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1 // This file is part of Eigen, a lightweight C++ template library
2 // for linear algebra.
3 //
4 // This Source Code Form is subject to the terms of the Mozilla
5 // Public License v. 2.0. If a copy of the MPL was not distributed
6 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
7 //
8 // The conversion routines are Copyright (c) Fabian Giesen, 2016.
9 // The original license follows:
10 //
11 // Copyright (c) Fabian Giesen, 2016
12 // All rights reserved.
13 // Redistribution and use in source and binary forms, with or without
14 // modification, are permitted.
15 // THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
16 // "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
17 // LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
18 // A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
19 // HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
20 // SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
21 // LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
22 // DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
23 // THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
24 // (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
25 // OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
26 
27 
28 // Standard 16-bit float type, mostly useful for GPUs. Defines a new
29 // type Eigen::half (inheriting from CUDA's __half struct) with
30 // operator overloads such that it behaves basically as an arithmetic
31 // type. It will be quite slow on CPUs (so it is recommended to stay
32 // in fp32 for CPUs, except for simple parameter conversions, I/O
33 // to disk and the likes), but fast on GPUs.
34 
35 
36 #ifndef EIGEN_HALF_CUDA_H
37 #define EIGEN_HALF_CUDA_H
38 
39 #if __cplusplus > 199711L
40 #define EIGEN_EXPLICIT_CAST(tgt_type) explicit operator tgt_type()
41 #else
42 #define EIGEN_EXPLICIT_CAST(tgt_type) operator tgt_type()
43 #endif
44 
45 
46 namespace Eigen {
47 
48 struct half;
49 
50 namespace half_impl {
51 
52 #if !defined(EIGEN_HAS_CUDA_FP16)
53 
54 // Make our own __half definition that is similar to CUDA's.
55 struct __half {
__half__half56   EIGEN_DEVICE_FUNC __half() {}
__half__half57   explicit EIGEN_DEVICE_FUNC __half(unsigned short raw) : x(raw) {}
58   unsigned short x;
59 };
60 
61 #endif
62 
63 EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC __half raw_uint16_to_half(unsigned short x);
64 EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC __half float_to_half_rtne(float ff);
65 EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC float half_to_float(__half h);
66 
67 struct half_base : public __half {
half_basehalf_base68   EIGEN_DEVICE_FUNC half_base() {}
half_basehalf_base69   EIGEN_DEVICE_FUNC half_base(const half_base& h) : __half(h) {}
half_basehalf_base70   EIGEN_DEVICE_FUNC half_base(const __half& h) : __half(h) {}
71 };
72 
73 } // namespace half_impl
74 
75 // Class definition.
76 struct half : public half_impl::half_base {
77   #if !defined(EIGEN_HAS_CUDA_FP16)
78     typedef half_impl::__half __half;
79   #endif
80 
halfhalf81   EIGEN_DEVICE_FUNC half() {}
82 
halfhalf83   EIGEN_DEVICE_FUNC half(const __half& h) : half_impl::half_base(h) {}
halfhalf84   EIGEN_DEVICE_FUNC half(const half& h) : half_impl::half_base(h) {}
85 
halfhalf86   explicit EIGEN_DEVICE_FUNC half(bool b)
87       : half_impl::half_base(half_impl::raw_uint16_to_half(b ? 0x3c00 : 0)) {}
88   template<class T>
halfhalf89   explicit EIGEN_DEVICE_FUNC half(const T& val)
90       : half_impl::half_base(half_impl::float_to_half_rtne(static_cast<float>(val))) {}
halfhalf91   explicit EIGEN_DEVICE_FUNC half(float f)
92       : half_impl::half_base(half_impl::float_to_half_rtne(f)) {}
93 
EIGEN_EXPLICIT_CASThalf94   EIGEN_DEVICE_FUNC EIGEN_EXPLICIT_CAST(bool) const {
95     // +0.0 and -0.0 become false, everything else becomes true.
