BFloat16.h

/* Copyright 2017 The TensorFlow Authors. All Rights Reserved.

Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at

    http://www.apache.org/licenses/LICENSE-2.0

Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
==============================================================================*/

#ifndef EIGEN_BFLOAT16_H
#define EIGEN_BFLOAT16_H

// IWYU pragma: private
#include "../../InternalHeaderCheck.h"

#if defined(EIGEN_HAS_HIP_BF16)
// When compiling with GPU support, the "hip_bfloat16" base class as well as
// some other routines are defined in the GPU compiler header files
// (hip_bfloat16.h), and they are not tagged constexpr
// As a consequence, we get compile failures when compiling Eigen with
// GPU support. Hence the need to disable EIGEN_CONSTEXPR when building
// Eigen with GPU support
#pragma push_macro("EIGEN_CONSTEXPR")
#undef EIGEN_CONSTEXPR
#define EIGEN_CONSTEXPR
#endif

#define BF16_PACKET_FUNCTION(PACKET_F, PACKET_BF16, METHOD)                                         \
  template <>                                                                                       \
  EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED PACKET_BF16 METHOD<PACKET_BF16>( \
      const PACKET_BF16& _x) {                                                                      \
    return F32ToBf16(METHOD<PACKET_F>(Bf16ToF32(_x)));                                              \
  }

// Only use HIP GPU bf16 in kernels
#if defined(EIGEN_HAS_HIP_BF16) && defined(EIGEN_GPU_COMPILE_PHASE)
#define EIGEN_USE_HIP_BF16
#endif

namespace Eigen {

struct bfloat16;

namespace numext {
template <>
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC Eigen::bfloat16 bit_cast<Eigen::bfloat16, uint16_t>(const uint16_t& src);

template <>
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC uint16_t bit_cast<uint16_t, Eigen::bfloat16>(const Eigen::bfloat16& src);
}  // namespace numext
namespace bfloat16_impl {

#if defined(EIGEN_USE_HIP_BF16)

struct __bfloat16_raw : public hip_bfloat16 {
  EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR __bfloat16_raw() {}
  EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR __bfloat16_raw(hip_bfloat16 hb) : hip_bfloat16(hb) {}
  explicit EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR __bfloat16_raw(unsigned short raw) : hip_bfloat16(raw) {}
};

#else

// Make our own __bfloat16_raw definition.
struct __bfloat16_raw {
#if defined(EIGEN_HAS_HIP_BF16) && !defined(EIGEN_GPU_COMPILE_PHASE)
  EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR __bfloat16_raw() {}
#else
  EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR __bfloat16_raw() : value(0) {}
#endif
  explicit EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR __bfloat16_raw(unsigned short raw) : value(raw) {}
  unsigned short value;
};

#endif  // defined(EIGEN_USE_HIP_BF16)

EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR __bfloat16_raw raw_uint16_to_bfloat16(unsigned short value);
template <bool AssumeArgumentIsNormalOrInfinityOrZero>
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC __bfloat16_raw float_to_bfloat16_rtne(float ff);
// Forward declarations of template specializations, to avoid Visual C++ 2019 errors, saying:
// > error C2908: explicit specialization; 'float_to_bfloat16_rtne' has already been instantiated
template <>
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC __bfloat16_raw float_to_bfloat16_rtne<false>(float ff);
template <>
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC __bfloat16_raw float_to_bfloat16_rtne<true>(float ff);
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC float bfloat16_to_float(__bfloat16_raw h);

struct bfloat16_base : public __bfloat16_raw {
  EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR bfloat16_base() {}
  EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR bfloat16_base(const __bfloat16_raw& h) : __bfloat16_raw(h) {}
};

}  // namespace bfloat16_impl

// Class definition.
struct bfloat16 : public bfloat16_impl::bfloat16_base {
  typedef bfloat16_impl::__bfloat16_raw __bfloat16_raw;

  EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR bfloat16() {}

  EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR bfloat16(const __bfloat16_raw& h) : bfloat16_impl::bfloat16_base(h) {}

  explicit EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR bfloat16(bool b)
      : bfloat16_impl::bfloat16_base(bfloat16_impl::raw_uint16_to_bfloat16(b ? 0x3f80 : 0)) {}

  template <class T>
  explicit EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR bfloat16(T val)
      : bfloat16_impl::bfloat16_base(
            bfloat16_impl::float_to_bfloat16_rtne<internal::is_integral<T>::value>(static_cast<float>(val))) {}

  explicit EIGEN_DEVICE_FUNC bfloat16(float f)
      : bfloat16_impl::bfloat16_base(bfloat16_impl::float_to_bfloat16_rtne<false>(f)) {}

  // Following the convention of numpy, converting between complex and
  // float will lead to loss of imag value.
  template <typename RealScalar>
  explicit EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR bfloat16(const std::complex<RealScalar>& val)
      : bfloat16_impl::bfloat16_base(bfloat16_impl::float_to_bfloat16_rtne<false>(static_cast<float>(val.real()))) {}

