test/packetmath.cpp

// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2008-2009 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/.

#include "packetmath_test_shared.h"
#include "random_without_cast_overflow.h"

template <typename T>
inline T REF_ADD(const T& a, const T& b) {
  return a + b;
}
template <typename T>
inline T REF_SUB(const T& a, const T& b) {
  return a - b;
}
template <typename T>
inline T REF_MUL(const T& a, const T& b) {
  return a * b;
}
template <typename T>
inline T REF_DIV(const T& a, const T& b) {
  return a / b;
}
template <typename T>
inline T REF_ABS_DIFF(const T& a, const T& b) {
  return a > b ? a - b : b - a;
}

// Specializations for bool.
template <>
inline bool REF_ADD(const bool& a, const bool& b) {
  return a || b;
}
template <>
inline bool REF_SUB(const bool& a, const bool& b) {
  return a ^ b;
}
template <>
inline bool REF_MUL(const bool& a, const bool& b) {
  return a && b;
}

// Uses pcast to cast from one array to another.
template <typename SrcPacket, typename TgtPacket, int SrcCoeffRatio, int TgtCoeffRatio>
struct pcast_array;

template <typename SrcPacket, typename TgtPacket, int TgtCoeffRatio>
struct pcast_array<SrcPacket, TgtPacket, 1, TgtCoeffRatio> {
  typedef typename internal::unpacket_traits<SrcPacket>::type SrcScalar;
  typedef typename internal::unpacket_traits<TgtPacket>::type TgtScalar;
  static void cast(const SrcScalar* src, size_t size, TgtScalar* dst) {
    static const int SrcPacketSize = internal::unpacket_traits<SrcPacket>::size;
    static const int TgtPacketSize = internal::unpacket_traits<TgtPacket>::size;
    size_t i;
    for (i = 0; i < size && i + SrcPacketSize <= size; i += TgtPacketSize) {
      internal::pstoreu(dst + i, internal::pcast<SrcPacket, TgtPacket>(internal::ploadu<SrcPacket>(src + i)));
    }
    // Leftovers that cannot be loaded into a packet.
    for (; i < size; ++i) {
      dst[i] = static_cast<TgtScalar>(src[i]);
    }
  }
};

template <typename SrcPacket, typename TgtPacket>
struct pcast_array<SrcPacket, TgtPacket, 2, 1> {
  static void cast(const typename internal::unpacket_traits<SrcPacket>::type* src, size_t size,
                   typename internal::unpacket_traits<TgtPacket>::type* dst) {
    static const int SrcPacketSize = internal::unpacket_traits<SrcPacket>::size;
    static const int TgtPacketSize = internal::unpacket_traits<TgtPacket>::size;
    for (size_t i = 0; i < size; i += TgtPacketSize) {
      SrcPacket a = internal::ploadu<SrcPacket>(src + i);
      SrcPacket b = internal::ploadu<SrcPacket>(src + i + SrcPacketSize);
      internal::pstoreu(dst + i, internal::pcast<SrcPacket, TgtPacket>(a, b));
    }
  }
};

template <typename SrcPacket, typename TgtPacket>
struct pcast_array<SrcPacket, TgtPacket, 4, 1> {
  static void cast(const typename internal::unpacket_traits<SrcPacket>::type* src, size_t size,
                   typename internal::unpacket_traits<TgtPacket>::type* dst) {
    static const int SrcPacketSize = internal::unpacket_traits<SrcPacket>::size;
    static const int TgtPacketSize = internal::unpacket_traits<TgtPacket>::size;
    for (size_t i = 0; i < size; i += TgtPacketSize) {
      SrcPacket a = internal::ploadu<SrcPacket>(src + i);
      SrcPacket b = internal::ploadu<SrcPacket>(src + i + SrcPacketSize);
      SrcPacket c = internal::ploadu<SrcPacket>(src + i + 2 * SrcPacketSize);
      SrcPacket d = internal::ploadu<SrcPacket>(src + i + 3 * SrcPacketSize);
      internal::pstoreu(dst + i, internal::pcast<SrcPacket, TgtPacket>(a, b, c, d));
    }
  }
};

template <typename SrcPacket, typename TgtPacket>
struct pcast_array<SrcPacket, TgtPacket, 8, 1> {
  static void cast(const typename internal::unpacket_traits<SrcPacket>::type* src, size_t size,
                   typename internal::unpacket_traits<TgtPacket>::type* dst) {
    static const int SrcPacketSize = internal::unpacket_traits<SrcPacket>::size;
    static const int TgtPacketSize = internal::unpacket_traits<TgtPacket>::size;
    for (size_t i = 0; i < size; i += TgtPacketSize) {
      SrcPacket a = internal::ploadu<SrcPacket>(src + i);
      SrcPacket b = internal::ploadu<SrcPacket>(src + i + SrcPacketSize);
      SrcPacket c = internal::ploadu<SrcPacket>(src + i + 2 * SrcPacketSize);
      SrcPacket d = internal::ploadu<SrcPacket>(src + i + 3 * SrcPacketSize);
      SrcPacket e = internal::ploadu<SrcPacket>(src + i + 4 * SrcPacketSize);
      SrcPacket f = internal::ploadu<SrcPacket>(src + i + 5 * SrcPacketSize);
      SrcPacket g = internal::ploadu<SrcPacket>(src + i + 6 * SrcPacketSize);
      SrcPacket h = internal::ploadu<SrcPacket>(src + i + 7 * SrcPacketSize);
      internal::pstoreu(dst + i, internal::pcast<SrcPacket, TgtPacket>(a, b, c, d, e, f, g, h));
    }
  }
};

template <typename SrcPacket, typename TgtPacket, int SrcCoeffRatio, int TgtCoeffRatio, bool CanCast = false>
struct test_cast_helper;

template <typename SrcPacket, typename TgtPacket, int SrcCoeffRatio, int TgtCoeffRatio>
struct test_cast_helper<SrcPacket, TgtPacket, SrcCoeffRatio, TgtCoeffRatio, false> {
  static void run() {}
};

template <typename SrcPacket, typename TgtPacket, int SrcCoeffRatio, int TgtCoeffRatio>
struct test_cast_helper<SrcPacket, TgtPacket, SrcCoeffRatio, TgtCoeffRatio, true> {
  static void run() {
    typedef typename internal::unpacket_traits<SrcPacket>::type SrcScalar;
    typedef typename internal::unpacket_traits<TgtPacket>::type TgtScalar;
    static const int SrcPacketSize = internal::unpacket_traits<SrcPacket>::size;
    static const int TgtPacketSize = internal::unpacket_traits<TgtPacket>::size;
    static const int BlockSize = SrcPacketSize * SrcCoeffRatio;
    eigen_assert(BlockSize == TgtPacketSize * TgtCoeffRatio && "Packet sizes and cast ratios are mismatched.");

    static const int DataSize = 10 * BlockSize;
    EIGEN_ALIGN_MAX SrcScalar data1[DataSize];
    EIGEN_ALIGN_MAX TgtScalar data2[DataSize];
    EIGEN_ALIGN_MAX TgtScalar ref[DataSize];

    // Construct a packet of scalars that will not overflow when casting
    for (int i = 0; i < DataSize; ++i) {
      data1[i] = internal::random_without_cast_overflow<SrcScalar, TgtScalar>::value();
    }

    for (int i = 0; i < DataSize; ++i) {
      ref[i] = static_cast<const TgtScalar>(data1[i]);
    }

    pcast_array<SrcPacket, TgtPacket, SrcCoeffRatio, TgtCoeffRatio>::cast(data1, DataSize, data2);