96     return (x & 0x7fff) != 0;
97   }
EIGEN_EXPLICIT_CASThalf98   EIGEN_DEVICE_FUNC EIGEN_EXPLICIT_CAST(signed char) const {
99     return static_cast<signed char>(half_impl::half_to_float(*this));
100   }
EIGEN_EXPLICIT_CASThalf101   EIGEN_DEVICE_FUNC EIGEN_EXPLICIT_CAST(unsigned char) const {
102     return static_cast<unsigned char>(half_impl::half_to_float(*this));
103   }
EIGEN_EXPLICIT_CASThalf104   EIGEN_DEVICE_FUNC EIGEN_EXPLICIT_CAST(short) const {
105     return static_cast<short>(half_impl::half_to_float(*this));
106   }
EIGEN_EXPLICIT_CASThalf107   EIGEN_DEVICE_FUNC EIGEN_EXPLICIT_CAST(unsigned short) const {
108     return static_cast<unsigned short>(half_impl::half_to_float(*this));
109   }
EIGEN_EXPLICIT_CASThalf110   EIGEN_DEVICE_FUNC EIGEN_EXPLICIT_CAST(int) const {
111     return static_cast<int>(half_impl::half_to_float(*this));
112   }
EIGEN_EXPLICIT_CASThalf113   EIGEN_DEVICE_FUNC EIGEN_EXPLICIT_CAST(unsigned int) const {
114     return static_cast<unsigned int>(half_impl::half_to_float(*this));
115   }
EIGEN_EXPLICIT_CASThalf116   EIGEN_DEVICE_FUNC EIGEN_EXPLICIT_CAST(long) const {
117     return static_cast<long>(half_impl::half_to_float(*this));
118   }
EIGEN_EXPLICIT_CASThalf119   EIGEN_DEVICE_FUNC EIGEN_EXPLICIT_CAST(unsigned long) const {
120     return static_cast<unsigned long>(half_impl::half_to_float(*this));
121   }
EIGEN_EXPLICIT_CASThalf122   EIGEN_DEVICE_FUNC EIGEN_EXPLICIT_CAST(long long) const {
123     return static_cast<long long>(half_impl::half_to_float(*this));
124   }
EIGEN_EXPLICIT_CASThalf125   EIGEN_DEVICE_FUNC EIGEN_EXPLICIT_CAST(unsigned long long) const {
126     return static_cast<unsigned long long>(half_to_float(*this));
127   }
EIGEN_EXPLICIT_CASThalf128   EIGEN_DEVICE_FUNC EIGEN_EXPLICIT_CAST(float) const {
129     return half_impl::half_to_float(*this);
130   }
EIGEN_EXPLICIT_CASThalf131   EIGEN_DEVICE_FUNC EIGEN_EXPLICIT_CAST(double) const {
132     return static_cast<double>(half_impl::half_to_float(*this));
133   }
134 
135   EIGEN_DEVICE_FUNC half& operator=(const half& other) {
136     x = other.x;
137     return *this;
138   }
139 };
140 
141 namespace half_impl {
142 
143 #if defined(EIGEN_HAS_CUDA_FP16) && defined(__CUDA_ARCH__) && __CUDA_ARCH__ >= 530
144 
145 // Intrinsics for native fp16 support. Note that on current hardware,
146 // these are no faster than fp32 arithmetic (you need to use the half2
147 // versions to get the ALU speed increased), but you do save the
148 // conversion steps back and forth.
149 
150 __device__ half operator + (const half& a, const half& b) {
151   return __hadd(a, b);
152 }
153 __device__ half operator * (const half& a, const half& b) {
154   return __hmul(a, b);
155 }
156 __device__ half operator - (const half& a, const half& b) {
157   return __hsub(a, b);
158 }
159 __device__ half operator / (const half& a, const half& b) {
160   float num = __half2float(a);
161   float denom = __half2float(b);
162   return __float2half(num / denom);
163 }
164 __device__ half operator - (const half& a) {
165   return __hneg(a);
166 }
167 __device__ half& operator += (half& a, const half& b) {
168   a = a + b;
169   return a;
170 }
171 __device__ half& operator *= (half& a, const half& b) {
172   a = a * b;
173   return a;
174 }
175 __device__ half& operator -= (half& a, const half& b) {
176   a = a - b;
177   return a;
178 }
179 __device__ half& operator /= (half& a, const half& b) {
180   a = a / b;
181   return a;
182 }
183 __device__ bool operator == (const half& a, const half& b) {
184   return __heq(a, b);
185 }
186 __device__ bool operator != (const half& a, const half& b) {
187   return __hne(a, b);
188 }
189 __device__ bool operator < (const half& a, const half& b) {
190   return __hlt(a, b);
191 }
192 __device__ bool operator <= (const half& a, const half& b) {
193   return __hle(a, b);
194 }
195 __device__ bool operator > (const half& a, const half& b) {
196   return __hgt(a, b);
197 }
198 __device__ bool operator >= (const half& a, const half& b) {
199   return __hge(a, b);
200 }
201 
202 #else  // Emulate support for half floats
203 
204 // Definitions for CPUs and older CUDA, mostly working through conversion
205 // to/from fp32.
