Special function implementations for half/bfloat16 packets.

Current implementations fail to consider half-float packets, only
half-float scalars.  Added specializations for packets on AVX, AVX512 and
NEON.  Added tests to `special_packetmath`.

The current `special_functions` tests would fail for half and bfloat16 due to
lack of precision. The NEON tests also fail with precision issues and
due to different handling of `sqrt(inf)`, so special functions bessel, ndtri
have been disabled.

Tested with AVX, AVX512.

11 files changed, 309 insertions, 95 deletions

diff --git a/unsupported/Eigen/SpecialFunctions b/unsupported/Eigen/SpecialFunctions
index dda6618de..f6a2460e6 100644
--- a/unsupported/Eigen/SpecialFunctions
+++ b/unsupported/Eigen/SpecialFunctions
@@ -61,23 +61,36 @@ namespace Eigen {
 }
 
 #include "src/SpecialFunctions/BesselFunctionsImpl.h"
-#include "src/SpecialFunctions/BesselFunctionsPacketMath.h"
 #include "src/SpecialFunctions/BesselFunctionsBFloat16.h"
 #include "src/SpecialFunctions/BesselFunctionsHalf.h"
+#include "src/SpecialFunctions/BesselFunctionsPacketMath.h"
 #include "src/SpecialFunctions/BesselFunctionsFunctors.h"
 #include "src/SpecialFunctions/BesselFunctionsArrayAPI.h"
 #include "src/SpecialFunctions/SpecialFunctionsImpl.h"
 #if defined(EIGEN_HIPCC)
 #include "src/SpecialFunctions/HipVectorCompatibility.h"
 #endif
-#include "src/SpecialFunctions/SpecialFunctionsPacketMath.h"
 #include "src/SpecialFunctions/SpecialFunctionsBFloat16.h"
 #include "src/SpecialFunctions/SpecialFunctionsHalf.h"
+#include "src/SpecialFunctions/SpecialFunctionsPacketMath.h"
 #include "src/SpecialFunctions/SpecialFunctionsFunctors.h"
 #include "src/SpecialFunctions/SpecialFunctionsArrayAPI.h"
 
+#if defined EIGEN_VECTORIZE_AVX512
+  #include "src/SpecialFunctions/arch/AVX/BesselFunctions.h"
+  #include "src/SpecialFunctions/arch/AVX/SpecialFunctions.h"
+  #include "src/SpecialFunctions/arch/AVX512/BesselFunctions.h"
+  #include "src/SpecialFunctions/arch/AVX512/SpecialFunctions.h"
+#elif defined EIGEN_VECTORIZE_AVX
+  #include "src/SpecialFunctions/arch/AVX/BesselFunctions.h"
+  #include "src/SpecialFunctions/arch/AVX/SpecialFunctions.h"
+#elif defined EIGEN_VECTORIZE_NEON
+  #include "src/SpecialFunctions/arch/NEON/BesselFunctions.h"
+  #include "src/SpecialFunctions/arch/NEON/SpecialFunctions.h"
+#endif
+
 #if defined EIGEN_VECTORIZE_GPU
-  #include "src/SpecialFunctions/arch/GPU/GpuSpecialFunctions.h"
+  #include "src/SpecialFunctions/arch/GPU/SpecialFunctions.h"
 #endif
 
 namespace Eigen {
diff --git a/unsupported/Eigen/src/SpecialFunctions/BesselFunctionsImpl.h b/unsupported/Eigen/src/SpecialFunctions/BesselFunctionsImpl.h
index a9b6ad940..24812be1b 100644
--- a/unsupported/Eigen/src/SpecialFunctions/BesselFunctionsImpl.h
+++ b/unsupported/Eigen/src/SpecialFunctions/BesselFunctionsImpl.h
@@ -46,7 +46,7 @@ struct bessel_i0e_retval {
   typedef Scalar type;
 };
 
-template <typename T, typename ScalarType>
+template <typename T, typename ScalarType = typename unpacket_traits<T>::type>
 struct generic_i0e {
   EIGEN_DEVICE_FUNC
   static EIGEN_STRONG_INLINE T run(const T&) {
@@ -201,11 +201,11 @@ struct generic_i0e<T, double> {
   }
 };
 
-template <typename Scalar>
+template <typename T>
 struct bessel_i0e_impl {
   EIGEN_DEVICE_FUNC
-  static EIGEN_STRONG_INLINE Scalar run(const Scalar x) {
-    return generic_i0e<Scalar, Scalar>::run(x);
+  static EIGEN_STRONG_INLINE T run(const T x) {
+    return generic_i0e<T>::run(x);
   }
 };
 
@@ -214,7 +214,7 @@ struct bessel_i0_retval {
   typedef Scalar type;
 };
 
