVICE_FUNC inline void pbroadcast2(const typename unpacket_traits<Packet>::type* a, Packet& a0, Packet& a1) {
  a0 = pload1<Packet>(a + 0);
  a1 = pload1<Packet>(a + 1);
}

/** \internal \brief Returns a packet with coefficients (a,a+1,...,a+packet_size-1). */
template <typename Packet>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Packet plset(const typename unpacket_traits<Packet>::type& a) {
  return a;
}

/** \internal \returns a packet with constant coefficients \a a, e.g.: (x, 0, x, 0),
     where x is the value of all 1-bits. */
template <typename Packet>
EIGEN_DEVICE_FUNC inline Packet peven_mask(const Packet& /*a*/) {
  typedef typename unpacket_traits<Packet>::type Scalar;
  const size_t n = unpacket_traits<Packet>::size;
  EIGEN_ALIGN_TO_BOUNDARY(sizeof(Packet)) Scalar elements[n];
  for (size_t i = 0; i < n; ++i) {
    memset(elements + i, ((i & 1) == 0 ? 0xff : 0), sizeof(Scalar));
  }
  return ploadu<Packet>(elements);
}

/** \internal copy the packet \a from to \a *to, \a to must be properly aligned */
template <typename Scalar, typename Packet>
EIGEN_DEVICE_FUNC inline void pstore(Scalar* to, const Packet& from) {
  (*to) = from;
}

/** \internal copy n elements of the packet \a from to \a *to, \a to must be properly aligned
 * offset indicates the starting element in which to store and
 * offset + n <= unpacket_traits::size */
template <typename Scalar, typename Packet>
EIGEN_DEVICE_FUNC inline void pstore_partial(Scalar* to, const Packet& from, const Index n, const Index offset = 0) {
  const Index packet_size = unpacket_traits<Packet>::size;
  eigen_assert(n + offset <= packet_size && "number of elements plus offset will write past end of packet");
  EIGEN_ALIGN_MAX Scalar elements[packet_size];
  pstore<Scalar>(elements, from);
  for (Index i = 0; i < numext::mini(n, packet_size - offset); i++) {
    to[i] = elements[i + offset];
  }
}

/** \internal copy the packet \a from to \a *to, (un-aligned store) */
template <typename Scalar, typename Packet>
EIGEN_DEVICE_FUNC inline void pstoreu(Scalar* to, const Packet& from) {
  (*to) = from;
}

/** \internal copy n elements of the packet \a from to \a *to, (un-aligned store) */
template <typename Scalar, typename Packet>
EIGEN_DEVICE_FUNC inline void pstoreu_partial(Scalar* to, const Packet& from, const Index n, const Index offset = 0) {
  const Index packet_size = unpacket_traits<Packet>::size;
  eigen_assert(n + offset <= packet_size && "number of elements plus offset will write past end of packet");
  EIGEN_ALIGN_MAX Scalar elements[packet_size];
  pstore<Scalar>(elements, from);
  for (Index i = 0; i < numext::mini(n, packet_size - offset); i++) {
    to[i] = elements[i + offset];
  }
}

/** \internal copy the packet \a from to \a *to, (un-aligned store with a mask)
 * There is no generic implementation. We only have implementations for specialized
 * cases. Generic case should not be called.
 */
template <typename Scalar, typename Packet>
EIGEN_DEVICE_FUNC inline std::enable_if_t<unpacket_traits<Packet>::masked_store_available, void> pstoreu(
    Scalar* to, const Packet& from, typename unpacket_traits<Packet>::mask_t umask);

template <typename Scalar, typename Packet>
EIGEN_DEVICE_FUNC inline Packet pgather(const Scalar* from, Index /*stride*/) {
  return ploadu<Packet>(from);
}

template <typename Scalar, typename Packet>
EIGEN_DEVICE_FUNC inline Packet pgather_partial(const Scalar* from, Index stride, const Index n) {
  const Index packet_size = unpacket_traits<Packet>::size;
  EIGEN_ALIGN_MAX Scalar elements[packet_size] = {Scalar(0)};
  for (Index i = 0; i < numext::mini(n, packet_size); i++) {
    elements[i] = from[i * stride];
  }
  return pload<Packet>(elements);
}

template <typename Scalar, typename Packet>
EIGEN_DEVICE_FUNC inline void pscatter(Scalar* to, const Packet& from, Index /*stride*/) {
  pstore(to, from);
}

template <typename Scalar, typename Packet>
EIGEN_DEVICE_FUNC inline void pscatter_partial(Scalar* to, const Packet& from, Index stride, const Index n) {
  const Index packet_size = unpacket_traits<Packet>::size;
  EIGEN_ALIGN_MAX Scalar elements[packet_size];
  pstore<Scalar>(elements, from);
  for (Index i = 0; i < numext::mini(n, packet_size); i++) {
    to[i * stride] = elements[i];
  }
}