206 
207 EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC half operator + (const half& a, const half& b) {
208   return half(float(a) + float(b));
209 }
210 EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC half operator * (const half& a, const half& b) {
211   return half(float(a) * float(b));
212 }
213 EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC half operator - (const half& a, const half& b) {
214   return half(float(a) - float(b));
215 }
216 EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC half operator / (const half& a, const half& b) {
217   return half(float(a) / float(b));
218 }
219 EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC half operator - (const half& a) {
220   half result;
221   result.x = a.x ^ 0x8000;
222   return result;
223 }
224 EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC half& operator += (half& a, const half& b) {
225   a = half(float(a) + float(b));
226   return a;
227 }
228 EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC half& operator *= (half& a, const half& b) {
229   a = half(float(a) * float(b));
230   return a;
231 }
232 EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC half& operator -= (half& a, const half& b) {
233   a = half(float(a) - float(b));
234   return a;
235 }
236 EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC half& operator /= (half& a, const half& b) {
237   a = half(float(a) / float(b));
238   return a;
239 }
240 EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bool operator == (const half& a, const half& b) {
241   return float(a) == float(b);
242 }
243 EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bool operator != (const half& a, const half& b) {
244   return float(a) != float(b);
245 }
246 EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bool operator < (const half& a, const half& b) {
247   return float(a) < float(b);
248 }
249 EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bool operator <= (const half& a, const half& b) {
250   return float(a) <= float(b);
251 }
252 EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bool operator > (const half& a, const half& b) {
253   return float(a) > float(b);
254 }
255 EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bool operator >= (const half& a, const half& b) {
256   return float(a) >= float(b);
257 }
258 
259 #endif  // Emulate support for half floats
260 
261 // Division by an index. Do it in full float precision to avoid accuracy
262 // issues in converting the denominator to half.
263 EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC half operator / (const half& a, Index b) {
264   return half(static_cast<float>(a) / static_cast<float>(b));
265 }
266 
267 // Conversion routines, including fallbacks for the host or older CUDA.
268 // Note that newer Intel CPUs (Haswell or newer) have vectorized versions of
269 // these in hardware. If we need more performance on older/other CPUs, they are
270 // also possible to vectorize directly.
271 
raw_uint16_to_half(unsigned short x)272 EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC __half raw_uint16_to_half(unsigned short x) {
273   __half h;
274   h.x = x;
275   return h;
276 }
277 
278 union FP32 {
279   unsigned int u;
280   float f;
281 };
282 
float_to_half_rtne(float ff)283 EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC __half float_to_half_rtne(float ff) {
284 #if defined(EIGEN_HAS_CUDA_FP16) && defined(__CUDA_ARCH__) && __CUDA_ARCH__ >= 300
285   return __float2half(ff);
286 
287 #elif defined(EIGEN_HAS_FP16_C)
288   __half h;
289   h.x = _cvtss_sh(ff, 0);
290   return h;
291 
292 #else
293   FP32 f; f.f = ff;
294 
295   const FP32 f32infty = { 255 << 23 };
296   const FP32 f16max = { (127 + 16) << 23 };
297   const FP32 denorm_magic = { ((127 - 15) + (23 - 10) + 1) << 23 };
298   unsigned int sign_mask = 0x80000000u;
299   __half o;
300   o.x = static_cast<unsigned short>(0x0u);
301 
302   unsigned int sign = f.u & sign_mask;
303   f.u ^= sign;
304 
305   // NOTE all the integer compares in this function can be safely
306   // compiled into signed compares since all operands are below
307   // 0x80000000. Important if you want fast straight SSE2 code
308   // (since there's no unsigned PCMPGTD).
309 
310   if (f.u >= f16max.u) {  // result is Inf or NaN (all exponent bits set)
311     o.x = (f.u > f32infty.u) ? 0x7e00 : 0x7c00; // NaN->qNaN and Inf->Inf
312   } else {  // (De)normalized number or zero
313     if (f.u < (113 << 23)) {  // resulting FP16 is subnormal or zero
314       // use a magic value to align our 10 mantissa bits at the bottom of
315       // the float. as long as FP addition is round-to-nearest-even this
316       // just works.