-template <typename T, typename ScalarType>
+template <typename T, typename ScalarType = typename unpacket_traits<T>::type>
 struct generic_i0 {
   EIGEN_DEVICE_FUNC
   static EIGEN_STRONG_INLINE T run(const T& x) {
@@ -224,11 +224,11 @@ struct generic_i0 {
   }
 };
 
-template <typename Scalar>
+template <typename T>
 struct bessel_i0_impl {
   EIGEN_DEVICE_FUNC
-  static EIGEN_STRONG_INLINE Scalar run(const Scalar x) {
-    return generic_i0<Scalar, Scalar>::run(x);
+  static EIGEN_STRONG_INLINE T run(const T x) {
+    return generic_i0<T>::run(x);
   }
 };
 
@@ -237,7 +237,7 @@ struct bessel_i1e_retval {
   typedef Scalar type;
 };
 
-template <typename T, typename ScalarType>
+template <typename T, typename ScalarType = typename unpacket_traits<T>::type >
 struct generic_i1e {
   EIGEN_DEVICE_FUNC
   static EIGEN_STRONG_INLINE T run(const T&) {
@@ -396,20 +396,20 @@ struct generic_i1e<T, double> {
   }
 };
 
-template <typename Scalar>
+template <typename T>
 struct bessel_i1e_impl {
   EIGEN_DEVICE_FUNC
-  static EIGEN_STRONG_INLINE Scalar run(const Scalar x) {
-    return generic_i1e<Scalar, Scalar>::run(x);
+  static EIGEN_STRONG_INLINE T run(const T x) {
+    return generic_i1e<T>::run(x);
   }
 };
 
-template <typename Scalar>
+template <typename T>
 struct bessel_i1_retval {
-  typedef Scalar type;
+  typedef T type;
 };
 
-template <typename T, typename ScalarType>
+template <typename T, typename ScalarType = typename unpacket_traits<T>::type>
 struct generic_i1 {
   EIGEN_DEVICE_FUNC
   static EIGEN_STRONG_INLINE T run(const T& x) {
@@ -419,20 +419,20 @@ struct generic_i1 {
   }
 };
 
-template <typename Scalar>
+template <typename T>
 struct bessel_i1_impl {
   EIGEN_DEVICE_FUNC
-  static EIGEN_STRONG_INLINE Scalar run(const Scalar x) {
-    return generic_i1<Scalar, Scalar>::run(x);
+  static EIGEN_STRONG_INLINE T run(const T x) {
+    return generic_i1<T>::run(x);
   }
 };
 
-template <typename Scalar>
+template <typename T>
 struct bessel_k0e_retval {
-  typedef Scalar type;
+  typedef T type;
 };
 
-template <typename T, typename ScalarType>
+template <typename T, typename ScalarType = typename unpacket_traits<T>::type>
 struct generic_k0e {
   EIGEN_DEVICE_FUNC
   static EIGEN_STRONG_INLINE T run(const T&) {
@@ -582,20 +582,20 @@ struct generic_k0e<T, double> {
   }
 };
 
-template <typename Scalar>
+template <typename T>
 struct bessel_k0e_impl {
   EIGEN_DEVICE_FUNC
-  static EIGEN_STRONG_INLINE Scalar run(const Scalar x) {
-    return generic_k0e<Scalar, Scalar>::run(x);
+  static EIGEN_STRONG_INLINE T run(const T x) {
+    return generic_k0e<T>::run(x);
   }
 };
 
-template <typename Scalar>
+template <typename T>
 struct bessel_k0_retval {
-  typedef Scalar type;
+  typedef T type;
 };
 
-template <typename T, typename ScalarType>
+template <typename T, typename ScalarType = typename unpacket_traits<T>::type>
 struct generic_k0 {
   EIGEN_DEVICE_FUNC
   static EIGEN_STRONG_INLINE T run(const T&) {
@@ -754,20 +754,20 @@ struct generic_k0<T, double> {
   }
 };
 
-template <typename Scalar>
+template <typename T>
 struct bessel_k0_impl {
   EIGEN_DEVICE_FUNC
-  static EIGEN_STRONG_INLINE Scalar run(const Scalar x) {
-    return generic_k0<Scalar, Scalar>::run(x);
+  static EIGEN_STRONG_INLINE T run(const T x) {
+    return generic_k0<T>::run(x);
   }
 };
 
-template <typename Scalar>
+template <typename T>
 struct bessel_k1e_retval {
-  typedef Scalar type;
+  typedef T type;
 };
 
-template <typename T, typename ScalarType>
+template <typename T, typename ScalarType = typename unpacket_traits<T>::type>
 struct generic_k1e {
   EIGEN_DEVICE_FUNC
   static EIGEN_STRONG_INLINE T run(const T&) {
@@ -910,20 +910,20 @@ struct generic_k1e<T, double> {
   }
 };
 
