diff --git a/Eigen/src/Core/arch/AVX/MathFunctions.h b/Eigen/src/Core/arch/AVX/MathFunctions.h
index aecfdd6ad..3031e7233 100644
--- a/Eigen/src/Core/arch/AVX/MathFunctions.h
+++ b/Eigen/src/Core/arch/AVX/MathFunctions.h
@@ -309,13 +309,44 @@ template <> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED
 Packet4d psqrt<Packet4d>(const Packet4d& x) {
   return _mm256_sqrt_pd(x);
 }
+#if EIGEN_FAST_MATH
 
+template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED
+Packet4f prsqrt<Packet4f>(const Packet4f& _x) {
+  _EIGEN_DECLARE_CONST_Packet4f_FROM_INT(inf, 0x7f800000);
+  _EIGEN_DECLARE_CONST_Packet4f_FROM_INT(nan, 0x7fc00000);
+  _EIGEN_DECLARE_CONST_Packet4f(one_point_five, 1.5f);
+  _EIGEN_DECLARE_CONST_Packet4f(minus_half, -0.5f);
+  _EIGEN_DECLARE_CONST_Packet4f_FROM_INT(flt_min, 0x00800000);
+
+  Packet4f neg_half = pmul(_x, p4f_minus_half);
+
+  // select only the inverse sqrt of positive normal inputs (denormals are
+  // flushed to zero and cause infs as well).
+  Packet4f le_zero_mask = _mm_cmple_ps(_x, p4f_flt_min);
+  Packet4f x = _mm_andnot_ps(le_zero_mask, _mm_rsqrt_ps(_x));
+
+  // Fill in NaNs and Infs for the negative/zero entries.
+  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));
+
+  // Do a single step of Newton's iteration.
+  x = pmul(x, pmadd(neg_half, pmul(x, x), p4f_one_point_five));
+
+  // Insert NaNs and Infs in all the right places.
+  return _mm_or_ps(x, infs_and_nans);
+}
+
+#else
 template <> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED
 Packet8f prsqrt<Packet8f>(const Packet8f& x) {
   _EIGEN_DECLARE_CONST_Packet8f(one, 1.0f);
   return _mm256_div_ps(p8f_one, _mm256_sqrt_ps(x));
 }
 #endif
+
 template <> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED
 Packet4d prsqrt<Packet4d>(const Packet4d& x) {
   _EIGEN_DECLARE_CONST_Packet4d(one, 1.0);
diff --git a/Eigen/src/Core/arch/SSE/MathFunctions.h b/Eigen/src/Core/arch/SSE/MathFunctions.h
index f9cb93bfc..3b8b7303f 100644
--- a/Eigen/src/Core/arch/SSE/MathFunctions.h
+++ b/Eigen/src/Core/arch/SSE/MathFunctions.h
@@ -470,12 +470,46 @@ Packet4f psqrt<Packet4f>(const Packet4f& x) { return _mm_sqrt_ps(x); }
 template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED
 Packet2d psqrt<Packet2d>(const Packet2d& x) { return _mm_sqrt_pd(x); }
 
+#if EIGEN_FAST_MATH
+
+template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED
+Packet4f prsqrt<Packet4f>(const Packet4f& _x) {
+  _EIGEN_DECLARE_CONST_Packet4f_FROM_INT(inf, 0x7f800000);
+  _EIGEN_DECLARE_CONST_Packet4f_FROM_INT(nan, 0x7fc00000);
+  _EIGEN_DECLARE_CONST_Packet4f(one_point_five, 1.5f);
+  _EIGEN_DECLARE_CONST_Packet4f(minus_half, -0.5f);
+  _EIGEN_DECLARE_CONST_Packet4f_FROM_INT(flt_min, 0x00800000);
+
+  Packet4f neg_half = pmul(_x, p4f_minus_half);
+
+  // select only the inverse sqrt of positive normal inputs (denormals are
+  // flushed to zero and cause infs as well).
+  Packet4f le_zero_mask = _mm_cmple_ps(_x, p4f_flt_min);
+  Packet4f x = _mm_andnot_ps(le_zero_mask, _mm_rsqrt_ps(_x));
+
+  // Fill in NaNs and Infs for the negative/zero entries.
+  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));
+
+  // Do a single step of Newton's iteration.
+  x = pmul(x, pmadd(neg_half, pmul(x, x), p4f_one_point_five));
+
+  // Insert NaNs and Infs in all the right places.
+  return _mm_or_ps(x, infs_and_nans);
+}
+
+#else
+
 template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED
 Packet4f prsqrt<Packet4f>(const Packet4f& x) {
   // Unfortunately we can't use the much faster mm_rqsrt_ps since it only provides an approximation.
   return _mm_div_ps(pset1<Packet4f>(1.0f), _mm_sqrt_ps(x));
 }
 
+#endif
+
 template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED
 Packet2d prsqrt<Packet2d>(const Packet2d& x) {
   // Unfortunately we can't use the much faster mm_rqsrt_pd since it only provides an approximation.