Line data Source code
1 : // Branch-free implementation of half-precision (16 bit) floating point
2 : // Copyright 2006 Mike Acton <macton@gmail.com>
3 : //
4 : // Permission is hereby granted, free of charge, to any person obtaining a
5 : // copy of this software and associated documentation files (the "Software"),
6 : // to deal in the Software without restriction, including without limitation
7 : // the rights to use, copy, modify, merge, publish, distribute, sublicense,
8 : // and/or sell copies of the Software, and to permit persons to whom the
9 : // Software is furnished to do so, subject to the following conditions:
10 : //
11 : // The above copyright notice and this permission notice shall be included
12 : // in all copies or substantial portions of the Software.
13 : //
14 : // THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
15 : // IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
16 : // FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
17 : // AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
18 : // LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
19 : // OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
20 : // THE SOFTWARE
21 : //
22 : // Half-precision floating point format
23 : // ------------------------------------
24 : //
25 : // | Field | Last | First | Note
26 : // |----------|------|-------|----------
27 : // | Sign | 15 | 15 |
28 : // | Exponent | 14 | 10 | Bias = 15
29 : // | Mantissa | 9 | 0 |
30 : //
31 : // Compiling
32 : // ---------
33 : //
34 : // Preferred compile flags for GCC:
35 : // -O3 -fstrict-aliasing -std=c99 -pedantic -Wall -Wstrict-aliasing
36 : //
37 : // This file is a C99 source file, intended to be compiled with a C99
38 : // compliant compiler. However, for the moment it remains combatible
39 : // with C++98. Therefore if you are using a compiler that poorly implements
40 : // C standards (e.g. MSVC), it may be compiled as C++. This is not
41 : // guaranteed for future versions.
42 : //
43 :
44 : #include "half.h"
45 :
46 : // Load immediate
47 27648000 : static inline uint32_t _uint32_li( uint32_t a )
48 : {
49 27648000 : return (a);
50 : }
51 :
52 : // Decrement
53 2304000 : static inline uint32_t _uint32_dec( uint32_t a )
54 : {
55 2304000 : return (a - 1);
56 : }
57 :
58 : // Increment
59 0 : static inline uint32_t _uint32_inc( uint32_t a )
60 : {
61 0 : return (a + 1);
62 : }
63 :
64 : // Complement
65 0 : static inline uint32_t _uint32_not( uint32_t a )
66 : {
67 0 : return (~a);
68 : }
69 :
70 : // Negate
71 4608000 : static inline uint32_t _uint32_neg( uint32_t a )
72 : {
73 : #pragma warning (disable: 4146)
74 4608000 : return (-a);
75 : #pragma warning( default: 4146)
76 : }
77 :
78 : // Extend sign
79 11520000 : static inline uint32_t _uint32_ext( uint32_t a )
80 : {
81 11520000 : return (((int32_t)a)>>31);
82 : }
83 :
84 : // And
85 23040000 : static inline uint32_t _uint32_and( uint32_t a, uint32_t b )
86 : {
87 23040000 : return (a & b);
88 : }
89 :
90 : // Exclusive Or
91 0 : static inline uint32_t _uint32_xor( uint32_t a, uint32_t b )
92 : {
93 0 : return (a ^ b);
94 : }
95 :
96 : // And with Complement
97 16128000 : static inline uint32_t _uint32_andc( uint32_t a, uint32_t b )
98 : {
99 16128000 : return (a & ~b);
100 : }
101 :
102 : // Or
103 18432000 : static inline uint32_t _uint32_or( uint32_t a, uint32_t b )
104 : {
105 18432000 : return (a | b);
106 : }
107 :
108 : // Shift Right Logical
109 0 : static inline uint32_t _uint32_srl( uint32_t a, int sa )
110 : {
111 0 : return (a >> sa);
112 : }
113 :
114 : // Shift Left Logical
115 11520000 : static inline uint32_t _uint32_sll( uint32_t a, int sa )
116 : {
117 11520000 : return (a << sa);
118 : }
119 :
120 : // Add
121 2304000 : static inline uint32_t _uint32_add( uint32_t a, uint32_t b )
122 : {
123 2304000 : return (a + b);
124 : }
125 :
126 : // Subtract
127 6912000 : static inline uint32_t _uint32_sub( uint32_t a, uint32_t b )
128 : {
129 6912000 : return (a - b);
130 : }
131 :
132 : // Multiply
133 0 : static inline uint32_t _uint32_mul( uint32_t a, uint32_t b )
134 : {
135 0 : return (a * b);
136 : }
137 :
138 : // Select on Sign bit
139 9216000 : static inline uint32_t _uint32_sels( uint32_t test, uint32_t a, uint32_t b )
140 : {
141 9216000 : const uint32_t mask = _uint32_ext( test );
142 9216000 : const uint32_t sel_a = _uint32_and( a, mask );
143 9216000 : const uint32_t sel_b = _uint32_andc( b, mask );
144 9216000 : const uint32_t result = _uint32_or( sel_a, sel_b );
145 :
146 9216000 : return (result);
147 : }
148 :
149 : // Select Bits on mask
