1 // This file is part of Eigen, a lightweight C++ template library
2 // for linear algebra.
3 //
4 // Copyright (C) 2007 Julien Pommier
5 // Copyright (C) 2009 Gael Guennebaud <gael.guennebaud@inria.fr>
6 //
7 // This Source Code Form is subject to the terms of the Mozilla
8 // Public License v. 2.0. If a copy of the MPL was not distributed
9 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
10
11 /* The sin, cos, exp, and log functions of this file come from
12 * Julien Pommier's sse math library: http://gruntthepeon.free.fr/ssemath/
13 */
14
15 #ifndef EIGEN_MATH_FUNCTIONS_SSE_H
16 #define EIGEN_MATH_FUNCTIONS_SSE_H
17
18 namespace Eigen {
19
20 namespace internal {
21
22 template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED
23 Packet4f plog<Packet4f>(const Packet4f& _x)
24 {
25 Packet4f x = _x;
26 _EIGEN_DECLARE_CONST_Packet4f(1 , 1.0f);
27 _EIGEN_DECLARE_CONST_Packet4f(half, 0.5f);
28 _EIGEN_DECLARE_CONST_Packet4i(0x7f, 0x7f);
29
30 _EIGEN_DECLARE_CONST_Packet4f_FROM_INT(inv_mant_mask, ~0x7f800000);
31
32 /* the smallest non denormalized float number */
33 _EIGEN_DECLARE_CONST_Packet4f_FROM_INT(min_norm_pos, 0x00800000);
34 _EIGEN_DECLARE_CONST_Packet4f_FROM_INT(minus_inf, 0xff800000);//-1.f/0.f);
35
36 /* natural logarithm computed for 4 simultaneous float
37 return NaN for x <= 0
38 */
39 _EIGEN_DECLARE_CONST_Packet4f(cephes_SQRTHF, 0.707106781186547524f);
40 _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p0, 7.0376836292E-2f);
41 _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p1, - 1.1514610310E-1f);
42 _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p2, 1.1676998740E-1f);
43 _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p3, - 1.2420140846E-1f);
44 _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p4, + 1.4249322787E-1f);
45 _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p5, - 1.6668057665E-1f);
46 _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p6, + 2.0000714765E-1f);
47 _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p7, - 2.4999993993E-1f);
48 _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p8, + 3.3333331174E-1f);
49 _EIGEN_DECLARE_CONST_Packet4f(cephes_log_q1, -2.12194440e-4f);
50 _EIGEN_DECLARE_CONST_Packet4f(cephes_log_q2, 0.693359375f);
51
52
53 Packet4i emm0;
54
55 Packet4f invalid_mask = _mm_cmpnge_ps(x, _mm_setzero_ps()); // not greater equal is true if x is NaN
56 Packet4f iszero_mask = _mm_cmpeq_ps(x, _mm_setzero_ps());
57
58 x = pmax(x, p4f_min_norm_pos); /* cut off denormalized stuff */
59 emm0 = _mm_srli_epi32(_mm_castps_si128(x), 23);
60
61 /* keep only the fractional part */
62 x = _mm_and_ps(x, p4f_inv_mant_mask);
63 x = _mm_or_ps(x, p4f_half);
64
65 emm0 = _mm_sub_epi32(emm0, p4i_0x7f);
66 Packet4f e = padd(Packet4f(_mm_cvtepi32_ps(emm0)), p4f_1);
67
68 /* part2:
69 if( x < SQRTHF ) {
70 e -= 1;
71 x = x + x - 1.0;
72 } else { x = x - 1.0; }
73 */
74 Packet4f mask = _mm_cmplt_ps(x, p4f_cephes_SQRTHF);
