1 // This file is part of Eigen, a lightweight C++ template library
2 // for linear algebra.
3 //
4 // Copyright (C) 2014 Pedro Gonnet (pedro.gonnet@gmail.com)
5 //
6 // This Source Code Form is subject to the terms of the Mozilla
7 // Public License v. 2.0. If a copy of the MPL was not distributed
8 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
9
10 #ifndef EIGEN_MATH_FUNCTIONS_AVX_H
11 #define EIGEN_MATH_FUNCTIONS_AVX_H
12
13 /* The sin, cos, exp, and log functions of this file are loosely derived from
14 * Julien Pommier's sse math library: http://gruntthepeon.free.fr/ssemath/
15 */
16
17 namespace Eigen {
18
19 namespace internal {
20
pshiftleft(Packet8i v,int n)21 inline Packet8i pshiftleft(Packet8i v, int n)
22 {
23 #ifdef EIGEN_VECTORIZE_AVX2
24 return _mm256_slli_epi32(v, n);
25 #else
26 __m128i lo = _mm_slli_epi32(_mm256_extractf128_si256(v, 0), n);
27 __m128i hi = _mm_slli_epi32(_mm256_extractf128_si256(v, 1), n);
28 return _mm256_insertf128_si256(_mm256_castsi128_si256(lo), (hi), 1);
29 #endif
30 }
31
pshiftright(Packet8f v,int n)32 inline Packet8f pshiftright(Packet8f v, int n)
33 {
34 #ifdef EIGEN_VECTORIZE_AVX2
35 return _mm256_cvtepi32_ps(_mm256_srli_epi32(_mm256_castps_si256(v), n));
36 #else
37 __m128i lo = _mm_srli_epi32(_mm256_extractf128_si256(_mm256_castps_si256(v), 0), n);
38 __m128i hi = _mm_srli_epi32(_mm256_extractf128_si256(_mm256_castps_si256(v), 1), n);
39 return _mm256_cvtepi32_ps(_mm256_insertf128_si256(_mm256_castsi128_si256(lo), (hi), 1));
40 #endif
41 }
42
43 // Sine function
44 // Computes sin(x) by wrapping x to the interval [-Pi/4,3*Pi/4] and
45 // evaluating interpolants in [-Pi/4,Pi/4] or [Pi/4,3*Pi/4]. The interpolants
46 // are (anti-)symmetric and thus have only odd/even coefficients
47 template <>
48 EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet8f
49 psin<Packet8f>(const Packet8f& _x) {
50 Packet8f x = _x;
51
52 // Some useful values.
53 _EIGEN_DECLARE_CONST_Packet8i(one, 1);
54 _EIGEN_DECLARE_CONST_Packet8f(one, 1.0f);
55 _EIGEN_DECLARE_CONST_Packet8f(two, 2.0f);
56 _EIGEN_DECLARE_CONST_Packet8f(one_over_four, 0.25f);
57 _EIGEN_DECLARE_CONST_Packet8f(one_over_pi, 3.183098861837907e-01f);
58 _EIGEN_DECLARE_CONST_Packet8f(neg_pi_first, -3.140625000000000e+00f);
59 _EIGEN_DECLARE_CONST_Packet8f(neg_pi_second, -9.670257568359375e-04f);
60 _EIGEN_DECLARE_CONST_Packet8f(neg_pi_third, -6.278329571784980e-07f);
61 _EIGEN_DECLARE_CONST_Packet8f(four_over_pi, 1.273239544735163e+00f);
62
63 // Map x from [-Pi/4,3*Pi/4] to z in [-1,3] and subtract the shifted period.
64 Packet8f z = pmul(x, p8f_one_over_pi);
65 Packet8f shift = _mm256_floor_ps(padd(z, p8f_one_over_four));
66 x = pmadd(shift, p8f_neg_pi_first, x);
67 x = pmadd(shift, p8f_neg_pi_second, x);
68 x = pmadd(shift, p8f_neg_pi_third, x);
69 z = pmul(x, p8f_four_over_pi);
70
71 // Make a mask for the entries that need flipping, i.e. wherever the shift
72 // is odd.