317       f.f += denorm_magic.f;
318 
319       // and one integer subtract of the bias later, we have our final float!
320       o.x = static_cast<unsigned short>(f.u - denorm_magic.u);
321     } else {
322       unsigned int mant_odd = (f.u >> 13) & 1; // resulting mantissa is odd
323 
324       // update exponent, rounding bias part 1
325       f.u += ((unsigned int)(15 - 127) << 23) + 0xfff;
326       // rounding bias part 2
327       f.u += mant_odd;
328       // take the bits!
329       o.x = static_cast<unsigned short>(f.u >> 13);
330     }
331   }
332 
333   o.x |= static_cast<unsigned short>(sign >> 16);
334   return o;
335 #endif
336 }
337 
half_to_float(__half h)338 EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC float half_to_float(__half h) {
339 #if defined(EIGEN_HAS_CUDA_FP16) && defined(__CUDA_ARCH__) && __CUDA_ARCH__ >= 300
340   return __half2float(h);
341 
342 #elif defined(EIGEN_HAS_FP16_C)
343   return _cvtsh_ss(h.x);
344 
345 #else
346   const FP32 magic = { 113 << 23 };
347   const unsigned int shifted_exp = 0x7c00 << 13; // exponent mask after shift
348   FP32 o;
349 
350   o.u = (h.x & 0x7fff) << 13;             // exponent/mantissa bits
351   unsigned int exp = shifted_exp & o.u;   // just the exponent
352   o.u += (127 - 15) << 23;                // exponent adjust
353 
354   // handle exponent special cases
355   if (exp == shifted_exp) {     // Inf/NaN?
356     o.u += (128 - 16) << 23;    // extra exp adjust
357   } else if (exp == 0) {        // Zero/Denormal?
358     o.u += 1 << 23;             // extra exp adjust
359     o.f -= magic.f;             // renormalize
360   }
361 
362   o.u |= (h.x & 0x8000) << 16;    // sign bit
363   return o.f;
364 #endif
365 }
366 
367 // --- standard functions ---
368 
369 EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bool (isinf)(const half& a) {
370   return (a.x & 0x7fff) == 0x7c00;
371 }
372 EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bool (isnan)(const half& a) {
373 #if defined(EIGEN_HAS_CUDA_FP16) && defined(__CUDA_ARCH__) && __CUDA_ARCH__ >= 530
374   return __hisnan(a);
375 #else
376   return (a.x & 0x7fff) > 0x7c00;
377 #endif
378 }
379 EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC bool (isfinite)(const half& a) {
380   return !(isinf EIGEN_NOT_A_MACRO (a)) && !(isnan EIGEN_NOT_A_MACRO (a));
381 }
382 
abs(const half & a)383 EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC half abs(const half& a) {
384   half result;
385   result.x = a.x & 0x7FFF;
386   return result;
387 }
exp(const half & a)388 EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC half exp(const half& a) {
389   return half(::expf(float(a)));
390 }
log(const half & a)391 EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC half log(const half& a) {
392 #if defined(EIGEN_HAS_CUDA_FP16) && defined __CUDACC_VER__ && __CUDACC_VER__ >= 80000 && defined(__CUDA_ARCH__) && __CUDA_ARCH__ >= 530
393   return Eigen::half(::hlog(a));
394 #else
395   return half(::logf(float(a)));
396 #endif
397 }
log1p(const half & a)398 EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC half log1p(const half& a) {
399   return half(numext::log1p(float(a)));
400 }
log10(const half & a)401 EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC half log10(const half& a) {
402   return half(::log10f(float(a)));
403 }
sqrt(const half & a)404 EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC half sqrt(const half& a) {
405   return half(::sqrtf(float(a)));
406 }
pow(const half & a,const half & b)407 EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC half pow(const half& a, const half& b) {
408   return half(::powf(float(a), float(b)));
409 }
sin(const half & a)410 EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC half sin(const half& a) {
411   return half(::sinf(float(a)));
412 }
cos(const half & a)413 EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC half cos(const half& a) {
414   return half(::cosf(float(a)));
415 }
tan(const half & a)416 EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC half tan(const half& a) {
417   return half(::tanf(float(a)));
418 }
tanh(const half & a)419 EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC half tanh(const half& a) {
420   return half(::tanhf(float(a)));
421 }
floor(const half & a)422 EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC half floor(const half& a) {
423   return half(::floorf(float(a)));
424 }
ceil(const half & a)425 EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC half ceil(const half& a) {
426   return half(::ceilf(float(a)));
427 }
428 