  EIGEN_DEVICE_FUNC operator float() const {  // NOLINT: Allow implicit conversion to float, because it is lossless.
    return bfloat16_impl::bfloat16_to_float(*this);
  }
};

// TODO(majnemer): Get rid of this once we can rely on C++17 inline variables do
// solve the ODR issue.
namespace bfloat16_impl {
template <typename = void>
struct numeric_limits_bfloat16_impl {
  static EIGEN_CONSTEXPR const bool is_specialized = true;
  static EIGEN_CONSTEXPR const bool is_signed = true;
  static EIGEN_CONSTEXPR const bool is_integer = false;
  static EIGEN_CONSTEXPR const bool is_exact = false;
  static EIGEN_CONSTEXPR const bool has_infinity = true;
  static EIGEN_CONSTEXPR const bool has_quiet_NaN = true;
  static EIGEN_CONSTEXPR const bool has_signaling_NaN = true;
  EIGEN_DIAGNOSTICS(push)
  EIGEN_DISABLE_DEPRECATED_WARNING
  static EIGEN_CONSTEXPR const std::float_denorm_style has_denorm = std::denorm_present;
  static EIGEN_CONSTEXPR const bool has_denorm_loss = false;
  EIGEN_DIAGNOSTICS(pop)
  static EIGEN_CONSTEXPR const std::float_round_style round_style = std::numeric_limits<float>::round_style;
  static EIGEN_CONSTEXPR const bool is_iec559 = true;
  // The C++ standard defines this as "true if the set of values representable
  // by the type is finite." BFloat16 has finite precision.
  static EIGEN_CONSTEXPR const bool is_bounded = true;
  static EIGEN_CONSTEXPR const bool is_modulo = false;
  static EIGEN_CONSTEXPR const int digits = 8;
  static EIGEN_CONSTEXPR const int digits10 = 2;
  static EIGEN_CONSTEXPR const int max_digits10 = 4;
  static EIGEN_CONSTEXPR const int radix = std::numeric_limits<float>::radix;
  static EIGEN_CONSTEXPR const int min_exponent = std::numeric_limits<float>::min_exponent;
  static EIGEN_CONSTEXPR const int min_exponent10 = std::numeric_limits<float>::min_exponent10;
  static EIGEN_CONSTEXPR const int max_exponent = std::numeric_limits<float>::max_exponent;
  static EIGEN_CONSTEXPR const int max_exponent10 = std::numeric_limits<float>::max_exponent10;
  static EIGEN_CONSTEXPR const bool traps = std::numeric_limits<float>::traps;
  // IEEE754: "The implementer shall choose how tininess is detected, but shall
  // detect tininess in the same way for all operations in radix two"
  static EIGEN_CONSTEXPR const bool tinyness_before = std::numeric_limits<float>::tinyness_before;

  static EIGEN_CONSTEXPR Eigen::bfloat16(min)() { return Eigen::bfloat16_impl::raw_uint16_to_bfloat16(0x0080); }
  static EIGEN_CONSTEXPR Eigen::bfloat16 lowest() { return Eigen::bfloat16_impl::raw_uint16_to_bfloat16(0xff7f); }
  static EIGEN_CONSTEXPR Eigen::bfloat16(max)() { return Eigen::bfloat16_impl::raw_uint16_to_bfloat16(0x7f7f); }
  static EIGEN_CONSTEXPR Eigen::bfloat16 epsilon() { return Eigen::bfloat16_impl::raw_uint16_to_bfloat16(0x3c00); }
  static EIGEN_CONSTEXPR Eigen::bfloat16 round_error() { return Eigen::bfloat16_impl::raw_uint16_to_bfloat16(0x3f00); }
  static EIGEN_CONSTEXPR Eigen::bfloat16 infinity() { return Eigen::bfloat16_impl::raw_uint16_to_bfloat16(0x7f80); }
  static EIGEN_CONSTEXPR Eigen::bfloat16 quiet_NaN() { return Eigen::bfloat16_impl::raw_uint16_to_bfloat16(0x7fc0); }
  static EIGEN_CONSTEXPR Eigen::bfloat16 signaling_NaN() {
    return Eigen::bfloat16_impl::raw_uint16_to_bfloat16(0x7fa0);
  }
  static EIGEN_CONSTEXPR Eigen::bfloat16 denorm_min() { return Eigen::bfloat16_impl::raw_uint16_to_bfloat16(0x0001); }
};