73 Packet8i shift_ints = _mm256_cvtps_epi32(shift);
74 Packet8i shift_isodd = _mm256_castps_si256(_mm256_and_ps(_mm256_castsi256_ps(shift_ints), _mm256_castsi256_ps(p8i_one)));
75 Packet8i sign_flip_mask = pshiftleft(shift_isodd, 31);
76
77 // Create a mask for which interpolant to use, i.e. if z > 1, then the mask
78 // is set to ones for that entry.
79 Packet8f ival_mask = _mm256_cmp_ps(z, p8f_one, _CMP_GT_OQ);
80
81 // Evaluate the polynomial for the interval [1,3] in z.
82 _EIGEN_DECLARE_CONST_Packet8f(coeff_right_0, 9.999999724233232e-01f);
83 _EIGEN_DECLARE_CONST_Packet8f(coeff_right_2, -3.084242535619928e-01f);
84 _EIGEN_DECLARE_CONST_Packet8f(coeff_right_4, 1.584991525700324e-02f);
85 _EIGEN_DECLARE_CONST_Packet8f(coeff_right_6, -3.188805084631342e-04f);
86 Packet8f z_minus_two = psub(z, p8f_two);
87 Packet8f z_minus_two2 = pmul(z_minus_two, z_minus_two);
88 Packet8f right = pmadd(p8f_coeff_right_6, z_minus_two2, p8f_coeff_right_4);
89 right = pmadd(right, z_minus_two2, p8f_coeff_right_2);
90 right = pmadd(right, z_minus_two2, p8f_coeff_right_0);
91
92 // Evaluate the polynomial for the interval [-1,1] in z.
93 _EIGEN_DECLARE_CONST_Packet8f(coeff_left_1, 7.853981525427295e-01f);
94 _EIGEN_DECLARE_CONST_Packet8f(coeff_left_3, -8.074536727092352e-02f);
95 _EIGEN_DECLARE_CONST_Packet8f(coeff_left_5, 2.489871967827018e-03f);
96 _EIGEN_DECLARE_CONST_Packet8f(coeff_left_7, -3.587725841214251e-05f);
97 Packet8f z2 = pmul(z, z);
98 Packet8f left = pmadd(p8f_coeff_left_7, z2, p8f_coeff_left_5);
99 left = pmadd(left, z2, p8f_coeff_left_3);
100 left = pmadd(left, z2, p8f_coeff_left_1);
101 left = pmul(left, z);
102
103 // Assemble the results, i.e. select the left and right polynomials.
104 left = _mm256_andnot_ps(ival_mask, left);
105 right = _mm256_and_ps(ival_mask, right);
106 Packet8f res = _mm256_or_ps(left, right);
107
108 // Flip the sign on the odd intervals and return the result.
109 res = _mm256_xor_ps(res, _mm256_castsi256_ps(sign_flip_mask));
110 return res;
111 }
112
113 // Natural logarithm
114 // Computes log(x) as log(2^e * m) = C*e + log(m), where the constant C =log(2)
115 // and m is in the range [sqrt(1/2),sqrt(2)). In this range, the logarithm can
116 // be easily approximated by a polynomial centered on m=1 for stability.
117 // TODO(gonnet): Further reduce the interval allowing for lower-degree
118 // polynomial interpolants -> ... -> profit!
119 template <>
120 EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet8f
121 plog<Packet8f>(const Packet8f& _x) {
122 Packet8f x = _x;
123 _EIGEN_DECLARE_CONST_Packet8f(1, 1.0f);
124 _EIGEN_DECLARE_CONST_Packet8f(half, 0.5f);
125 _EIGEN_DECLARE_CONST_Packet8f(126f, 126.0f);
126
127 _EIGEN_DECLARE_CONST_Packet8f_FROM_INT(inv_mant_mask, ~0x7f800000);
128
129 // The smallest non denormalized float number.
130 _EIGEN_DECLARE_CONST_Packet8f_FROM_INT(min_norm_pos, 0x00800000);
131 _EIGEN_DECLARE_CONST_Packet8f_FROM_INT(minus_inf, 0xff800000);
132
133 // Polynomial coefficients.
