text: // This file is part of Eigen, a lightweight C++ template library
  // for linear algebra.
  //
  // 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/.
  //
  // The conversion routines are Copyright (c) Fabian Giesen, 2016.
  // The original license follows:
  //
  // Copyright (c) Fabian Giesen, 2016
  // All rights reserved.
  // Redistribution and use in source and binary forms, with or without
  // modification, are permitted.
  // THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
  // "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
  // LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
  // A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
  // HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
  // SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
  // LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
  // DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
  // THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
  // (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
  // OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
  
  
  // Standard 16-bit float type, mostly useful for GPUs. Defines a new
  // type Eigen::half (inheriting from CUDA's __half struct) with
  // operator overloads such that it behaves basically as an arithmetic
  // type. It will be quite slow on CPUs (so it is recommended to stay
  // in float32_bits for CPUs, except for simple parameter conversions, I/O
  // to disk and the likes), but fast on GPUs.
  
  
  #ifndef EIGEN_HALF_CUDA_H
  #define EIGEN_HALF_CUDA_H
  
  #if __cplusplus > 199711L
  #define EIGEN_EXPLICIT_CAST(tgt_type) explicit operator tgt_type()
  #else
  #define EIGEN_EXPLICIT_CAST(tgt_type) operator tgt_type()
  #endif
  
  #include <sstream>
  
  namespace Eigen {
  
  struct half;
  
  namespace half_impl {
  
  #if !defined(EIGEN_HAS_CUDA_FP16)
  // Make our own __half_raw definition that is similar to CUDA's.
  struct __half_raw {
    EIGEN_DEVICE_FUNC __half_raw() : x(0) {}
    explicit EIGEN_DEVICE_FUNC __half_raw(unsigned short raw) : x(raw) {}
    unsigned short x;
  };
  #elif defined(EIGEN_CUDACC_VER) && EIGEN_CUDACC_VER < 90000
  // In CUDA < 9.0, __half is the equivalent of CUDA 9's __half_raw
  typedef __half __half_raw;
  #endif
  
  EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC __half_raw raw_uint16_to_half(unsigned short x);
  EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC __half_raw float_to_half_rtne(float ff);
  EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC float half_to_float(__half_raw h);
  
  struct half_base : public __half_raw {
    EIGEN_DEVICE_FUNC half_base() {}
    EIGEN_DEVICE_FUNC half_base(const half_base& h) : __half_raw(h) {}
    EIGEN_DEVICE_FUNC half_base(const __half_raw& h) : __half_raw(h) {}
  #if defined(EIGEN_HAS_CUDA_FP16) && defined(EIGEN_CUDACC_VER) && EIGEN_CUDACC_VER >= 90000
    EIGEN_DEVICE_FUNC half_base(const __half& h) : __half_raw(*(__half_raw*)&h) {}
  #endif
  };
  
  } // namespace half_impl
  
  // Class definition.
  struct half : public half_impl::half_base {
    #if !defined(EIGEN_HAS_CUDA_FP16) || (defined(EIGEN_CUDACC_VER) && EIGEN_CUDACC_VER < 90000)
      typedef half_impl::__half_raw __half_raw;
    #endif
  
    EIGEN_DEVICE_FUNC half() {}
  
    EIGEN_DEVICE_FUNC half(const __half_raw& h) : half_impl::half_base(h) {}
    EIGEN_DEVICE_FUNC half(const half& h) : half_impl::half_base(h) {}
  #if defined(EIGEN_HAS_CUDA_FP16) && defined(EIGEN_CUDACC_VER) && EIGEN_CUDACC_VER >= 90000
    EIGEN_DEVICE_FUNC half(const __half& h) : half_impl::half_base(h) {}
  #endif
  
    explicit EIGEN_DEVICE_FUNC half(bool b)
        : half_impl::half_base(half_impl::raw_uint16_to_half(b ? 0x3c00 : 0)) {}
    template<class T>
    explicit EIGEN_DEVICE_FUNC half(const T& val)
        : half_impl::half_base(half_impl::float_to_half_rtne(static_cast<float>(val))) {}
    explicit EIGEN_DEVICE_FUNC half(float f)
        : half_impl::half_base(half_impl::float_to_half_rtne(f)) {}
  