75 Packet4f tmp = pand(x, mask);
76 x = psub(x, p4f_1);
77 e = psub(e, pand(p4f_1, mask));
78 x = padd(x, tmp);
79
80 Packet4f x2 = pmul(x,x);
81 Packet4f x3 = pmul(x2,x);
82
83 Packet4f y, y1, y2;
84 y = pmadd(p4f_cephes_log_p0, x, p4f_cephes_log_p1);
85 y1 = pmadd(p4f_cephes_log_p3, x, p4f_cephes_log_p4);
86 y2 = pmadd(p4f_cephes_log_p6, x, p4f_cephes_log_p7);
87 y = pmadd(y , x, p4f_cephes_log_p2);
88 y1 = pmadd(y1, x, p4f_cephes_log_p5);
89 y2 = pmadd(y2, x, p4f_cephes_log_p8);
90 y = pmadd(y, x3, y1);
91 y = pmadd(y, x3, y2);
92 y = pmul(y, x3);
93
94 y1 = pmul(e, p4f_cephes_log_q1);
95 tmp = pmul(x2, p4f_half);
96 y = padd(y, y1);
97 x = psub(x, tmp);
98 y2 = pmul(e, p4f_cephes_log_q2);
99 x = padd(x, y);
100 x = padd(x, y2);
101 // negative arg will be NAN, 0 will be -INF
102 return _mm_or_ps(_mm_andnot_ps(iszero_mask, _mm_or_ps(x, invalid_mask)),
103 _mm_and_ps(iszero_mask, p4f_minus_inf));
104 }
105
106 template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED
107 Packet4f pexp<Packet4f>(const Packet4f& _x)
108 {
109 Packet4f x = _x;
110 _EIGEN_DECLARE_CONST_Packet4f(1 , 1.0f);
111 _EIGEN_DECLARE_CONST_Packet4f(half, 0.5f);
112 _EIGEN_DECLARE_CONST_Packet4i(0x7f, 0x7f);
113
114
115 _EIGEN_DECLARE_CONST_Packet4f(exp_hi, 88.3762626647950f);
116 _EIGEN_DECLARE_CONST_Packet4f(exp_lo, -88.3762626647949f);
117
118 _EIGEN_DECLARE_CONST_Packet4f(cephes_LOG2EF, 1.44269504088896341f);
119 _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_C1, 0.693359375f);
120 _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_C2, -2.12194440e-4f);
121
122 _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_p0, 1.9875691500E-4f);
123 _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_p1, 1.3981999507E-3f);
124 _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_p2, 8.3334519073E-3f);
125 _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_p3, 4.1665795894E-2f);
126 _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_p4, 1.6666665459E-1f);
127 _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_p5, 5.0000001201E-1f);
128
129 Packet4f tmp, fx;
130 Packet4i emm0;
131
132 // clamp x
133 x = pmax(pmin(x, p4f_exp_hi), p4f_exp_lo);
134
135 /* express exp(x) as exp(g + n*log(2)) */
136 fx = pmadd(x, p4f_cephes_LOG2EF, p4f_half);
137
138 #ifdef EIGEN_VECTORIZE_SSE4_1
139 fx = _mm_floor_ps(fx);
140 #else
141 emm0 = _mm_cvttps_epi32(fx);
142 tmp = _mm_cvtepi32_ps(emm0);
143 /* if greater, substract 1 */
144 Packet4f mask = _mm_cmpgt_ps(tmp, fx);
145 mask = _mm_and_ps(mask, p4f_1);
146 fx = psub(tmp, mask);
147 #endif
148
149 tmp = pmul(fx, p4f_cephes_exp_C1);
150 Packet4f z = pmul(fx, p4f_cephes_exp_C2);
151 x = psub(x, tmp);
152 x = psub(x, z);
153
154 z = pmul(x,x);
155
156 Packet4f y = p4f_cephes_exp_p0;
157 y = pmadd(y, x, p4f_cephes_exp_p1);
158 y = pmadd(y, x, p4f_cephes_exp_p2);
159 y = pmadd(y, x, p4f_cephes_exp_p3);
160 y = pmadd(y, x, p4f_cephes_exp_p4);
