eigen/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 "main.h"
#include "unsupported/Eigen/SpecialFunctions"
#include <typeinfo>

#if defined __GNUC__ && __GNUC__>=6
  #pragma GCC diagnostic ignored "-Wignored-attributes"
#endif
// using namespace Eigen;

#ifdef EIGEN_VECTORIZE_SSE
const bool g_vectorize_sse = true;
#else
const bool g_vectorize_sse = false;
#endif

bool g_first_pass = true;

namespace Eigen {
namespace internal {

template<typename T> T negate(const T& x) { return -x; }

template<typename T>
Map<const Array<unsigned char,sizeof(T),1> >
bits(const T& x) {
  return Map<const Array<unsigned char,sizeof(T),1> >(reinterpret_cast<const unsigned char *>(&x));
}

// The following implement bitwise operations on floating point types
template<typename T,typename Bits,typename Func>
T apply_bit_op(Bits a, Bits b, Func f) {
  Array<unsigned char,sizeof(T),1> res;
  for(Index i=0; i<res.size();++i) res[i] = f(a[i],b[i]);
  return *reinterpret_cast<T*>(&res);
}

#define EIGEN_TEST_MAKE_BITWISE2(OP,FUNC,T)             \
  template<> T EIGEN_CAT(p,OP)(const T& a,const T& b) { \
    return apply_bit_op<T>(bits(a),bits(b),FUNC);     \
  }

#define EIGEN_TEST_MAKE_BITWISE(OP,FUNC)                  \
  EIGEN_TEST_MAKE_BITWISE2(OP,FUNC,float)                 \
  EIGEN_TEST_MAKE_BITWISE2(OP,FUNC,double)                \
  EIGEN_TEST_MAKE_BITWISE2(OP,FUNC,half)                  \
  EIGEN_TEST_MAKE_BITWISE2(OP,FUNC,std::complex<float>)   \
  EIGEN_TEST_MAKE_BITWISE2(OP,FUNC,std::complex<double>)

EIGEN_TEST_MAKE_BITWISE(xor,std::bit_xor<unsigned char>())
EIGEN_TEST_MAKE_BITWISE(and,std::bit_and<unsigned char>())
EIGEN_TEST_MAKE_BITWISE(or, std::bit_or<unsigned char>())
struct bit_andnot{
  template<typename T> T
  operator()(T a, T b) const { return a & (~b); }
};
EIGEN_TEST_MAKE_BITWISE(andnot, bit_andnot())
template<typename T>
bool biteq(T a, T b) {
  return (bits(a) == bits(b)).all();
}

}
}

// NOTE: we disable inlining for this function to workaround a GCC issue when using -O3 and the i387 FPU.
template<typename Scalar> EIGEN_DONT_INLINE
bool isApproxAbs(const Scalar& a, const Scalar& b, const typename NumTraits<Scalar>::Real& refvalue)
{
  return internal::isMuchSmallerThan(a-b, refvalue);
}

template<typename Scalar> bool areApproxAbs(const Scalar* a, const Scalar* b, int size, const typename NumTraits<Scalar>::Real& refvalue)
{
  for (int i=0; i<size; ++i)
  {
    if (!isApproxAbs(a[i],b[i],refvalue))
    {
      std::cout << "ref: [" << Map<const Matrix<Scalar,1,Dynamic> >(a,size) << "]" << " != vec: [" << Map<const Matrix<Scalar,1,Dynamic> >(b,size) << "]\n";
      return false;
    }
  }
  return true;
}

template<typename Scalar> bool areApprox(const Scalar* a, const Scalar* b, int size)
{
  for (int i=0; i<size; ++i)
  {
    if ((!internal::biteq(a[i],b[i])) && a[i]!=b[i] && !internal::isApprox(a[i],b[i]))
    {
      std::cout << "ref: [" << Map<const Matrix<Scalar,1,Dynamic> >(a,size) << "]" << " != vec: [" << Map<const Matrix<Scalar,1,Dynamic> >(b,size) << "]\n";
      return false;
    }
  }
  return true;
}