half(min)429 EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC half (min)(const half& a, const half& b) {
430 #if defined(EIGEN_HAS_CUDA_FP16) && defined(__CUDA_ARCH__) && __CUDA_ARCH__ >= 530
431   return __hlt(b, a) ? b : a;
432 #else
433   const float f1 = static_cast<float>(a);
434   const float f2 = static_cast<float>(b);
435   return f2 < f1 ? b : a;
436 #endif
437 }
half(max)438 EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC half (max)(const half& a, const half& b) {
439 #if defined(EIGEN_HAS_CUDA_FP16) && defined(__CUDA_ARCH__) && __CUDA_ARCH__ >= 530
440   return __hlt(a, b) ? b : a;
441 #else
442   const float f1 = static_cast<float>(a);
443   const float f2 = static_cast<float>(b);
444   return f1 < f2 ? b : a;
445 #endif
446 }
447 
448 EIGEN_ALWAYS_INLINE std::ostream& operator << (std::ostream& os, const half& v) {
449   os << static_cast<float>(v);
450   return os;
451 }
452 
453 } // end namespace half_impl
454 
455 // import Eigen::half_impl::half into Eigen namespace
456 // using half_impl::half;
457 
458 namespace internal {
459 
460 template<>
461 struct random_default_impl<half, false, false>
462 {
463   static inline half run(const half& x, const half& y)
464   {
465     return x + (y-x) * half(float(std::rand()) / float(RAND_MAX));
466   }
467   static inline half run()
468   {
469     return run(half(-1.f), half(1.f));
470   }
471 };
472 
473 template<> struct is_arithmetic<half> { enum { value = true }; };
474 
475 } // end namespace internal
476 
477 }  // end namespace Eigen
478 
479 namespace std {
480 template<>
481 struct numeric_limits<Eigen::half> {
482   static const bool is_specialized = true;
483   static const bool is_signed = true;
484   static const bool is_integer = false;
485   static const bool is_exact = false;
486   static const bool has_infinity = true;
487   static const bool has_quiet_NaN = true;
488   static const bool has_signaling_NaN = true;
489   static const float_denorm_style has_denorm = denorm_present;
490   static const bool has_denorm_loss = false;
491   static const std::float_round_style round_style = std::round_to_nearest;
492   static const bool is_iec559 = false;
493   static const bool is_bounded = false;
494   static const bool is_modulo = false;
495   static const int digits = 11;
496   static const int digits10 = 2;
497   //static const int max_digits10 = ;
498   static const int radix = 2;
499   static const int min_exponent = -13;
500   static const int min_exponent10 = -4;
501   static const int max_exponent = 16;
502   static const int max_exponent10 = 4;
503   static const bool traps = true;
504   static const bool tinyness_before = false;
505 
506   static Eigen::half (min)() { return Eigen::half_impl::raw_uint16_to_half(0x400); }
507   static Eigen::half lowest() { return Eigen::half_impl::raw_uint16_to_half(0xfbff); }
508   static Eigen::half (max)() { return Eigen::half_impl::raw_uint16_to_half(0x7bff); }
509   static Eigen::half epsilon() { return Eigen::half_impl::raw_uint16_to_half(0x0800); }
510   static Eigen::half round_error() { return Eigen::half(0.5); }
511   static Eigen::half infinity() { return Eigen::half_impl::raw_uint16_to_half(0x7c00); }
512   static Eigen::half quiet_NaN() { return Eigen::half_impl::raw_uint16_to_half(0x7e00); }
513   static Eigen::half signaling_NaN() { return Eigen::half_impl::raw_uint16_to_half(0x7e00); }
514   static Eigen::half denorm_min() { return Eigen::half_impl::raw_uint16_to_half(0x1); }
515 };
516 }
517 
518 namespace Eigen {
519 
520 template<> struct NumTraits<Eigen::half>
521     : GenericNumTraits<Eigen::half>
522 {
523   enum {
524     IsSigned = true,
525     IsInteger = false,
526     IsComplex = false,
527     RequireInitialization = false
528   };
529 
530   EIGEN_DEVICE_FUNC static EIGEN_STRONG_INLINE Eigen::half epsilon() {
531     return half_impl::raw_uint16_to_half(0x0800);
532   }
533   EIGEN_DEVICE_FUNC static EIGEN_STRONG_INLINE Eigen::half dummy_precision() { return Eigen::half(1e-2f); }
534   EIGEN_DEVICE_FUNC static EIGEN_STRONG_INLINE Eigen::half highest() {
535     return half_impl::raw_uint16_to_half(0x7bff);
536   }
537   EIGEN_DEVICE_FUNC static EIGEN_STRONG_INLINE Eigen::half lowest() {
538     return half_impl::raw_uint16_to_half(0xfbff);
539   }
540   EIGEN_DEVICE_FUNC static EIGEN_STRONG_INLINE Eigen::half infinity() {
541     return half_impl::raw_uint16_to_half(0x7c00);
542   }
543   EIGEN_DEVICE_FUNC static EIGEN_STRONG_INLINE Eigen::half quiet_NaN() {
544     return half_impl::raw_uint16_to_half(0x7c01);
545   }
546 };
547 
548 } // end namespace Eigen
549 
550 // C-like standard mathematical functions and trancendentals.