-template <typename Scalar>
+template <typename T>
 struct bessel_k1e_impl {
   EIGEN_DEVICE_FUNC
-  static EIGEN_STRONG_INLINE Scalar run(const Scalar x) {
-    return generic_k1e<Scalar, Scalar>::run(x);
+  static EIGEN_STRONG_INLINE T run(const T x) {
+    return generic_k1e<T>::run(x);
   }
 };
 
-template <typename Scalar>
+template <typename T>
 struct bessel_k1_retval {
-  typedef Scalar type;
+  typedef T type;
 };
 
-template <typename T, typename ScalarType>
+template <typename T, typename ScalarType = typename unpacket_traits<T>::type>
 struct generic_k1 {
   EIGEN_DEVICE_FUNC
   static EIGEN_STRONG_INLINE T run(const T&) {
@@ -1076,20 +1076,20 @@ struct generic_k1<T, double> {
   }
 };
 
-template <typename Scalar>
+template <typename T>
 struct bessel_k1_impl {
   EIGEN_DEVICE_FUNC
-  static EIGEN_STRONG_INLINE Scalar run(const Scalar x) {
-    return generic_k1<Scalar, Scalar>::run(x);
+  static EIGEN_STRONG_INLINE T run(const T x) {
+    return generic_k1<T>::run(x);
   }
 };
 
-template <typename Scalar>
+template <typename T>
 struct bessel_j0_retval {
-  typedef Scalar type;
+  typedef T type;
 };
 
-template <typename T, typename ScalarType>
+template <typename T, typename ScalarType = typename unpacket_traits<T>::type>
 struct generic_j0 {
   EIGEN_DEVICE_FUNC
   static EIGEN_STRONG_INLINE T run(const T&) {
@@ -1276,20 +1276,20 @@ struct generic_j0<T, double> {
   }
 };
 
-template <typename Scalar>
+template <typename T>
 struct bessel_j0_impl {
   EIGEN_DEVICE_FUNC
-  static EIGEN_STRONG_INLINE Scalar run(const Scalar x) {
-    return generic_j0<Scalar, Scalar>::run(x);
+  static EIGEN_STRONG_INLINE T run(const T x) {
+    return generic_j0<T>::run(x);
   }
 };
 
-template <typename Scalar>
+template <typename T>
 struct bessel_y0_retval {
-  typedef Scalar type;
+  typedef T type;
 };
 
-template <typename T, typename ScalarType>
+template <typename T, typename ScalarType = typename unpacket_traits<T>::type>
 struct generic_y0 {
   EIGEN_DEVICE_FUNC
   static EIGEN_STRONG_INLINE T run(const T&) {
@@ -1474,20 +1474,20 @@ struct generic_y0<T, double> {
   }
 };
 
-template <typename Scalar>
+template <typename T>
 struct bessel_y0_impl {
   EIGEN_DEVICE_FUNC
-  static EIGEN_STRONG_INLINE Scalar run(const Scalar x) {
-    return generic_y0<Scalar, Scalar>::run(x);
+  static EIGEN_STRONG_INLINE T run(const T x) {
+    return generic_y0<T>::run(x);
   }
 };
 
-template <typename Scalar>
+template <typename T>
 struct bessel_j1_retval {
-  typedef Scalar type;
+  typedef T type;
 };
 
-template <typename T, typename ScalarType>
+template <typename T, typename ScalarType = typename unpacket_traits<T>::type>
 struct generic_j1 {
   EIGEN_DEVICE_FUNC
   static EIGEN_STRONG_INLINE T run(const T&) {
@@ -1665,20 +1665,20 @@ struct generic_j1<T, double> {
   }
 };
 
-template <typename Scalar>
+template <typename T>
 struct bessel_j1_impl {
   EIGEN_DEVICE_FUNC
-  static EIGEN_STRONG_INLINE Scalar run(const Scalar x) {
-    return generic_j1<Scalar, Scalar>::run(x);
+  static EIGEN_STRONG_INLINE T run(const T x) {
+    return generic_j1<T>::run(x);
   }
 };
 
-template <typename Scalar>
+template <typename T>
 struct bessel_y1_retval {
-  typedef Scalar type;
+  typedef T type;
 };
 
-template <typename T, typename ScalarType>
+template <typename T, typename ScalarType = typename unpacket_traits<T>::type>
 struct generic_y1 {
   EIGEN_DEVICE_FUNC
   static EIGEN_STRONG_INLINE T run(const T&) {
@@ -1868,11 +1868,11 @@ struct generic_y1<T, double> {
   }
 };
 