150 0 : static inline uint32_t _uint32_selb( uint32_t mask, uint32_t a, uint32_t b )
151 : {
152 0 : const uint32_t sel_a = _uint32_and( a, mask );
153 0 : const uint32_t sel_b = _uint32_andc( b, mask );
154 0 : const uint32_t result = _uint32_or( sel_a, sel_b );
155 :
156 0 : return (result);
157 : }
158 :
159 : // Load Immediate
160 0 : static inline uint16_t _uint16_li( uint16_t a )
161 : {
162 0 : return (a);
163 : }
164 :
165 : // Extend sign
166 0 : static inline uint16_t _uint16_ext( uint16_t a )
167 : {
168 0 : return (((int16_t)a)>>15);
169 : }
170 :
171 : // Negate
172 0 : static inline uint16_t _uint16_neg( uint16_t a )
173 : {
174 0 : return (-a);
175 : }
176 :
177 : #ifndef __GNUC__
178 : // Complement
179 : static inline uint16_t _uint16_not( uint16_t a )
180 : {
181 : return (~a);
182 : }
183 : // Add then Mask
184 : static inline uint16_t _uint16_addm( uint16_t a, uint16_t b, uint16_t mask )
185 : {
186 : return ((a + b) & mask);
187 : }
188 : #endif
189 :
190 : // Decrement
191 0 : static inline uint16_t _uint16_dec( uint16_t a )
192 : {
193 0 : return (a - 1);
194 : }
195 :
196 : // Shift Left Logical
197 0 : static inline uint16_t _uint16_sll( uint16_t a, int sa )
198 : {
199 0 : return (a << sa);
200 : }
201 :
202 : // Shift Right Logical
203 0 : static inline uint16_t _uint16_srl( uint16_t a, int sa )
204 : {
205 0 : return (a >> sa);
206 : }
207 :
208 : // Add
209 0 : static inline uint16_t _uint16_add( uint16_t a, uint16_t b )
210 : {
211 0 : return (a + b);
212 : }
213 :
214 : // Subtract
215 0 : static inline uint16_t _uint16_sub( uint16_t a, uint16_t b )
216 : {
217 0 : return (a - b);
218 : }
219 :
220 : // And
221 0 : static inline uint16_t _uint16_and( uint16_t a, uint16_t b )
222 : {
223 0 : return (a & b);
224 : }
225 :
226 : // Or
227 0 : static inline uint16_t _uint16_or( uint16_t a, uint16_t b )
228 : {
229 0 : return (a | b);
230 : }
231 :
232 : // Exclusive Or
233 0 : static inline uint16_t _uint16_xor( uint16_t a, uint16_t b )
234 : {
235 0 : return (a ^ b);
236 : }
237 :
238 : // And with Complement
239 0 : static inline uint16_t _uint16_andc( uint16_t a, uint16_t b )
240 : {
241 0 : return (a & ~b);
242 : }
243 :
244 : // And then Shift Right Logical
245 0 : static inline uint16_t _uint16_andsrl( uint16_t a, uint16_t b, int sa )
246 : {
247 0 : return ((a & b) >> sa);
248 : }
249 :
250 : // Shift Right Logical then Mask
251 0 : static inline uint16_t _uint16_srlm( uint16_t a, int sa, uint16_t mask )
252 : {
253 0 : return ((a >> sa) & mask);
254 : }
255 :
256 : // Select on Sign bit
257 0 : static inline uint16_t _uint16_sels( uint16_t test, uint16_t a, uint16_t b )
258 : {
259 0 : const uint16_t mask = _uint16_ext( test );
260 0 : const uint16_t sel_a = _uint16_and( a, mask );
261 0 : const uint16_t sel_b = _uint16_andc( b, mask );
262 0 : const uint16_t result = _uint16_or( sel_a, sel_b );
263 :
264 0 : return (result);
265 : }
266 :
267 : // Count Leading Zeros
268 2304000 : static inline uint32_t _uint32_cntlz( uint32_t x )
269 : {
270 : #ifdef __GNUC__
271 : /* NOTE: __builtin_clz is undefined for x == 0 */
272 : /* On PowerPC, this will map to insn: cntlzw */
273 : /* On Pentium, this will map to insn: clz */
274 2304000 : uint32_t is_x_nez_msb = _uint32_neg( x );
275 2304000 : uint32_t nlz = __builtin_clz( x );
276 2304000 : uint32_t result = _uint32_sels( is_x_nez_msb, nlz, 0x00000020 );
277 2304000 : return (result);
278 : #else
279 : const uint32_t x0 = _uint32_srl( x, 1 );
280 : const uint32_t x1 = _uint32_or( x, x0 );
281 : const uint32_t x2 = _uint32_srl( x1, 2 );
282 : const uint32_t x3 = _uint32_or( x1, x2 );
283 : const uint32_t x4 = _uint32_srl( x3, 4 );
284 : const uint32_t x5 = _uint32_or( x3, x4 );
285 : const uint32_t x6 = _uint32_srl( x5, 8 );
286 : const uint32_t x7 = _uint32_or( x5, x6 );
287 : const uint32_t x8 = _uint32_srl( x7, 16 );
288 : const uint32_t x9 = _uint32_or( x7, x8 );
289 : const uint32_t xA = _uint32_not( x9 );
290 : const uint32_t xB = _uint32_srl( xA, 1 );
291 : const uint32_t xC = _uint32_and( xB, 0x55555555 );
292 : const uint32_t xD = _uint32_sub( xA, xC );
293 : const uint32_t xE = _uint32_and( xD, 0x33333333 );
294 : const uint32_t xF = _uint32_srl( xD, 2 );
295 : const uint32_t x10 = _uint32_and( xF, 0x33333333 );
296 : const uint32_t x11 = _uint32_add( xE, x10 );
297 : const uint32_t x12 = _uint32_srl( x11, 4 );
298 : const uint32_t x13 = _uint32_add( x11, x12 );
299 : const uint32_t x14 = _uint32_and( x13, 0x0f0f0f0f );
300 : const uint32_t x15 = _uint32_srl( x14, 8 );
301 : const uint32_t x16 = _uint32_add( x14, x15 );