    VERIFY(test::areApprox(ref, data2, DataSize) && "internal::pcast<>");
  }
};

template <typename SrcPacket, typename TgtPacket>
struct test_cast {
  static void run() {
    typedef typename internal::unpacket_traits<SrcPacket>::type SrcScalar;
    typedef typename internal::unpacket_traits<TgtPacket>::type TgtScalar;
    typedef typename internal::type_casting_traits<SrcScalar, TgtScalar> TypeCastingTraits;
    static const int SrcCoeffRatio = TypeCastingTraits::SrcCoeffRatio;
    static const int TgtCoeffRatio = TypeCastingTraits::TgtCoeffRatio;
    static const int SrcPacketSize = internal::unpacket_traits<SrcPacket>::size;
    static const int TgtPacketSize = internal::unpacket_traits<TgtPacket>::size;
    static const bool HasCast =
        internal::unpacket_traits<SrcPacket>::vectorizable && internal::unpacket_traits<TgtPacket>::vectorizable &&
        TypeCastingTraits::VectorizedCast && (SrcPacketSize * SrcCoeffRatio == TgtPacketSize * TgtCoeffRatio);
    test_cast_helper<SrcPacket, TgtPacket, SrcCoeffRatio, TgtCoeffRatio, HasCast>::run();
  }
};

template <typename SrcPacket, typename TgtScalar,
          typename TgtPacket = typename internal::packet_traits<TgtScalar>::type,
          bool Vectorized = internal::packet_traits<TgtScalar>::Vectorizable,
          bool HasHalf = !internal::is_same<typename internal::unpacket_traits<TgtPacket>::half, TgtPacket>::value>
struct test_cast_runner;

template <typename SrcPacket, typename TgtScalar, typename TgtPacket>
struct test_cast_runner<SrcPacket, TgtScalar, TgtPacket, true, false> {
  static void run() { test_cast<SrcPacket, TgtPacket>::run(); }
};

template <typename SrcPacket, typename TgtScalar, typename TgtPacket>
struct test_cast_runner<SrcPacket, TgtScalar, TgtPacket, true, true> {
  static void run() {
    test_cast<SrcPacket, TgtPacket>::run();
    test_cast_runner<SrcPacket, TgtScalar, typename internal::unpacket_traits<TgtPacket>::half>::run();
  }
};

template <typename SrcPacket, typename TgtScalar, typename TgtPacket>
struct test_cast_runner<SrcPacket, TgtScalar, TgtPacket, false, false> {
  static void run() {}
};

template <typename Scalar, typename Packet, typename EnableIf = void>
struct packetmath_pcast_ops_runner {
  static void run() {
    test_cast_runner<Packet, float>::run();
    test_cast_runner<Packet, double>::run();
    test_cast_runner<Packet, int8_t>::run();
    test_cast_runner<Packet, uint8_t>::run();
    test_cast_runner<Packet, int16_t>::run();
    test_cast_runner<Packet, uint16_t>::run();
    test_cast_runner<Packet, int32_t>::run();
    test_cast_runner<Packet, uint32_t>::run();
    test_cast_runner<Packet, int64_t>::run();
    test_cast_runner<Packet, uint64_t>::run();
    test_cast_runner<Packet, bool>::run();
    test_cast_runner<Packet, std::complex<float> >::run();
    test_cast_runner<Packet, std::complex<double> >::run();
    test_cast_runner<Packet, half>::run();
    test_cast_runner<Packet, bfloat16>::run();
  }
};

// Only some types support cast from std::complex<>.
template <typename Scalar, typename Packet>
struct packetmath_pcast_ops_runner<Scalar, Packet, typename internal::enable_if<NumTraits<Scalar>::IsComplex>::type> {
  static void run() {
    test_cast_runner<Packet, std::complex<float> >::run();
    test_cast_runner<Packet, std::complex<double> >::run();
    test_cast_runner<Packet, half>::run();
    test_cast_runner<Packet, bfloat16>::run();
  }
};

template <typename Scalar, typename Packet>
void packetmath_boolean_mask_ops() {
  const int PacketSize = internal::unpacket_traits<Packet>::size;
  const int size = 2 * PacketSize;
  EIGEN_ALIGN_MAX Scalar data1[size];
  EIGEN_ALIGN_MAX Scalar data2[size];
  EIGEN_ALIGN_MAX Scalar ref[size];

  for (int i = 0; i < size; ++i) {
    data1[i] = internal::random<Scalar>();
  }
  CHECK_CWISE1(internal::ptrue, internal::ptrue);
  CHECK_CWISE2_IF(true, internal::pandnot, internal::pandnot);
  for (int i = 0; i < PacketSize; ++i) {
    data1[i] = Scalar(i);
    data1[i + PacketSize] = internal::random<bool>() ? data1[i] : Scalar(0);
  }

  CHECK_CWISE2_IF(true, internal::pcmp_eq, internal::pcmp_eq);

  //Test (-0) == (0) for signed operations
  for (int i = 0; i < PacketSize; ++i) {
    data1[i] = Scalar(-0.0);
    data1[i + PacketSize] = internal::random<bool>() ? data1[i] : Scalar(0);
  }
  CHECK_CWISE2_IF(true, internal::pcmp_eq, internal::pcmp_eq);

  //Test NaN
  for (int i = 0; i < PacketSize; ++i) {
    data1[i] = std::numeric_limits<Scalar>::quiet_NaN();
    data1[i + PacketSize] = internal::random<bool>() ? data1[i] : Scalar(0);
  }
  CHECK_CWISE2_IF(true, internal::pcmp_eq, internal::pcmp_eq);
}

// Packet16b representing bool does not support ptrue, pandnot or pcmp_eq, since the scalar path
// (for some compilers) compute the bitwise and with 0x1 of the results to keep the value in [0,1].
template<>
void packetmath_boolean_mask_ops<bool, internal::packet_traits<bool>::type>() {}

template <typename Scalar, typename Packet>
void packetmath_minus_zero_add() {
  const int PacketSize = internal::unpacket_traits<Packet>::size;
  const int size = 2 * PacketSize;
  EIGEN_ALIGN_MAX Scalar data1[size];
  EIGEN_ALIGN_MAX Scalar data2[size];
  EIGEN_ALIGN_MAX Scalar ref[size];

  for (int i = 0; i < PacketSize; ++i) {
    data1[i] = Scalar(-0.0);
    data1[i + PacketSize] = Scalar(-0.0);
  }
  CHECK_CWISE2_IF(internal::packet_traits<Scalar>::HasAdd, REF_ADD, internal::padd);
}


template <typename Scalar, typename Packet>
void packetmath() {
  typedef internal::packet_traits<Scalar> PacketTraits;
  const int PacketSize = internal::unpacket_traits<Packet>::size;
  typedef typename NumTraits<Scalar>::Real RealScalar;

  if (g_first_pass)
    std::cerr << "=== Testing packet of type '" << typeid(Packet).name() << "' and scalar type '"
              << typeid(Scalar).name() << "' and size '" << PacketSize << "' ===\n";

  const int max_size = PacketSize > 4 ? PacketSize : 4;
  const int size = PacketSize * max_size;
  EIGEN_ALIGN_MAX Scalar data1[size];
  EIGEN_ALIGN_MAX Scalar data2[size];
  EIGEN_ALIGN_MAX Scalar data3[size];
  EIGEN_ALIGN_MAX Scalar ref[size];
  RealScalar refvalue = RealScalar(0);
  for (int i = 0; i < size; ++i) {
    data1[i] = internal::random<Scalar>() / RealScalar(PacketSize);
    data2[i] = internal::random<Scalar>() / RealScalar(PacketSize);
    refvalue = (std::max)(refvalue, numext::abs(data1[i]));
  }