-template <typename Scalar>
+template <typename T>
 struct bessel_y1_impl {
   EIGEN_DEVICE_FUNC
-  static EIGEN_STRONG_INLINE Scalar run(const Scalar x) {
-    return generic_y1<Scalar, Scalar>::run(x);
+  static EIGEN_STRONG_INLINE T run(const T x) {
+    return generic_y1<T>::run(x);
   }
 };
 
diff --git a/unsupported/Eigen/src/SpecialFunctions/BesselFunctionsPacketMath.h b/unsupported/Eigen/src/SpecialFunctions/BesselFunctionsPacketMath.h
index efc6d9c8f..943d10f6a 100644
--- a/unsupported/Eigen/src/SpecialFunctions/BesselFunctionsPacketMath.h
+++ b/unsupported/Eigen/src/SpecialFunctions/BesselFunctionsPacketMath.h
@@ -19,8 +19,7 @@ namespace internal {
 template <typename Packet>
 EIGEN_DEVICE_FUNC EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS
 Packet pbessel_i0(const Packet& x) {
-  typedef typename unpacket_traits<Packet>::type ScalarType;
-  using internal::generic_i0; return generic_i0<Packet, ScalarType>::run(x);
+  return numext::bessel_i0(x);
 }
 
 /** \internal \returns the exponentially scaled modified Bessel function of
@@ -28,8 +27,7 @@ Packet pbessel_i0(const Packet& x) {
 template <typename Packet>
 EIGEN_DEVICE_FUNC EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS
 Packet pbessel_i0e(const Packet& x) {
-  typedef typename unpacket_traits<Packet>::type ScalarType;
-  using internal::generic_i0e; return generic_i0e<Packet, ScalarType>::run(x);
+  return numext::bessel_i0e(x);
 }
 
 /** \internal \returns the exponentially scaled modified Bessel function of
@@ -37,8 +35,7 @@ Packet pbessel_i0e(const Packet& x) {
 template <typename Packet>
 EIGEN_DEVICE_FUNC EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS
 Packet pbessel_i1(const Packet& x) {
-  typedef typename unpacket_traits<Packet>::type ScalarType;
-  using internal::generic_i1; return generic_i1<Packet, ScalarType>::run(x);
+  return numext::bessel_i1(x);
 }
 
 /** \internal \returns the exponentially scaled modified Bessel function of
@@ -46,8 +43,7 @@ Packet pbessel_i1(const Packet& x) {
 template <typename Packet>
 EIGEN_DEVICE_FUNC EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS
 Packet pbessel_i1e(const Packet& x) {
-  typedef typename unpacket_traits<Packet>::type ScalarType;
-  using internal::generic_i1e; return generic_i1e<Packet, ScalarType>::run(x);
+  return numext::bessel_i1e(x);
 }
 
 /** \internal \returns the exponentially scaled modified Bessel function of
@@ -55,8 +51,7 @@ Packet pbessel_i1e(const Packet& x) {
 template <typename Packet>
 EIGEN_DEVICE_FUNC EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS
 Packet pbessel_j0(const Packet& x) {
-  typedef typename unpacket_traits<Packet>::type ScalarType;
-  using internal::generic_j0; return generic_j0<Packet, ScalarType>::run(x);
+  return numext::bessel_j0(x);
 }
 
 /** \internal \returns the exponentially scaled modified Bessel function of
@@ -64,8 +59,7 @@ Packet pbessel_j0(const Packet& x) {
 template <typename Packet>
 EIGEN_DEVICE_FUNC EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS
 Packet pbessel_j1(const Packet& x) {
-  typedef typename unpacket_traits<Packet>::type ScalarType;
-  using internal::generic_j1; return generic_j1<Packet, ScalarType>::run(x);
+  return numext::bessel_j1(x);
 }
 
 /** \internal \returns the exponentially scaled modified Bessel function of
@@ -73,8 +67,7 @@ Packet pbessel_j1(const Packet& x) {
 template <typename Packet>
 EIGEN_DEVICE_FUNC EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS
 Packet pbessel_y0(const Packet& x) {
-  typedef typename unpacket_traits<Packet>::type ScalarType;
-  using internal::generic_y0; return generic_y0<Packet, ScalarType>::run(x);
+  return numext::bessel_y0(x);
 }
 
 /** \internal \returns the exponentially scaled modified Bessel function of
@@ -82,8 +75,7 @@ Packet pbessel_y0(const Packet& x) {
 template <typename Packet>
 EIGEN_DEVICE_FUNC EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS
 Packet pbessel_y1(const Packet& x) {
-  typedef typename unpacket_traits<Packet>::type ScalarType;
-  using internal::generic_y1; return generic_y1<Packet, ScalarType>::run(x);
+  return numext::bessel_y1(x);
 }
 