    EIGEN_DEVICE_FUNC EIGEN_EXPLICIT_CAST(bool) const {
      // +0.0 and -0.0 become false, everything else becomes true.
      return (x & 0x7fff) != 0;
    }
    EIGEN_DEVICE_FUNC EIGEN_EXPLICIT_CAST(signed char) const {
      return static_cast<signed char>(half_impl::half_to_float(*this));
    }
    EIGEN_DEVICE_FUNC EIGEN_EXPLICIT_CAST(unsigned char) const {
      return static_cast<unsigned char>(half_impl::half_to_float(*this));
    }
    EIGEN_DEVICE_FUNC EIGEN_EXPLICIT_CAST(short) const {
      return static_cast<short>(half_impl::half_to_float(*this));
    }
    EIGEN_DEVICE_FUNC EIGEN_EXPLICIT_CAST(unsigned short) const {
      return static_cast<unsigned short>(half_impl::half_to_float(*this));
    }
    EIGEN_DEVICE_FUNC EIGEN_EXPLICIT_CAST(int) const {
      return static_cast<int>(half_impl::half_to_float(*this));
    }
    EIGEN_DEVICE_FUNC EIGEN_EXPLICIT_CAST(unsigned int) const {
      return static_cast<unsigned int>(half_impl::half_to_float(*this));
    }
    EIGEN_DEVICE_FUNC EIGEN_EXPLICIT_CAST(long) const {
      return static_cast<long>(half_impl::half_to_float(*this));
    }
    EIGEN_DEVICE_FUNC EIGEN_EXPLICIT_CAST(unsigned long) const {
      return static_cast<unsigned long>(half_impl::half_to_float(*this));
    }
    EIGEN_DEVICE_FUNC EIGEN_EXPLICIT_CAST(long long) const {
      return static_cast<long long>(half_impl::half_to_float(*this));
    }
    EIGEN_DEVICE_FUNC EIGEN_EXPLICIT_CAST(unsigned long long) const {
      return static_cast<unsigned long long>(half_to_float(*this));
    }
    EIGEN_DEVICE_FUNC EIGEN_EXPLICIT_CAST(float) const {
      return half_impl::half_to_float(*this);
    }
    EIGEN_DEVICE_FUNC EIGEN_EXPLICIT_CAST(double) const {
      return static_cast<double>(half_impl::half_to_float(*this));
    }
  
    EIGEN_DEVICE_FUNC half& operator=(const half& other) {
      x = other.x;
      return *this;
    }
  };
  
  } // end namespace Eigen
  
  namespace std {
  template<>
  struct numeric_limits<Eigen::half> {
    static const bool is_specialized = true;
    static const bool is_signed = true;
    static const bool is_integer = false;
    static const bool is_exact = false;
    static const bool has_infinity = true;
    static const bool has_quiet_NaN = true;
    static const bool has_signaling_NaN = true;
    static const float_denorm_style has_denorm = denorm_present;
    static const bool has_denorm_loss = false;
    static const std::float_round_style round_style = std::round_to_nearest;
    static const bool is_iec559 = false;
    static const bool is_bounded = false;
    static const bool is_modulo = false;
    static const int digits = 11;
    static const int digits10 = 3;      // according to http://half.sourceforge.net/structstd_1_1numeric__limits_3_01half__float_1_1half_01_4.html
    static const int max_digits10 = 5;  // according to http://half.sourceforge.net/structstd_1_1numeric__limits_3_01half__float_1_1half_01_4.html
    static const int radix = 2;
    static const int min_exponent = -13;
    static const int min_exponent10 = -4;
    static const int max_exponent = 16;
    static const int max_exponent10 = 4;
    static const bool traps = true;
    static const bool tinyness_before = false;
  
    static Eigen::half (min)() { return Eigen::half_impl::raw_uint16_to_half(0x400); }
    static Eigen::half lowest() { return Eigen::half_impl::raw_uint16_to_half(0xfbff); }
    static Eigen::half (max)() { return Eigen::half_impl::raw_uint16_to_half(0x7bff); }
    static Eigen::half epsilon() { return Eigen::half_impl::raw_uint16_to_half(0x0800); }
    static Eigen::half round_error() { return Eigen::half(0.5); }
    static Eigen::half infinity() { return Eigen::half_impl::raw_uint16_to_half(0x7c00); }
    static Eigen::half quiet_NaN() { return Eigen::half_impl::raw_uint16_to_half(0x7e00); }
    static Eigen::half signaling_NaN() { return Eigen::half_impl::raw_uint16_to_half(0x7e00); }
    static Eigen::half denorm_min() { return Eigen::half_impl::raw_uint16_to_half(0x1); }
  };
  