  internal::pstore(data2, internal::pload<Packet>(data1));
  VERIFY(test::areApprox(data1, data2, PacketSize) && "aligned load/store");

  for (int offset = 0; offset < PacketSize; ++offset) {
    internal::pstore(data2, internal::ploadu<Packet>(data1 + offset));
    VERIFY(test::areApprox(data1 + offset, data2, PacketSize) && "internal::ploadu");
  }

  for (int offset = 0; offset < PacketSize; ++offset) {
    internal::pstoreu(data2 + offset, internal::pload<Packet>(data1));
    VERIFY(test::areApprox(data1, data2 + offset, PacketSize) && "internal::pstoreu");
  }

  if (internal::unpacket_traits<Packet>::masked_load_available) {
    test::packet_helper<internal::unpacket_traits<Packet>::masked_load_available, Packet> h;
    unsigned long long max_umask = (0x1ull << PacketSize);

    for (int offset = 0; offset < PacketSize; ++offset) {
      for (unsigned long long umask = 0; umask < max_umask; ++umask) {
        h.store(data2, h.load(data1 + offset, umask));
        for (int k = 0; k < PacketSize; ++k) data3[k] = ((umask & (0x1ull << k)) >> k) ? data1[k + offset] : Scalar(0);
        VERIFY(test::areApprox(data3, data2, PacketSize) && "internal::ploadu masked");
      }
    }
  }

  if (internal::unpacket_traits<Packet>::masked_store_available) {
    test::packet_helper<internal::unpacket_traits<Packet>::masked_store_available, Packet> h;
    unsigned long long max_umask = (0x1ull << PacketSize);

    for (int offset = 0; offset < PacketSize; ++offset) {
      for (unsigned long long umask = 0; umask < max_umask; ++umask) {
        internal::pstore(data2, internal::pset1<Packet>(Scalar(0)));
        h.store(data2, h.loadu(data1 + offset), umask);
        for (int k = 0; k < PacketSize; ++k) data3[k] = ((umask & (0x1ull << k)) >> k) ? data1[k + offset] : Scalar(0);
        VERIFY(test::areApprox(data3, data2, PacketSize) && "internal::pstoreu masked");
      }
    }
  }

  VERIFY((!PacketTraits::Vectorizable) || PacketTraits::HasAdd);
  VERIFY((!PacketTraits::Vectorizable) || PacketTraits::HasSub);
  VERIFY((!PacketTraits::Vectorizable) || PacketTraits::HasMul);

  CHECK_CWISE2_IF(PacketTraits::HasAdd, REF_ADD, internal::padd);
  CHECK_CWISE2_IF(PacketTraits::HasSub, REF_SUB, internal::psub);
  CHECK_CWISE2_IF(PacketTraits::HasMul, REF_MUL, internal::pmul);
  CHECK_CWISE2_IF(PacketTraits::HasDiv, REF_DIV, internal::pdiv);

  if (PacketTraits::HasNegate) CHECK_CWISE1(internal::negate, internal::pnegate);
  CHECK_CWISE1(numext::conj, internal::pconj);

  for (int offset = 0; offset < 3; ++offset) {
    for (int i = 0; i < PacketSize; ++i) ref[i] = data1[offset];
    internal::pstore(data2, internal::pset1<Packet>(data1[offset]));
    VERIFY(test::areApprox(ref, data2, PacketSize) && "internal::pset1");
  }

  {
    for (int i = 0; i < PacketSize * 4; ++i) ref[i] = data1[i / PacketSize];
    Packet A0, A1, A2, A3;
    internal::pbroadcast4<Packet>(data1, A0, A1, A2, A3);
    internal::pstore(data2 + 0 * PacketSize, A0);
    internal::pstore(data2 + 1 * PacketSize, A1);
    internal::pstore(data2 + 2 * PacketSize, A2);
    internal::pstore(data2 + 3 * PacketSize, A3);
    VERIFY(test::areApprox(ref, data2, 4 * PacketSize) && "internal::pbroadcast4");
  }

  {
    for (int i = 0; i < PacketSize * 2; ++i) ref[i] = data1[i / PacketSize];
    Packet A0, A1;
    internal::pbroadcast2<Packet>(data1, A0, A1);
    internal::pstore(data2 + 0 * PacketSize, A0);
    internal::pstore(data2 + 1 * PacketSize, A1);
    VERIFY(test::areApprox(ref, data2, 2 * PacketSize) && "internal::pbroadcast2");
  }

  VERIFY(internal::isApprox(data1[0], internal::pfirst(internal::pload<Packet>(data1))) && "internal::pfirst");

  if (PacketSize > 1) {
    // apply different offsets to check that ploaddup is robust to unaligned inputs
    for (int offset = 0; offset < 4; ++offset) {
      for (int i = 0; i < PacketSize / 2; ++i) ref[2 * i + 0] = ref[2 * i + 1] = data1[offset + i];
      internal::pstore(data2, internal::ploaddup<Packet>(data1 + offset));
      VERIFY(test::areApprox(ref, data2, PacketSize) && "ploaddup");
    }
  }

  if (PacketSize > 2) {
    // apply different offsets to check that ploadquad is robust to unaligned inputs
    for (int offset = 0; offset < 4; ++offset) {
      for (int i = 0; i < PacketSize / 4; ++i)
        ref[4 * i + 0] = ref[4 * i + 1] = ref[4 * i + 2] = ref[4 * i + 3] = data1[offset + i];
      internal::pstore(data2, internal::ploadquad<Packet>(data1 + offset));
      VERIFY(test::areApprox(ref, data2, PacketSize) && "ploadquad");
    }
  }

  ref[0] = Scalar(0);
  for (int i = 0; i < PacketSize; ++i) ref[0] += data1[i];
  VERIFY(test::isApproxAbs(ref[0], internal::predux(internal::pload<Packet>(data1)), refvalue) && "internal::predux");

  if (PacketSize == 8 && internal::unpacket_traits<typename internal::unpacket_traits<Packet>::half>::size ==
                             4)  // so far, predux_half_downto4 is only required in such a case
  {
    int HalfPacketSize = PacketSize > 4 ? PacketSize / 2 : PacketSize;
    for (int i = 0; i < HalfPacketSize; ++i) ref[i] = Scalar(0);
    for (int i = 0; i < PacketSize; ++i) ref[i % HalfPacketSize] += data1[i];
    internal::pstore(data2, internal::predux_half_dowto4(internal::pload<Packet>(data1)));
    VERIFY(test::areApprox(ref, data2, HalfPacketSize) && "internal::predux_half_dowto4");
  }

  ref[0] = Scalar(1);
  for (int i = 0; i < PacketSize; ++i) ref[0] = REF_MUL(ref[0], data1[i]);
  VERIFY(internal::isApprox(ref[0], internal::predux_mul(internal::pload<Packet>(data1))) && "internal::predux_mul");

  for (int i = 0; i < PacketSize; ++i) ref[i] = data1[PacketSize - i - 1];
  internal::pstore(data2, internal::preverse(internal::pload<Packet>(data1)));
  VERIFY(test::areApprox(ref, data2, PacketSize) && "internal::preverse");

  internal::PacketBlock<Packet> kernel;
  for (int i = 0; i < PacketSize; ++i) {
    kernel.packet[i] = internal::pload<Packet>(data1 + i * PacketSize);
  }
  ptranspose(kernel);
  for (int i = 0; i < PacketSize; ++i) {
    internal::pstore(data2, kernel.packet[i]);
    for (int j = 0; j < PacketSize; ++j) {
      VERIFY(test::isApproxAbs(data2[j], data1[i + j * PacketSize], refvalue) && "ptranspose");
    }
  }

  if (PacketTraits::HasBlend) {
    Packet thenPacket = internal::pload<Packet>(data1);
    Packet elsePacket = internal::pload<Packet>(data2);
    EIGEN_ALIGN_MAX internal::Selector<PacketSize> selector;
    for (int i = 0; i < PacketSize; ++i) {
      selector.select[i] = i;
    }