161 y = pmadd(y, x, p4f_cephes_exp_p5);
162 y = pmadd(y, z, x);
163 y = padd(y, p4f_1);
164
165 // build 2^n
166 emm0 = _mm_cvttps_epi32(fx);
167 emm0 = _mm_add_epi32(emm0, p4i_0x7f);
168 emm0 = _mm_slli_epi32(emm0, 23);
169 return pmax(pmul(y, Packet4f(_mm_castsi128_ps(emm0))), _x);
170 }
171 template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED
172 Packet2d pexp<Packet2d>(const Packet2d& _x)
173 {
174 Packet2d x = _x;
175
176 _EIGEN_DECLARE_CONST_Packet2d(1 , 1.0);
177 _EIGEN_DECLARE_CONST_Packet2d(2 , 2.0);
178 _EIGEN_DECLARE_CONST_Packet2d(half, 0.5);
179
180 _EIGEN_DECLARE_CONST_Packet2d(exp_hi, 709.437);
181 _EIGEN_DECLARE_CONST_Packet2d(exp_lo, -709.436139303);
182
183 _EIGEN_DECLARE_CONST_Packet2d(cephes_LOG2EF, 1.4426950408889634073599);
184
185 _EIGEN_DECLARE_CONST_Packet2d(cephes_exp_p0, 1.26177193074810590878e-4);
186 _EIGEN_DECLARE_CONST_Packet2d(cephes_exp_p1, 3.02994407707441961300e-2);
187 _EIGEN_DECLARE_CONST_Packet2d(cephes_exp_p2, 9.99999999999999999910e-1);
188
189 _EIGEN_DECLARE_CONST_Packet2d(cephes_exp_q0, 3.00198505138664455042e-6);
190 _EIGEN_DECLARE_CONST_Packet2d(cephes_exp_q1, 2.52448340349684104192e-3);
191 _EIGEN_DECLARE_CONST_Packet2d(cephes_exp_q2, 2.27265548208155028766e-1);
192 _EIGEN_DECLARE_CONST_Packet2d(cephes_exp_q3, 2.00000000000000000009e0);
193
194 _EIGEN_DECLARE_CONST_Packet2d(cephes_exp_C1, 0.693145751953125);
195 _EIGEN_DECLARE_CONST_Packet2d(cephes_exp_C2, 1.42860682030941723212e-6);
196 static const __m128i p4i_1023_0 = _mm_setr_epi32(1023, 1023, 0, 0);
197
198 Packet2d tmp, fx;
199 Packet4i emm0;
200
201 // clamp x
202 x = pmax(pmin(x, p2d_exp_hi), p2d_exp_lo);
203 /* express exp(x) as exp(g + n*log(2)) */
204 fx = pmadd(p2d_cephes_LOG2EF, x, p2d_half);
205
206 #ifdef EIGEN_VECTORIZE_SSE4_1
207 fx = _mm_floor_pd(fx);
208 #else
209 emm0 = _mm_cvttpd_epi32(fx);
210 tmp = _mm_cvtepi32_pd(emm0);
211 /* if greater, substract 1 */
212 Packet2d mask = _mm_cmpgt_pd(tmp, fx);
213 mask = _mm_and_pd(mask, p2d_1);
214 fx = psub(tmp, mask);
215 #endif
216
217 tmp = pmul(fx, p2d_cephes_exp_C1);
218 Packet2d z = pmul(fx, p2d_cephes_exp_C2);
219 x = psub(x, tmp);
220 x = psub(x, z);
221
222 Packet2d x2 = pmul(x,x);
223
224 Packet2d px = p2d_cephes_exp_p0;
225 px = pmadd(px, x2, p2d_cephes_exp_p1);
226 px = pmadd(px, x2, p2d_cephes_exp_p2);
227 px = pmul (px, x);
228
229 Packet2d qx = p2d_cephes_exp_q0;
230 qx = pmadd(qx, x2, p2d_cephes_exp_q1);
231 qx = pmadd(qx, x2, p2d_cephes_exp_q2);
232 qx = pmadd(qx, x2, p2d_cephes_exp_q3);
233
234 x = pdiv(px,psub(qx,px));
235 x = pmadd(p2d_2,x,p2d_1);
236
237 // build 2^n
238 emm0 = _mm_cvttpd_epi32(fx);
239 emm0 = _mm_add_epi32(emm0, p4i_1023_0);
240 emm0 = _mm_slli_epi32(emm0, 20);
241 emm0 = _mm_shuffle_epi32(emm0, _MM_SHUFFLE(1,2,0,3));
242 return pmax(pmul(x, Packet2d(_mm_castsi128_pd(emm0))), _x);
243 }
244
245 /* evaluation of 4 sines at onces, using SSE2 intrinsics.