134 _EIGEN_DECLARE_CONST_Packet8f(cephes_SQRTHF, 0.707106781186547524f);
135 _EIGEN_DECLARE_CONST_Packet8f(cephes_log_p0, 7.0376836292E-2f);
136 _EIGEN_DECLARE_CONST_Packet8f(cephes_log_p1, -1.1514610310E-1f);
137 _EIGEN_DECLARE_CONST_Packet8f(cephes_log_p2, 1.1676998740E-1f);
138 _EIGEN_DECLARE_CONST_Packet8f(cephes_log_p3, -1.2420140846E-1f);
139 _EIGEN_DECLARE_CONST_Packet8f(cephes_log_p4, +1.4249322787E-1f);
140 _EIGEN_DECLARE_CONST_Packet8f(cephes_log_p5, -1.6668057665E-1f);
141 _EIGEN_DECLARE_CONST_Packet8f(cephes_log_p6, +2.0000714765E-1f);
142 _EIGEN_DECLARE_CONST_Packet8f(cephes_log_p7, -2.4999993993E-1f);
143 _EIGEN_DECLARE_CONST_Packet8f(cephes_log_p8, +3.3333331174E-1f);
144 _EIGEN_DECLARE_CONST_Packet8f(cephes_log_q1, -2.12194440e-4f);
145 _EIGEN_DECLARE_CONST_Packet8f(cephes_log_q2, 0.693359375f);
146
147 Packet8f invalid_mask = _mm256_cmp_ps(x, _mm256_setzero_ps(), _CMP_NGE_UQ); // not greater equal is true if x is NaN
148 Packet8f iszero_mask = _mm256_cmp_ps(x, _mm256_setzero_ps(), _CMP_EQ_OQ);
149
150 // Truncate input values to the minimum positive normal.
151 x = pmax(x, p8f_min_norm_pos);
152
153 Packet8f emm0 = pshiftright(x,23);
154 Packet8f e = _mm256_sub_ps(emm0, p8f_126f);
155
156 // Set the exponents to -1, i.e. x are in the range [0.5,1).
157 x = _mm256_and_ps(x, p8f_inv_mant_mask);
158 x = _mm256_or_ps(x, p8f_half);
159
160 // part2: Shift the inputs from the range [0.5,1) to [sqrt(1/2),sqrt(2))
161 // and shift by -1. The values are then centered around 0, which improves
162 // the stability of the polynomial evaluation.
163 // if( x < SQRTHF ) {
164 // e -= 1;
165 // x = x + x - 1.0;
166 // } else { x = x - 1.0; }
167 Packet8f mask = _mm256_cmp_ps(x, p8f_cephes_SQRTHF, _CMP_LT_OQ);
168 Packet8f tmp = _mm256_and_ps(x, mask);
169 x = psub(x, p8f_1);
170 e = psub(e, _mm256_and_ps(p8f_1, mask));
171 x = padd(x, tmp);
172
173 Packet8f x2 = pmul(x, x);
174 Packet8f x3 = pmul(x2, x);
175
176 // Evaluate the polynomial approximant of degree 8 in three parts, probably
177 // to improve instruction-level parallelism.
178 Packet8f y, y1, y2;
179 y = pmadd(p8f_cephes_log_p0, x, p8f_cephes_log_p1);
180 y1 = pmadd(p8f_cephes_log_p3, x, p8f_cephes_log_p4);
181 y2 = pmadd(p8f_cephes_log_p6, x, p8f_cephes_log_p7);
182 y = pmadd(y, x, p8f_cephes_log_p2);
183 y1 = pmadd(y1, x, p8f_cephes_log_p5);
184 y2 = pmadd(y2, x, p8f_cephes_log_p8);
185 y = pmadd(y, x3, y1);
186 y = pmadd(y, x3, y2);
187 y = pmul(y, x3);
188
189 // Add the logarithm of the exponent back to the result of the interpolation.
190 y1 = pmul(e, p8f_cephes_log_q1);
191 tmp = pmul(x2, p8f_half);
192 y = padd(y, y1);
193 x = psub(x, tmp);
194 y2 = pmul(e, p8f_cephes_log_q2);
195 x = padd(x, y);
196 x = padd(x, y2);
197
198 // Filter out invalid inputs, i.e. negative arg will be NAN, 0 will be -INF.