#define CHECK_CWISE1(REFOP, POP) { \
  for (int i=0; i<PacketSize; ++i) \
    ref[i] = REFOP(data1[i]); \
  internal::pstore(data2, POP(internal::pload<Packet>(data1))); \
  VERIFY(areApprox(ref, data2, PacketSize) && #POP); \
}

template<bool Cond,typename Packet>
struct packet_helper
{
  template<typename T>
  inline Packet load(const T* from) const { return internal::pload<Packet>(from); }

  template<typename T>
  inline Packet loadu(const T* from) const { return internal::ploadu<Packet>(from); }

  template<typename T>
  inline Packet load(const T* from, unsigned long long umask) const { return internal::ploadu<Packet>(from, umask); }

  template<typename T>
  inline void store(T* to, const Packet& x) const { internal::pstore(to,x); }

  template<typename T>
  inline void store(T* to, const Packet& x, unsigned long long umask) const { internal::pstoreu(to, x, umask); }
};

template<typename Packet>
struct packet_helper<false,Packet>
{
  template<typename T>
  inline T load(const T* from) const { return *from; }

  template<typename T>
  inline T loadu(const T* from) const { return *from; }

  template<typename T>
  inline T load(const T* from, unsigned long long) const { return *from; }

  template<typename T>
  inline void store(T* to, const T& x) const { *to = x; }

  template<typename T>
  inline void store(T* to, const T& x, unsigned long long) const { *to = x; }
};

#define CHECK_CWISE1_IF(COND, REFOP, POP) if(COND) { \
  packet_helper<COND,Packet> h; \
  for (int i=0; i<PacketSize; ++i) \
    ref[i] = REFOP(data1[i]); \
  h.store(data2, POP(h.load(data1))); \
  VERIFY(areApprox(ref, data2, PacketSize) && #POP); \
}

#define CHECK_CWISE2_IF(COND, REFOP, POP) if(COND) { \
  packet_helper<COND,Packet> h; \
  for (int i=0; i<PacketSize; ++i) \
    ref[i] = REFOP(data1[i], data1[i+PacketSize]); \
  h.store(data2, POP(h.load(data1),h.load(data1+PacketSize))); \
  VERIFY(areApprox(ref, data2, PacketSize) && #POP); \
}

#define CHECK_CWISE3_IF(COND, REFOP, POP) if (COND) {                      \
  packet_helper<COND, Packet> h;                                           \
  for (int i = 0; i < PacketSize; ++i)                                     \
    ref[i] =                                                               \
        REFOP(data1[i], data1[i + PacketSize], data1[i + 2 * PacketSize]); \
  h.store(data2, POP(h.load(data1), h.load(data1 + PacketSize),            \
                     h.load(data1 + 2 * PacketSize)));                     \
  VERIFY(areApprox(ref, data2, PacketSize) && #POP);                       \
}

#define REF_ADD(a,b) ((a)+(b))
#define REF_SUB(a,b) ((a)-(b))
#define REF_MUL(a,b) ((a)*(b))
#define REF_DIV(a,b) ((a)/(b))

template<typename Scalar,typename Packet> void packetmath()
{
  using std::abs;
  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 Packet packets[PacketSize*2];
  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,abs(data1[i]));
  }