/** \internal tries to do cache prefetching of \a addr */
template <typename Scalar>
EIGEN_DEVICE_FUNC inline void prefetch(const Scalar* addr) {
#if defined(EIGEN_HIP_DEVICE_COMPILE)
  // do nothing
#elif defined(EIGEN_CUDA_ARCH)
#if defined(__LP64__) || EIGEN_OS_WIN64
  // 64-bit pointer operand constraint for inlined asm
  asm(" prefetch.L1 [ %1 ];" : "=l"(addr) : "l"(addr));
#else
  // 32-bit pointer operand constraint for inlined asm
  asm(" prefetch.L1 [ %1 ];" : "=r"(addr) : "r"(addr));
#endif
#elif (!EIGEN_COMP_MSVC) && (EIGEN_COMP_GNUC || EIGEN_COMP_CLANG || EIGEN_COMP_ICC)
  __builtin_prefetch(addr);
#endif
}

/** \internal \returns the reversed elements of \a a*/
template <typename Packet>
EIGEN_DEVICE_FUNC inline Packet preverse(const Packet& a) {
  return a;
}

/** \internal \returns \a a with real and imaginary part flipped (for complex type only) */
template <typename Packet>
EIGEN_DEVICE_FUNC inline Packet pcplxflip(const Packet& a) {
  return Packet(numext::imag(a), numext::real(a));
}

/**************************
 * Special math functions
 ***************************/

/** \internal \returns the sine of \a a (coeff-wise) */
template <typename Packet>
EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet psin(const Packet& a) {
  EIGEN_USING_STD(sin);
  return sin(a);
}

/** \internal \returns the cosine of \a a (coeff-wise) */
template <typename Packet>
EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet pcos(const Packet& a) {
  EIGEN_USING_STD(cos);
  return cos(a);
}

/** \internal \returns the tan of \a a (coeff-wise) */
template <typename Packet>
EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet ptan(const Packet& a) {
  EIGEN_USING_STD(tan);
  return tan(a);
}

/** \internal \returns the arc sine of \a a (coeff-wise) */
template <typename Packet>
EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet pasin(const Packet& a) {
  EIGEN_USING_STD(asin);
  return asin(a);
}

/** \internal \returns the arc cosine of \a a (coeff-wise) */
template <typename Packet>
EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet pacos(const Packet& a) {
  EIGEN_USING_STD(acos);
  return acos(a);
}

/** \internal \returns the hyperbolic sine of \a a (coeff-wise) */
template <typename Packet>
EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet psinh(const Packet& a) {
  EIGEN_USING_STD(sinh);
  return sinh(a);
}

/** \internal \returns the hyperbolic cosine of \a a (coeff-wise) */
template <typename Packet>
EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet pcosh(const Packet& a) {
  EIGEN_USING_STD(cosh);
  return cosh(a);
}

/** \internal \returns the arc tangent of \a a (coeff-wise) */
template <typename Packet>
EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet patan(const Packet& a) {
  EIGEN_USING_STD(atan);
  return atan(a);
}

/** \internal \returns the hyperbolic tan of \a a (coeff-wise) */
template <typename Packet>
EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet ptanh(const Packet& a) {
  EIGEN_USING_STD(tanh);
  return tanh(a);
}

/** \internal \returns the arc tangent of \a a (coeff-wise) */
template <typename Packet>
EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet patanh(const Packet& a) {
  EIGEN_USING_STD(atanh);
  return atanh(a);
}

/** \internal \returns the exp of \a a (coeff-wise) */
template <typename Packet>
EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet pexp(const Packet& a) {
  EIGEN_USING_STD(exp);
  return exp(a);
}

/** \internal \returns the expm1 of \a a (coeff-wise) */
template <typename Packet>
EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet pexpm1(const Packet& a) {
  return numext::expm1(a);
}