302 : const uint32_t x17 = _uint32_srl( x16, 16 );
303 : const uint32_t x18 = _uint32_add( x16, x17 );
304 : const uint32_t x19 = _uint32_and( x18, 0x0000003f );
305 : return ( x19 );
306 : #endif
307 : }
308 :
309 : // Count Leading Zeros
310 0 : static inline uint16_t _uint16_cntlz( uint16_t x )
311 : {
312 : #ifdef __GNUC__
313 0 : uint16_t nlz32 = (uint16_t)_uint32_cntlz( (uint32_t)x );
314 0 : uint32_t nlz = _uint32_sub( nlz32, 16 );
315 0 : return (nlz);
316 : #else
317 : const uint16_t x0 = _uint16_srl( x, 1 );
318 : const uint16_t x1 = _uint16_or( x, x0 );
319 : const uint16_t x2 = _uint16_srl( x1, 2 );
320 : const uint16_t x3 = _uint16_or( x1, x2 );
321 : const uint16_t x4 = _uint16_srl( x3, 4 );
322 : const uint16_t x5 = _uint16_or( x3, x4 );
323 : const uint16_t x6 = _uint16_srl( x5, 8 );
324 : const uint16_t x7 = _uint16_or( x5, x6 );
325 : const uint16_t x8 = _uint16_not( x7 );
326 : const uint16_t x9 = _uint16_srlm( x8, 1, 0x5555 );
327 : const uint16_t xA = _uint16_sub( x8, x9 );
328 : const uint16_t xB = _uint16_and( xA, 0x3333 );
329 : const uint16_t xC = _uint16_srlm( xA, 2, 0x3333 );
330 : const uint16_t xD = _uint16_add( xB, xC );
331 : const uint16_t xE = _uint16_srl( xD, 4 );
332 : const uint16_t xF = _uint16_addm( xD, xE, 0x0f0f );
333 : const uint16_t x10 = _uint16_srl( xF, 8 );
334 : const uint16_t x11 = _uint16_addm( xF, x10, 0x001f );
335 : return ( x11 );
336 : #endif
337 : }
338 :
339 : uint16_t
340 0 : half_from_float( float f_ )
341 : {
342 0 : const uint32_t f = *reinterpret_cast< uint32_t* >( &f_ );
343 0 : const uint32_t one = _uint32_li( 0x00000001 );
344 0 : const uint32_t f_s_mask = _uint32_li( 0x80000000 );
345 0 : const uint32_t f_e_mask = _uint32_li( 0x7f800000 );
346 0 : const uint32_t f_m_mask = _uint32_li( 0x007fffff );
347 0 : const uint32_t f_m_hidden_bit = _uint32_li( 0x00800000 );
348 0 : const uint32_t f_m_round_bit = _uint32_li( 0x00001000 );
349 0 : const uint32_t f_snan_mask = _uint32_li( 0x7fc00000 );
350 0 : const uint32_t f_e_pos = _uint32_li( 0x00000017 );
351 0 : const uint32_t h_e_pos = _uint32_li( 0x0000000a );
352 0 : const uint32_t h_e_mask = _uint32_li( 0x00007c00 );
353 0 : const uint32_t h_snan_mask = _uint32_li( 0x00007e00 );
354 0 : const uint32_t h_e_mask_value = _uint32_li( 0x0000001f );
355 0 : const uint32_t f_h_s_pos_offset = _uint32_li( 0x00000010 );
356 0 : const uint32_t f_h_bias_offset = _uint32_li( 0x00000070 );
357 0 : const uint32_t f_h_m_pos_offset = _uint32_li( 0x0000000d );
358 0 : const uint32_t h_nan_min = _uint32_li( 0x00007c01 );
359 0 : const uint32_t f_h_e_biased_flag = _uint32_li( 0x0000008f );
360 0 : const uint32_t f_s = _uint32_and( f, f_s_mask );
361 0 : const uint32_t f_e = _uint32_and( f, f_e_mask );
362 0 : const uint16_t h_s = _uint32_srl( f_s, f_h_s_pos_offset );
363 0 : const uint32_t f_m = _uint32_and( f, f_m_mask );
364 0 : const uint16_t f_e_amount = _uint32_srl( f_e, f_e_pos );
365 0 : const uint32_t f_e_half_bias = _uint32_sub( f_e_amount, f_h_bias_offset );
366 0 : const uint32_t f_snan = _uint32_and( f, f_snan_mask );
367 0 : const uint32_t f_m_round_mask = _uint32_and( f_m, f_m_round_bit );
368 0 : const uint32_t f_m_round_offset = _uint32_sll( f_m_round_mask, one );
369 0 : const uint32_t f_m_rounded = _uint32_add( f_m, f_m_round_offset );
370 0 : const uint32_t f_m_denorm_sa = _uint32_sub( one, f_e_half_bias );
371 0 : const uint32_t f_m_with_hidden = _uint32_or( f_m_rounded, f_m_hidden_bit );
372 0 : const uint32_t f_m_denorm = _uint32_srl( f_m_with_hidden, f_m_denorm_sa );
373 0 : const uint32_t h_m_denorm = _uint32_srl( f_m_denorm, f_h_m_pos_offset );
374 0 : const uint32_t f_m_rounded_overflow = _uint32_and( f_m_rounded, f_m_hidden_bit );
375 0 : const uint32_t m_nan = _uint32_srl( f_m, f_h_m_pos_offset );
376 0 : const uint32_t h_em_nan = _uint32_or( h_e_mask, m_nan );
377 0 : const uint32_t h_e_norm_overflow_offset = _uint32_inc( f_e_half_bias );
378 0 : const uint32_t h_e_norm_overflow = _uint32_sll( h_e_norm_overflow_offset, h_e_pos );
379 0 : const uint32_t h_e_norm = _uint32_sll( f_e_half_bias, h_e_pos );
380 0 : const uint32_t h_m_norm = _uint32_srl( f_m_rounded, f_h_m_pos_offset );
381 0 : const uint32_t h_em_norm = _uint32_or( h_e_norm, h_m_norm );
382 0 : const uint32_t is_h_ndenorm_msb = _uint32_sub( f_h_bias_offset, f_e_amount );
383 0 : const uint32_t is_f_e_flagged_msb = _uint32_sub( f_h_e_biased_flag, f_e_half_bias );
384 0 : const uint32_t is_h_denorm_msb = _uint32_not( is_h_ndenorm_msb );
385 0 : const uint32_t is_f_m_eqz_msb = _uint32_dec( f_m );