199 return _mm256_or_ps(
200 _mm256_andnot_ps(iszero_mask, _mm256_or_ps(x, invalid_mask)),
201 _mm256_and_ps(iszero_mask, p8f_minus_inf));
202 }
203
204 // Exponential function. Works by writing "x = m*log(2) + r" where
205 // "m = floor(x/log(2)+1/2)" and "r" is the remainder. The result is then
206 // "exp(x) = 2^m*exp(r)" where exp(r) is in the range [-1,1).
207 template <>
208 EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet8f
209 pexp<Packet8f>(const Packet8f& _x) {
210 _EIGEN_DECLARE_CONST_Packet8f(1, 1.0f);
211 _EIGEN_DECLARE_CONST_Packet8f(half, 0.5f);
212 _EIGEN_DECLARE_CONST_Packet8f(127, 127.0f);
213
214 _EIGEN_DECLARE_CONST_Packet8f(exp_hi, 88.3762626647950f);
215 _EIGEN_DECLARE_CONST_Packet8f(exp_lo, -88.3762626647949f);
216
217 _EIGEN_DECLARE_CONST_Packet8f(cephes_LOG2EF, 1.44269504088896341f);
218
219 _EIGEN_DECLARE_CONST_Packet8f(cephes_exp_p0, 1.9875691500E-4f);
220 _EIGEN_DECLARE_CONST_Packet8f(cephes_exp_p1, 1.3981999507E-3f);
221 _EIGEN_DECLARE_CONST_Packet8f(cephes_exp_p2, 8.3334519073E-3f);
222 _EIGEN_DECLARE_CONST_Packet8f(cephes_exp_p3, 4.1665795894E-2f);
223 _EIGEN_DECLARE_CONST_Packet8f(cephes_exp_p4, 1.6666665459E-1f);
224 _EIGEN_DECLARE_CONST_Packet8f(cephes_exp_p5, 5.0000001201E-1f);
225
226 // Clamp x.
227 Packet8f x = pmax(pmin(_x, p8f_exp_hi), p8f_exp_lo);
228
229 // Express exp(x) as exp(m*ln(2) + r), start by extracting
230 // m = floor(x/ln(2) + 0.5).
231 Packet8f m = _mm256_floor_ps(pmadd(x, p8f_cephes_LOG2EF, p8f_half));
232
233 // Get r = x - m*ln(2). If no FMA instructions are available, m*ln(2) is
234 // subtracted out in two parts, m*C1+m*C2 = m*ln(2), to avoid accumulating
235 // truncation errors. Note that we don't use the "pmadd" function here to
236 // ensure that a precision-preserving FMA instruction is used.
237 #ifdef EIGEN_VECTORIZE_FMA
238 _EIGEN_DECLARE_CONST_Packet8f(nln2, -0.6931471805599453f);
239 Packet8f r = _mm256_fmadd_ps(m, p8f_nln2, x);
240 #else
241 _EIGEN_DECLARE_CONST_Packet8f(cephes_exp_C1, 0.693359375f);
242 _EIGEN_DECLARE_CONST_Packet8f(cephes_exp_C2, -2.12194440e-4f);
243 Packet8f r = psub(x, pmul(m, p8f_cephes_exp_C1));
244 r = psub(r, pmul(m, p8f_cephes_exp_C2));
245 #endif
246
247 Packet8f r2 = pmul(r, r);
248
249 // TODO(gonnet): Split into odd/even polynomials and try to exploit
250 // instruction-level parallelism.
251 Packet8f y = p8f_cephes_exp_p0;
252 y = pmadd(y, r, p8f_cephes_exp_p1);
253 y = pmadd(y, r, p8f_cephes_exp_p2);
254 y = pmadd(y, r, p8f_cephes_exp_p3);
255 y = pmadd(y, r, p8f_cephes_exp_p4);
256 y = pmadd(y, r, p8f_cephes_exp_p5);
257 y = pmadd(y, r2, r);
258 y = padd(y, p8f_1);
259
260 // Build emm0 = 2^m.
261 Packet8i emm0 = _mm256_cvttps_epi32(padd(m, p8f_127));
262 emm0 = pshiftleft(emm0, 23);
263
264 // Return 2^m * exp(r).