    Packet blend = internal::pblend(selector, thenPacket, elsePacket);
    EIGEN_ALIGN_MAX Scalar result[size];
    internal::pstore(result, blend);
    for (int i = 0; i < PacketSize; ++i) {
      VERIFY(test::isApproxAbs(result[i], (selector.select[i] ? data1[i] : data2[i]), refvalue));
    }
  }

  {
    for (int i = 0; i < PacketSize; ++i) {
      // "if" mask
      unsigned char v = internal::random<bool>() ? 0xff : 0;
      char* bytes = (char*)(data1 + i);
      for (int k = 0; k < int(sizeof(Scalar)); ++k) {
        bytes[k] = v;
      }
      // "then" packet
      data1[i + PacketSize] = internal::random<Scalar>();
      // "else" packet
      data1[i + 2 * PacketSize] = internal::random<Scalar>();
    }
    CHECK_CWISE3_IF(true, internal::pselect, internal::pselect);
  }

  CHECK_CWISE1_IF(PacketTraits::HasSqrt, numext::sqrt, internal::psqrt);

  for (int i = 0; i < size; ++i) {
    data1[i] = internal::random<Scalar>();
  }
  CHECK_CWISE1(internal::pzero, internal::pzero);
  CHECK_CWISE2_IF(true, internal::por, internal::por);
  CHECK_CWISE2_IF(true, internal::pxor, internal::pxor);
  CHECK_CWISE2_IF(true, internal::pand, internal::pand);

  packetmath_boolean_mask_ops<Scalar, Packet>();
  packetmath_pcast_ops_runner<Scalar, Packet>::run();
  packetmath_minus_zero_add<Scalar, Packet>();
}

template <typename Scalar, typename Packet>
void packetmath_real() {
  typedef internal::packet_traits<Scalar> PacketTraits;
  const int PacketSize = internal::unpacket_traits<Packet>::size;

  const int size = PacketSize * 4;
  EIGEN_ALIGN_MAX Scalar data1[PacketSize * 4];
  EIGEN_ALIGN_MAX Scalar data2[PacketSize * 4];
  EIGEN_ALIGN_MAX Scalar ref[PacketSize * 4];

  for (int i = 0; i < size; ++i) {
    data1[i] = Scalar(internal::random<double>(0, 1) * std::pow(10., internal::random<double>(-6, 6)));
    data2[i] = Scalar(internal::random<double>(0, 1) * std::pow(10., internal::random<double>(-6, 6)));
  }

  if (internal::random<float>(0, 1) < 0.1f) data1[internal::random<int>(0, PacketSize)] = Scalar(0);

  CHECK_CWISE1_IF(PacketTraits::HasLog, std::log, internal::plog);
  CHECK_CWISE1_IF(PacketTraits::HasRsqrt, 1 / std::sqrt, internal::prsqrt);

  for (int i = 0; i < size; ++i) {
    data1[i] = Scalar(internal::random<double>(-1, 1) * std::pow(10., internal::random<double>(-3, 3)));
    data2[i] = Scalar(internal::random<double>(-1, 1) * std::pow(10., internal::random<double>(-3, 3)));
  }
  CHECK_CWISE1_IF(PacketTraits::HasSin, std::sin, internal::psin);
  CHECK_CWISE1_IF(PacketTraits::HasCos, std::cos, internal::pcos);
  CHECK_CWISE1_IF(PacketTraits::HasTan, std::tan, internal::ptan);

  CHECK_CWISE1_IF(PacketTraits::HasRound, numext::round, internal::pround);
  CHECK_CWISE1_IF(PacketTraits::HasCeil, numext::ceil, internal::pceil);
  CHECK_CWISE1_IF(PacketTraits::HasFloor, numext::floor, internal::pfloor);
  CHECK_CWISE1_IF(PacketTraits::HasRint, numext::rint, internal::print);

  // See bug 1785.
  for (int i = 0; i < size; ++i) {
    data1[i] = Scalar(-1.5 + i);
    data2[i] = Scalar(-1.5 + i);
  }
  CHECK_CWISE1_IF(PacketTraits::HasRound, numext::round, internal::pround);
  CHECK_CWISE1_IF(PacketTraits::HasRint, numext::rint, internal::print);

  for (int i = 0; i < size; ++i) {
    data1[i] = Scalar(internal::random<double>(-1, 1));
    data2[i] = Scalar(internal::random<double>(-1, 1));
  }
  CHECK_CWISE1_IF(PacketTraits::HasASin, std::asin, internal::pasin);
  CHECK_CWISE1_IF(PacketTraits::HasACos, std::acos, internal::pacos);

  for (int i = 0; i < size; ++i) {
    data1[i] = Scalar(internal::random<double>(-87, 88));
    data2[i] = Scalar(internal::random<double>(-87, 88));
  }
  CHECK_CWISE1_IF(PacketTraits::HasExp, std::exp, internal::pexp);
  for (int i = 0; i < size; ++i) {
    data1[i] = Scalar(internal::random<double>(-1, 1) * std::pow(10., internal::random<double>(-6, 6)));
    data2[i] = Scalar(internal::random<double>(-1, 1) * std::pow(10., internal::random<double>(-6, 6)));
  }
  data1[0] = Scalar(1e-20);
  CHECK_CWISE1_IF(PacketTraits::HasTanh, std::tanh, internal::ptanh);
  if (PacketTraits::HasExp && PacketSize >= 2) {
    const Scalar small = std::numeric_limits<Scalar>::epsilon();
    data1[0] = std::numeric_limits<Scalar>::quiet_NaN();
    data1[1] = small;
    test::packet_helper<PacketTraits::HasExp, Packet> h;
    h.store(data2, internal::pexp(h.load(data1)));
    VERIFY((numext::isnan)(data2[0]));
    // TODO(rmlarsen): Re-enable for bfloat16.
    if (!internal::is_same<Scalar, bfloat16>::value) {
      VERIFY_IS_EQUAL(std::exp(small), data2[1]);
    }

    data1[0] = -small;
    data1[1] = Scalar(0);
    h.store(data2, internal::pexp(h.load(data1)));
    // TODO(rmlarsen): Re-enable for bfloat16.
    if (!internal::is_same<Scalar, bfloat16>::value) {
      VERIFY_IS_EQUAL(std::exp(-small), data2[0]);
    }
    VERIFY_IS_EQUAL(std::exp(Scalar(0)), data2[1]);

    data1[0] = (std::numeric_limits<Scalar>::min)();
    data1[1] = -(std::numeric_limits<Scalar>::min)();
    h.store(data2, internal::pexp(h.load(data1)));
    VERIFY_IS_EQUAL(std::exp((std::numeric_limits<Scalar>::min)()), data2[0]);
    VERIFY_IS_EQUAL(std::exp(-(std::numeric_limits<Scalar>::min)()), data2[1]);