  // 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<>
  struct numeric_limits<const Eigen::half> : numeric_limits<Eigen::half> {};
  template<>
  struct numeric_limits<volatile Eigen::half> : numeric_limits<Eigen::half> {};
  template<>
  struct numeric_limits<const volatile Eigen::half> : numeric_limits<Eigen::half> {};
  } // end namespace std
  
  namespace Eigen {
  
  namespace half_impl {
  
  #if defined(EIGEN_HAS_CUDA_FP16) && defined(EIGEN_CUDA_ARCH) && EIGEN_CUDA_ARCH >= 530
  
  // Intrinsics for native fp16 support. Note that on current hardware,
  // these are no faster than float32_bits arithmetic (you need to use the half2
  // versions to get the ALU speed increased), but you do save the
  // conversion steps back and forth.
  
  EIGEN_STRONG_INLINE __device__ half operator + (const half& a, const half& b) {
    return __hadd(a, b);
  }
  EIGEN_STRONG_INLINE __device__ half operator * (const half& a, const half& b) {
    return __hmul(a, b);
  }
  EIGEN_STRONG_INLINE __device__ half operator - (const half& a, const half& b) {
    return __hsub(a, b);
  }
  EIGEN_STRONG_INLINE __device__ half operator / (const half& a, const half& b) {
    float num = __half2float(a);
    float denom = __half2float(b);
    return __float2half(num / denom);
  }
  EIGEN_STRONG_INLINE __device__ half operator - (const half& a) {
    return __hneg(a);
  }
  EIGEN_STRONG_INLINE __device__ half& operator += (half& a, const half& b) {
    a = a + b;
    return a;
  }
  EIGEN_STRONG_INLINE __device__ half& operator *= (half& a, const half& b) {
    a = a * b;
    return a;
  }
  EIGEN_STRONG_INLINE __device__ half& operator -= (half& a, const half& b) {
    a = a - b;
    return a;
  }
  EIGEN_STRONG_INLINE __device__ half& operator /= (half& a, const half& b) {
    a = a / b;
    return a;
  }
  EIGEN_STRONG_INLINE __device__ bool operator == (const half& a, const half& b) {
    return __heq(a, b);
  }
  EIGEN_STRONG_INLINE __device__ bool operator != (const half& a, const half& b) {
    return __hne(a, b);
  }
  EIGEN_STRONG_INLINE __device__ bool operator < (const half& a, const half& b) {
    return __hlt(a, b);
  }
  EIGEN_STRONG_INLINE __device__ bool operator <= (const half& a, const half& b) {
    return __hle(a, b);
  }
  EIGEN_STRONG_INLINE __device__ bool operator > (const half& a, const half& b) {
    return __hgt(a, b);
  }
  EIGEN_STRONG_INLINE __device__ bool operator >= (const half& a, const half& b) {
    return __hge(a, b);
  }
  
  #else  // Emulate support for half floats
  
  // Definitions for CPUs and older CUDA, mostly working through conversion
  // to/from float32_bits.
  
  EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC half operator + (const half& a, const half& b) {
    return half(float(a) + float(b));
  }
  EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC half operator * (const half& a, const half& b) {
    return half(float(a) * float(b));
  }
  EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC half operator - (const half& a, const half& b) {
    return half(float(a) - float(b));
  }
  EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC half operator / (const half& a, const half& b) {
    return half(float(a) / float(b));
  }
  EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC half operator - (const half& a) {
    half result;
    result.x = a.x ^ 0x8000;
    return result;
  }
  EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC half& operator += (half& a, const half& b) {
    a = half(float(a) + float(b));
    return a;
  }
  EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC half& operator *= (half& a, const half& b) {
    a = half(float(a) * float(b));
    return a;
  }
  EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC half& operator -= (half& a, const half& b) {
    a = half(float(a) - float(b));
    return a;
  }
  EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC half& operator /= (half& a, const half& b) {
    a = half(float(a) / float(b));
    return a;
  }
  EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bool operator == (const half& a, const half& b) {
    return numext::equal_strict(float(a),float(b));
  }
  EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bool operator != (const half& a, const half& b) {
    return numext::not_equal_strict(float(a), float(b));
  }
  EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bool operator < (const half& a, const half& b) {
    return float(a) < float(b);
  }
  EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bool operator <= (const half& a, const half& b) {
    return float(a) <= float(b);
  }
  EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bool operator > (const half& a, const half& b) {
    return float(a) > float(b);
  }
  EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bool operator >= (const half& a, const half& b) {
    return float(a) >= float(b);
  }
  
  #endif  // Emulate support for half floats
  
  // Division by an index. Do it in full float precision to avoid accuracy
  // issues in converting the denominator to half.
  EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC half operator / (const half& a, Index b) {
    return half(static_cast<float>(a) / static_cast<float>(b));
  }
  
  // Conversion routines, including fallbacks for the host or older CUDA.
  // Note that newer Intel CPUs (Haswell or newer) have vectorized versions of
  // these in hardware. If we need more performance on older/other CPUs, they are
  // also possible to vectorize directly.
  
  EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC __half_raw raw_uint16_to_half(unsigned short x) {
    __half_raw h;
    h.x = x;
    return h;
  }
  
  union float32_bits {
    unsigned int u;
    float f;
  };
  
  EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC __half_raw float_to_half_rtne(float ff) {
  #if defined(EIGEN_HAS_CUDA_FP16) && defined(EIGEN_CUDA_ARCH) && EIGEN_CUDA_ARCH >= 300
    __half tmp_ff = __float2half(ff);
    return *(__half_raw*)&tmp_ff;
  
  #elif defined(EIGEN_HAS_FP16_C)
    __half_raw h;
    h.x = _cvtss_sh(ff, 0);
    return h;
  
  #else
    float32_bits f; f.f = ff;
  
    const float32_bits f32infty = { 255 << 23 };
    const float32_bits f16max = { (127 + 16) << 23 };
    const float32_bits denorm_magic = { ((127 - 15) + (23 - 10) + 1) << 23 };
    unsigned int sign_mask = 0x80000000u;
    __half_raw o;
    o.x = static_cast<unsigned short>(0x0u);
  
    unsigned int sign = f.u & sign_mask;
    f.u ^= sign;
  
    // NOTE all the integer compares in this function can be safely
    // compiled into signed compares since all operands are below
    // 0x80000000. Important if you want fast straight SSE2 code
    // (since there's no unsigned PCMPGTD).
  
    if (f.u >= f16max.u) {  // result is Inf or NaN (all exponent bits set)
      o.x = (f.u > f32infty.u) ? 0x7e00 : 0x7c00; // NaN->qNaN and Inf->Inf
    } else {  // (De)normalized number or zero
      if (f.u < (113 << 23)) {  // resulting FP16 is subnormal or zero
        // use a magic value to align our 10 mantissa bits at the bottom of
        // the float. as long as FP addition is round-to-nearest-even this
        // just works.
        f.f += denorm_magic.f;
  
        // and one integer subtract of the bias later, we have our final float!
        o.x = static_cast<unsigned short>(f.u - denorm_magic.u);
      } else {
        unsigned int mant_odd = (f.u >> 13) & 1; // resulting mantissa is odd
  
        // update exponent, rounding bias part 1
        f.u += ((unsigned int)(15 - 127) << 23) + 0xfff;
        // rounding bias part 2
        f.u += mant_odd;
        // take the bits!
        o.x = static_cast<unsigned short>(f.u >> 13);
      }
    }
  
    o.x |= static_cast<unsigned short>(sign >> 16);
    return o;
  #endif
  }
  
  EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC float half_to_float(__half_raw h) {
  #if defined(EIGEN_HAS_CUDA_FP16) && defined(EIGEN_CUDA_ARCH) && EIGEN_CUDA_ARCH >= 300
    return __half2float(h);
  
  #elif defined(EIGEN_HAS_FP16_C)
    return _cvtsh_ss(h.x);
  
  #else
    const float32_bits magic = { 113 << 23 };
    const unsigned int shifted_exp = 0x7c00 << 13; // exponent mask after shift
    float32_bits o;
  
    o.u = (h.x & 0x7fff) << 13;             // exponent/mantissa bits
    unsigned int exp = shifted_exp & o.u;   // just the exponent
    o.u += (127 - 15) << 23;                // exponent adjust
  
    // handle exponent special cases
    if (exp == shifted_exp) {     // Inf/NaN?
      o.u += (128 - 16) << 23;    // extra exp adjust
    } else if (exp == 0) {        // Zero/Denormal?
      o.u += 1 << 23;             // extra exp adjust
      o.f -= magic.f;             // renormalize
    }
  
    o.u |= (h.x & 0x8000) << 16;    // sign bit
    return o.f;
  #endif
  }
  
  // --- standard functions ---
  
  EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bool (isinf)(const half& a) {
    return (a.x & 0x7fff) == 0x7c00;
  }
  EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bool (isnan)(const half& a) {
  #if defined(EIGEN_HAS_CUDA_FP16) && defined(EIGEN_CUDA_ARCH) && EIGEN_CUDA_ARCH >= 530
    return __hisnan(a);
  #else
    return (a.x & 0x7fff) > 0x7c00;
  #endif
  }
  EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bool (isfinite)(const half& a) {
    return !(isinf EIGEN_NOT_A_MACRO (a)) && !(isnan EIGEN_NOT_A_MACRO (a));
  }
  
  EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC half abs(const half& a) {
    half result;
    result.x = a.x & 0x7FFF;
    return result;
  }
  EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC half exp(const half& a) {
  #if EIGEN_CUDACC_VER >= 80000 && defined EIGEN_CUDA_ARCH && EIGEN_CUDA_ARCH >= 530
    return half(hexp(a));
  #else
     return half(::expf(float(a)));
  #endif
  }
  EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC half log(const half& a) {
  #if defined(EIGEN_HAS_CUDA_FP16) && EIGEN_CUDACC_VER >= 80000 && defined(EIGEN_CUDA_ARCH) && EIGEN_CUDA_ARCH >= 530
    return half(::hlog(a));
  #else
    return half(::logf(float(a)));
  #endif
  }
  EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC half log1p(const half& a) {
    return half(numext::log1p(float(a)));
  }
  EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC half log10(const half& a) {
    return half(::log10f(float(a)));
  }
  EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC half sqrt(const half& a) {
  #if EIGEN_CUDACC_VER >= 80000 && defined EIGEN_CUDA_ARCH && EIGEN_CUDA_ARCH >= 530
    return half(hsqrt(a));
  #else
      return half(::sqrtf(float(a)));
  #endif
  }
  EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC half pow(const half& a, const half& b) {
    return half(::powf(float(a), float(b)));
  }
  EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC half sin(const half& a) {
    return half(::sinf(float(a)));
  }
  EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC half cos(const half& a) {
    return half(::cosf(float(a)));
  }
  EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC half tan(const half& a) {
    return half(::tanf(float(a)));
  }
  EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC half tanh(const half& a) {
    return half(::tanhf(float(a)));
  }
  EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC half floor(const half& a) {
  #if EIGEN_CUDACC_VER >= 80000 && defined EIGEN_CUDA_ARCH && EIGEN_CUDA_ARCH >= 300
    return half(hfloor(a));
  #else
    return half(::floorf(float(a)));
  #endif
  }
  EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC half ceil(const half& a) {
  #if EIGEN_CUDACC_VER >= 80000 && defined EIGEN_CUDA_ARCH && EIGEN_CUDA_ARCH >= 300
    return half(hceil(a));
  #else
    return half(::ceilf(float(a)));
  #endif
  }
  
  EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC half (min)(const half& a, const half& b) {
  #if defined(EIGEN_HAS_CUDA_FP16) && defined(EIGEN_CUDA_ARCH) && EIGEN_CUDA_ARCH >= 530
    return __hlt(b, a) ? b : a;
  #else
    const float f1 = static_cast<float>(a);
    const float f2 = static_cast<float>(b);
    return f2 < f1 ? b : a;
  #endif
  }
  EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC half (max)(const half& a, const half& b) {
  #if defined(EIGEN_HAS_CUDA_FP16) && defined(EIGEN_CUDA_ARCH) && EIGEN_CUDA_ARCH >= 530
    return __hlt(a, b) ? b : a;
  #else
    const float f1 = static_cast<float>(a);
    const float f2 = static_cast<float>(b);
    return f1 < f2 ? b : a;
  #endif
  }
  
  EIGEN_ALWAYS_INLINE std::ostream& operator << (std::ostream& os, const half& v) {
    os << static_cast<float>(v);
    return os;
  }
  
  } // end namespace half_impl
  
  // import Eigen::half_impl::half into Eigen namespace
  // using half_impl::half;
  
  namespace internal {
  
  template<>
  struct random_default_impl<half, false, false>
  {
    static inline half run(const half& x, const half& y)
    {
      return x + (y-x) * half(float(std::rand()) / float(RAND_MAX));
    }
    static inline half run()
    {
      return run(half(-1.f), half(1.f));
    }
  };
  
  template<> struct is_arithmetic<half> { enum { value = true }; };
  
  } // end namespace internal
  
  template<> struct NumTraits<Eigen::half>
      : GenericNumTraits<Eigen::half>
  {
    enum {
      IsSigned = true,
      IsInteger = false,
      IsComplex = false,
      RequireInitialization = false
    };
  