386 0 : const uint32_t is_h_nan_eqz_msb = _uint32_dec( m_nan );
387 0 : const uint32_t is_f_inf_msb = _uint32_and( is_f_e_flagged_msb, is_f_m_eqz_msb );
388 0 : const uint32_t is_f_nan_underflow_msb = _uint32_and( is_f_e_flagged_msb, is_h_nan_eqz_msb );
389 0 : const uint32_t is_e_overflow_msb = _uint32_sub( h_e_mask_value, f_e_half_bias );
390 0 : const uint32_t is_h_inf_msb = _uint32_or( is_e_overflow_msb, is_f_inf_msb );
391 0 : const uint32_t is_f_nsnan_msb = _uint32_sub( f_snan, f_snan_mask );
392 0 : const uint32_t is_m_norm_overflow_msb = _uint32_neg( f_m_rounded_overflow );
393 0 : const uint32_t is_f_snan_msb = _uint32_not( is_f_nsnan_msb );
394 0 : const uint32_t h_em_overflow_result = _uint32_sels( is_m_norm_overflow_msb, h_e_norm_overflow, h_em_norm );
395 0 : const uint32_t h_em_nan_result = _uint32_sels( is_f_e_flagged_msb, h_em_nan, h_em_overflow_result );
396 0 : const uint32_t h_em_nan_underflow_result = _uint32_sels( is_f_nan_underflow_msb, h_nan_min, h_em_nan_result );
397 0 : const uint32_t h_em_inf_result = _uint32_sels( is_h_inf_msb, h_e_mask, h_em_nan_underflow_result );
398 0 : const uint32_t h_em_denorm_result = _uint32_sels( is_h_denorm_msb, h_m_denorm, h_em_inf_result );
399 0 : const uint32_t h_em_snan_result = _uint32_sels( is_f_snan_msb, h_snan_mask, h_em_denorm_result );
400 0 : const uint32_t h_result = _uint32_or( h_s, h_em_snan_result );
401 :
402 0 : return (uint16_t)(h_result);
403 : }
404 :
405 2304000 : float half_to_float( uint16_t h )
406 : {
407 2304000 : const uint32_t h_e_mask = _uint32_li( 0x00007c00 );
408 2304000 : const uint32_t h_m_mask = _uint32_li( 0x000003ff );
409 2304000 : const uint32_t h_s_mask = _uint32_li( 0x00008000 );
410 2304000 : const uint32_t h_f_s_pos_offset = _uint32_li( 0x00000010 );
411 2304000 : const uint32_t h_f_e_pos_offset = _uint32_li( 0x0000000d );
412 2304000 : const uint32_t h_f_bias_offset = _uint32_li( 0x0001c000 );
413 2304000 : const uint32_t f_e_mask = _uint32_li( 0x7f800000 );
414 2304000 : const uint32_t f_m_mask = _uint32_li( 0x007fffff );
415 2304000 : const uint32_t h_f_e_denorm_bias = _uint32_li( 0x0000007e );
416 2304000 : const uint32_t h_f_m_denorm_sa_bias = _uint32_li( 0x00000008 );
417 2304000 : const uint32_t f_e_pos = _uint32_li( 0x00000017 );
418 2304000 : const uint32_t h_e_mask_minus_one = _uint32_li( 0x00007bff );
419 2304000 : const uint32_t h_e = _uint32_and( h, h_e_mask );
420 2304000 : const uint32_t h_m = _uint32_and( h, h_m_mask );
421 2304000 : const uint32_t h_s = _uint32_and( h, h_s_mask );
422 2304000 : const uint32_t h_e_f_bias = _uint32_add( h_e, h_f_bias_offset );
423 2304000 : const uint32_t h_m_nlz = _uint32_cntlz( h_m );
424 2304000 : const uint32_t f_s = _uint32_sll( h_s, h_f_s_pos_offset );
425 2304000 : const uint32_t f_e = _uint32_sll( h_e_f_bias, h_f_e_pos_offset );
426 2304000 : const uint32_t f_m = _uint32_sll( h_m, h_f_e_pos_offset );
427 2304000 : const uint32_t f_em = _uint32_or( f_e, f_m );
428 2304000 : const uint32_t h_f_m_sa = _uint32_sub( h_m_nlz, h_f_m_denorm_sa_bias );
429 2304000 : const uint32_t f_e_denorm_unpacked = _uint32_sub( h_f_e_denorm_bias, h_f_m_sa );
430 2304000 : const uint32_t h_f_m = _uint32_sll( h_m, h_f_m_sa );
431 2304000 : const uint32_t f_m_denorm = _uint32_and( h_f_m, f_m_mask );
432 2304000 : const uint32_t f_e_denorm = _uint32_sll( f_e_denorm_unpacked, f_e_pos );
433 2304000 : const uint32_t f_em_denorm = _uint32_or( f_e_denorm, f_m_denorm );
434 2304000 : const uint32_t f_em_nan = _uint32_or( f_e_mask, f_m );
435 2304000 : const uint32_t is_e_eqz_msb = _uint32_dec( h_e );
436 2304000 : const uint32_t is_m_nez_msb = _uint32_neg( h_m );
437 2304000 : const uint32_t is_e_flagged_msb = _uint32_sub( h_e_mask_minus_one, h_e );
438 2304000 : const uint32_t is_zero_msb = _uint32_andc( is_e_eqz_msb, is_m_nez_msb );
439 2304000 : const uint32_t is_inf_msb = _uint32_andc( is_e_flagged_msb, is_m_nez_msb );
440 2304000 : const uint32_t is_denorm_msb = _uint32_and( is_m_nez_msb, is_e_eqz_msb );
441 2304000 : const uint32_t is_nan_msb = _uint32_and( is_e_flagged_msb, is_m_nez_msb );
442 2304000 : const uint32_t is_zero = _uint32_ext( is_zero_msb );
443 2304000 : const uint32_t f_zero_result = _uint32_andc( f_em, is_zero );
444 2304000 : const uint32_t f_denorm_result = _uint32_sels( is_denorm_msb, f_em_denorm, f_zero_result );
445 2304000 : const uint32_t f_inf_result = _uint32_sels( is_inf_msb, f_e_mask, f_denorm_result );
446 2304000 : const uint32_t f_nan_result = _uint32_sels( is_nan_msb, f_em_nan, f_inf_result );
447 2304000 : const uint32_t f_result = _uint32_or( f_s, f_nan_result );