    data1[0] = std::numeric_limits<Scalar>::denorm_min();
    data1[1] = -std::numeric_limits<Scalar>::denorm_min();
    h.store(data2, internal::pexp(h.load(data1)));
    VERIFY_IS_EQUAL(std::exp(std::numeric_limits<Scalar>::denorm_min()), data2[0]);
    VERIFY_IS_EQUAL(std::exp(-std::numeric_limits<Scalar>::denorm_min()), data2[1]);
  }

  if (PacketTraits::HasTanh) {
    // NOTE this test migh fail with GCC prior to 6.3, see MathFunctionsImpl.h for details.
    data1[0] = std::numeric_limits<Scalar>::quiet_NaN();
    test::packet_helper<internal::packet_traits<Scalar>::HasTanh, Packet> h;
    h.store(data2, internal::ptanh(h.load(data1)));
    VERIFY((numext::isnan)(data2[0]));
  }

  if (PacketTraits::HasExp) {
    internal::scalar_logistic_op<Scalar> logistic;
    for (int i = 0; i < size; ++i) {
      data1[i] = Scalar(internal::random<double>(-20, 20));
    }

    test::packet_helper<PacketTraits::HasExp, Packet> h;
    h.store(data2, logistic.packetOp(h.load(data1)));
    for (int i = 0; i < PacketSize; ++i) {
      VERIFY_IS_APPROX(data2[i], logistic(data1[i]));
    }
  }

#if EIGEN_HAS_C99_MATH && (__cplusplus > 199711L)
  data1[0] = std::numeric_limits<Scalar>::infinity();
  data1[1] = Scalar(-1);
  CHECK_CWISE1_IF(PacketTraits::HasLog1p, std::log1p, internal::plog1p);
  data1[0] = std::numeric_limits<Scalar>::infinity();
  data1[1] = -std::numeric_limits<Scalar>::infinity();
  CHECK_CWISE1_IF(PacketTraits::HasExpm1, std::expm1, internal::pexpm1);
#endif

  if (PacketSize >= 2) {
    data1[0] = std::numeric_limits<Scalar>::quiet_NaN();
    data1[1] = std::numeric_limits<Scalar>::epsilon();
    if (PacketTraits::HasLog) {
      test::packet_helper<PacketTraits::HasLog, Packet> h;
      h.store(data2, internal::plog(h.load(data1)));
      VERIFY((numext::isnan)(data2[0]));
      VERIFY_IS_EQUAL(std::log(std::numeric_limits<Scalar>::epsilon()), data2[1]);

      data1[0] = -std::numeric_limits<Scalar>::epsilon();
      data1[1] = Scalar(0);
      h.store(data2, internal::plog(h.load(data1)));
      VERIFY((numext::isnan)(data2[0]));
      VERIFY_IS_EQUAL(std::log(Scalar(0)), data2[1]);

      data1[0] = (std::numeric_limits<Scalar>::min)();
      data1[1] = -(std::numeric_limits<Scalar>::min)();
      h.store(data2, internal::plog(h.load(data1)));
      VERIFY_IS_EQUAL(std::log((std::numeric_limits<Scalar>::min)()), data2[0]);
      VERIFY((numext::isnan)(data2[1]));

      // Note: 32-bit arm always flushes denorms to zero.
#if !EIGEN_ARCH_ARM
      if (std::numeric_limits<Scalar>::has_denorm == std::denorm_present) {
        data1[0] = std::numeric_limits<Scalar>::denorm_min();
        data1[1] = -std::numeric_limits<Scalar>::denorm_min();
        h.store(data2, internal::plog(h.load(data1)));
        // TODO(rmlarsen): Reenable.
        //        VERIFY_IS_EQUAL(std::log(std::numeric_limits<Scalar>::denorm_min()), data2[0]);
        VERIFY((numext::isnan)(data2[1]));
      }
#endif

      data1[0] = Scalar(-1.0f);
      h.store(data2, internal::plog(h.load(data1)));
      VERIFY((numext::isnan)(data2[0]));

      data1[0] = std::numeric_limits<Scalar>::infinity();
      h.store(data2, internal::plog(h.load(data1)));
      VERIFY((numext::isinf)(data2[0]));
    }
    if (PacketTraits::HasLog1p) {
      test::packet_helper<PacketTraits::HasLog1p, Packet> h;
      data1[0] = Scalar(-2);
      data1[1] = -std::numeric_limits<Scalar>::infinity();
      h.store(data2, internal::plog1p(h.load(data1)));
      VERIFY((numext::isnan)(data2[0]));
      VERIFY((numext::isnan)(data2[1]));
    }
    if (PacketTraits::HasSqrt) {
      test::packet_helper<PacketTraits::HasSqrt, Packet> h;
      data1[0] = Scalar(-1.0f);
      if (std::numeric_limits<Scalar>::has_denorm == std::denorm_present) {
        data1[1] = -std::numeric_limits<Scalar>::denorm_min();
      } else {
        data1[1] = -std::numeric_limits<Scalar>::epsilon();
      }
      h.store(data2, internal::psqrt(h.load(data1)));
      VERIFY((numext::isnan)(data2[0]));
      VERIFY((numext::isnan)(data2[1]));
    }
    // TODO(rmlarsen): Re-enable for bfloat16.
    if (PacketTraits::HasCos && !internal::is_same<Scalar, bfloat16>::value) {
      test::packet_helper<PacketTraits::HasCos, Packet> h;
      for (Scalar k = Scalar(1); k < Scalar(10000) / std::numeric_limits<Scalar>::epsilon(); k *= Scalar(2)) {
        for (int k1 = 0; k1 <= 1; ++k1) {
          data1[0] = Scalar((2 * double(k) + k1) * double(EIGEN_PI) / 2 * internal::random<double>(0.8, 1.2));
          data1[1] = Scalar((2 * double(k) + 2 + k1) * double(EIGEN_PI) / 2 * internal::random<double>(0.8, 1.2));
          h.store(data2, internal::pcos(h.load(data1)));
          h.store(data2 + PacketSize, internal::psin(h.load(data1)));
          VERIFY(data2[0] <= Scalar(1.) && data2[0] >= Scalar(-1.));
          VERIFY(data2[1] <= Scalar(1.) && data2[1] >= Scalar(-1.));
          VERIFY(data2[PacketSize + 0] <= Scalar(1.) && data2[PacketSize + 0] >= Scalar(-1.));
          VERIFY(data2[PacketSize + 1] <= Scalar(1.) && data2[PacketSize + 1] >= Scalar(-1.));

          VERIFY_IS_APPROX(data2[0], std::cos(data1[0]));
          VERIFY_IS_APPROX(data2[1], std::cos(data1[1]));
          VERIFY_IS_APPROX(data2[PacketSize + 0], std::sin(data1[0]));
          VERIFY_IS_APPROX(data2[PacketSize + 1], std::sin(data1[1]));