/** \internal \returns the log of \a a (coeff-wise) */
template <typename Packet>
EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet plog(const Packet& a) {
  EIGEN_USING_STD(log);
  return log(a);
}

/** \internal \returns the log1p of \a a (coeff-wise) */
template <typename Packet>
EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet plog1p(const Packet& a) {
  return numext::log1p(a);
}

/** \internal \returns the log10 of \a a (coeff-wise) */
template <typename Packet>
EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet plog10(const Packet& a) {
  EIGEN_USING_STD(log10);
  return log10(a);
}

/** \internal \returns the log10 of \a a (coeff-wise) */
template <typename Packet>
EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet plog2(const Packet& a) {
  using Scalar = typename internal::unpacket_traits<Packet>::type;
  using RealScalar = typename NumTraits<Scalar>::Real;
  return pmul(pset1<Packet>(Scalar(RealScalar(EIGEN_LOG2E))), plog(a));
}

/** \internal \returns the square-root of \a a (coeff-wise) */
template <typename Packet>
EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet psqrt(const Packet& a) {
  return numext::sqrt(a);
}

/** \internal \returns the cube-root of \a a (coeff-wise) */
template <typename Packet>
EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet pcbrt(const Packet& a) {
  return numext::cbrt(a);
}

/** \internal \returns the rounded value of \a a (coeff-wise) */
template <typename Packet>
EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet pround(const Packet& a) {
  using numext::round;
  return round(a);
}

/** \internal \returns the floor of \a a (coeff-wise) */
template <typename Packet>
EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet pfloor(const Packet& a) {
  using numext::floor;
  return floor(a);
}

/** \internal \returns the rounded value of \a a (coeff-wise) with current
 * rounding mode */
template <typename Packet>
EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet print(const Packet& a) {
  using numext::rint;
  return rint(a);
}

/** \internal \returns the ceil of \a a (coeff-wise) */
template <typename Packet>
EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet pceil(const Packet& a) {
  using numext::ceil;
  return ceil(a);
}

template <typename Packet, typename EnableIf = void>
struct psign_impl {
  static EIGEN_DEVICE_FUNC inline Packet run(const Packet& a) { return numext::sign(a); }
};

/** \internal \returns the sign of \a a (coeff-wise) */
template <typename Packet>
EIGEN_DEVICE_FUNC inline Packet psign(const Packet& a) {
  return psign_impl<Packet>::run(a);
}

template <>
EIGEN_DEVICE_FUNC inline bool psign(const bool& a) {
  return a;
}

/** \internal \returns the first element of a packet */
template <typename Packet>
EIGEN_DEVICE_FUNC inline typename unpacket_traits<Packet>::type pfirst(const Packet& a) {
  return a;
}

/** \internal \returns the sum of the elements of upper and lower half of \a a if \a a is larger than 4.
 * For a packet {a0, a1, a2, a3, a4, a5, a6, a7}, it returns a half packet {a0+a4, a1+a5, a2+a6, a3+a7}
 * For packet-size smaller or equal to 4, this boils down to a noop.
 */
template <typename Packet>
EIGEN_DEVICE_FUNC inline std::conditional_t<(unpacket_traits<Packet>::size % 8) == 0,
                                            typename unpacket_traits<Packet>::half, Packet>
predux_half_dowto4(const Packet& a) {
  return a;
}

// Slow generic implementation of Packet reduction.
template <typename Packet, typename Op>
EIGEN_DEVICE_FUNC inline typename unpacket_traits<Packet>::type predux_helper(const Packet& a, Op op) {
  typedef typename unpacket_traits<Packet>::type Scalar;
  const size_t n = unpacket_traits<Packet>::size;
  EIGEN_ALIGN_TO_BOUNDARY(sizeof(Packet)) Scalar elements[n];
  pstoreu<Scalar>(elements, a);
  for (size_t k = n / 2; k > 0; k /= 2) {
    for (size_t i = 0; i < k; ++i) {
      elements[i] = op(elements[i], elements[i + k]);
    }
  }
  return elements[0];
}

/** \internal \returns the sum of the elements of \a a*/
template <typename Packet>
EIGEN_DEVICE_FUNC inline typename unpacket_traits<Packet>::type predux(const Packet& a) {
  return a;
}

/** \internal \returns the product of the elements of \a a */
template <typename Packet>
EIGEN_DEVICE_FUNC inline typename unpacket_traits<Packet>::type predux_mul(const Packet& a) {
  typedef typename unpacket_traits<Packet>::type Scalar;
  return predux_helper(a, EIGEN_BINARY_OP_NAN_PROPAGATION(Scalar, (pmul<Scalar>)));
}