246
247 The code is the exact rewriting of the cephes sinf function.
248 Precision is excellent as long as x < 8192 (I did not bother to
249 take into account the special handling they have for greater values
250 -- it does not return garbage for arguments over 8192, though, but
251 the extra precision is missing).
252
253 Note that it is such that sinf((float)M_PI) = 8.74e-8, which is the
254 surprising but correct result.
255 */
256
257 template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED
258 Packet4f psin<Packet4f>(const Packet4f& _x)
259 {
260 Packet4f x = _x;
261 _EIGEN_DECLARE_CONST_Packet4f(1 , 1.0f);
262 _EIGEN_DECLARE_CONST_Packet4f(half, 0.5f);
263
264 _EIGEN_DECLARE_CONST_Packet4i(1, 1);
265 _EIGEN_DECLARE_CONST_Packet4i(not1, ~1);
266 _EIGEN_DECLARE_CONST_Packet4i(2, 2);
267 _EIGEN_DECLARE_CONST_Packet4i(4, 4);
268
269 _EIGEN_DECLARE_CONST_Packet4f_FROM_INT(sign_mask, 0x80000000);
270
271 _EIGEN_DECLARE_CONST_Packet4f(minus_cephes_DP1,-0.78515625f);
272 _EIGEN_DECLARE_CONST_Packet4f(minus_cephes_DP2, -2.4187564849853515625e-4f);
273 _EIGEN_DECLARE_CONST_Packet4f(minus_cephes_DP3, -3.77489497744594108e-8f);
274 _EIGEN_DECLARE_CONST_Packet4f(sincof_p0, -1.9515295891E-4f);
275 _EIGEN_DECLARE_CONST_Packet4f(sincof_p1, 8.3321608736E-3f);
276 _EIGEN_DECLARE_CONST_Packet4f(sincof_p2, -1.6666654611E-1f);
277 _EIGEN_DECLARE_CONST_Packet4f(coscof_p0, 2.443315711809948E-005f);
278 _EIGEN_DECLARE_CONST_Packet4f(coscof_p1, -1.388731625493765E-003f);
279 _EIGEN_DECLARE_CONST_Packet4f(coscof_p2, 4.166664568298827E-002f);
280 _EIGEN_DECLARE_CONST_Packet4f(cephes_FOPI, 1.27323954473516f); // 4 / M_PI
281
282 Packet4f xmm1, xmm2, xmm3, sign_bit, y;
283
284 Packet4i emm0, emm2;
285 sign_bit = x;
286 /* take the absolute value */
287 x = pabs(x);
288
289 /* take the modulo */
290
291 /* extract the sign bit (upper one) */
292 sign_bit = _mm_and_ps(sign_bit, p4f_sign_mask);
293
294 /* scale by 4/Pi */
295 y = pmul(x, p4f_cephes_FOPI);
296
297 /* store the integer part of y in mm0 */
298 emm2 = _mm_cvttps_epi32(y);
299 /* j=(j+1) & (~1) (see the cephes sources) */
300 emm2 = _mm_add_epi32(emm2, p4i_1);
301 emm2 = _mm_and_si128(emm2, p4i_not1);
302 y = _mm_cvtepi32_ps(emm2);
303 /* get the swap sign flag */
304 emm0 = _mm_and_si128(emm2, p4i_4);
305 emm0 = _mm_slli_epi32(emm0, 29);