    EIGEN_DEVICE_FUNC static EIGEN_STRONG_INLINE Eigen::half epsilon() {
      return half_impl::raw_uint16_to_half(0x0800);
    }
    EIGEN_DEVICE_FUNC static EIGEN_STRONG_INLINE Eigen::half dummy_precision() { return Eigen::half(1e-2f); }
    EIGEN_DEVICE_FUNC static EIGEN_STRONG_INLINE Eigen::half highest() {
      return half_impl::raw_uint16_to_half(0x7bff);
    }
    EIGEN_DEVICE_FUNC static EIGEN_STRONG_INLINE Eigen::half lowest() {
      return half_impl::raw_uint16_to_half(0xfbff);
    }
    EIGEN_DEVICE_FUNC static EIGEN_STRONG_INLINE Eigen::half infinity() {
      return half_impl::raw_uint16_to_half(0x7c00);
    }
    EIGEN_DEVICE_FUNC static EIGEN_STRONG_INLINE Eigen::half quiet_NaN() {
      return half_impl::raw_uint16_to_half(0x7c01);
    }
  };
  
  } // end namespace Eigen
  
  // C-like standard mathematical functions and trancendentals.
  EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC Eigen::half fabsh(const Eigen::half& a) {
    Eigen::half result;
    result.x = a.x & 0x7FFF;
    return result;
  }
  EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC Eigen::half exph(const Eigen::half& a) {
    return Eigen::half(::expf(float(a)));
  }
  EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC Eigen::half logh(const Eigen::half& a) {
  #if EIGEN_CUDACC_VER >= 80000 && defined(EIGEN_CUDA_ARCH) && EIGEN_CUDA_ARCH >= 530
    return Eigen::half(::hlog(a));
  #else
    return Eigen::half(::logf(float(a)));
  #endif
  }
  EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC Eigen::half sqrth(const Eigen::half& a) {
    return Eigen::half(::sqrtf(float(a)));
  }
  EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC Eigen::half powh(const Eigen::half& a, const Eigen::half& b) {
    return Eigen::half(::powf(float(a), float(b)));
  }
  EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC Eigen::half floorh(const Eigen::half& a) {
    return Eigen::half(::floorf(float(a)));
  }
  EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC Eigen::half ceilh(const Eigen::half& a) {
    return Eigen::half(::ceilf(float(a)));
  }
  
  namespace std {
  
  #if __cplusplus > 199711L
  template <>
  struct hash<Eigen::half> {
    EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE std::size_t operator()(const Eigen::half& a) const {
      return static_cast<std::size_t>(a.x);
    }
  };
  #endif
  
  } // end namespace std
  
  
  // Add the missing shfl_xor intrinsic
  #if defined(EIGEN_CUDA_ARCH) && EIGEN_CUDA_ARCH >= 300
  __device__ EIGEN_STRONG_INLINE Eigen::half __shfl_xor(Eigen::half var, int laneMask, int width=warpSize) {
    #if EIGEN_CUDACC_VER < 90000
    return static_cast<Eigen::half>(__shfl_xor(static_cast<float>(var), laneMask, width));
    #else
    return static_cast<Eigen::half>(__shfl_xor_sync(0xFFFFFFFF, static_cast<float>(var), laneMask, width));
    #endif
  }
  #endif
  
  // ldg() has an overload for __half_raw, but we also need one for Eigen::half.
  #if defined(EIGEN_CUDA_ARCH) && EIGEN_CUDA_ARCH >= 350
  EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC Eigen::half __ldg(const Eigen::half* ptr) {
    return Eigen::half_impl::raw_uint16_to_half(
        __ldg(reinterpret_cast<const unsigned short*>(ptr)));
  }
  #endif
  
  
  #if defined(EIGEN_CUDA_ARCH)
  namespace Eigen {
  namespace numext {
  
  template<>
  EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE
  bool (isnan)(const Eigen::half& h) {
    return (half_impl::isnan)(h);
  }
  
  template<>
  EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE
  bool (isinf)(const Eigen::half& h) {
    return (half_impl::isinf)(h);
  }
  
  template<>
  EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE
  bool (isfinite)(const Eigen::half& h) {
    return (half_impl::isfinite)(h);
  }
  
  } // namespace Eigen
  }  // namespace numext
  #endif
  
  #endif // EIGEN_HALF_CUDA_H
  