/** \internal \returns the min of the elements of \a a */
template <typename Packet>
EIGEN_DEVICE_FUNC inline typename unpacket_traits<Packet>::type predux_min(const Packet& a) {
  typedef typename unpacket_traits<Packet>::type Scalar;
  return predux_helper(a, EIGEN_BINARY_OP_NAN_PROPAGATION(Scalar, (pmin<PropagateFast, Scalar>)));
}

template <int NaNPropagation, typename Packet>
EIGEN_DEVICE_FUNC inline typename unpacket_traits<Packet>::type predux_min(const Packet& a) {
  typedef typename unpacket_traits<Packet>::type Scalar;
  return predux_helper(a, EIGEN_BINARY_OP_NAN_PROPAGATION(Scalar, (pmin<NaNPropagation, Scalar>)));
}

/** \internal \returns the min of the elements of \a a */
template <typename Packet>
EIGEN_DEVICE_FUNC inline typename unpacket_traits<Packet>::type predux_max(const Packet& a) {
  typedef typename unpacket_traits<Packet>::type Scalar;
  return predux_helper(a, EIGEN_BINARY_OP_NAN_PROPAGATION(Scalar, (pmax<PropagateFast, Scalar>)));
}

template <int NaNPropagation, typename Packet>
EIGEN_DEVICE_FUNC inline typename unpacket_traits<Packet>::type predux_max(const Packet& a) {
  typedef typename unpacket_traits<Packet>::type Scalar;
  return predux_helper(a, EIGEN_BINARY_OP_NAN_PROPAGATION(Scalar, (pmax<NaNPropagation, Scalar>)));
}

#undef EIGEN_BINARY_OP_NAN_PROPAGATION

/** \internal \returns true if all coeffs of \a a means "true"
 * It is supposed to be called on values returned by pcmp_*.
 */
// not needed yet
// template<typename Packet> EIGEN_DEVICE_FUNC inline bool predux_all(const Packet& a)
// { return bool(a); }

/** \internal \returns true if any coeffs of \a a means "true"
 * It is supposed to be called on values returned by pcmp_*.
 */
template <typename Packet>
EIGEN_DEVICE_FUNC inline bool predux_any(const Packet& a) {
  // Dirty but generic implementation where "true" is assumed to be non 0 and all the sames.
  // It is expected that "true" is either:
  //  - Scalar(1)
  //  - bits full of ones (NaN for floats),
  //  - or first bit equals to 1 (1 for ints, smallest denormal for floats).
  // For all these cases, taking the sum is just fine, and this boils down to a no-op for scalars.
  typedef typename unpacket_traits<Packet>::type Scalar;
  return numext::not_equal_strict(predux(a), Scalar(0));
}

/***************************************************************************
 * The following functions might not have to be overwritten for vectorized types
 ***************************************************************************/

// FMA instructions.
/** \internal \returns a * b + c (coeff-wise) */
template <typename Packet>
EIGEN_DEVICE_FUNC inline Packet pmadd(const Packet& a, const Packet& b, const Packet& c) {
  return padd(pmul(a, b), c);
}

/** \internal \returns a * b - c (coeff-wise) */
template <typename Packet>
EIGEN_DEVICE_FUNC inline Packet pmsub(const Packet& a, const Packet& b, const Packet& c) {
  return psub(pmul(a, b), c);
}

/** \internal \returns -(a * b) + c (coeff-wise) */
template <typename Packet>
EIGEN_DEVICE_FUNC inline Packet pnmadd(const Packet& a, const Packet& b, const Packet& c) {
  return psub(c, pmul(a, b));
}

/** \internal \returns -((a * b + c) (coeff-wise) */
template <typename Packet>
EIGEN_DEVICE_FUNC inline Packet pnmsub(const Packet& a, const Packet& b, const Packet& c) {
  return pnegate(pmadd(a, b, c));
}

/** \internal copy a packet with constant coefficient \a a (e.g., [a,a,a,a]) to \a *to. \a to must be 16 bytes aligned
 */
// NOTE: this function must really be templated on the packet type (think about different packet types for the same
// scalar type)
template <typename Packet>
inline void pstore1(typename unpacket_traits<Packet>::type* to, const typename unpacket_traits<Packet>::type& a) {
  pstore(to, pset1<Packet>(a));
}