306 /* get the polynom selection mask
307 there is one polynom for 0 <= x <= Pi/4
308 and another one for Pi/4<x<=Pi/2
309
310 Both branches will be computed.
311 */
312 emm2 = _mm_and_si128(emm2, p4i_2);
313 emm2 = _mm_cmpeq_epi32(emm2, _mm_setzero_si128());
314
315 Packet4f swap_sign_bit = _mm_castsi128_ps(emm0);
316 Packet4f poly_mask = _mm_castsi128_ps(emm2);
317 sign_bit = _mm_xor_ps(sign_bit, swap_sign_bit);
318
319 /* The magic pass: "Extended precision modular arithmetic"
320 x = ((x - y * DP1) - y * DP2) - y * DP3; */
321 xmm1 = pmul(y, p4f_minus_cephes_DP1);
322 xmm2 = pmul(y, p4f_minus_cephes_DP2);
323 xmm3 = pmul(y, p4f_minus_cephes_DP3);
324 x = padd(x, xmm1);
325 x = padd(x, xmm2);
326 x = padd(x, xmm3);
327
328 /* Evaluate the first polynom (0 <= x <= Pi/4) */
329 y = p4f_coscof_p0;
330 Packet4f z = _mm_mul_ps(x,x);
331
332 y = pmadd(y, z, p4f_coscof_p1);
333 y = pmadd(y, z, p4f_coscof_p2);
334 y = pmul(y, z);
335 y = pmul(y, z);
336 Packet4f tmp = pmul(z, p4f_half);
337 y = psub(y, tmp);
338 y = padd(y, p4f_1);
339
340 /* Evaluate the second polynom (Pi/4 <= x <= 0) */
341
342 Packet4f y2 = p4f_sincof_p0;
343 y2 = pmadd(y2, z, p4f_sincof_p1);
344 y2 = pmadd(y2, z, p4f_sincof_p2);
345 y2 = pmul(y2, z);
346 y2 = pmul(y2, x);
347 y2 = padd(y2, x);
348
349 /* select the correct result from the two polynoms */
350 y2 = _mm_and_ps(poly_mask, y2);
351 y = _mm_andnot_ps(poly_mask, y);
352 y = _mm_or_ps(y,y2);
353 /* update the sign */
354 return _mm_xor_ps(y, sign_bit);
355 }
356
357 /* almost the same as psin */
358 template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED
359 Packet4f pcos<Packet4f>(const Packet4f& _x)
360 {
361 Packet4f x = _x;
362 _EIGEN_DECLARE_CONST_Packet4f(1 , 1.0f);
363 _EIGEN_DECLARE_CONST_Packet4f(half, 0.5f);
364
365 _EIGEN_DECLARE_CONST_Packet4i(1, 1);
366 _EIGEN_DECLARE_CONST_Packet4i(not1, ~1);
367 _EIGEN_DECLARE_CONST_Packet4i(2, 2);
368 _EIGEN_DECLARE_CONST_Packet4i(4, 4);
369
370 _EIGEN_DECLARE_CONST_Packet4f(minus_cephes_DP1,-0.78515625f);
371 _EIGEN_DECLARE_CONST_Packet4f(minus_cephes_DP2, -2.4187564849853515625e-4f);
372 _EIGEN_DECLARE_CONST_Packet4f(minus_cephes_DP3, -3.77489497744594108e-8f);
373 _EIGEN_DECLARE_CONST_Packet4f(sincof_p0, -1.9515295891E-4f);
374 _EIGEN_DECLARE_CONST_Packet4f(sincof_p1, 8.3321608736E-3f);
375 _EIGEN_DECLARE_CONST_Packet4f(sincof_p2, -1.6666654611E-1f);
376 _EIGEN_DECLARE_CONST_Packet4f(coscof_p0, 2.443315711809948E-005f);
377 _EIGEN_DECLARE_CONST_Packet4f(coscof_p1, -1.388731625493765E-003f);
378 _EIGEN_DECLARE_CONST_Packet4f(coscof_p2, 4.166664568298827E-002f);
379 _EIGEN_DECLARE_CONST_Packet4f(cephes_FOPI, 1.27323954473516f); // 4 / M_PI
380
381 Packet4f xmm1, xmm2, xmm3, y;
382 Packet4i emm0, emm2;
383
384 x = pabs(x);
385
386 /* scale by 4/Pi */
387 y = pmul(x, p4f_cephes_FOPI);
388
389 /* get the integer part of y */