 /** \internal \returns the exponentially scaled modified Bessel function of
@@ -91,8 +83,7 @@ Packet pbessel_y1(const Packet& x) {
 template <typename Packet>
 EIGEN_DEVICE_FUNC EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS
 Packet pbessel_k0(const Packet& x) {
-  typedef typename unpacket_traits<Packet>::type ScalarType;
-  using internal::generic_k0; return generic_k0<Packet, ScalarType>::run(x);
+  return numext::bessel_k0(x);
 }
 
 /** \internal \returns the exponentially scaled modified Bessel function of
@@ -100,8 +91,7 @@ Packet pbessel_k0(const Packet& x) {
 template <typename Packet>
 EIGEN_DEVICE_FUNC EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS
 Packet pbessel_k0e(const Packet& x) {
-  typedef typename unpacket_traits<Packet>::type ScalarType;
-  using internal::generic_k0e; return generic_k0e<Packet, ScalarType>::run(x);
+  return numext::bessel_k0e(x);
 }
 
 /** \internal \returns the exponentially scaled modified Bessel function of
@@ -109,8 +99,7 @@ Packet pbessel_k0e(const Packet& x) {
 template <typename Packet>
 EIGEN_DEVICE_FUNC EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS
 Packet pbessel_k1(const Packet& x) {
-  typedef typename unpacket_traits<Packet>::type ScalarType;
-  using internal::generic_k1; return generic_k1<Packet, ScalarType>::run(x);
+  return numext::bessel_k1(x);
 }
 
 /** \internal \returns the exponentially scaled modified Bessel function of
@@ -118,8 +107,7 @@ Packet pbessel_k1(const Packet& x) {
 template <typename Packet>
 EIGEN_DEVICE_FUNC EIGEN_DECLARE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS
 Packet pbessel_k1e(const Packet& x) {
-  typedef typename unpacket_traits<Packet>::type ScalarType;
-  using internal::generic_k1e; return generic_k1e<Packet, ScalarType>::run(x);
+  return numext::bessel_k1e(x);
 }
 