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

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

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

  if (internal::unpacket_traits<Packet>::masked_load_available)
  {
    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(areApprox(data3, data2, PacketSize) && "internal::ploadu masked");
      }
    }
  }

  if (internal::unpacket_traits<Packet>::masked_store_available)
  {
    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(areApprox(data3, data2, PacketSize) && "internal::pstoreu masked");
      }
    }
  }

  for (int offset=0; offset<PacketSize; ++offset)
  {
    #define MIN(A,B) (A<B?A:B)
    packets[0] = internal::pload<Packet>(data1);
    packets[1] = internal::pload<Packet>(data1+PacketSize);
         if (offset==0) internal::palign<0>(packets[0], packets[1]);
    else if (offset==1) internal::palign<MIN(1,PacketSize-1)>(packets[0], packets[1]);
    else if (offset==2) internal::palign<MIN(2,PacketSize-1)>(packets[0], packets[1]);
    else if (offset==3) internal::palign<MIN(3,PacketSize-1)>(packets[0], packets[1]);
    else if (offset==4) internal::palign<MIN(4,PacketSize-1)>(packets[0], packets[1]);
    else if (offset==5) internal::palign<MIN(5,PacketSize-1)>(packets[0], packets[1]);
    else if (offset==6) internal::palign<MIN(6,PacketSize-1)>(packets[0], packets[1]);
    else if (offset==7) internal::palign<MIN(7,PacketSize-1)>(packets[0], packets[1]);
    else if (offset==8) internal::palign<MIN(8,PacketSize-1)>(packets[0], packets[1]);
    else if (offset==9) internal::palign<MIN(9,PacketSize-1)>(packets[0], packets[1]);
    else if (offset==10) internal::palign<MIN(10,PacketSize-1)>(packets[0], packets[1]);
    else if (offset==11) internal::palign<MIN(11,PacketSize-1)>(packets[0], packets[1]);
    else if (offset==12) internal::palign<MIN(12,PacketSize-1)>(packets[0], packets[1]);
    else if (offset==13) internal::palign<MIN(13,PacketSize-1)>(packets[0], packets[1]);
    else if (offset==14) internal::palign<MIN(14,PacketSize-1)>(packets[0], packets[1]);
    else if (offset==15) internal::palign<MIN(15,PacketSize-1)>(packets[0], packets[1]);
    internal::pstore(data2, packets[0]);

    for (int i=0; i<PacketSize; ++i)
      ref[i] = data1[i+offset];

    // palign is not used anymore, so let's just put a warning if it fails
    ++g_test_level;
    VERIFY(areApprox(ref, data2, PacketSize) && "internal::palign");
    --g_test_level;
  }

  VERIFY((!PacketTraits::Vectorizable) || PacketTraits::HasAdd);
  VERIFY((!PacketTraits::Vectorizable) || PacketTraits::HasSub);
  VERIFY((!PacketTraits::Vectorizable) || PacketTraits::HasMul);
  VERIFY((!PacketTraits::Vectorizable) || PacketTraits::HasNegate);
  // Disabled as it is not clear why it would be mandatory to support division.
  //VERIFY((internal::is_same<Scalar,int>::value) || (!PacketTraits::Vectorizable) || PacketTraits::HasDiv);

  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);

  CHECK_CWISE1(internal::pnot, internal::pnot);
  CHECK_CWISE1(internal::pzero, internal::pzero);
  CHECK_CWISE1(internal::ptrue, internal::ptrue);
  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(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(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(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(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(areApprox(ref, data2, PacketSize) && "ploadquad");
    }
  }

  ref[0] = Scalar(0);
  for (int i=0; i<PacketSize; ++i)
    ref[0] += data1[i];
  VERIFY(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(areApprox(ref, data2, HalfPacketSize) && "internal::predux_half_dowto4");
  }