390 emm2 = _mm_cvttps_epi32(y);
391 /* j=(j+1) & (~1) (see the cephes sources) */
392 emm2 = _mm_add_epi32(emm2, p4i_1);
393 emm2 = _mm_and_si128(emm2, p4i_not1);
394 y = _mm_cvtepi32_ps(emm2);
395
396 emm2 = _mm_sub_epi32(emm2, p4i_2);
397
398 /* get the swap sign flag */
399 emm0 = _mm_andnot_si128(emm2, p4i_4);
400 emm0 = _mm_slli_epi32(emm0, 29);
401 /* get the polynom selection mask */
402 emm2 = _mm_and_si128(emm2, p4i_2);
403 emm2 = _mm_cmpeq_epi32(emm2, _mm_setzero_si128());
404
405 Packet4f sign_bit = _mm_castsi128_ps(emm0);
406 Packet4f poly_mask = _mm_castsi128_ps(emm2);
407
408 /* The magic pass: "Extended precision modular arithmetic"
409 x = ((x - y * DP1) - y * DP2) - y * DP3; */
410 xmm1 = pmul(y, p4f_minus_cephes_DP1);
411 xmm2 = pmul(y, p4f_minus_cephes_DP2);
412 xmm3 = pmul(y, p4f_minus_cephes_DP3);
413 x = padd(x, xmm1);
414 x = padd(x, xmm2);
415 x = padd(x, xmm3);
416
417 /* Evaluate the first polynom (0 <= x <= Pi/4) */
418 y = p4f_coscof_p0;
419 Packet4f z = pmul(x,x);
420
421 y = pmadd(y,z,p4f_coscof_p1);
422 y = pmadd(y,z,p4f_coscof_p2);
423 y = pmul(y, z);
424 y = pmul(y, z);
425 Packet4f tmp = _mm_mul_ps(z, p4f_half);
426 y = psub(y, tmp);
427 y = padd(y, p4f_1);
428
429 /* Evaluate the second polynom (Pi/4 <= x <= 0) */
430 Packet4f y2 = p4f_sincof_p0;
431 y2 = pmadd(y2, z, p4f_sincof_p1);
432 y2 = pmadd(y2, z, p4f_sincof_p2);
433 y2 = pmul(y2, z);
434 y2 = pmadd(y2, x, x);
435
436 /* select the correct result from the two polynoms */
437 y2 = _mm_and_ps(poly_mask, y2);
438 y = _mm_andnot_ps(poly_mask, y);
439 y = _mm_or_ps(y,y2);
440
441 /* update the sign */
442 return _mm_xor_ps(y, sign_bit);
443 }
444
445 #if EIGEN_FAST_MATH
446
447 // Functions for sqrt.
448 // The EIGEN_FAST_MATH version uses the _mm_rsqrt_ps approximation and one step
449 // of Newton's method, at a cost of 1-2 bits of precision as opposed to the
450 // exact solution. It does not handle +inf, or denormalized numbers correctly.
451 // The main advantage of this approach is not just speed, but also the fact that
452 // it can be inlined and pipelined with other computations, further reducing its
453 // effective latency. This is similar to Quake3's fast inverse square root.
454 // For detail see here: http://www.beyond3d.com/content/articles/8/
455 template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED
456 Packet4f psqrt<Packet4f>(const Packet4f& _x)
457 {
458 Packet4f half = pmul(_x, pset1<Packet4f>(.5f));
459 Packet4f denormal_mask = _mm_and_ps(
460 _mm_cmpge_ps(_x, _mm_setzero_ps()),
461 _mm_cmplt_ps(_x, pset1<Packet4f>((std::numeric_limits<float>::min)())));
462
463 // Compute approximate reciprocal sqrt.
464 Packet4f x = _mm_rsqrt_ps(_x);
465 // Do a single step of Newton's iteration.
466 x = pmul(x, psub(pset1<Packet4f>(1.5f), pmul(half, pmul(x,x))));
467 // Flush results for denormals to zero.