 } // end namespace internal
diff --git a/unsupported/Eigen/src/SpecialFunctions/SpecialFunctionsImpl.h b/unsupported/Eigen/src/SpecialFunctions/SpecialFunctionsImpl.h
index 648eb053e..cfc13aff7 100644
--- a/unsupported/Eigen/src/SpecialFunctions/SpecialFunctionsImpl.h
+++ b/unsupported/Eigen/src/SpecialFunctions/SpecialFunctionsImpl.h
@@ -348,7 +348,7 @@ EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE T generic_fast_erf_float(const T& a_x) {
 template <typename T>
 struct erf_impl {
   EIGEN_DEVICE_FUNC
-  static EIGEN_STRONG_INLINE T run(const T x) {
+  static EIGEN_STRONG_INLINE T run(const T& x) {
     return generic_fast_erf_float(x);
   }
 };
@@ -490,7 +490,8 @@ struct erfc_impl<double> {
 template<typename T>
 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE T flipsign(
     const T& should_flipsign, const T& x) {
-  const T sign_mask = pset1<T>(-0.0);
+  typedef typename unpacket_traits<T>::type Scalar;
+  const T sign_mask = pset1<T>(Scalar(-0.0));
   T sign_bit = pand<T>(should_flipsign, sign_mask);
   return pxor<T>(sign_bit, x);
 }
diff --git a/unsupported/Eigen/src/SpecialFunctions/arch/AVX/BesselFunctions.h b/unsupported/Eigen/src/SpecialFunctions/arch/AVX/BesselFunctions.h
new file mode 100644
index 000000000..2d7669209
--- /dev/null
+++ b/unsupported/Eigen/src/SpecialFunctions/arch/AVX/BesselFunctions.h
@@ -0,0 +1,46 @@
+#ifndef EIGEN_AVX_BESSELFUNCTIONS_H
+#define EIGEN_AVX_BESSELFUNCTIONS_H
+
+namespace Eigen {
+namespace internal {
+
+F16_PACKET_FUNCTION(Packet8f, Packet8h, pbessel_i0)
+BF16_PACKET_FUNCTION(Packet8f, Packet8bf, pbessel_i0)
+
+F16_PACKET_FUNCTION(Packet8f, Packet8h, pbessel_i0e)
+BF16_PACKET_FUNCTION(Packet8f, Packet8bf, pbessel_i0e)
+
+F16_PACKET_FUNCTION(Packet8f, Packet8h, pbessel_i1)
+BF16_PACKET_FUNCTION(Packet8f, Packet8bf, pbessel_i1)
+
+F16_PACKET_FUNCTION(Packet8f, Packet8h, pbessel_i1e)
+BF16_PACKET_FUNCTION(Packet8f, Packet8bf, pbessel_i1e)
+
+F16_PACKET_FUNCTION(Packet8f, Packet8h, pbessel_j0)
+BF16_PACKET_FUNCTION(Packet8f, Packet8bf, pbessel_j0)
+
+F16_PACKET_FUNCTION(Packet8f, Packet8h, pbessel_j1)
+BF16_PACKET_FUNCTION(Packet8f, Packet8bf, pbessel_j1)
+
+F16_PACKET_FUNCTION(Packet8f, Packet8h, pbessel_k0)
+BF16_PACKET_FUNCTION(Packet8f, Packet8bf, pbessel_k0)
+
+F16_PACKET_FUNCTION(Packet8f, Packet8h, pbessel_k0e)
+BF16_PACKET_FUNCTION(Packet8f, Packet8bf, pbessel_k0e)
+
+F16_PACKET_FUNCTION(Packet8f, Packet8h, pbessel_k1)
+BF16_PACKET_FUNCTION(Packet8f, Packet8bf, pbessel_k1)
+
+F16_PACKET_FUNCTION(Packet8f, Packet8h, pbessel_k1e)
+BF16_PACKET_FUNCTION(Packet8f, Packet8bf, pbessel_k1e)
+
+F16_PACKET_FUNCTION(Packet8f, Packet8h, pbessel_y0)
+BF16_PACKET_FUNCTION(Packet8f, Packet8bf, pbessel_y0)
+
+F16_PACKET_FUNCTION(Packet8f, Packet8h, pbessel_y1)
+BF16_PACKET_FUNCTION(Packet8f, Packet8bf, pbessel_y1)
+
+}  // namespace internal
+}  // namespace Eigen
+
+#endif  // EIGEN_AVX_BESSELFUNCTIONS_H
diff --git a/unsupported/Eigen/src/SpecialFunctions/arch/AVX/SpecialFunctions.h b/unsupported/Eigen/src/SpecialFunctions/arch/AVX/SpecialFunctions.h
new file mode 100644
index 000000000..35e62a8ac
--- /dev/null
+++ b/unsupported/Eigen/src/SpecialFunctions/arch/AVX/SpecialFunctions.h
@@ -0,0 +1,16 @@
+#ifndef EIGEN_AVX_SPECIALFUNCTIONS_H
+#define EIGEN_AVX_SPECIALFUNCTIONS_H
+
+namespace Eigen {
+namespace internal {
+
+F16_PACKET_FUNCTION(Packet8f, Packet8h, perf)
+BF16_PACKET_FUNCTION(Packet8f, Packet8bf, perf)
+
+F16_PACKET_FUNCTION(Packet8f, Packet8h, pndtri)
+BF16_PACKET_FUNCTION(Packet8f, Packet8bf, pndtri)
+
+}  // namespace internal
+}  // namespace Eigen
+
+#endif  // EIGEN_AVX_SPECIAL_FUNCTIONS_H