          VERIFY_IS_APPROX(numext::abs2(data2[0]) + numext::abs2(data2[PacketSize + 0]), Scalar(1));
          VERIFY_IS_APPROX(numext::abs2(data2[1]) + numext::abs2(data2[PacketSize + 1]), Scalar(1));
        }
      }

      data1[0] = std::numeric_limits<Scalar>::infinity();
      data1[1] = -std::numeric_limits<Scalar>::infinity();
      h.store(data2, internal::psin(h.load(data1)));
      VERIFY((numext::isnan)(data2[0]));
      VERIFY((numext::isnan)(data2[1]));

      h.store(data2, internal::pcos(h.load(data1)));
      VERIFY((numext::isnan)(data2[0]));
      VERIFY((numext::isnan)(data2[1]));

      data1[0] = std::numeric_limits<Scalar>::quiet_NaN();
      h.store(data2, internal::psin(h.load(data1)));
      VERIFY((numext::isnan)(data2[0]));
      h.store(data2, internal::pcos(h.load(data1)));
      VERIFY((numext::isnan)(data2[0]));

      data1[0] = -Scalar(0.);
      h.store(data2, internal::psin(h.load(data1)));
      VERIFY(internal::biteq(data2[0], data1[0]));
      h.store(data2, internal::pcos(h.load(data1)));
      VERIFY_IS_EQUAL(data2[0], Scalar(1));
    }
  }
}

#define CAST_CHECK_CWISE1_IF(COND, REFOP, POP, SCALAR, REFTYPE) if(COND) { \
  test::packet_helper<COND,Packet> h; \
  for (int i=0; i<PacketSize; ++i) \
    ref[i] = SCALAR(REFOP(static_cast<REFTYPE>(data1[i]))); \
  h.store(data2, POP(h.load(data1))); \
  VERIFY(test::areApprox(ref, data2, PacketSize) && #POP); \
}

template <>
void packetmath_real<bfloat16, typename internal::packet_traits<bfloat16>::type>(){
  typedef internal::packet_traits<bfloat16> PacketTraits;
  typedef internal::packet_traits<bfloat16>::type Packet;

  const int PacketSize = internal::unpacket_traits<Packet>::size;
  const int size = PacketSize * 4;
  EIGEN_ALIGN_MAX bfloat16 data1[PacketSize * 4];
  EIGEN_ALIGN_MAX bfloat16 data2[PacketSize * 4];
  EIGEN_ALIGN_MAX bfloat16 ref[PacketSize * 4];

  for (int i = 0; i < size; ++i) {
    data1[i] = bfloat16(internal::random<float>(0, 1) * std::pow(float(10), internal::random<float>(-6, 6)));
    data2[i] = bfloat16(internal::random<float>(0, 1) * std::pow(float(10), internal::random<float>(-6, 6)));
    data1[i] = bfloat16(0);
  }

  if (internal::random<float>(0, 1) < 0.1f) data1[internal::random<int>(0, PacketSize)] = bfloat16(0);

  CAST_CHECK_CWISE1_IF(PacketTraits::HasLog, std::log, internal::plog, bfloat16, float);
  CAST_CHECK_CWISE1_IF(PacketTraits::HasRsqrt, float(1) / std::sqrt, internal::prsqrt, bfloat16, float);

  for (int i = 0; i < size; ++i) {
    data1[i] = bfloat16(internal::random<float>(-1, 1) * std::pow(float(10), internal::random<float>(-3, 3)));
    data2[i] = bfloat16(internal::random<float>(-1, 1) * std::pow(float(10), internal::random<float>(-3, 3)));
  }
  CAST_CHECK_CWISE1_IF(PacketTraits::HasSin, std::sin, internal::psin, bfloat16, float);
  CAST_CHECK_CWISE1_IF(PacketTraits::HasCos, std::cos, internal::pcos, bfloat16, float);
  CAST_CHECK_CWISE1_IF(PacketTraits::HasTan, std::tan, internal::ptan, bfloat16, float);

  CAST_CHECK_CWISE1_IF(PacketTraits::HasRound, numext::round, internal::pround, bfloat16, float);
  CAST_CHECK_CWISE1_IF(PacketTraits::HasCeil, numext::ceil, internal::pceil, bfloat16, float);
  CAST_CHECK_CWISE1_IF(PacketTraits::HasFloor, numext::floor, internal::pfloor, bfloat16, float);

  for (int i = 0; i < size; ++i) {
    data1[i] = bfloat16(-1.5 + i);
    data2[i] = bfloat16(-1.5 + i);
  }
  CAST_CHECK_CWISE1_IF(PacketTraits::HasRound, numext::round, internal::pround, bfloat16, float);

  for (int i = 0; i < size; ++i) {
    data1[i] = bfloat16(internal::random<float>(-87, 88));
    data2[i] = bfloat16(internal::random<float>(-87, 88));
  }
  CAST_CHECK_CWISE1_IF(PacketTraits::HasExp, std::exp, internal::pexp, bfloat16, float);

}

template <typename Scalar>
Scalar propagate_nan_max(const Scalar& a, const Scalar& b) {
  if ((numext::isnan)(a)) return a;
  if ((numext::isnan)(b)) return b;
  return (numext::maxi)(a,b);
}

template <typename Scalar>
Scalar propagate_nan_min(const Scalar& a, const Scalar& b) {
  if ((numext::isnan)(a)) return a;
  if ((numext::isnan)(b)) return b;
  return (numext::mini)(a,b);
}

template <typename Scalar, typename Packet>
void packetmath_notcomplex() {
  typedef internal::packet_traits<Scalar> PacketTraits;
  const int PacketSize = internal::unpacket_traits<Packet>::size;

  EIGEN_ALIGN_MAX Scalar data1[PacketSize * 4];
  EIGEN_ALIGN_MAX Scalar data2[PacketSize * 4];
  EIGEN_ALIGN_MAX Scalar ref[PacketSize * 4];

  Array<Scalar, Dynamic, 1>::Map(data1, PacketSize * 4).setRandom();

  VERIFY((!PacketTraits::Vectorizable) || PacketTraits::HasMin);
  VERIFY((!PacketTraits::Vectorizable) || PacketTraits::HasMax);

  CHECK_CWISE2_IF(PacketTraits::HasMin, (std::min), internal::pmin);
  CHECK_CWISE2_IF(PacketTraits::HasMax, (std::max), internal::pmax);
#if EIGEN_HAS_CXX11_MATH
  using std::fmin;
  using std::fmax;
#else
  using ::fmin;
  using ::fmax;
#endif
  CHECK_CWISE2_IF(PacketTraits::HasMin, fmin, (internal::pmin<PropagateNumbers>));
  CHECK_CWISE2_IF(PacketTraits::HasMax, fmax, internal::pmax<PropagateNumbers>);
  CHECK_CWISE1(numext::abs, internal::pabs);
  CHECK_CWISE2_IF(PacketTraits::HasAbsDiff, REF_ABS_DIFF, internal::pabsdiff);

  ref[0] = data1[0];
  for (int i = 0; i < PacketSize; ++i) ref[0] = internal::pmin(ref[0], data1[i]);
  VERIFY(internal::isApprox(ref[0], internal::predux_min(internal::pload<Packet>(data1))) && "internal::predux_min");
  ref[0] = data1[0];
  for (int i = 0; i < PacketSize; ++i) ref[0] = internal::pmax(ref[0], data1[i]);
  VERIFY(internal::isApprox(ref[0], internal::predux_max(internal::pload<Packet>(data1))) && "internal::predux_max");

  for (int i = 0; i < PacketSize; ++i) ref[i] = data1[0] + Scalar(i);
  internal::pstore(data2, internal::plset<Packet>(data1[0]));
  VERIFY(test::areApprox(ref, data2, PacketSize) && "internal::plset");

