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Fix a bug in the implementation of Carmack's fast sqrt algorithm in Eigen (enabled by EIGEN_FAST_MATH), which causes the vectorized parts of the computation to return -0.0 instead of NaN for negative arguments.

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Benchmark speed in Giga-sqrts/s
Intel(R) Xeon(R) CPU E5-1650 v3 @ 3.50GHz
-----------------------------------------
                    SSE        AVX
Fast=1              2.529G     4.380G
Fast=0              1.944G     1.898G
Fast=1 fixed        2.214G     3.739G

This table illustrates the worst case in terms speed impact: It was measured by repeatedly computing the sqrt of an n=4096 float vector that fits in L1 cache. For large vectors the operation becomes memory bound and the differences between the different versions almost negligible.
This commit is contained in:
Rasmus Munk Larsen 2016-10-04 14:22:56 -07:00
parent 6af5ac7e27
commit 3ed67cb0bb
3 changed files with 34 additions and 40 deletions
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24
Eigen/src/Core/arch/AVX/MathFunctions.h
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@ -362,23 +362,17 @@ pexp<Packet4d>(const Packet4d& _x) {
template <>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet8f
psqrt<Packet8f>(const Packet8f& _x) {
_EIGEN_DECLARE_CONST_Packet8f(one_point_five, 1.5f);
_EIGEN_DECLARE_CONST_Packet8f(minus_half, -0.5f);
_EIGEN_DECLARE_CONST_Packet8f_FROM_INT(flt_min, 0x00800000);
Packet8f neg_half = pmul(_x, p8f_minus_half);
// select only the inverse sqrt of positive normal inputs (denormals are
// flushed to zero and cause infs as well).
Packet8f non_zero_mask = _mm256_cmp_ps(_x, p8f_flt_min, _CMP_GE_OQ);
Packet8f x = _mm256_and_ps(non_zero_mask, _mm256_rsqrt_ps(_x));
Packet8f half = pmul(_x, pset1<Packet8f>(.5f));
Packet8f denormal_mask = _mm256_and_ps(
_mm256_cmpge_ps(_x, _mm256_setzero_ps()),
_mm256_cmplt_ps(_x, pset1<Packet8f>((std::numeric_limits<float>::min)())));
// Compute approximate reciprocal sqrt.
Packet8f x = _mm256_rsqrt_ps(_x);
// Do a single step of Newton's iteration.
x = pmul(x, pmadd(neg_half, pmul(x, x), p8f_one_point_five));
// Multiply the original _x by it's reciprocal square root to extract the
// square root.
return pmul(_x, x);
x = pmul(x, psub(pset1<Packet8f>(1.5f), pmul(half, pmul(x,x))));
// Flush results for denormals to zero.
return _mm256_andnot_ps(denormal_mask, pmul(_x,x));
}
#else
template <> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED

19
Eigen/src/Core/arch/SSE/MathFunctions.h
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@ -32,7 +32,7 @@ Packet4f plog<Packet4f>(const Packet4f& _x)
/* the smallest non denormalized float number */
_EIGEN_DECLARE_CONST_Packet4f_FROM_INT(min_norm_pos, 0x00800000);
_EIGEN_DECLARE_CONST_Packet4f_FROM_INT(minus_inf, 0xff800000);//-1.f/0.f);
/* natural logarithm computed for 4 simultaneous float
return NaN for x <= 0
*/
@ -451,18 +451,21 @@ template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED
Packet4f psqrt<Packet4f>(const Packet4f& _x)
{
Packet4f half = pmul(_x, pset1<Packet4f>(.5f));
Packet4f denormal_mask = _mm_and_ps(
_mm_cmpge_ps(_x, _mm_setzero_ps()),
_mm_cmplt_ps(_x, pset1<Packet4f>((std::numeric_limits<float>::min)())));
/* select only the inverse sqrt of non-zero inputs */
Packet4f non_zero_mask = _mm_cmpge_ps(_x, pset1<Packet4f>((std::numeric_limits<float>::min)()));
Packet4f x = _mm_and_ps(non_zero_mask, _mm_rsqrt_ps(_x));
// Compute approximate reciprocal sqrt.
Packet4f x = _mm_rsqrt_ps(_x);
// Do a single step of Newton's iteration.
x = pmul(x, psub(pset1<Packet4f>(1.5f), pmul(half, pmul(x,x))));
return pmul(_x,x);
// Flush results for denormals to zero.
return _mm_andnot_ps(denormal_mask, pmul(_x,x));
}
#else
template<>EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED
template<>EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED
Packet4f psqrt<Packet4f>(const Packet4f& x) { return _mm_sqrt_ps(x); }
#endif
@ -491,7 +494,7 @@ Packet4f prsqrt<Packet4f>(const Packet4f& _x) {
Packet4f neg_mask = _mm_cmplt_ps(_x, _mm_setzero_ps());
Packet4f zero_mask = _mm_andnot_ps(neg_mask, le_zero_mask);
Packet4f infs_and_nans = _mm_or_ps(_mm_and_ps(neg_mask, p4f_nan),
_mm_and_ps(zero_mask, p4f_inf));
_mm_and_ps(zero_mask, p4f_inf));
// Do a single step of Newton's iteration.
x = pmul(x, pmadd(neg_half, pmul(x, x), p4f_one_point_five));

31
test/packetmath.cpp
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@ -193,7 +193,7 @@ template<typename Scalar> void packetmath()
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];
@ -203,9 +203,9 @@ template<typename Scalar> void packetmath()
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)
@ -315,7 +315,7 @@ template<typename Scalar> void packetmath_real()
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);
@ -440,12 +440,9 @@ template<typename Scalar> void packetmath_real()
data1[0] = Scalar(-1.0f);
h.store(data2, internal::plog(h.load(data1)));
VERIFY((numext::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
}
}
@ -459,7 +456,7 @@ template<typename Scalar> void packetmath_notcomplex()
EIGEN_ALIGN_MAX Scalar data1[PacketTraits::size*4];
EIGEN_ALIGN_MAX Scalar data2[PacketTraits::size*4];
EIGEN_ALIGN_MAX Scalar ref[PacketTraits::size*4];
Array<Scalar,Dynamic,1>::Map(data1, PacketTraits::size*4).setRandom();
ref[0] = data1[0];
@ -478,7 +475,7 @@ template<typename Scalar> void packetmath_notcomplex()
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]));
@ -490,12 +487,12 @@ template<typename Scalar,bool ConjLhs,bool ConjRhs> void test_conj_helper(Scalar
typedef internal::packet_traits<Scalar> PacketTraits;
typedef typename PacketTraits::type Packet;
const int PacketSize = PacketTraits::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]);
@ -503,7 +500,7 @@ template<typename Scalar,bool ConjLhs,bool ConjRhs> void test_conj_helper(Scalar
}
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];
@ -531,12 +528,12 @@ template<typename Scalar> void packetmath_complex()
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]));
@ -556,9 +553,9 @@ template<typename Scalar> void packetmath_scatter_gather()
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*sizeof(Packet));
Packet packet = internal::pload<Packet>(data1);
@ -594,7 +591,7 @@ void test_packetmath()
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>() );