551 EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC Eigen::half fabsh(const Eigen::half& a) {
552   Eigen::half result;
553   result.x = a.x & 0x7FFF;
554   return result;
555 }
556 EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC Eigen::half exph(const Eigen::half& a) {
557   return Eigen::half(::expf(float(a)));
558 }
559 EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC Eigen::half logh(const Eigen::half& a) {
560 #if defined __CUDACC_VER__ && __CUDACC_VER__ >= 80000 && defined(__CUDA_ARCH__) && __CUDA_ARCH__ >= 530
561   return Eigen::half(::hlog(a));
562 #else
563   return Eigen::half(::logf(float(a)));
564 #endif
565 }
566 EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC Eigen::half sqrth(const Eigen::half& a) {
567   return Eigen::half(::sqrtf(float(a)));
568 }
569 EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC Eigen::half powh(const Eigen::half& a, const Eigen::half& b) {
570   return Eigen::half(::powf(float(a), float(b)));
571 }
572 EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC Eigen::half floorh(const Eigen::half& a) {
573   return Eigen::half(::floorf(float(a)));
574 }
575 EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC Eigen::half ceilh(const Eigen::half& a) {
576   return Eigen::half(::ceilf(float(a)));
577 }
578 
579 namespace std {
580 
581 #if __cplusplus > 199711L
582 template <>
583 struct hash<Eigen::half> {
584   EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE std::size_t operator()(const Eigen::half& a) const {
585     return static_cast<std::size_t>(a.x);
586   }
587 };
588 #endif
589 
590 } // end namespace std
591 
592 
593 // Add the missing shfl_xor intrinsic
594 #if defined(__CUDA_ARCH__) && __CUDA_ARCH__ >= 300
595 __device__ EIGEN_STRONG_INLINE Eigen::half __shfl_xor(Eigen::half var, int laneMask, int width=warpSize) {
596   return static_cast<Eigen::half>(__shfl_xor(static_cast<float>(var), laneMask, width));
597 }
598 #endif
599 
600 // ldg() has an overload for __half, but we also need one for Eigen::half.
601 #if defined(__CUDA_ARCH__) && __CUDA_ARCH__ >= 350
602 EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC Eigen::half __ldg(const Eigen::half* ptr) {
603   return Eigen::half_impl::raw_uint16_to_half(
604       __ldg(reinterpret_cast<const unsigned short*>(ptr)));
605 }
606 #endif
607 
608 
609 #if defined(__CUDA_ARCH__)
610 namespace Eigen {
611 namespace numext {
612 
613 template<>
614 EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE
615 bool (isnan)(const Eigen::half& h) {
616   return (half_impl::isnan)(h);
617 }
618 
619 template<>
620 EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE
621 bool (isinf)(const Eigen::half& h) {
622   return (half_impl::isinf)(h);
623 }
624 
625 template<>
626 EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE
627 bool (isfinite)(const Eigen::half& h) {
628   return (half_impl::isfinite)(h);
629 }
630 
631 } // namespace Eigen
632 }  // namespace numext
633 #endif
634 
635 #endif // EIGEN_HALF_CUDA_H
636