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

  if (PacketTraits::HasReduxp)
  {
    for (int j=0; j<PacketSize; ++j)
    {
      ref[j] = Scalar(0);
      for (int i=0; i<PacketSize; ++i)
        ref[j] += data1[i+j*PacketSize];
      packets[j] = internal::pload<Packet>(data1+j*PacketSize);
    }
    internal::pstore(data2, internal::preduxp(packets));
    VERIFY(areApproxAbs(ref, data2, PacketSize, refvalue) && "internal::preduxp");
  }

  for (int i=0; i<PacketSize; ++i)
    ref[i] = data1[PacketSize-i-1];
  internal::pstore(data2, internal::preverse(internal::pload<Packet>(data1)));
  VERIFY(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(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(isApproxAbs(result[i], (selector.select[i] ? data1[i] : data2[i]), refvalue));
    }
  }

  if (PacketTraits::HasBlend || g_vectorize_sse) {
    // pinsertfirst
    for (int i=0; i<PacketSize; ++i)
      ref[i] = data1[i];
    Scalar s = internal::random<Scalar>();
    ref[0] = s;
    internal::pstore(data2, internal::pinsertfirst(internal::pload<Packet>(data1),s));
    VERIFY(areApprox(ref, data2, PacketSize) && "internal::pinsertfirst");
  }

  if (PacketTraits::HasBlend || g_vectorize_sse) {
    // pinsertlast
    for (int i=0; i<PacketSize; ++i)
      ref[i] = data1[i];
    Scalar s = internal::random<Scalar>();
    ref[PacketSize-1] = s;
    internal::pstore(data2, internal::pinsertlast(internal::pload<Packet>(data1),s));
    VERIFY(areApprox(ref, data2, PacketSize) && "internal::pinsertlast");
  }

  {
    for (int i=0; i<PacketSize; ++i)
    {
      data1[i] = internal::random<Scalar>();
      unsigned char v = internal::random<bool>() ? 0xff : 0;
      char* bytes = (char*)(data1+PacketSize+i);
      for(int k=0; k<int(sizeof(Scalar)); ++k) {
        bytes[k] = v;
      }
    }
    CHECK_CWISE2_IF(true, internal::por, internal::por);
    CHECK_CWISE2_IF(true, internal::pxor, internal::pxor);
    CHECK_CWISE2_IF(true, internal::pand, internal::pand);
    CHECK_CWISE2_IF(true, internal::pandnot, internal::pandnot);
  }
  {
    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);
  }

  {
    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);
  }
}

template<typename Scalar,typename Packet> void packetmath_real()
{
  using std::abs;
  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] = internal::random<Scalar>(-1,1) * std::pow(Scalar(10), internal::random<Scalar>(-3,3));
    data2[i] = internal::random<Scalar>(-1,1) * std::pow(Scalar(10), internal::random<Scalar>(-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);

  for (int i=0; i<size; ++i)
  {
    data1[i] = internal::random<Scalar>(-1,1);
    data2[i] = internal::random<Scalar>(-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] = internal::random<Scalar>(-87,88);
    data2[i] = internal::random<Scalar>(-87,88);
  }
  CHECK_CWISE1_IF(PacketTraits::HasExp, std::exp, internal::pexp);
  for (int i=0; i<size; ++i)
  {
    data1[i] = internal::random<Scalar>(-1,1) * std::pow(Scalar(10), internal::random<Scalar>(-6,6));
    data2[i] = internal::random<Scalar>(-1,1) * std::pow(Scalar(10), internal::random<Scalar>(-6,6));
  }
  CHECK_CWISE1_IF(PacketTraits::HasTanh, std::tanh, internal::ptanh);
  if(PacketTraits::HasExp && PacketSize>=2)
  {
    data1[0] = std::numeric_limits<Scalar>::quiet_NaN();
    data1[1] = std::numeric_limits<Scalar>::epsilon();
    packet_helper<PacketTraits::HasExp,Packet> h;
    h.store(data2, internal::pexp(h.load(data1)));
    VERIFY((numext::isnan)(data2[0]));
    VERIFY_IS_EQUAL(std::exp(std::numeric_limits<Scalar>::epsilon()), data2[1]);

    data1[0] = -std::numeric_limits<Scalar>::epsilon();
    data1[1] = 0;
    h.store(data2, internal::pexp(h.load(data1)));
    VERIFY_IS_EQUAL(std::exp(-std::numeric_limits<Scalar>::epsilon()), 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();
    packet_helper<internal::packet_traits<Scalar>::HasTanh,Packet> h;
    h.store(data2, internal::ptanh(h.load(data1)));
    VERIFY((numext::isnan)(data2[0]));
  }

