eigen/test/packetmath.cpp
Benoit Steiner 3625734bc8 Moved some utilities to TensorMeta.h to make it easier to reuse them accross several tensor operations.
Created the TensorDimensionList class to encode the list of all the dimensions of a tensor of rank n. This could be done using TensorIndexList, however TensorIndexList require cxx11 which isn't yet supported as widely as we'd like.
2015-06-29 10:49:55 -07:00

540 lines
20 KiB
C++

// 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"
// using namespace Eigen;
namespace Eigen {
namespace internal {
template<typename T> T negate(const T& x) { return -x; }
}
}
template<typename Scalar> 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 << "[" << Map<const Matrix<Scalar,1,Dynamic> >(a,size) << "]" << " != " << 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 (a[i]!=b[i] && !internal::isApprox(a[i],b[i]))
{
std::cout << "[" << Map<const Matrix<Scalar,1,Dynamic> >(a,size) << "]" << " != " << Map<const Matrix<Scalar,1,Dynamic> >(b,size) << "\n";
return false;
}
}
return true;
}
#define CHECK_CWISE2(REFOP, POP) { \
for (int i=0; i<PacketSize; ++i) \
ref[i] = REFOP(data1[i], data1[i+PacketSize]); \
internal::pstore(data2, POP(internal::pload<Packet>(data1), internal::pload<Packet>(data1+PacketSize))); \
VERIFY(areApprox(ref, data2, PacketSize) && #POP); \
}
#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 void store(T* to, const Packet& x) const { internal::pstore(to,x); }
};
template<typename Packet>
struct packet_helper<false,Packet>
{
template<typename T>
inline T load(const T* from) const { return *from; }
template<typename T>
inline void store(T* to, const T& x) 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 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> void packetmath()
{
using std::abs;
typedef typename internal::packet_traits<Scalar>::type Packet;
const int PacketSize = internal::packet_traits<Scalar>::size;
typedef typename NumTraits<Scalar>::Real RealScalar;
const int max_size = PacketSize > 4 ? PacketSize : 4;
const int size = PacketSize*max_size;
EIGEN_ALIGN_DEFAULT Scalar data1[size];
EIGEN_ALIGN_DEFAULT Scalar data2[size];
EIGEN_ALIGN_DEFAULT Packet packets[PacketSize*2];
EIGEN_ALIGN_DEFAULT Scalar ref[size];
RealScalar refvalue = 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");
}
for (int offset=0; offset<PacketSize; ++offset)
{
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<1>(packets[0], packets[1]);
else if (offset==2) internal::palign<2>(packets[0], packets[1]);
else if (offset==3) internal::palign<3>(packets[0], packets[1]);
else if (offset==4) internal::palign<4>(packets[0], packets[1]);
else if (offset==5) internal::palign<5>(packets[0], packets[1]);
else if (offset==6) internal::palign<6>(packets[0], packets[1]);
else if (offset==7) internal::palign<7>(packets[0], packets[1]);
internal::pstore(data2, packets[0]);
for (int i=0; i<PacketSize; ++i)
ref[i] = data1[i+offset];
VERIFY(areApprox(ref, data2, PacketSize) && "internal::palign");
}
CHECK_CWISE2(REF_ADD, internal::padd);
CHECK_CWISE2(REF_SUB, internal::psub);
CHECK_CWISE2(REF_MUL, internal::pmul);
#if !defined(EIGEN_VECTORIZE_ALTIVEC) && !defined(EIGEN_VECTORIZE_VSX)
if (!internal::is_same<Scalar,int>::value)
CHECK_CWISE2(REF_DIV, internal::pdiv);
#endif
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)
{
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)
{
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] = 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");
{
for (int i=0; i<4; ++i)
ref[i] = 0;
for (int i=0; i<PacketSize; ++i)
ref[i%4] += data1[i];
internal::pstore(data2, internal::predux4(internal::pload<Packet>(data1)));
VERIFY(areApprox(ref, data2, PacketSize>4?PacketSize/2:PacketSize) && "internal::predux4");
}
ref[0] = 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");
for (int j=0; j<PacketSize; ++j)
{
ref[j] = 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 (internal::packet_traits<Scalar>::HasBlend) {
Packet thenPacket = internal::pload<Packet>(data1);
Packet elsePacket = internal::pload<Packet>(data2);
EIGEN_ALIGN_DEFAULT internal::Selector<PacketSize> selector;
for (int i = 0; i < PacketSize; ++i) {
selector.select[i] = i;
}
Packet blend = internal::pblend(selector, thenPacket, elsePacket);
EIGEN_ALIGN_DEFAULT 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));
}
}
}
template<typename Scalar> void packetmath_real()
{
using std::abs;
typedef typename internal::packet_traits<Scalar>::type Packet;
const int PacketSize = internal::packet_traits<Scalar>::size;
const int size = PacketSize*4;
EIGEN_ALIGN_DEFAULT Scalar data1[internal::packet_traits<Scalar>::size*4];
EIGEN_ALIGN_DEFAULT Scalar data2[internal::packet_traits<Scalar>::size*4];
EIGEN_ALIGN_DEFAULT Scalar ref[internal::packet_traits<Scalar>::size*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(internal::packet_traits<Scalar>::HasSin, std::sin, internal::psin);