template <typename T>
EIGEN_CONSTEXPR const bool numeric_limits_bfloat16_impl<T>::is_specialized;
template <typename T>
EIGEN_CONSTEXPR const bool numeric_limits_bfloat16_impl<T>::is_signed;
template <typename T>
EIGEN_CONSTEXPR const bool numeric_limits_bfloat16_impl<T>::is_integer;
template <typename T>
EIGEN_CONSTEXPR const bool numeric_limits_bfloat16_impl<T>::is_exact;
template <typename T>
EIGEN_CONSTEXPR const bool numeric_limits_bfloat16_impl<T>::has_infinity;
template <typename T>
EIGEN_CONSTEXPR const bool numeric_limits_bfloat16_impl<T>::has_quiet_NaN;
template <typename T>
EIGEN_CONSTEXPR const bool numeric_limits_bfloat16_impl<T>::has_signaling_NaN;
EIGEN_DIAGNOSTICS(push)
EIGEN_DISABLE_DEPRECATED_WARNING
template <typename T>
EIGEN_CONSTEXPR const std::float_denorm_style numeric_limits_bfloat16_impl<T>::has_denorm;
template <typename T>
EIGEN_CONSTEXPR const bool numeric_limits_bfloat16_impl<T>::has_denorm_loss;
EIGEN_DIAGNOSTICS(pop)
template <typename T>
EIGEN_CONSTEXPR const std::float_round_style numeric_limits_bfloat16_impl<T>::round_style;
template <typename T>
EIGEN_CONSTEXPR const bool numeric_limits_bfloat16_impl<T>::is_iec559;
template <typename T>
EIGEN_CONSTEXPR const bool numeric_limits_bfloat16_impl<T>::is_bounded;
template <typename T>
EIGEN_CONSTEXPR const bool numeric_limits_bfloat16_impl<T>::is_modulo;
template <typename T>
EIGEN_CONSTEXPR const int numeric_limits_bfloat16_impl<T>::digits;
template <typename T>
EIGEN_CONSTEXPR const int numeric_limits_bfloat16_impl<T>::digits10;
template <typename T>
EIGEN_CONSTEXPR const int numeric_limits_bfloat16_impl<T>::max_digits10;
template <typename T>
EIGEN_CONSTEXPR const int numeric_limits_bfloat16_impl<T>::radix;
template <typename T>
EIGEN_CONSTEXPR const int numeric_limits_bfloat16_impl<T>::min_exponent;
template <typename T>
EIGEN_CONSTEXPR const int numeric_limits_bfloat16_impl<T>::min_exponent10;
template <typename T>
EIGEN_CONSTEXPR const int numeric_limits_bfloat16_impl<T>::max_exponent;
template <typename T>
EIGEN_CONSTEXPR const int numeric_limits_bfloat16_impl<T>::max_exponent10;
template <typename T>
EIGEN_CONSTEXPR const bool numeric_limits_bfloat16_impl<T>::traps;
template <typename T>
EIGEN_CONSTEXPR const bool numeric_limits_bfloat16_impl<T>::tinyness_before;
}  // end namespace bfloat16_impl
}  // end namespace Eigen

namespace std {
// If std::numeric_limits<T> is specialized, should also specialize
// std::numeric_limits<const T>, std::numeric_limits<volatile T>, and
// std::numeric_limits<const volatile T>
// https://stackoverflow.com/a/16519653/
template <>
class numeric_limits<Eigen::bfloat16> : public Eigen::bfloat16_impl::numeric_limits_bfloat16_impl<> {};
template <>
class numeric_limits<const Eigen::bfloat16> : public numeric_limits<Eigen::bfloat16> {};
template <>
class numeric_limits<volatile Eigen::bfloat16> : public numeric_limits<Eigen::bfloat16> {};
template <>
class numeric_limits<const volatile Eigen::bfloat16> : public numeric_limits<Eigen::bfloat16> {};
}  // end namespace std

namespace Eigen {

namespace bfloat16_impl {

// We need to distinguish ‘clang as the CUDA compiler’ from ‘clang as the host compiler,
// invoked by NVCC’ (e.g. on MacOS). The former needs to see both host and device implementation
// of the functions, while the latter can only deal with one of them.
#if !defined(EIGEN_HAS_NATIVE_BF16) || (EIGEN_COMP_CLANG && !EIGEN_COMP_NVCC)  // Emulate support for bfloat16 floats

#if EIGEN_COMP_CLANG && defined(EIGEN_CUDACC)
// We need to provide emulated *host-side* BF16 operators for clang.
#pragma push_macro("EIGEN_DEVICE_FUNC")
#undef EIGEN_DEVICE_FUNC
#if (defined(EIGEN_HAS_GPU_BF16) && defined(EIGEN_HAS_NATIVE_BF16))
#define EIGEN_DEVICE_FUNC __host__
#else  // both host and device need emulated ops.
#define EIGEN_DEVICE_FUNC __host__ __device__
#endif
#endif

// Definitions for CPUs, mostly working through conversion
// to/from fp32.

EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bfloat16 operator+(const bfloat16& a, const bfloat16& b) {
  return bfloat16(float(a) + float(b));
}
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bfloat16 operator+(const bfloat16& a, const int& b) {
  return bfloat16(float(a) + static_cast<float>(b));
}
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bfloat16 operator+(const int& a, const bfloat16& b) {
  return bfloat16(static_cast<float>(a) + float(b));
}
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bfloat16 operator*(const bfloat16& a, const bfloat16& b) {
  return bfloat16(float(a) * float(b));
}
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bfloat16 operator-(const bfloat16& a, const bfloat16& b) {
  return bfloat16(float(a) - float(b));
}
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bfloat16 operator/(const bfloat16& a, const bfloat16& b) {
  return bfloat16(float(a) / float(b));
}
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bfloat16 operator-(const bfloat16& a) {
  numext::uint16_t x = numext::bit_cast<uint16_t>(a) ^ 0x8000;
  return numext::bit_cast<bfloat16>(x);
}
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bfloat16& operator+=(bfloat16& a, const bfloat16& b) {
  a = bfloat16(float(a) + float(b));
  return a;
}
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bfloat16& operator*=(bfloat16& a, const bfloat16& b) {
  a = bfloat16(float(a) * float(b));
  return a;
}
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bfloat16& operator-=(bfloat16& a, const bfloat16& b) {
  a = bfloat16(float(a) - float(b));
  return a;
}
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bfloat16& operator/=(bfloat16& a, const bfloat16& b) {
  a = bfloat16(float(a) / float(b));
  return a;
}
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bfloat16 operator++(bfloat16& a) {
  a += bfloat16(1);
  return a;
}
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bfloat16 operator--(bfloat16& a) {
  a -= bfloat16(1);
  return a;
}
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bfloat16 operator++(bfloat16& a, int) {
  bfloat16 original_value = a;
  ++a;
  return original_value;
}
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bfloat16 operator--(bfloat16& a, int) {
  bfloat16 original_value = a;
  --a;
  return original_value;
}
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bool operator==(const bfloat16& a, const bfloat16& b) {
  return numext::equal_strict(float(a), float(b));
}
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bool operator!=(const bfloat16& a, const bfloat16& b) {
  return numext::not_equal_strict(float(a), float(b));
}
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bool operator<(const bfloat16& a, const bfloat16& b) {
  return float(a) < float(b);
}
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bool operator<=(const bfloat16& a, const bfloat16& b) {
  return float(a) <= float(b);
}
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bool operator>(const bfloat16& a, const bfloat16& b) {
  return float(a) > float(b);
}
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bool operator>=(const bfloat16& a, const bfloat16& b) {
  return float(a) >= float(b);
}

#if EIGEN_COMP_CLANG && defined(EIGEN_CUDACC)
#pragma pop_macro("EIGEN_DEVICE_FUNC")
#endif
#endif  // Emulate support for bfloat16 floats

// Division by an index. Do it in full float precision to avoid accuracy
// issues in converting the denominator to bfloat16.
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bfloat16 operator/(const bfloat16& a, Index b) {
  return bfloat16(static_cast<float>(a) / static_cast<float>(b));
}

EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC __bfloat16_raw truncate_to_bfloat16(const float v) {
#if defined(EIGEN_USE_HIP_BF16)
  return __bfloat16_raw(__bfloat16_raw::round_to_bfloat16(v, __bfloat16_raw::truncate));
#else
  __bfloat16_raw output;
  if (numext::isnan EIGEN_NOT_A_MACRO(v)) {
    output.value = std::signbit(v) ? 0xFFC0 : 0x7FC0;
    return output;
  }
  output.value = static_cast<numext::uint16_t>(numext::bit_cast<numext::uint32_t>(v) >> 16);
  return output;
#endif
}

EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR __bfloat16_raw raw_uint16_to_bfloat16(numext::uint16_t value) {
#if defined(EIGEN_USE_HIP_BF16)
  __bfloat16_raw bf;
  bf.data = value;
  return bf;
#else
  return __bfloat16_raw(value);
#endif
}

EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR numext::uint16_t raw_bfloat16_as_uint16(
    const __bfloat16_raw& bf) {
#if defined(EIGEN_USE_HIP_BF16)
  return bf.data;
#else
  return bf.value;
#endif
}

// float_to_bfloat16_rtne template specialization that does not make any
// assumption about the value of its function argument (ff).
template <>
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC __bfloat16_raw float_to_bfloat16_rtne<false>(float ff) {
#if defined(EIGEN_USE_HIP_BF16)
  return __bfloat16_raw(__bfloat16_raw::round_to_bfloat16(ff));
#else
  __bfloat16_raw output;