/** \internal \returns a packet version of \a *from.
 * The pointer \a from must be aligned on a \a Alignment bytes boundary. */
template <typename Packet, int Alignment>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE Packet ploadt(const typename unpacket_traits<Packet>::type* from) {
  if (Alignment >= unpacket_traits<Packet>::alignment)
    return pload<Packet>(from);
  else
    return ploadu<Packet>(from);
}

/** \internal \returns n elements of a packet version of \a *from.
 * The pointer \a from must be aligned on a \a Alignment bytes boundary. */
template <typename Packet, int Alignment>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE Packet ploadt_partial(const typename unpacket_traits<Packet>::type* from,
                                                            const Index n, const Index offset = 0) {
  if (Alignment >= unpacket_traits<Packet>::alignment)
    return pload_partial<Packet>(from, n, offset);
  else
    return ploadu_partial<Packet>(from, n, offset);
}

/** \internal copy the packet \a from to \a *to.
 * The pointer \a from must be aligned on a \a Alignment bytes boundary. */
template <typename Scalar, typename Packet, int Alignment>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE void pstoret(Scalar* to, const Packet& from) {
  if (Alignment >= unpacket_traits<Packet>::alignment)
    pstore(to, from);
  else
    pstoreu(to, from);
}

/** \internal copy n elements of the packet \a from to \a *to.
 * The pointer \a from must be aligned on a \a Alignment bytes boundary. */
template <typename Scalar, typename Packet, int Alignment>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE void pstoret_partial(Scalar* to, const Packet& from, const Index n,
                                                           const Index offset = 0) {
  if (Alignment >= unpacket_traits<Packet>::alignment)
    pstore_partial(to, from, n, offset);
  else
    pstoreu_partial(to, from, n, offset);
}

/** \internal \returns a packet version of \a *from.
 * Unlike ploadt, ploadt_ro takes advantage of the read-only memory path on the
 * hardware if available to speedup the loading of data that won't be modified
 * by the current computation.
 */
template <typename Packet, int LoadMode>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE Packet ploadt_ro(const typename unpacket_traits<Packet>::type* from) {
  return ploadt<Packet, LoadMode>(from);
}

/***************************************************************************
 * Fast complex products (GCC generates a function call which is very slow)
 ***************************************************************************/

// Eigen+CUDA does not support complexes.
#if !defined(EIGEN_GPUCC)

template <>
inline std::complex<float> pmul(const std::complex<float>& a, const std::complex<float>& b) {
  return std::complex<float>(a.real() * b.real() - a.imag() * b.imag(), a.imag() * b.real() + a.real() * b.imag());
}

template <>
inline std::complex<double> pmul(const std::complex<double>& a, const std::complex<double>& b) {
  return std::complex<double>(a.real() * b.real() - a.imag() * b.imag(), a.imag() * b.real() + a.real() * b.imag());
}

#endif

/***************************************************************************
 * PacketBlock, that is a collection of N packets where the number of words
 * in the packet is a multiple of N.
 ***************************************************************************/
template <typename Packet, int N = unpacket_traits<Packet>::size>
struct PacketBlock {
  Packet packet[N];
};

template <typename Packet>
EIGEN_DEVICE_FUNC inline void ptranspose(PacketBlock<Packet, 1>& /*kernel*/) {
  // Nothing to do in the scalar case, i.e. a 1x1 matrix.
}

/***************************************************************************
 * Selector, i.e. vector of N boolean values used to select (i.e. blend)
 * words from 2 packets.
 ***************************************************************************/
template <size_t N>
struct Selector {
  bool select[N];
};

template <typename Packet>
EIGEN_DEVICE_FUNC inline Packet pblend(const Selector<unpacket_traits<Packet>::size>& ifPacket,
                                       const Packet& thenPacket, const Packet& elsePacket) {
  return ifPacket.select[0] ? thenPacket : elsePacket;
}

/** \internal \returns 1 / a (coeff-wise) */
template <typename Packet>
EIGEN_DEVICE_FUNC inline Packet preciprocal(const Packet& a) {
  using Scalar = typename unpacket_traits<Packet>::type;
  return pdiv(pset1<Packet>(Scalar(1)), a);
}