448 :
449 2304000 : return *reinterpret_cast< const float* >( &f_result );
450 : }
451 :
452 : // half_add
453 : // --------
454 : //
455 : // (SUM) uint16_t z = half_add( x, y );
456 : // (DIFFERENCE) uint16_t z = half_add( x, -y );
457 : //
458 : // * Difference of ZEROs is always +ZERO
459 : // * Sum round with guard + round + sticky bit (grs)
460 : // * QNaN + <x> = QNaN
461 : // * <x> + +INF = +INF
462 : // * <x> - -INF = -INF
463 : // * INF - INF = SNaN
464 : //
465 : // Will have exactly (0 ulps difference) the same result as:
466 : // (Round up)
467 : //
468 : // union FLOAT_32
469 : // {
470 : // float f32;
471 : // uint32_t u32;
472 : // };
473 : //
474 : // union FLOAT_32 fx = { .u32 = half_to_float( x ) };
475 : // union FLOAT_32 fy = { .u32 = half_to_float( y ) };
476 : // union FLOAT_32 fz = { .f32 = fx.f32 + fy.f32 };
477 : // uint16_t z = float_to_half( fz );
478 : //
479 : uint16_t
480 0 : half_add( uint16_t x, uint16_t y )
481 : {
482 0 : const uint16_t one = _uint16_li( 0x0001 );
483 0 : const uint16_t msb_to_lsb_sa = _uint16_li( 0x000f );
484 0 : const uint16_t h_s_mask = _uint16_li( 0x8000 );
485 0 : const uint16_t h_e_mask = _uint16_li( 0x7c00 );
486 0 : const uint16_t h_m_mask = _uint16_li( 0x03ff );
487 0 : const uint16_t h_m_msb_mask = _uint16_li( 0x2000 );
488 0 : const uint16_t h_m_msb_sa = _uint16_li( 0x000d );
489 0 : const uint16_t h_m_hidden = _uint16_li( 0x0400 );
490 0 : const uint16_t h_e_pos = _uint16_li( 0x000a );
491 0 : const uint16_t h_e_bias_minus_one = _uint16_li( 0x000e );
492 0 : const uint16_t h_m_grs_carry = _uint16_li( 0x4000 );
493 0 : const uint16_t h_m_grs_carry_pos = _uint16_li( 0x000e );
494 0 : const uint16_t h_grs_size = _uint16_li( 0x0003 );
495 0 : const uint16_t h_snan = _uint16_li( 0xfe00 );
496 0 : const uint16_t h_e_mask_minus_one = _uint16_li( 0x7bff );
497 0 : const uint16_t h_grs_round_carry = _uint16_sll( one, h_grs_size );
498 0 : const uint16_t h_grs_round_mask = _uint16_sub( h_grs_round_carry, one );
499 0 : const uint16_t x_e = _uint16_and( x, h_e_mask );
500 0 : const uint16_t y_e = _uint16_and( y, h_e_mask );
501 0 : const uint16_t is_y_e_larger_msb = _uint16_sub( x_e, y_e );
502 0 : const uint16_t a = _uint16_sels( is_y_e_larger_msb, y, x);
503 0 : const uint16_t a_s = _uint16_and( a, h_s_mask );
504 0 : const uint16_t a_e = _uint16_and( a, h_e_mask );
505 0 : const uint16_t a_m_no_hidden_bit = _uint16_and( a, h_m_mask );
506 0 : const uint16_t a_em_no_hidden_bit = _uint16_or( a_e, a_m_no_hidden_bit );
507 0 : const uint16_t b = _uint16_sels( is_y_e_larger_msb, x, y);
508 0 : const uint16_t b_s = _uint16_and( b, h_s_mask );
509 0 : const uint16_t b_e = _uint16_and( b, h_e_mask );
510 0 : const uint16_t b_m_no_hidden_bit = _uint16_and( b, h_m_mask );
511 0 : const uint16_t b_em_no_hidden_bit = _uint16_or( b_e, b_m_no_hidden_bit );
512 0 : const uint16_t is_diff_sign_msb = _uint16_xor( a_s, b_s );
513 0 : const uint16_t is_a_inf_msb = _uint16_sub( h_e_mask_minus_one, a_em_no_hidden_bit );
514 0 : const uint16_t is_b_inf_msb = _uint16_sub( h_e_mask_minus_one, b_em_no_hidden_bit );
515 0 : const uint16_t is_undenorm_msb = _uint16_dec( a_e );
516 0 : const uint16_t is_undenorm = _uint16_ext( is_undenorm_msb );
517 0 : const uint16_t is_both_inf_msb = _uint16_and( is_a_inf_msb, is_b_inf_msb );
518 0 : const uint16_t is_invalid_inf_op_msb = _uint16_and( is_both_inf_msb, b_s );
519 0 : const uint16_t is_a_e_nez_msb = _uint16_neg( a_e );
520 0 : const uint16_t is_b_e_nez_msb = _uint16_neg( b_e );
521 0 : const uint16_t is_a_e_nez = _uint16_ext( is_a_e_nez_msb );
522 0 : const uint16_t is_b_e_nez = _uint16_ext( is_b_e_nez_msb );
523 0 : const uint16_t a_m_hidden_bit = _uint16_and( is_a_e_nez, h_m_hidden );
524 0 : const uint16_t b_m_hidden_bit = _uint16_and( is_b_e_nez, h_m_hidden );
525 0 : const uint16_t a_m_no_grs = _uint16_or( a_m_no_hidden_bit, a_m_hidden_bit );
526 0 : const uint16_t b_m_no_grs = _uint16_or( b_m_no_hidden_bit, b_m_hidden_bit );
527 0 : const uint16_t diff_e = _uint16_sub( a_e, b_e );
528 0 : const uint16_t a_e_unbias = _uint16_sub( a_e, h_e_bias_minus_one );
529 0 : const uint16_t a_m = _uint16_sll( a_m_no_grs, h_grs_size );
530 0 : const uint16_t a_e_biased = _uint16_srl( a_e, h_e_pos );
531 0 : const uint16_t m_sa_unbias = _uint16_srl( a_e_unbias, h_e_pos );
532 0 : const uint16_t m_sa_default = _uint16_srl( diff_e, h_e_pos );
533 0 : const uint16_t m_sa_unbias_mask = _uint16_andc( is_a_e_nez_msb, is_b_e_nez_msb );
534 0 : const uint16_t m_sa = _uint16_sels( m_sa_unbias_mask, m_sa_unbias, m_sa_default );