  if (numext::isnan EIGEN_NOT_A_MACRO(ff)) {
    // If the value is a NaN, squash it to a qNaN with msb of fraction set,
    // this makes sure after truncation we don't end up with an inf.
    //
    // qNaN magic: All exponent bits set + most significant bit of fraction
    // set.
    output.value = std::signbit(ff) ? 0xFFC0 : 0x7FC0;
  } else {
    // Fast rounding algorithm that rounds a half value to nearest even. This
    // reduces expected error when we convert a large number of floats. Here
    // is how it works:
    //
    // Definitions:
    // To convert a float 32 to bfloat16, a float 32 can be viewed as 32 bits
    // with the following tags:
    //
    // Sign |  Exp (8 bits) | Frac (23 bits)
    //  S     EEEEEEEE         FFFFFFLRTTTTTTTTTTTTTTT
    //
    //  S: Sign bit.
    //  E: Exponent bits.
    //  F: First 6 bits of fraction.
    //  L: Least significant bit of resulting bfloat16 if we truncate away the
    //  rest of the float32. This is also the 7th bit of fraction
    //  R: Rounding bit, 8th bit of fraction.
    //  T: Sticky bits, rest of fraction, 15 bits.
    //
    // To round half to nearest even, there are 3 cases where we want to round
    // down (simply truncate the result of the bits away, which consists of
    // rounding bit and sticky bits) and two cases where we want to round up
    // (truncate then add one to the result).
    //
    // The fast converting algorithm simply adds lsb (L) to 0x7fff (15 bits of
    // 1s) as the rounding bias, adds the rounding bias to the input, then
    // truncates the last 16 bits away.
    //
    // To understand how it works, we can analyze this algorithm case by case:
    //
    // 1. L = 0, R = 0:
    //   Expect: round down, this is less than half value.
    //
    //   Algorithm:
    //   - Rounding bias: 0x7fff + 0 = 0x7fff
    //   - Adding rounding bias to input may create any carry, depending on
    //   whether there is any value set to 1 in T bits.
    //   - R may be set to 1 if there is a carry.
    //   - L remains 0.
    //   - Note that this case also handles Inf and -Inf, where all fraction
    //   bits, including L, R and Ts are all 0. The output remains Inf after
    //   this algorithm.
    //
    // 2. L = 1, R = 0:
    //   Expect: round down, this is less than half value.
    //
    //   Algorithm:
    //   - Rounding bias: 0x7fff + 1 = 0x8000
    //   - Adding rounding bias to input doesn't change sticky bits but
    //   adds 1 to rounding bit.
    //   - L remains 1.
    //
    // 3. L = 0, R = 1, all of T are 0:
    //   Expect: round down, this is exactly at half, the result is already
    //   even (L=0).
    //
    //   Algorithm:
    //   - Rounding bias: 0x7fff + 0 = 0x7fff
    //   - Adding rounding bias to input sets all sticky bits to 1, but
    //   doesn't create a carry.
    //   - R remains 1.
    //   - L remains 0.
    //
    // 4. L = 1, R = 1:
    //   Expect: round up, this is exactly at half, the result needs to be
    //   round to the next even number.
    //
    //   Algorithm:
    //   - Rounding bias: 0x7fff + 1 = 0x8000
    //   - Adding rounding bias to input doesn't change sticky bits, but
    //   creates a carry from rounding bit.
    //   - The carry sets L to 0, creates another carry bit and propagate
    //   forward to F bits.
    //   - If all the F bits are 1, a carry then propagates to the exponent
    //   bits, which then creates the minimum value with the next exponent
    //   value. Note that we won't have the case where exponents are all 1,
    //   since that's either a NaN (handled in the other if condition) or inf
    //   (handled in case 1).
    //
    // 5. L = 0, R = 1, any of T is 1:
    //   Expect: round up, this is greater than half.
    //
    //   Algorithm:
    //   - Rounding bias: 0x7fff + 0 = 0x7fff
    //   - Adding rounding bias to input creates a carry from sticky bits,
    //   sets rounding bit to 0, then create another carry.
    //   - The second carry sets L to 1.
    //
    // Examples:
    //
    //  Exact half value that is already even:
    //    Input:
    //    Sign |  Exp (8 bit)     | Frac (first 7 bit) | Frac (last 16 bit)
    //     S     E E E E E E E E      F F F F F F L     RTTTTTTTTTTTTTTT
    //     0     0 0 0 0 0 0 0 0      0 0 0 0 0 1 0     1000000000000000
    //
    //     This falls into case 3. We truncate the rest of 16 bits and no
    //     carry is created into F and L:
    //
    //    Output:
    //    Sign |  Exp (8 bit)     | Frac (first 7 bit)
    //     S     E E E E E E E E      F F F F F F L
    //     0     0 0 0 0 0 0 0 0      0 0 0 0 0 1 0
    //
    //  Exact half value, round to next even number:
    //    Input:
    //    Sign |  Exp (8 bit)     | Frac (first 7 bit) | Frac (last 16 bit)
    //     S     E E E E E E E E      F F F F F F L     RTTTTTTTTTTTTTTT
    //     0     0 0 0 0 0 0 0 0      0 0 0 0 0 0 1     1000000000000000
    //
    //     This falls into case 4. We create a carry from R and T,
    //     which then propagates into L and F:
    //
    //    Output:
    //    Sign |  Exp (8 bit)     | Frac (first 7 bit)
    //     S     E E E E E E E E      F F F F F F L
    //     0     0 0 0 0 0 0 0 0      0 0 0 0 0 1 0
    //
    //
    //  Max denormal value round to min normal value:
    //    Input:
    //    Sign |  Exp (8 bit)     | Frac (first 7 bit) | Frac (last 16 bit)
    //     S     E E E E E E E E      F F F F F F L     RTTTTTTTTTTTTTTT
    //     0     0 0 0 0 0 0 0 0      1 1 1 1 1 1 1     1111111111111111
    //
    //     This falls into case 4. We create a carry from R and T,
    //     propagate into L and F, which then propagates into exponent
    //     bits:
    //
    //    Output:
    //    Sign |  Exp (8 bit)     | Frac (first 7 bit)
    //     S     E E E E E E E E      F F F F F F L
    //     0     0 0 0 0 0 0 0 1      0 0 0 0 0 0 0
    //
    //  Max normal value round to Inf:
    //    Input:
    //    Sign |  Exp (8 bit)     | Frac (first 7 bit) | Frac (last 16 bit)
    //     S     E E E E E E E E      F F F F F F L     RTTTTTTTTTTTTTTT
    //     0     1 1 1 1 1 1 1 0      1 1 1 1 1 1 1     1111111111111111
    //
    //     This falls into case 4. We create a carry from R and T,
    //     propagate into L and F, which then propagates into exponent
    //     bits:
    //
    //    Sign |  Exp (8 bit)     | Frac (first 7 bit)
    //     S     E E E E E E E E      F F F F F F L
    //     0     1 1 1 1 1 1 1 1      0 0 0 0 0 0 0