  {
    unsigned char* data1_bits = reinterpret_cast<unsigned char*>(data1);
    // predux_all - not needed yet
    // for (unsigned int i=0; i<PacketSize*sizeof(Scalar); ++i) data1_bits[i] = 0xff;
    // VERIFY(internal::predux_all(internal::pload<Packet>(data1)) && "internal::predux_all(1111)");
    // for(int k=0; k<PacketSize; ++k)
    // {
    //   for (unsigned int i=0; i<sizeof(Scalar); ++i) data1_bits[k*sizeof(Scalar)+i] = 0x0;
    //   VERIFY( (!internal::predux_all(internal::pload<Packet>(data1))) && "internal::predux_all(0101)");
    //   for (unsigned int i=0; i<sizeof(Scalar); ++i) data1_bits[k*sizeof(Scalar)+i] = 0xff;
    // }

    // predux_any
    for (unsigned int i = 0; i < PacketSize * sizeof(Scalar); ++i) data1_bits[i] = 0x0;
    VERIFY((!internal::predux_any(internal::pload<Packet>(data1))) && "internal::predux_any(0000)");
    for (int k = 0; k < PacketSize; ++k) {
      for (unsigned int i = 0; i < sizeof(Scalar); ++i) data1_bits[k * sizeof(Scalar) + i] = 0xff;
      VERIFY(internal::predux_any(internal::pload<Packet>(data1)) && "internal::predux_any(0101)");
      for (unsigned int i = 0; i < sizeof(Scalar); ++i) data1_bits[k * sizeof(Scalar) + i] = 0x00;
    }
  }


  // Test NaN propagation.
  if (!NumTraits<Scalar>::IsInteger) {
    // Test reductions with no NaNs.
    ref[0] = data1[0];
    for (int i = 0; i < PacketSize; ++i) ref[0] = internal::pmin<PropagateNumbers>(ref[0], data1[i]);
    VERIFY(internal::isApprox(ref[0], internal::predux_min<PropagateNumbers>(internal::pload<Packet>(data1))) && "internal::predux_min<PropagateNumbers>");
    ref[0] = data1[0];
    for (int i = 0; i < PacketSize; ++i) ref[0] = internal::pmin<PropagateNaN>(ref[0], data1[i]);
    VERIFY(internal::isApprox(ref[0], internal::predux_min<PropagateNaN>(internal::pload<Packet>(data1))) && "internal::predux_min<PropagateNaN>");
    ref[0] = data1[0];
    for (int i = 0; i < PacketSize; ++i) ref[0] = internal::pmax<PropagateNumbers>(ref[0], data1[i]);
    VERIFY(internal::isApprox(ref[0], internal::predux_max<PropagateNumbers>(internal::pload<Packet>(data1))) && "internal::predux_max<PropagateNumbers>");
    ref[0] = data1[0];
    for (int i = 0; i < PacketSize; ++i) ref[0] = internal::pmax<PropagateNaN>(ref[0], data1[i]);
    VERIFY(internal::isApprox(ref[0], internal::predux_max<PropagateNaN>(internal::pload<Packet>(data1))) && "internal::predux_max<PropagateNumbers>");
    // A single NaN.
    const size_t index = std::numeric_limits<size_t>::quiet_NaN() % PacketSize;
    data1[index] = std::numeric_limits<Scalar>::quiet_NaN();
    VERIFY(PacketSize==1 || !(numext::isnan)(internal::predux_min<PropagateNumbers>(internal::pload<Packet>(data1))));
    VERIFY((numext::isnan)(internal::predux_min<PropagateNaN>(internal::pload<Packet>(data1))));
    VERIFY(PacketSize==1 || !(numext::isnan)(internal::predux_max<PropagateNumbers>(internal::pload<Packet>(data1))));
    VERIFY((numext::isnan)(internal::predux_max<PropagateNaN>(internal::pload<Packet>(data1))));
    // All NaNs.
    for (int i = 0; i < 4 * PacketSize; ++i) data1[i] = std::numeric_limits<Scalar>::quiet_NaN();
    VERIFY((numext::isnan)(internal::predux_min<PropagateNumbers>(internal::pload<Packet>(data1))));
    VERIFY((numext::isnan)(internal::predux_min<PropagateNaN>(internal::pload<Packet>(data1))));
    VERIFY((numext::isnan)(internal::predux_max<PropagateNumbers>(internal::pload<Packet>(data1))));
    VERIFY((numext::isnan)(internal::predux_max<PropagateNaN>(internal::pload<Packet>(data1))));

    // Test NaN propagation for coefficient-wise min and max.
    for (int i = 0; i < PacketSize; ++i) {
      data1[i] = internal::random<bool>() ? std::numeric_limits<Scalar>::quiet_NaN() : Scalar(0);
      data1[i + PacketSize] = internal::random<bool>() ? std::numeric_limits<Scalar>::quiet_NaN() : Scalar(0);
    }
    // Note: NaN propagation is implementation defined for pmin/pmax, so we do not test it here.
    CHECK_CWISE2_IF(PacketTraits::HasMin, fmin, (internal::pmin<PropagateNumbers>));
    CHECK_CWISE2_IF(PacketTraits::HasMax, fmax, internal::pmax<PropagateNumbers>);
    CHECK_CWISE2_IF(PacketTraits::HasMin, propagate_nan_min, (internal::pmin<PropagateNaN>));
    CHECK_CWISE2_IF(PacketTraits::HasMax, propagate_nan_max, internal::pmax<PropagateNaN>);
  }
}

template <>
void packetmath_notcomplex<bfloat16, typename internal::packet_traits<bfloat16>::type>(){
  typedef bfloat16 Scalar;
  typedef internal::packet_traits<bfloat16>::type Packet;
  typedef internal::packet_traits<Scalar> PacketTraits;
  const int PacketSize = internal::unpacket_traits<Packet>::size;

  EIGEN_ALIGN_MAX Scalar data1[PacketSize * 4];
  EIGEN_ALIGN_MAX Scalar data2[PacketSize * 4];
  EIGEN_ALIGN_MAX Scalar ref[PacketSize * 4];
  Array<Scalar, Dynamic, 1>::Map(data1, PacketSize * 4).setRandom();

  ref[0] = data1[0];
  for (int i = 0; i < PacketSize; ++i) ref[0] = (std::min)(ref[0], data1[i]);
  VERIFY(internal::isApprox(ref[0], internal::predux_min(internal::pload<Packet>(data1))) && "internal::predux_min");

  VERIFY((!PacketTraits::Vectorizable) || PacketTraits::HasMin);
  VERIFY((!PacketTraits::Vectorizable) || PacketTraits::HasMax);

  CHECK_CWISE2_IF(PacketTraits::HasMin, (std::min), internal::pmin);
  CHECK_CWISE2_IF(PacketTraits::HasMax, (std::max), internal::pmax);
  CHECK_CWISE1(numext::abs, internal::pabs);
  CHECK_CWISE2_IF(PacketTraits::HasAbsDiff, REF_ABS_DIFF, internal::pabsdiff);

  ref[0] = data1[0];
  for (int i = 0; i < PacketSize; ++i) ref[0] = (std::max)(ref[0], data1[i]);
  VERIFY(internal::isApprox(ref[0], internal::predux_max(internal::pload<Packet>(data1))) && "internal::predux_max");