#if EIGEN_HAS_C99_MATH
  {
    data1[0] = std::numeric_limits<Scalar>::quiet_NaN();
    packet_helper<internal::packet_traits<Scalar>::HasLGamma,Packet> h;
    h.store(data2, internal::plgamma(h.load(data1)));
    VERIFY((numext::isnan)(data2[0]));
  }
  if (internal::packet_traits<Scalar>::HasErf) {
    data1[0] = std::numeric_limits<Scalar>::quiet_NaN();
    packet_helper<internal::packet_traits<Scalar>::HasErf,Packet> h;
    h.store(data2, internal::perf(h.load(data1)));
    VERIFY((numext::isnan)(data2[0]));
  }
  {
    data1[0] = std::numeric_limits<Scalar>::quiet_NaN();
    packet_helper<internal::packet_traits<Scalar>::HasErfc,Packet> h;
    h.store(data2, internal::perfc(h.load(data1)));
    VERIFY((numext::isnan)(data2[0]));
  }
  {
    for (int i=0; i<size; ++i) {
      data1[i] = internal::random<Scalar>(0,1);
    }
    CHECK_CWISE1_IF(internal::packet_traits<Scalar>::HasNdtri, numext::ndtri, internal::pndtri);
  }
#endif  // EIGEN_HAS_C99_MATH

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

  if(internal::random<float>(0,1)<0.1f)
     data1[internal::random<int>(0, PacketSize)] = 0;
  CHECK_CWISE1_IF(PacketTraits::HasSqrt, std::sqrt, internal::psqrt);
  CHECK_CWISE1_IF(PacketTraits::HasLog, std::log, internal::plog);
  CHECK_CWISE1_IF(PacketTraits::HasBessel, numext::bessel_i0, internal::pbessel_i0);
  CHECK_CWISE1_IF(PacketTraits::HasBessel, numext::bessel_i0e, internal::pbessel_i0e);
  CHECK_CWISE1_IF(PacketTraits::HasBessel, numext::bessel_i1, internal::pbessel_i1);
  CHECK_CWISE1_IF(PacketTraits::HasBessel, numext::bessel_i1e, internal::pbessel_i1e);
  CHECK_CWISE1_IF(PacketTraits::HasBessel, numext::bessel_j0, internal::pbessel_j0);
  CHECK_CWISE1_IF(PacketTraits::HasBessel, numext::bessel_j1, internal::pbessel_j1);

  data1[0] = std::numeric_limits<Scalar>::infinity();
  CHECK_CWISE1_IF(PacketTraits::HasRsqrt, Scalar(1)/std::sqrt, internal::prsqrt);

  // Use a smaller data range for the positive bessel operations as these
  // can have much more error at very small and very large values.
  for (int i=0; i<size; ++i) {
      data1[i] = internal::random<Scalar>(0.01,1) * std::pow(
          Scalar(10), internal::random<Scalar>(-1,2));
      data2[i] = internal::random<Scalar>(0.01,1) * std::pow(
          Scalar(10), internal::random<Scalar>(-1,2));
  }
  CHECK_CWISE1_IF(PacketTraits::HasBessel, numext::bessel_y0, internal::pbessel_y0);
  CHECK_CWISE1_IF(PacketTraits::HasBessel, numext::bessel_y1, internal::pbessel_y1);
  CHECK_CWISE1_IF(PacketTraits::HasBessel, numext::bessel_k0, internal::pbessel_k0);
  CHECK_CWISE1_IF(PacketTraits::HasBessel, numext::bessel_k0e, internal::pbessel_k0e);
  CHECK_CWISE1_IF(PacketTraits::HasBessel, numext::bessel_k1, internal::pbessel_k1);
  CHECK_CWISE1_IF(PacketTraits::HasBessel, numext::bessel_k1e, internal::pbessel_k1e);