    // At this point, ff must be either a normal float, or +/-infinity.
    output = float_to_bfloat16_rtne<true>(ff);
  }
  return output;
#endif
}

// float_to_bfloat16_rtne template specialization that assumes that its function
// argument (ff) is either a normal floating point number, or +/-infinity, or
// zero. Used to improve the runtime performance of conversion from an integer
// type to bfloat16.
template <>
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC __bfloat16_raw float_to_bfloat16_rtne<true>(float ff) {
#if defined(EIGEN_USE_HIP_BF16)
  return __bfloat16_raw(__bfloat16_raw::round_to_bfloat16(ff));
#else
  numext::uint32_t input = numext::bit_cast<numext::uint32_t>(ff);
  __bfloat16_raw output;

  // Least significant bit of resulting bfloat.
  numext::uint32_t lsb = (input >> 16) & 1;
  numext::uint32_t rounding_bias = 0x7fff + lsb;
  input += rounding_bias;
  output.value = static_cast<numext::uint16_t>(input >> 16);
  return output;
#endif
}

EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC float bfloat16_to_float(__bfloat16_raw h) {
#if defined(EIGEN_USE_HIP_BF16)
  return static_cast<float>(h);
#else
  return numext::bit_cast<float>(static_cast<numext::uint32_t>(h.value) << 16);
#endif
}

// --- standard functions ---

EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bool(isinf)(const bfloat16& a) {
  EIGEN_USING_STD(isinf);
#if defined(EIGEN_USE_HIP_BF16)
  return (isinf)(a);  // Uses HIP hip_bfloat16 isinf operator
#else
  return (isinf)(float(a));
#endif
}
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bool(isnan)(const bfloat16& a) {
  EIGEN_USING_STD(isnan);
#if defined(EIGEN_USE_HIP_BF16)
  return (isnan)(a);  // Uses HIP hip_bfloat16 isnan operator
#else
  return (isnan)(float(a));
#endif
}
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bool(isfinite)(const bfloat16& a) {
  return !(isinf EIGEN_NOT_A_MACRO(a)) && !(isnan EIGEN_NOT_A_MACRO(a));
}

EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bfloat16 abs(const bfloat16& a) {
  numext::uint16_t x = numext::bit_cast<numext::uint16_t>(a) & 0x7FFF;
  return numext::bit_cast<bfloat16>(x);
}
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bfloat16 exp(const bfloat16& a) { return bfloat16(::expf(float(a))); }
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bfloat16 exp2(const bfloat16& a) { return bfloat16(::exp2f(float(a))); }
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bfloat16 expm1(const bfloat16& a) { return bfloat16(numext::expm1(float(a))); }
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bfloat16 log(const bfloat16& a) { return bfloat16(::logf(float(a))); }
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bfloat16 log1p(const bfloat16& a) { return bfloat16(numext::log1p(float(a))); }
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bfloat16 log10(const bfloat16& a) { return bfloat16(::log10f(float(a))); }
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bfloat16 log2(const bfloat16& a) {
  return bfloat16(static_cast<float>(EIGEN_LOG2E) * ::logf(float(a)));
}
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bfloat16 sqrt(const bfloat16& a) { return bfloat16(::sqrtf(float(a))); }
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bfloat16 pow(const bfloat16& a, const bfloat16& b) {
  return bfloat16(::powf(float(a), float(b)));
}
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bfloat16 atan2(const bfloat16& a, const bfloat16& b) {
  return bfloat16(::atan2f(float(a), float(b)));
}
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bfloat16 sin(const bfloat16& a) { return bfloat16(::sinf(float(a))); }
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bfloat16 cos(const bfloat16& a) { return bfloat16(::cosf(float(a))); }
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bfloat16 tan(const bfloat16& a) { return bfloat16(::tanf(float(a))); }
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bfloat16 asin(const bfloat16& a) { return bfloat16(::asinf(float(a))); }
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bfloat16 acos(const bfloat16& a) { return bfloat16(::acosf(float(a))); }
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bfloat16 atan(const bfloat16& a) { return bfloat16(::atanf(float(a))); }
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bfloat16 sinh(const bfloat16& a) { return bfloat16(::sinhf(float(a))); }
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bfloat16 cosh(const bfloat16& a) { return bfloat16(::coshf(float(a))); }
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bfloat16 tanh(const bfloat16& a) { return bfloat16(::tanhf(float(a))); }
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bfloat16 asinh(const bfloat16& a) { return bfloat16(::asinhf(float(a))); }
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bfloat16 acosh(const bfloat16& a) { return bfloat16(::acoshf(float(a))); }
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bfloat16 atanh(const bfloat16& a) { return bfloat16(::atanhf(float(a))); }
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bfloat16 floor(const bfloat16& a) { return bfloat16(::floorf(float(a))); }
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bfloat16 ceil(const bfloat16& a) { return bfloat16(::ceilf(float(a))); }
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bfloat16 rint(const bfloat16& a) { return bfloat16(::rintf(float(a))); }
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bfloat16 round(const bfloat16& a) { return bfloat16(::roundf(float(a))); }
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bfloat16 trunc(const bfloat16& a) { return bfloat16(::truncf(float(a))); }
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bfloat16 fmod(const bfloat16& a, const bfloat16& b) {
  return bfloat16(::fmodf(float(a), float(b)));
}

EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bfloat16(min)(const bfloat16& a, const bfloat16& b) {
  const float f1 = static_cast<float>(a);
  const float f2 = static_cast<float>(b);
  return f2 < f1 ? b : a;
}

EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bfloat16(max)(const bfloat16& a, const bfloat16& b) {
  const float f1 = static_cast<float>(a);
  const float f2 = static_cast<float>(b);
  return f1 < f2 ? b : a;
}

EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bfloat16 fmin(const bfloat16& a, const bfloat16& b) {
  const float f1 = static_cast<float>(a);
  const float f2 = static_cast<float>(b);
  return bfloat16(::fminf(f1, f2));
}

EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bfloat16 fmax(const bfloat16& a, const bfloat16& b) {
  const float f1 = static_cast<float>(a);
  const float f2 = static_cast<float>(b);
  return bfloat16(::fmaxf(f1, f2));
}

EIGEN_DEVICE_FUNC inline bfloat16 fma(const bfloat16& a, const bfloat16& b, const bfloat16& c) {
  // Emulate FMA via float.
  return bfloat16(numext::fma(static_cast<float>(a), static_cast<float>(b), static_cast<float>(c)));
}

#ifndef EIGEN_NO_IO
EIGEN_ALWAYS_INLINE std::ostream& operator<<(std::ostream& os, const bfloat16& v) {
  os << static_cast<float>(v);
  return os;
}
#endif

}  // namespace bfloat16_impl

namespace internal {

template <>
struct is_arithmetic<bfloat16> {
  enum { value = true };
};

template <>
struct random_impl<bfloat16> {
  enum : int { MantissaBits = 7 };
  using Impl = random_impl<float>;
  static EIGEN_DEVICE_FUNC inline bfloat16 run(const bfloat16& x, const bfloat16& y) {
    float result = Impl::run(x, y, MantissaBits);
    return bfloat16(result);
  }
  static EIGEN_DEVICE_FUNC inline bfloat16 run() {
    float result = Impl::run(MantissaBits);
    return bfloat16(result);
  }
};

}  // namespace internal

template <>
struct NumTraits<Eigen::bfloat16> : GenericNumTraits<Eigen::bfloat16> {
  enum { IsSigned = true, IsInteger = false, IsComplex = false, RequireInitialization = false };

  EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR static EIGEN_STRONG_INLINE Eigen::bfloat16 epsilon() {
    return bfloat16_impl::raw_uint16_to_bfloat16(0x3c00);
  }
  EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR static EIGEN_STRONG_INLINE Eigen::bfloat16 dummy_precision() {
    return bfloat16_impl::raw_uint16_to_bfloat16(0x3D4D);  // bfloat16(5e-2f);
  }
  EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR static EIGEN_STRONG_INLINE Eigen::bfloat16 highest() {
    return bfloat16_impl::raw_uint16_to_bfloat16(0x7F7F);
  }
  EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR static EIGEN_STRONG_INLINE Eigen::bfloat16 lowest() {
    return bfloat16_impl::raw_uint16_to_bfloat16(0xFF7F);
  }
  EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR static EIGEN_STRONG_INLINE Eigen::bfloat16 infinity() {
    return bfloat16_impl::raw_uint16_to_bfloat16(0x7f80);
  }
  EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR static EIGEN_STRONG_INLINE Eigen::bfloat16 quiet_NaN() {
    return bfloat16_impl::raw_uint16_to_bfloat16(0x7fc0);
  }
};

}  // namespace Eigen

#if defined(EIGEN_HAS_HIP_BF16)
#pragma pop_macro("EIGEN_CONSTEXPR")
#endif

namespace Eigen {
namespace numext {

template <>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE bool(isnan)(const Eigen::bfloat16& h) {
  return (bfloat16_impl::isnan)(h);
}

template <>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE bool(isinf)(const Eigen::bfloat16& h) {
  return (bfloat16_impl::isinf)(h);
}

template <>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE bool(isfinite)(const Eigen::bfloat16& h) {
  return (bfloat16_impl::isfinite)(h);
}

template <>
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC Eigen::bfloat16 bit_cast<Eigen::bfloat16, uint16_t>(const uint16_t& src) {
  return Eigen::bfloat16_impl::raw_uint16_to_bfloat16(src);
}

template <>
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC uint16_t bit_cast<uint16_t, Eigen::bfloat16>(const Eigen::bfloat16& src) {
  return Eigen::bfloat16_impl::raw_bfloat16_as_uint16(src);
}

EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bfloat16 nextafter(const bfloat16& from, const bfloat16& to) {
  if (numext::isnan EIGEN_NOT_A_MACRO(from)) {
    return from;
  }
  if (numext::isnan EIGEN_NOT_A_MACRO(to)) {
    return to;
  }
  if (from == to) {
    return to;
  }
  uint16_t from_bits = numext::bit_cast<uint16_t>(from);
  bool from_sign = from_bits >> 15;
  // Whether we are adjusting toward the infinity with the same sign as from.
  bool toward_inf = (to > from) == !from_sign;
  if (toward_inf) {
    ++from_bits;
  } else if ((from_bits & 0x7fff) == 0) {
    // Adjusting away from inf, but from is zero, so just toggle the sign.
    from_bits ^= 0x8000;
  } else {
    --from_bits;
  }
  return numext::bit_cast<bfloat16>(from_bits);
}

// Specialize multiply-add to match packet operations and reduce conversions to/from float.
template<>
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC Eigen::bfloat16 madd<Eigen::bfloat16>(const Eigen::bfloat16& x, const Eigen::bfloat16& y, const Eigen::bfloat16& z) {
  return Eigen::bfloat16(static_cast<float>(x) * static_cast<float>(y) + static_cast<float>(z));
}

}  // namespace numext
}  // namespace Eigen

#if EIGEN_HAS_STD_HASH
namespace std {
template <>
struct hash<Eigen::bfloat16> {
  EIGEN_STRONG_INLINE std::size_t operator()(const Eigen::bfloat16& a) const {
    return static_cast<std::size_t>(Eigen::numext::bit_cast<Eigen::numext::uint16_t>(a));
  }
};
}  // namespace std
#endif

// Add the missing shfl* intrinsics.
// The __shfl* functions are only valid on HIP or _CUDA_ARCH_ >= 300.
//   CUDA defines them for (__CUDA_ARCH__ >= 300 || !defined(__CUDA_ARCH__))
//
// HIP and CUDA prior to SDK 9.0 define
//    __shfl, __shfl_up, __shfl_down, __shfl_xor for int and float
// CUDA since 9.0 deprecates those and instead defines
//    __shfl_sync, __shfl_up_sync, __shfl_down_sync, __shfl_xor_sync,
//    with native support for __half and __nv_bfloat16
//
// Note that the following are __device__ - only functions.
#if defined(EIGEN_HIPCC)

#if defined(EIGEN_HAS_HIP_BF16)

__device__ EIGEN_STRONG_INLINE Eigen::bfloat16 __shfl(Eigen::bfloat16 var, int srcLane, int width = warpSize) {
  const int ivar = static_cast<int>(Eigen::numext::bit_cast<Eigen::numext::uint16_t>(var));
  return Eigen::numext::bit_cast<Eigen::bfloat16>(static_cast<Eigen::numext::uint16_t>(__shfl(ivar, srcLane, width)));
}

__device__ EIGEN_STRONG_INLINE Eigen::bfloat16 __shfl_up(Eigen::bfloat16 var, unsigned int delta,
                                                         int width = warpSize) {
  const int ivar = static_cast<int>(Eigen::numext::bit_cast<Eigen::numext::uint16_t>(var));
  return Eigen::numext::bit_cast<Eigen::bfloat16>(static_cast<Eigen::numext::uint16_t>(__shfl_up(ivar, delta, width)));
}

__device__ EIGEN_STRONG_INLINE Eigen::bfloat16 __shfl_down(Eigen::bfloat16 var, unsigned int delta,
                                                           int width = warpSize) {
  const int ivar = static_cast<int>(Eigen::numext::bit_cast<Eigen::numext::uint16_t>(var));
  return Eigen::numext::bit_cast<Eigen::bfloat16>(
      static_cast<Eigen::numext::uint16_t>(__shfl_down(ivar, delta, width)));
}

__device__ EIGEN_STRONG_INLINE Eigen::bfloat16 __shfl_xor(Eigen::bfloat16 var, int laneMask, int width = warpSize) {
  const int ivar = static_cast<int>(Eigen::numext::bit_cast<Eigen::numext::uint16_t>(var));
  return Eigen::numext::bit_cast<Eigen::bfloat16>(
      static_cast<Eigen::numext::uint16_t>(__shfl_xor(ivar, laneMask, width)));
}

#endif  // HIP

#endif  // __shfl*

#if defined(EIGEN_HIPCC)
EIGEN_STRONG_INLINE __device__ Eigen::bfloat16 __ldg(const Eigen::bfloat16* ptr) {
  return Eigen::bfloat16_impl::raw_uint16_to_bfloat16(
      __ldg(Eigen::numext::bit_cast<const Eigen::numext::uint16_t*>(ptr)));
}
#endif  // __ldg

#endif  // EIGEN_BFLOAT16_H