/** \internal \returns the reciprocal square-root of \a a (coeff-wise) */
template <typename Packet>
EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet prsqrt(const Packet& a) {
  return preciprocal<Packet>(psqrt(a));
}

template <typename Packet, bool IsScalar = is_scalar<Packet>::value,
          bool IsInteger = NumTraits<typename unpacket_traits<Packet>::type>::IsInteger>
struct psignbit_impl;
template <typename Packet, bool IsInteger>
struct psignbit_impl<Packet, true, IsInteger> {
  EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE static constexpr Packet run(const Packet& a) { return numext::signbit(a); }
};
template <typename Packet>
struct psignbit_impl<Packet, false, false> {
  // generic implementation if not specialized in PacketMath.h
  // slower than arithmetic shift
  typedef typename unpacket_traits<Packet>::type Scalar;
  EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE static Packet run(const Packet& a) {
    const Packet cst_pos_one = pset1<Packet>(Scalar(1));
    const Packet cst_neg_one = pset1<Packet>(Scalar(-1));
    return pcmp_eq(por(pand(a, cst_neg_one), cst_pos_one), cst_neg_one);
  }
};
template <typename Packet>
struct psignbit_impl<Packet, false, true> {
  // generic implementation for integer packets
  EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE static constexpr Packet run(const Packet& a) { return pcmp_lt(a, pzero(a)); }
};
/** \internal \returns the sign bit of \a a as a bitmask*/
template <typename Packet>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE constexpr Packet psignbit(const Packet& a) {
  return psignbit_impl<Packet>::run(a);
}

/** \internal \returns the 2-argument arc tangent of \a y and \a x (coeff-wise) */
template <typename Packet, std::enable_if_t<is_scalar<Packet>::value, int> = 0>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE Packet patan2(const Packet& y, const Packet& x) {
  return numext::atan2(y, x);
}

/** \internal \returns the 2-argument arc tangent of \a y and \a x (coeff-wise) */
template <typename Packet, std::enable_if_t<!is_scalar<Packet>::value, int> = 0>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE Packet patan2(const Packet& y, const Packet& x) {
  typedef typename internal::unpacket_traits<Packet>::type Scalar;

  // See https://en.cppreference.com/w/cpp/numeric/math/atan2
  // for how corner cases are supposed to be handled according to the
  // IEEE floating-point standard (IEC 60559).
  const Packet kSignMask = pset1<Packet>(-Scalar(0));
  const Packet kZero = pzero(x);
  const Packet kOne = pset1<Packet>(Scalar(1));
  const Packet kPi = pset1<Packet>(Scalar(EIGEN_PI));

  const Packet x_has_signbit = psignbit(x);
  const Packet y_signmask = pand(y, kSignMask);
  const Packet x_signmask = pand(x, kSignMask);
  const Packet result_signmask = pxor(y_signmask, x_signmask);
  const Packet shift = por(pand(x_has_signbit, kPi), y_signmask);

  const Packet x_and_y_are_same = pcmp_eq(pabs(x), pabs(y));
  const Packet x_and_y_are_zero = pcmp_eq(por(x, y), kZero);

  Packet arg = pdiv(y, x);
  arg = pselect(x_and_y_are_same, por(kOne, result_signmask), arg);
  arg = pselect(x_and_y_are_zero, result_signmask, arg);

  Packet result = patan(arg);
  result = padd(result, shift);
  return result;
}

/** \internal \returns the argument of \a a as a complex number */
template <typename Packet, std::enable_if_t<is_scalar<Packet>::value, int> = 0>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE Packet pcarg(const Packet& a) {
  return Packet(numext::arg(a));
}

/** \internal \returns the argument of \a a as a complex number */
template <typename Packet, std::enable_if_t<!is_scalar<Packet>::value, int> = 0>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE Packet pcarg(const Packet& a) {
  EIGEN_STATIC_ASSERT(NumTraits<typename unpacket_traits<Packet>::type>::IsComplex,
                      THIS METHOD IS FOR COMPLEX TYPES ONLY)
  using RealPacket = typename unpacket_traits<Packet>::as_real;
  // a                                              // r     i    r     i    ...
  RealPacket aflip = pcplxflip(a).v;                // i     r    i     r    ...
  RealPacket result = patan2(aflip, a.v);           // atan2 crap atan2 crap ...
  return (Packet)pand(result, peven_mask(result));  // atan2 0    atan2 0    ...
}

}  // end namespace internal

}  // end namespace Eigen

#endif  // EIGEN_GENERIC_PACKET_MATH_H