CHECK_CWISE1_IF(internal::packet_traits<Scalar>::HasCos, std::cos, internal::pcos);
CHECK_CWISE1_IF(internal::packet_traits<Scalar>::HasTan, std::tan, internal::ptan);
for (int i=0; i<size; ++i)
{
data1[i] = internal::random<Scalar>(-1,1);
data2[i] = internal::random<Scalar>(-1,1);
}
CHECK_CWISE1_IF(internal::packet_traits<Scalar>::HasASin, std::asin, internal::pasin);
CHECK_CWISE1_IF(internal::packet_traits<Scalar>::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(internal::packet_traits<Scalar>::HasExp, std::exp, internal::pexp);
{
data1[0] = std::numeric_limits<Scalar>::quiet_NaN();
data1[1] = std::numeric_limits<Scalar>::epsilon();
packet_helper<internal::packet_traits<Scalar>::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(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]);
}
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)<0.1)
data1[internal::random<int>(0, PacketSize)] = 0;
CHECK_CWISE1_IF(internal::packet_traits<Scalar>::HasSqrt, std::sqrt, internal::psqrt);
CHECK_CWISE1_IF(internal::packet_traits<Scalar>::HasLog, std::log, internal::plog);
{
data1[0] = std::numeric_limits<Scalar>::quiet_NaN();
data1[1] = std::numeric_limits<Scalar>::epsilon();
packet_helper<internal::packet_traits<Scalar>::HasLog,Packet> h;
h.store(data2, internal::plog(h.load(data1)));
VERIFY(std::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(std::isnan(data2[0]));
// VERIFY_IS_EQUAL(std::log(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(std::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(std::isnan(data2[1]));
data1[0] = -1.0f;
h.store(data2, internal::plog(h.load(data1)));
VERIFY(std::isnan(data2[0]));
#if !EIGEN_FAST_MATH
h.store(data2, internal::psqrt(h.load(data1)));
VERIFY(numext::isnan(data2[0]));
VERIFY(numext::isnan(data2[1]));
#endif
}
}
template<typename Scalar> void packetmath_notcomplex()
{
using std::abs;
typedef typename internal::packet_traits<Scalar>::type Packet;
const int PacketSize = internal::packet_traits<Scalar>::size;
EIGEN_ALIGN_DEFAULT Scalar data1[internal::packet_traits<Scalar>::size*4];
EIGEN_ALIGN_DEFAULT Scalar data2[internal::packet_traits<Scalar>::size*4];
EIGEN_ALIGN_DEFAULT Scalar ref[internal::packet_traits<Scalar>::size*4];
Array<Scalar,Dynamic,1>::Map(data1, internal::packet_traits<Scalar>::size*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");
CHECK_CWISE2((std::min), internal::pmin);
CHECK_CWISE2((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(data1[0]));
VERIFY(areApprox(ref, data2, PacketSize) && "internal::plset");
}
template<typename Scalar,bool ConjLhs,bool ConjRhs> void test_conj_helper(Scalar* data1, Scalar* data2, Scalar* ref, Scalar* pval)
{
typedef typename internal::packet_traits<Scalar>::type Packet;
const int PacketSize = internal::packet_traits<Scalar>::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> void packetmath_complex()
{
typedef typename internal::packet_traits<Scalar>::type Packet;
const int PacketSize = internal::packet_traits<Scalar>::size;
const int size = PacketSize*4;
EIGEN_ALIGN_DEFAULT Scalar data1[PacketSize*4];
EIGEN_ALIGN_DEFAULT Scalar data2[PacketSize*4];
EIGEN_ALIGN_DEFAULT Scalar ref[PacketSize*4];
EIGEN_ALIGN_DEFAULT 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,false,false> (data1,data2,ref,pval);
test_conj_helper<Scalar,false,true> (data1,data2,ref,pval);
test_conj_helper<Scalar,true,false> (data1,data2,ref,pval);
test_conj_helper<Scalar,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> void packetmath_scatter_gather() {
typedef typename internal::packet_traits<Scalar>::type Packet;
typedef typename NumTraits<Scalar>::Real RealScalar;
const int PacketSize = internal::packet_traits<Scalar>::size;
EIGEN_ALIGN_DEFAULT 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_DEFAULT Scalar buffer[PacketSize*20];
memset(buffer, 0, 20*sizeof(Packet));
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");
}
}
void test_packetmath()
{
for(int i = 0; i < g_repeat; i++) {
CALL_SUBTEST_1( packetmath<float>() );
CALL_SUBTEST_2( packetmath<double>() );
CALL_SUBTEST_3( packetmath<int>() );
CALL_SUBTEST_4( packetmath<std::complex<float> >() );
CALL_SUBTEST_5( packetmath<std::complex<double> >() );
CALL_SUBTEST_1( packetmath_notcomplex<float>() );
CALL_SUBTEST_2( packetmath_notcomplex<double>() );
CALL_SUBTEST_3( packetmath_notcomplex<int>() );
CALL_SUBTEST_1( packetmath_real<float>() );
CALL_SUBTEST_2( packetmath_real<double>() );
CALL_SUBTEST_4( packetmath_complex<std::complex<float> >() );
CALL_SUBTEST_5( packetmath_complex<std::complex<double> >() );
CALL_SUBTEST_1( packetmath_scatter_gather<float>() );
CALL_SUBTEST_2( packetmath_scatter_gather<double>() );
CALL_SUBTEST_3( packetmath_scatter_gather<int>() );
CALL_SUBTEST_4( packetmath_scatter_gather<std::complex<float> >() );
CALL_SUBTEST_5( packetmath_scatter_gather<std::complex<double> >() );
}
}