diff --git a/unsupported/Eigen/src/SpecialFunctions/arch/AVX512/BesselFunctions.h b/unsupported/Eigen/src/SpecialFunctions/arch/AVX512/BesselFunctions.h
new file mode 100644
index 000000000..7dd3c3e5b
--- /dev/null
+++ b/unsupported/Eigen/src/SpecialFunctions/arch/AVX512/BesselFunctions.h
@@ -0,0 +1,46 @@
+#ifndef EIGEN_AVX512_BESSELFUNCTIONS_H
+#define EIGEN_AVX512_BESSELFUNCTIONS_H
+
+namespace Eigen {
+namespace internal {
+
+F16_PACKET_FUNCTION(Packet16f, Packet16h, pbessel_i0)
+BF16_PACKET_FUNCTION(Packet16f, Packet16bf, pbessel_i0)
+
+F16_PACKET_FUNCTION(Packet16f, Packet16h, pbessel_i0e)
+BF16_PACKET_FUNCTION(Packet16f, Packet16bf, pbessel_i0e)
+
+F16_PACKET_FUNCTION(Packet16f, Packet16h, pbessel_i1)
+BF16_PACKET_FUNCTION(Packet16f, Packet16bf, pbessel_i1)
+
+F16_PACKET_FUNCTION(Packet16f, Packet16h, pbessel_i1e)
+BF16_PACKET_FUNCTION(Packet16f, Packet16bf, pbessel_i1e)
+
+F16_PACKET_FUNCTION(Packet16f, Packet16h, pbessel_j0)
+BF16_PACKET_FUNCTION(Packet16f, Packet16bf, pbessel_j0)
+
+F16_PACKET_FUNCTION(Packet16f, Packet16h, pbessel_j1)
+BF16_PACKET_FUNCTION(Packet16f, Packet16bf, pbessel_j1)
+
+F16_PACKET_FUNCTION(Packet16f, Packet16h, pbessel_k0)
+BF16_PACKET_FUNCTION(Packet16f, Packet16bf, pbessel_k0)
+
+F16_PACKET_FUNCTION(Packet16f, Packet16h, pbessel_k0e)
+BF16_PACKET_FUNCTION(Packet16f, Packet16bf, pbessel_k0e)
+
+F16_PACKET_FUNCTION(Packet16f, Packet16h, pbessel_k1)
+BF16_PACKET_FUNCTION(Packet16f, Packet16bf, pbessel_k1)
+
+F16_PACKET_FUNCTION(Packet16f, Packet16h, pbessel_k1e)
+BF16_PACKET_FUNCTION(Packet16f, Packet16bf, pbessel_k1e)
+
+F16_PACKET_FUNCTION(Packet16f, Packet16h, pbessel_y0)
+BF16_PACKET_FUNCTION(Packet16f, Packet16bf, pbessel_y0)
+
+F16_PACKET_FUNCTION(Packet16f, Packet16h, pbessel_y1)
+BF16_PACKET_FUNCTION(Packet16f, Packet16bf, pbessel_y1)
+
+}  // namespace internal
+}  // namespace Eigen
+
+#endif  // EIGEN_AVX512_BESSELFUNCTIONS_H
diff --git a/unsupported/Eigen/src/SpecialFunctions/arch/AVX512/SpecialFunctions.h b/unsupported/Eigen/src/SpecialFunctions/arch/AVX512/SpecialFunctions.h
new file mode 100644
index 000000000..79878f2b6
--- /dev/null
+++ b/unsupported/Eigen/src/SpecialFunctions/arch/AVX512/SpecialFunctions.h
@@ -0,0 +1,16 @@
+#ifndef EIGEN_AVX512_SPECIALFUNCTIONS_H
+#define EIGEN_AVX512_SPECIALFUNCTIONS_H
+
+namespace Eigen {
+namespace internal {
+
+F16_PACKET_FUNCTION(Packet16f, Packet16h, perf)
+BF16_PACKET_FUNCTION(Packet16f, Packet16bf, perf)
+
+F16_PACKET_FUNCTION(Packet16f, Packet16h, pndtri)
+BF16_PACKET_FUNCTION(Packet16f, Packet16bf, pndtri)
+
+}  // namespace internal
+}  // namespace Eigen
+
+#endif  // EIGEN_AVX512_SPECIAL_FUNCTIONS_H
diff --git a/unsupported/Eigen/src/SpecialFunctions/arch/GPU/GpuSpecialFunctions.h b/unsupported/Eigen/src/SpecialFunctions/arch/GPU/SpecialFunctions.h
index dd3bf4dd1..dd3bf4dd1 100644
--- a/unsupported/Eigen/src/SpecialFunctions/arch/GPU/GpuSpecialFunctions.h
+++ b/unsupported/Eigen/src/SpecialFunctions/arch/GPU/SpecialFunctions.h
diff --git a/unsupported/Eigen/src/SpecialFunctions/arch/NEON/BesselFunctions.h b/unsupported/Eigen/src/SpecialFunctions/arch/NEON/BesselFunctions.h
new file mode 100644
index 000000000..67433b057
--- /dev/null
+++ b/unsupported/Eigen/src/SpecialFunctions/arch/NEON/BesselFunctions.h
@@ -0,0 +1,54 @@
+#ifndef EIGEN_NEON_BESSELFUNCTIONS_H
+#define EIGEN_NEON_BESSELFUNCTIONS_H
+
+namespace Eigen {
+namespace internal {
+
+#if EIGEN_HAS_ARM64_FP16_VECTOR_ARITHMETIC
+
+#define NEON_HALF_TO_FLOAT_FUNCTIONS(METHOD)                            \
+template <> EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE                       \