535 0 : const uint16_t b_m_no_sticky = _uint16_sll( b_m_no_grs, h_grs_size );
536 0 : const uint16_t sh_m = _uint16_srl( b_m_no_sticky, m_sa );
537 0 : const uint16_t sticky_overflow = _uint16_sll( one, m_sa );
538 0 : const uint16_t sticky_mask = _uint16_dec( sticky_overflow );
539 0 : const uint16_t sticky_collect = _uint16_and( b_m_no_sticky, sticky_mask );
540 0 : const uint16_t is_sticky_set_msb = _uint16_neg( sticky_collect );
541 0 : const uint16_t sticky = _uint16_srl( is_sticky_set_msb, msb_to_lsb_sa);
542 0 : const uint16_t b_m = _uint16_or( sh_m, sticky );
543 0 : const uint16_t is_c_m_ab_pos_msb = _uint16_sub( b_m, a_m );
544 0 : const uint16_t c_inf = _uint16_or( a_s, h_e_mask );
545 0 : const uint16_t c_m_sum = _uint16_add( a_m, b_m );
546 0 : const uint16_t c_m_diff_ab = _uint16_sub( a_m, b_m );
547 0 : const uint16_t c_m_diff_ba = _uint16_sub( b_m, a_m );
548 0 : const uint16_t c_m_smag_diff = _uint16_sels( is_c_m_ab_pos_msb, c_m_diff_ab, c_m_diff_ba );
549 0 : const uint16_t c_s_diff = _uint16_sels( is_c_m_ab_pos_msb, a_s, b_s );
550 0 : const uint16_t c_s = _uint16_sels( is_diff_sign_msb, c_s_diff, a_s );
551 0 : const uint16_t c_m_smag_diff_nlz = _uint16_cntlz( c_m_smag_diff );
552 0 : const uint16_t diff_norm_sa = _uint16_sub( c_m_smag_diff_nlz, one );
553 0 : const uint16_t is_diff_denorm_msb = _uint16_sub( a_e_biased, diff_norm_sa );
554 0 : const uint16_t is_diff_denorm = _uint16_ext( is_diff_denorm_msb );
555 0 : const uint16_t is_a_or_b_norm_msb = _uint16_neg( a_e_biased );
556 0 : const uint16_t diff_denorm_sa = _uint16_dec( a_e_biased );
557 0 : const uint16_t c_m_diff_denorm = _uint16_sll( c_m_smag_diff, diff_denorm_sa );
558 0 : const uint16_t c_m_diff_norm = _uint16_sll( c_m_smag_diff, diff_norm_sa );
559 0 : const uint16_t c_e_diff_norm = _uint16_sub( a_e_biased, diff_norm_sa );
560 0 : const uint16_t c_m_diff_ab_norm = _uint16_sels( is_diff_denorm_msb, c_m_diff_denorm, c_m_diff_norm );
561 0 : const uint16_t c_e_diff_ab_norm = _uint16_andc( c_e_diff_norm, is_diff_denorm );
562 0 : const uint16_t c_m_diff = _uint16_sels( is_a_or_b_norm_msb, c_m_diff_ab_norm, c_m_smag_diff );
563 0 : const uint16_t c_e_diff = _uint16_sels( is_a_or_b_norm_msb, c_e_diff_ab_norm, a_e_biased );
564 0 : const uint16_t is_diff_eqz_msb = _uint16_dec( c_m_diff );
565 0 : const uint16_t is_diff_exactly_zero_msb = _uint16_and( is_diff_sign_msb, is_diff_eqz_msb );
566 0 : const uint16_t is_diff_exactly_zero = _uint16_ext( is_diff_exactly_zero_msb );
567 0 : const uint16_t c_m_added = _uint16_sels( is_diff_sign_msb, c_m_diff, c_m_sum );
568 0 : const uint16_t c_e_added = _uint16_sels( is_diff_sign_msb, c_e_diff, a_e_biased );
569 0 : const uint16_t c_m_carry = _uint16_and( c_m_added, h_m_grs_carry );
570 0 : const uint16_t is_c_m_carry_msb = _uint16_neg( c_m_carry );
571 0 : const uint16_t c_e_hidden_offset = _uint16_andsrl( c_m_added, h_m_grs_carry, h_m_grs_carry_pos );
572 0 : const uint16_t c_m_sub_hidden = _uint16_srl( c_m_added, one );
573 0 : const uint16_t c_m_no_hidden = _uint16_sels( is_c_m_carry_msb, c_m_sub_hidden, c_m_added );
574 0 : const uint16_t c_e_no_hidden = _uint16_add( c_e_added, c_e_hidden_offset );
575 0 : const uint16_t c_m_no_hidden_msb = _uint16_and( c_m_no_hidden, h_m_msb_mask );
576 0 : const uint16_t undenorm_m_msb_odd = _uint16_srl( c_m_no_hidden_msb, h_m_msb_sa );
577 0 : const uint16_t undenorm_fix_e = _uint16_and( is_undenorm, undenorm_m_msb_odd );
578 0 : const uint16_t c_e_fixed = _uint16_add( c_e_no_hidden, undenorm_fix_e );
579 0 : const uint16_t c_m_round_amount = _uint16_and( c_m_no_hidden, h_grs_round_mask );
580 0 : const uint16_t c_m_rounded = _uint16_add( c_m_no_hidden, c_m_round_amount );
581 0 : const uint16_t c_m_round_overflow = _uint16_andsrl( c_m_rounded, h_m_grs_carry, h_m_grs_carry_pos );
582 0 : const uint16_t c_e_rounded = _uint16_add( c_e_fixed, c_m_round_overflow );
583 0 : const uint16_t c_m_no_grs = _uint16_srlm( c_m_rounded, h_grs_size, h_m_mask );
584 0 : const uint16_t c_e = _uint16_sll( c_e_rounded, h_e_pos );
585 0 : const uint16_t c_em = _uint16_or( c_e, c_m_no_grs );
586 0 : const uint16_t c_normal = _uint16_or( c_s, c_em );
587 0 : const uint16_t c_inf_result = _uint16_sels( is_a_inf_msb, c_inf, c_normal );
588 0 : const uint16_t c_zero_result = _uint16_andc( c_inf_result, is_diff_exactly_zero );
589 0 : const uint16_t c_result = _uint16_sels( is_invalid_inf_op_msb, h_snan, c_zero_result );
590 :
591 0 : return (c_result);
592 : }
593 :
594 : // half_mul
595 : // --------
596 : //
597 : // May have 0 or 1 ulp difference from the following result:
598 : // (Round to nearest)
599 : // NOTE: Rounding mode differs between conversion and multiply
600 : //
601 : // union FLOAT_32
602 : // {
603 : // float f32;
604 : // uint32_t u32;
605 : // };
606 : //
607 : // union FLOAT_32 fx = { .u32 = half_to_float( x ) };
608 : // union FLOAT_32 fy = { .u32 = half_to_float( y ) };
609 : // union FLOAT_32 fz = { .f32 = fx.f32 * fy.f32 };
610 : // uint16_t z = float_to_half( fz );
611 : //
612 : uint16_t
613 0 : half_mul( uint16_t x, uint16_t y )
614 : {
615 0 : const uint32_t one = _uint32_li( 0x00000001 );
616 0 : const uint32_t h_s_mask = _uint32_li( 0x00008000 );
617 0 : const uint32_t h_e_mask = _uint32_li( 0x00007c00 );
618 0 : const uint32_t h_m_mask = _uint32_li( 0x000003ff );
619 0 : const uint32_t h_m_hidden = _uint32_li( 0x00000400 );
620 0 : const uint32_t h_e_pos = _uint32_li( 0x0000000a );
621 0 : const uint32_t h_e_bias = _uint32_li( 0x0000000f );
622 0 : const uint32_t h_m_bit_count = _uint32_li( 0x0000000a );
623 0 : const uint32_t h_m_bit_half_count = _uint32_li( 0x00000005 );
624 0 : const uint32_t h_nan_min = _uint32_li( 0x00007c01 );
625 0 : const uint32_t h_e_mask_minus_one = _uint32_li( 0x00007bff );
626 0 : const uint32_t h_snan = _uint32_li( 0x0000fe00 );
627 0 : const uint32_t m_round_overflow_bit = _uint32_li( 0x00000020 );
628 0 : const uint32_t m_hidden_bit = _uint32_li( 0x00100000 );
629 0 : const uint32_t a_s = _uint32_and( x, h_s_mask );
630 0 : const uint32_t b_s = _uint32_and( y, h_s_mask );
631 0 : const uint32_t c_s = _uint32_xor( a_s, b_s );
632 0 : const uint32_t x_e = _uint32_and( x, h_e_mask );
633 0 : const uint32_t x_e_eqz_msb = _uint32_dec( x_e );
634 0 : const uint32_t a = _uint32_sels( x_e_eqz_msb, y, x );
635 0 : const uint32_t b = _uint32_sels( x_e_eqz_msb, x, y );
636 0 : const uint32_t a_e = _uint32_and( a, h_e_mask );
637 0 : const uint32_t b_e = _uint32_and( b, h_e_mask );
638 0 : const uint32_t a_m = _uint32_and( a, h_m_mask );
639 0 : const uint32_t b_m = _uint32_and( b, h_m_mask );
640 0 : const uint32_t a_e_amount = _uint32_srl( a_e, h_e_pos );
641 0 : const uint32_t b_e_amount = _uint32_srl( b_e, h_e_pos );
642 0 : const uint32_t a_m_with_hidden = _uint32_or( a_m, h_m_hidden );
643 0 : const uint32_t b_m_with_hidden = _uint32_or( b_m, h_m_hidden );
644 0 : const uint32_t c_m_normal = _uint32_mul( a_m_with_hidden, b_m_with_hidden );
645 0 : const uint32_t c_m_denorm_biased = _uint32_mul( a_m_with_hidden, b_m );
646 0 : const uint32_t c_e_denorm_unbias_e = _uint32_sub( h_e_bias, a_e_amount );
647 0 : const uint32_t c_m_denorm_round_amount = _uint32_and( c_m_denorm_biased, h_m_mask );
648 0 : const uint32_t c_m_denorm_rounded = _uint32_add( c_m_denorm_biased, c_m_denorm_round_amount );
649 0 : const uint32_t c_m_denorm_inplace = _uint32_srl( c_m_denorm_rounded, h_m_bit_count );
650 0 : const uint32_t c_m_denorm_unbiased = _uint32_srl( c_m_denorm_inplace, c_e_denorm_unbias_e );
651 0 : const uint32_t c_m_denorm = _uint32_and( c_m_denorm_unbiased, h_m_mask );
652 0 : const uint32_t c_e_amount_biased = _uint32_add( a_e_amount, b_e_amount );
653 0 : const uint32_t c_e_amount_unbiased = _uint32_sub( c_e_amount_biased, h_e_bias );
654 0 : const uint32_t is_c_e_unbiased_underflow = _uint32_ext( c_e_amount_unbiased );
655 0 : const uint32_t c_e_underflow_half_sa = _uint32_neg( c_e_amount_unbiased );
656 0 : const uint32_t c_e_underflow_sa = _uint32_sll( c_e_underflow_half_sa, one );
657 0 : const uint32_t c_m_underflow = _uint32_srl( c_m_normal, c_e_underflow_sa );
658 0 : const uint32_t c_e_underflow_added = _uint32_andc( c_e_amount_unbiased, is_c_e_unbiased_underflow );
659 0 : const uint32_t c_m_underflow_added = _uint32_selb( is_c_e_unbiased_underflow, c_m_underflow, c_m_normal );
660 0 : const uint32_t is_mul_overflow_test = _uint32_and( c_e_underflow_added, m_round_overflow_bit );
661 0 : const uint32_t is_mul_overflow_msb = _uint32_neg( is_mul_overflow_test );
662 0 : const uint32_t c_e_norm_radix_corrected = _uint32_inc( c_e_underflow_added );
663 0 : const uint32_t c_m_norm_radix_corrected = _uint32_srl( c_m_underflow_added, one );
664 0 : const uint32_t c_m_norm_hidden_bit = _uint32_and( c_m_norm_radix_corrected, m_hidden_bit );
665 0 : const uint32_t is_c_m_norm_no_hidden_msb = _uint32_dec( c_m_norm_hidden_bit );
666 0 : const uint32_t c_m_norm_lo = _uint32_srl( c_m_norm_radix_corrected, h_m_bit_half_count );
667 0 : const uint32_t c_m_norm_lo_nlz = _uint16_cntlz( c_m_norm_lo );
668 0 : const uint32_t is_c_m_hidden_nunderflow_msb = _uint32_sub( c_m_norm_lo_nlz, c_e_norm_radix_corrected );