  {
    unsigned char* data1_bits = reinterpret_cast<unsigned char*>(data1);
    // predux_any
    for (unsigned int i = 0; i < PacketSize * sizeof(Scalar); ++i) data1_bits[i] = 0x0;
    VERIFY((!internal::predux_any(internal::pload<Packet>(data1))) && "internal::predux_any(0000)");
    for (int k = 0; k < PacketSize; ++k) {
      for (unsigned int i = 0; i < sizeof(Scalar); ++i) data1_bits[k * sizeof(Scalar) + i] = 0xff;
      VERIFY(internal::predux_any(internal::pload<Packet>(data1)) && "internal::predux_any(0101)");
      for (unsigned int i = 0; i < sizeof(Scalar); ++i) data1_bits[k * sizeof(Scalar) + i] = 0x00;
    }
  }
}

template <typename Scalar, typename Packet, bool ConjLhs, bool ConjRhs>
void test_conj_helper(Scalar* data1, Scalar* data2, Scalar* ref, Scalar* pval) {
  const int PacketSize = internal::unpacket_traits<Packet>::size;

  internal::conj_if<ConjLhs> cj0;
  internal::conj_if<ConjRhs> cj1;
  internal::conj_helper<Scalar, Scalar, ConjLhs, ConjRhs> cj;
  internal::conj_helper<Packet, Packet, ConjLhs, ConjRhs> pcj;

  for (int i = 0; i < PacketSize; ++i) {
    ref[i] = cj0(data1[i]) * cj1(data2[i]);
    VERIFY(internal::isApprox(ref[i], cj.pmul(data1[i], data2[i])) && "conj_helper pmul");
  }
  internal::pstore(pval, pcj.pmul(internal::pload<Packet>(data1), internal::pload<Packet>(data2)));
  VERIFY(test::areApprox(ref, pval, PacketSize) && "conj_helper pmul");

  for (int i = 0; i < PacketSize; ++i) {
    Scalar tmp = ref[i];
    ref[i] += cj0(data1[i]) * cj1(data2[i]);
    VERIFY(internal::isApprox(ref[i], cj.pmadd(data1[i], data2[i], tmp)) && "conj_helper pmadd");
  }
  internal::pstore(
      pval, pcj.pmadd(internal::pload<Packet>(data1), internal::pload<Packet>(data2), internal::pload<Packet>(pval)));
  VERIFY(test::areApprox(ref, pval, PacketSize) && "conj_helper pmadd");
}

template <typename Scalar, typename Packet>
void packetmath_complex() {
  const int PacketSize = internal::unpacket_traits<Packet>::size;

  const int size = PacketSize * 4;
  EIGEN_ALIGN_MAX Scalar data1[PacketSize * 4];
  EIGEN_ALIGN_MAX Scalar data2[PacketSize * 4];
  EIGEN_ALIGN_MAX Scalar ref[PacketSize * 4];
  EIGEN_ALIGN_MAX Scalar pval[PacketSize * 4];

  for (int i = 0; i < size; ++i) {
    data1[i] = internal::random<Scalar>() * Scalar(1e2);
    data2[i] = internal::random<Scalar>() * Scalar(1e2);
  }

  test_conj_helper<Scalar, Packet, false, false>(data1, data2, ref, pval);
  test_conj_helper<Scalar, Packet, false, true>(data1, data2, ref, pval);
  test_conj_helper<Scalar, Packet, true, false>(data1, data2, ref, pval);
  test_conj_helper<Scalar, Packet, true, true>(data1, data2, ref, pval);

  {
    for (int i = 0; i < PacketSize; ++i) ref[i] = Scalar(std::imag(data1[i]), std::real(data1[i]));
    internal::pstore(pval, internal::pcplxflip(internal::pload<Packet>(data1)));
    VERIFY(test::areApprox(ref, pval, PacketSize) && "pcplxflip");
  }
}

template <typename Scalar, typename Packet>
void packetmath_scatter_gather() {
  typedef typename NumTraits<Scalar>::Real RealScalar;
  const int PacketSize = internal::unpacket_traits<Packet>::size;
  EIGEN_ALIGN_MAX Scalar data1[PacketSize];
  RealScalar refvalue = RealScalar(0);
  for (int i = 0; i < PacketSize; ++i) {
    data1[i] = internal::random<Scalar>() / RealScalar(PacketSize);
  }

  int stride = internal::random<int>(1, 20);

  EIGEN_ALIGN_MAX Scalar buffer[PacketSize * 20];
  memset(buffer, 0, 20 * PacketSize * sizeof(Scalar));
  Packet packet = internal::pload<Packet>(data1);
  internal::pscatter<Scalar, Packet>(buffer, packet, stride);

  for (int i = 0; i < PacketSize * 20; ++i) {
    if ((i % stride) == 0 && i < stride * PacketSize) {
      VERIFY(test::isApproxAbs(buffer[i], data1[i / stride], refvalue) && "pscatter");
    } else {
      VERIFY(test::isApproxAbs(buffer[i], Scalar(0), refvalue) && "pscatter");
    }
  }

  for (int i = 0; i < PacketSize * 7; ++i) {
    buffer[i] = internal::random<Scalar>() / RealScalar(PacketSize);
  }
  packet = internal::pgather<Scalar, Packet>(buffer, 7);
  internal::pstore(data1, packet);
  for (int i = 0; i < PacketSize; ++i) {
    VERIFY(test::isApproxAbs(data1[i], buffer[i * 7], refvalue) && "pgather");
  }
}

namespace Eigen {
namespace test {

template <typename Scalar, typename PacketType>
struct runall<Scalar, PacketType, false, false> {  // i.e. float or double
  static void run() {
    packetmath<Scalar, PacketType>();
    packetmath_scatter_gather<Scalar, PacketType>();
    packetmath_notcomplex<Scalar, PacketType>();
    packetmath_real<Scalar, PacketType>();
  }
};

template <typename Scalar, typename PacketType>
struct runall<Scalar, PacketType, false, true> {  // i.e. int
  static void run() {
    packetmath<Scalar, PacketType>();
    packetmath_scatter_gather<Scalar, PacketType>();
    packetmath_notcomplex<Scalar, PacketType>();
  }
};

template <typename Scalar, typename PacketType>
struct runall<Scalar, PacketType, true, false> {  // i.e. complex
  static void run() {
    packetmath<Scalar, PacketType>();
    packetmath_scatter_gather<Scalar, PacketType>();
    packetmath_complex<Scalar, PacketType>();
  }
};

}  // namespace test
}  // namespace Eigen

EIGEN_DECLARE_TEST(packetmath) {
  g_first_pass = true;
  for (int i = 0; i < g_repeat; i++) {
    CALL_SUBTEST_1(test::runner<float>::run());
    CALL_SUBTEST_2(test::runner<double>::run());
    CALL_SUBTEST_3(test::runner<int8_t>::run());
    CALL_SUBTEST_4(test::runner<uint8_t>::run());
    CALL_SUBTEST_5(test::runner<int16_t>::run());
    CALL_SUBTEST_6(test::runner<uint16_t>::run());
    CALL_SUBTEST_7(test::runner<int32_t>::run());
    CALL_SUBTEST_8(test::runner<uint32_t>::run());
    CALL_SUBTEST_9(test::runner<int64_t>::run());
    CALL_SUBTEST_10(test::runner<uint64_t>::run());
    CALL_SUBTEST_11(test::runner<std::complex<float> >::run());
    CALL_SUBTEST_12(test::runner<std::complex<double> >::run());
#if defined(EIGEN_VECTORIZE_AVX)
    // AVX half packets not fully implemented.
    CALL_SUBTEST_13((packetmath<half, internal::packet_traits<half>::type>()));
#else
    CALL_SUBTEST_13(test::runner<half>::run());
#endif
    CALL_SUBTEST_14((packetmath<bool, internal::packet_traits<bool>::type>()));
    CALL_SUBTEST_15(test::runner<bfloat16>::run());
    g_first_pass = false;
  }
}