468 return _mm_andnot_ps(denormal_mask, pmul(_x,x));
469 }
470
471 #else
472
473 template<>EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED
474 Packet4f psqrt<Packet4f>(const Packet4f& x) { return _mm_sqrt_ps(x); }
475
476 #endif
477
478 template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED
479 Packet2d psqrt<Packet2d>(const Packet2d& x) { return _mm_sqrt_pd(x); }
480
481 #if EIGEN_FAST_MATH
482
483 template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED
484 Packet4f prsqrt<Packet4f>(const Packet4f& _x) {
485 _EIGEN_DECLARE_CONST_Packet4f_FROM_INT(inf, 0x7f800000);
486 _EIGEN_DECLARE_CONST_Packet4f_FROM_INT(nan, 0x7fc00000);
487 _EIGEN_DECLARE_CONST_Packet4f(one_point_five, 1.5f);
488 _EIGEN_DECLARE_CONST_Packet4f(minus_half, -0.5f);
489 _EIGEN_DECLARE_CONST_Packet4f_FROM_INT(flt_min, 0x00800000);
490
491 Packet4f neg_half = pmul(_x, p4f_minus_half);
492
493 // select only the inverse sqrt of positive normal inputs (denormals are
494 // flushed to zero and cause infs as well).
495 Packet4f le_zero_mask = _mm_cmple_ps(_x, p4f_flt_min);
496 Packet4f x = _mm_andnot_ps(le_zero_mask, _mm_rsqrt_ps(_x));
497
498 // Fill in NaNs and Infs for the negative/zero entries.
499 Packet4f neg_mask = _mm_cmplt_ps(_x, _mm_setzero_ps());
500 Packet4f zero_mask = _mm_andnot_ps(neg_mask, le_zero_mask);
501 Packet4f infs_and_nans = _mm_or_ps(_mm_and_ps(neg_mask, p4f_nan),
502 _mm_and_ps(zero_mask, p4f_inf));
503
504 // Do a single step of Newton's iteration.
505 x = pmul(x, pmadd(neg_half, pmul(x, x), p4f_one_point_five));
506
507 // Insert NaNs and Infs in all the right places.
508 return _mm_or_ps(x, infs_and_nans);
509 }
510
511 #else
512
513 template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED
514 Packet4f prsqrt<Packet4f>(const Packet4f& x) {
515 // Unfortunately we can't use the much faster mm_rqsrt_ps since it only provides an approximation.
516 return _mm_div_ps(pset1<Packet4f>(1.0f), _mm_sqrt_ps(x));
517 }
518
519 #endif
520
521 template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED
522 Packet2d prsqrt<Packet2d>(const Packet2d& x) {
523 // Unfortunately we can't use the much faster mm_rqsrt_pd since it only provides an approximation.
524 return _mm_div_pd(pset1<Packet2d>(1.0), _mm_sqrt_pd(x));
525 }
526
527 // Hyperbolic Tangent function.
528 template <>
529 EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet4f
530 ptanh<Packet4f>(const Packet4f& x) {
531 return internal::generic_fast_tanh_float(x);
532 }
533
534 } // end namespace internal
535
536 namespace numext {
537
538 template<>
539 EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE
sqrt(const float & x)540 float sqrt(const float &x)
541 {
542 return internal::pfirst(internal::Packet4f(_mm_sqrt_ss(_mm_set_ss(x))));
543 }
544
545 template<>
546 EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE
sqrt(const double & x)547 double sqrt(const double &x)
548 {
549 #if EIGEN_COMP_GNUC_STRICT
550 // This works around a GCC bug generating poor code for _mm_sqrt_pd
551 // See https://bitbucket.org/eigen/eigen/commits/14f468dba4d350d7c19c9b93072e19f7b3df563b
552 return internal::pfirst(internal::Packet2d(__builtin_ia32_sqrtsd(_mm_set_sd(x))));
553 #else
554 return internal::pfirst(internal::Packet2d(_mm_sqrt_pd(_mm_set_sd(x))));
555 #endif
556 }
557
558 } // end namespace numex
559
560 } // end namespace Eigen
561
562 #endif // EIGEN_MATH_FUNCTIONS_SSE_H
563