669 0 : const uint32_t is_c_m_hidden_underflow_msb = _uint32_not( is_c_m_hidden_nunderflow_msb );
670 0 : const uint32_t is_c_m_hidden_underflow = _uint32_ext( is_c_m_hidden_underflow_msb );
671 0 : const uint32_t c_m_hidden_underflow_normalized_sa = _uint32_srl( c_m_norm_lo_nlz, one );
672 0 : const uint32_t c_m_hidden_underflow_normalized = _uint32_sll( c_m_norm_radix_corrected, c_m_hidden_underflow_normalized_sa );
673 0 : const uint32_t c_m_hidden_normalized = _uint32_sll( c_m_norm_radix_corrected, c_m_norm_lo_nlz );
674 0 : const uint32_t c_e_hidden_normalized = _uint32_sub( c_e_norm_radix_corrected, c_m_norm_lo_nlz );
675 0 : const uint32_t c_e_hidden = _uint32_andc( c_e_hidden_normalized, is_c_m_hidden_underflow );
676 0 : const uint32_t c_m_hidden = _uint32_sels( is_c_m_hidden_underflow_msb, c_m_hidden_underflow_normalized, c_m_hidden_normalized );
677 0 : const uint32_t c_m_normalized = _uint32_sels( is_c_m_norm_no_hidden_msb, c_m_hidden, c_m_norm_radix_corrected );
678 0 : const uint32_t c_e_normalized = _uint32_sels( is_c_m_norm_no_hidden_msb, c_e_hidden, c_e_norm_radix_corrected );
679 0 : const uint32_t c_m_norm_round_amount = _uint32_and( c_m_normalized, h_m_mask );
680 0 : const uint32_t c_m_norm_rounded = _uint32_add( c_m_normalized, c_m_norm_round_amount );
681 0 : const uint32_t is_round_overflow_test = _uint32_and( c_e_normalized, m_round_overflow_bit );
682 0 : const uint32_t is_round_overflow_msb = _uint32_neg( is_round_overflow_test );
683 0 : const uint32_t c_m_norm_inplace = _uint32_srl( c_m_norm_rounded, h_m_bit_count );
684 0 : const uint32_t c_m = _uint32_and( c_m_norm_inplace, h_m_mask );
685 0 : const uint32_t c_e_norm_inplace = _uint32_sll( c_e_normalized, h_e_pos );
686 0 : const uint32_t c_e = _uint32_and( c_e_norm_inplace, h_e_mask );
687 0 : const uint32_t c_em_nan = _uint32_or( h_e_mask, a_m );
688 0 : const uint32_t c_nan = _uint32_or( a_s, c_em_nan );
689 0 : const uint32_t c_denorm = _uint32_or( c_s, c_m_denorm );
690 0 : const uint32_t c_inf = _uint32_or( c_s, h_e_mask );
691 0 : const uint32_t c_em_norm = _uint32_or( c_e, c_m );
692 0 : const uint32_t is_a_e_flagged_msb = _uint32_sub( h_e_mask_minus_one, a_e );
693 0 : const uint32_t is_b_e_flagged_msb = _uint32_sub( h_e_mask_minus_one, b_e );
694 0 : const uint32_t is_a_e_eqz_msb = _uint32_dec( a_e );
695 0 : const uint32_t is_a_m_eqz_msb = _uint32_dec( a_m );
696 0 : const uint32_t is_b_e_eqz_msb = _uint32_dec( b_e );
697 0 : const uint32_t is_b_m_eqz_msb = _uint32_dec( b_m );
698 0 : const uint32_t is_b_eqz_msb = _uint32_and( is_b_e_eqz_msb, is_b_m_eqz_msb );
699 0 : const uint32_t is_a_eqz_msb = _uint32_and( is_a_e_eqz_msb, is_a_m_eqz_msb );
700 0 : const uint32_t is_c_nan_via_a_msb = _uint32_andc( is_a_e_flagged_msb, is_b_e_flagged_msb );
701 0 : const uint32_t is_c_nan_via_b_msb = _uint32_andc( is_b_e_flagged_msb, is_b_m_eqz_msb );
702 0 : const uint32_t is_c_nan_msb = _uint32_or( is_c_nan_via_a_msb, is_c_nan_via_b_msb );
703 0 : const uint32_t is_c_denorm_msb = _uint32_andc( is_b_e_eqz_msb, is_a_e_flagged_msb );
704 0 : const uint32_t is_a_inf_msb = _uint32_and( is_a_e_flagged_msb, is_a_m_eqz_msb );
705 0 : const uint32_t is_c_snan_msb = _uint32_and( is_a_inf_msb, is_b_eqz_msb );
706 0 : const uint32_t is_c_nan_min_via_a_msb = _uint32_and( is_a_e_flagged_msb, is_b_eqz_msb );
707 0 : const uint32_t is_c_nan_min_via_b_msb = _uint32_and( is_b_e_flagged_msb, is_a_eqz_msb );
708 0 : const uint32_t is_c_nan_min_msb = _uint32_or( is_c_nan_min_via_a_msb, is_c_nan_min_via_b_msb );
709 0 : const uint32_t is_c_inf_msb = _uint32_or( is_a_e_flagged_msb, is_b_e_flagged_msb );
710 0 : const uint32_t is_overflow_msb = _uint32_or( is_round_overflow_msb, is_mul_overflow_msb );
711 0 : const uint32_t c_em_overflow_result = _uint32_sels( is_overflow_msb, h_e_mask, c_em_norm );
712 0 : const uint32_t c_common_result = _uint32_or( c_s, c_em_overflow_result );
713 0 : const uint32_t c_zero_result = _uint32_sels( is_b_eqz_msb, c_s, c_common_result );
714 0 : const uint32_t c_nan_result = _uint32_sels( is_c_nan_msb, c_nan, c_zero_result );
715 0 : const uint32_t c_nan_min_result = _uint32_sels( is_c_nan_min_msb, h_nan_min, c_nan_result );
716 0 : const uint32_t c_inf_result = _uint32_sels( is_c_inf_msb, c_inf, c_nan_min_result );
717 0 : const uint32_t c_denorm_result = _uint32_sels( is_c_denorm_msb, c_denorm, c_inf_result);
718 0 : const uint32_t c_result = _uint32_sels( is_c_snan_msb, h_snan, c_denorm_result );
719 :
720 0 : return (uint16_t)(c_result);
721 : }
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