+Packet8hf METHOD<Packet8hf>(const Packet8hf& x) {                       \
+  const Packet4f lo = METHOD<Packet4f>(vcvt_f32_f16(vget_low_f16(x)));  \
+  const Packet4f hi = METHOD<Packet4f>(vcvt_f32_f16(vget_high_f16(x))); \
+  return vcombine_f16(vcvt_f16_f32(lo), vcvt_f16_f32(hi));              \
+}                                                                       \
+                                                                        \
+template <> EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE                       \
+Packet4hf METHOD<Packet4hf>(const Packet4hf& x) {                       \
+  return vcvt_f16_f32(METHOD<Packet4f>(vcvt_f32_f16(x)));               \
+}
+
+NEON_HALF_TO_FLOAT_FUNCTIONS(pbessel_i0)
+NEON_HALF_TO_FLOAT_FUNCTIONS(pbessel_i0e)
+NEON_HALF_TO_FLOAT_FUNCTIONS(pbessel_i1)
+NEON_HALF_TO_FLOAT_FUNCTIONS(pbessel_i1e)
+NEON_HALF_TO_FLOAT_FUNCTIONS(pbessel_j0)
+NEON_HALF_TO_FLOAT_FUNCTIONS(pbessel_j1)
+NEON_HALF_TO_FLOAT_FUNCTIONS(pbessel_k0)
+NEON_HALF_TO_FLOAT_FUNCTIONS(pbessel_k0e)
+NEON_HALF_TO_FLOAT_FUNCTIONS(pbessel_k1)
+NEON_HALF_TO_FLOAT_FUNCTIONS(pbessel_k1e)
+NEON_HALF_TO_FLOAT_FUNCTIONS(pbessel_y0)
+NEON_HALF_TO_FLOAT_FUNCTIONS(pbessel_y1)
+
+#undef NEON_HALF_TO_FLOAT_FUNCTIONS
+#endif
+
+BF16_PACKET_FUNCTION(Packet4f, Packet4bf, pbessel_i0)
+BF16_PACKET_FUNCTION(Packet4f, Packet4bf, pbessel_i0e)
+BF16_PACKET_FUNCTION(Packet4f, Packet4bf, pbessel_i1)
+BF16_PACKET_FUNCTION(Packet4f, Packet4bf, pbessel_i1e)
+BF16_PACKET_FUNCTION(Packet4f, Packet4bf, pbessel_j0)
+BF16_PACKET_FUNCTION(Packet4f, Packet4bf, pbessel_j1)
+BF16_PACKET_FUNCTION(Packet4f, Packet4bf, pbessel_k0)
+BF16_PACKET_FUNCTION(Packet4f, Packet4bf, pbessel_k0e)
+BF16_PACKET_FUNCTION(Packet4f, Packet4bf, pbessel_k1)
+BF16_PACKET_FUNCTION(Packet4f, Packet4bf, pbessel_k1e)
+BF16_PACKET_FUNCTION(Packet4f, Packet4bf, pbessel_y0)
+BF16_PACKET_FUNCTION(Packet4f, Packet4bf, pbessel_y1)
+
+}  // namespace internal
+}  // namespace Eigen
+
+#endif  // EIGEN_NEON_BESSELFUNCTIONS_H
diff --git a/unsupported/Eigen/src/SpecialFunctions/arch/NEON/SpecialFunctions.h b/unsupported/Eigen/src/SpecialFunctions/arch/NEON/SpecialFunctions.h
new file mode 100644
index 000000000..f8dda28fc
--- /dev/null
+++ b/unsupported/Eigen/src/SpecialFunctions/arch/NEON/SpecialFunctions.h
@@ -0,0 +1,34 @@
+#ifndef EIGEN_NEON_SPECIALFUNCTIONS_H
+#define EIGEN_NEON_SPECIALFUNCTIONS_H
+
+namespace Eigen {
+namespace internal {
+
+#ifdef EIGEN_HAS_ARM64_FP16_VECTOR_ARITHMETIC
+
+#define NEON_HALF_TO_FLOAT_FUNCTIONS(METHOD)                            \
+template <> EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE                       \
+Packet8hf METHOD<Packet8hf>(const Packet8hf& x) {                       \
+  const Packet4f lo = METHOD<Packet4f>(vcvt_f32_f16(vget_low_f16(x)));  \
+  const Packet4f hi = METHOD<Packet4f>(vcvt_f32_f16(vget_high_f16(x))); \
+  return vcombine_f16(vcvt_f16_f32(lo), vcvt_f16_f32(hi));              \
+}                                                                       \
+                                                                        \
+template <> EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE                       \
+Packet4hf METHOD<Packet4hf>(const Packet4hf& x) {                       \
+  return vcvt_f16_f32(METHOD<Packet4f>(vcvt_f32_f16(x)));               \
+}
+
+NEON_HALF_TO_FLOAT_FUNCTIONS(perf)
+NEON_HALF_TO_FLOAT_FUNCTIONS(pndtri)
+
+#undef NEON_HALF_TO_FLOAT_FUNCTIONS
+#endif
+
+BF16_PACKET_FUNCTION(Packet4f, Packet4bf, perf)
+BF16_PACKET_FUNCTION(Packet4f, Packet4bf, pndtri)
+
+}  // namespace internal
+}  // namespace Eigen
+
+#endif  // EIGEN_NEON_SPECIALFUNCTIONS_H