#if EIGEN_HAS_C99_MATH && (__cplusplus > 199711L)
  CHECK_CWISE1_IF(internal::packet_traits<Scalar>::HasLGamma, std::lgamma, internal::plgamma);
  CHECK_CWISE1_IF(internal::packet_traits<Scalar>::HasErf, std::erf, internal::perf);
  CHECK_CWISE1_IF(internal::packet_traits<Scalar>::HasErfc, std::erfc, internal::perfc);
  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)
    {
      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] = 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]));

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

      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) {
      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)
    {
      packet_helper<PacketTraits::HasSqrt,Packet> h;
      data1[0] = Scalar(-1.0f);
      data1[1] = -std::numeric_limits<Scalar>::denorm_min();
      h.store(data2, internal::psqrt(h.load(data1)));
      VERIFY((numext::isnan)(data2[0]));
      VERIFY((numext::isnan)(data2[1]));
    }
    if(PacketTraits::HasCos)
    {
      packet_helper<PacketTraits::HasCos,Packet> h;
      for(Scalar k = 1; k<Scalar(10000)/std::numeric_limits<Scalar>::epsilon(); k*=2)
      {
        for(int k1=0;k1<=1; ++k1)
        {
          data1[0] = (2*k+k1  )*Scalar(EIGEN_PI)/2 * internal::random<Scalar>(0.8,1.2);
          data1[1] = (2*k+2+k1)*Scalar(EIGEN_PI)/2 * internal::random<Scalar>(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(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));
    }
  }
}

template<typename Scalar,typename Packet> void packetmath_notcomplex()
{
  using std::abs;
  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(abs, internal::pabs);

  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");

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

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(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(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(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 = 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(isApproxAbs(buffer[i], data1[i/stride], refvalue) && "pscatter");
    } else {
      VERIFY(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(isApproxAbs(data1[i], buffer[i*7], refvalue) && "pgather");
  }
}


template<
  typename Scalar,
  typename PacketType,
  bool IsComplex = NumTraits<Scalar>::IsComplex,
  bool IsInteger = NumTraits<Scalar>::IsInteger>
struct runall;

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>();
  }
};

template<
  typename Scalar,
  typename PacketType = typename internal::packet_traits<Scalar>::type,
  bool Vectorized = internal::packet_traits<Scalar>::Vectorizable,
  bool HasHalf = !internal::is_same<typename internal::unpacket_traits<PacketType>::half,PacketType>::value >
struct runner;

template<typename Scalar,typename PacketType>
struct runner<Scalar,PacketType,true,true>
{
  static void run() {
    runall<Scalar,PacketType>::run();
    runner<Scalar,typename internal::unpacket_traits<PacketType>::half>::run();
  }
};

template<typename Scalar,typename PacketType>
struct runner<Scalar,PacketType,true,false>
{
  static void run() {
    runall<Scalar,PacketType>::run();
    runall<Scalar,Scalar>::run();
  }
};

template<typename Scalar,typename PacketType>
struct runner<Scalar,PacketType,false,false>
{
  static void run() {
    runall<Scalar,PacketType>::run();
  }
};

EIGEN_DECLARE_TEST(packetmath)
{
  g_first_pass = true;
  for(int i = 0; i < g_repeat; i++) {

    CALL_SUBTEST_1( runner<float>::run() );
    CALL_SUBTEST_2( runner<double>::run() );
    CALL_SUBTEST_3( runner<int>::run() );
    CALL_SUBTEST_4( runner<std::complex<float> >::run() );
    CALL_SUBTEST_5( runner<std::complex<double> >::run() );
    CALL_SUBTEST_6(( packetmath<half,internal::packet_traits<half>::type>() ));
    g_first_pass = false;
  }
}