265 return pmax(pmul(y, _mm256_castsi256_ps(emm0)), _x);
266 }
267
268 // Hyperbolic Tangent function.
269 template <>
270 EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet8f
271 ptanh<Packet8f>(const Packet8f& x) {
272 return internal::generic_fast_tanh_float(x);
273 }
274
275 template <>
276 EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet4d
277 pexp<Packet4d>(const Packet4d& _x) {
278 Packet4d x = _x;
279
280 _EIGEN_DECLARE_CONST_Packet4d(1, 1.0);
281 _EIGEN_DECLARE_CONST_Packet4d(2, 2.0);
282 _EIGEN_DECLARE_CONST_Packet4d(half, 0.5);
283
284 _EIGEN_DECLARE_CONST_Packet4d(exp_hi, 709.437);
285 _EIGEN_DECLARE_CONST_Packet4d(exp_lo, -709.436139303);
286
287 _EIGEN_DECLARE_CONST_Packet4d(cephes_LOG2EF, 1.4426950408889634073599);
288
289 _EIGEN_DECLARE_CONST_Packet4d(cephes_exp_p0, 1.26177193074810590878e-4);
290 _EIGEN_DECLARE_CONST_Packet4d(cephes_exp_p1, 3.02994407707441961300e-2);
291 _EIGEN_DECLARE_CONST_Packet4d(cephes_exp_p2, 9.99999999999999999910e-1);
292
293 _EIGEN_DECLARE_CONST_Packet4d(cephes_exp_q0, 3.00198505138664455042e-6);
294 _EIGEN_DECLARE_CONST_Packet4d(cephes_exp_q1, 2.52448340349684104192e-3);
295 _EIGEN_DECLARE_CONST_Packet4d(cephes_exp_q2, 2.27265548208155028766e-1);
296 _EIGEN_DECLARE_CONST_Packet4d(cephes_exp_q3, 2.00000000000000000009e0);
297
298 _EIGEN_DECLARE_CONST_Packet4d(cephes_exp_C1, 0.693145751953125);
299 _EIGEN_DECLARE_CONST_Packet4d(cephes_exp_C2, 1.42860682030941723212e-6);
300 _EIGEN_DECLARE_CONST_Packet4i(1023, 1023);
301
302 Packet4d tmp, fx;
303
304 // clamp x
305 x = pmax(pmin(x, p4d_exp_hi), p4d_exp_lo);
306 // Express exp(x) as exp(g + n*log(2)).
307 fx = pmadd(p4d_cephes_LOG2EF, x, p4d_half);
308
309 // Get the integer modulus of log(2), i.e. the "n" described above.
310 fx = _mm256_floor_pd(fx);
311
312 // Get the remainder modulo log(2), i.e. the "g" described above. Subtract
313 // n*log(2) out in two steps, i.e. n*C1 + n*C2, C1+C2=log2 to get the last
314 // digits right.
315 tmp = pmul(fx, p4d_cephes_exp_C1);
316 Packet4d z = pmul(fx, p4d_cephes_exp_C2);
317 x = psub(x, tmp);
318 x = psub(x, z);
319
320 Packet4d x2 = pmul(x, x);
321
322 // Evaluate the numerator polynomial of the rational interpolant.
323 Packet4d px = p4d_cephes_exp_p0;
324 px = pmadd(px, x2, p4d_cephes_exp_p1);
325 px = pmadd(px, x2, p4d_cephes_exp_p2);
326 px = pmul(px, x);
327
328 // Evaluate the denominator polynomial of the rational interpolant.
329 Packet4d qx = p4d_cephes_exp_q0;
330 qx = pmadd(qx, x2, p4d_cephes_exp_q1);
331 qx = pmadd(qx, x2, p4d_cephes_exp_q2);
332 qx = pmadd(qx, x2, p4d_cephes_exp_q3);
333
334 // I don't really get this bit, copied from the SSE2 routines, so...
335 // TODO(gonnet): Figure out what is going on here, perhaps find a better
336 // rational interpolant?
337 x = _mm256_div_pd(px, psub(qx, px));
338 x = pmadd(p4d_2, x, p4d_1);
339
340 // Build e=2^n by constructing the exponents in a 128-bit vector and
341 // shifting them to where they belong in double-precision values.
342 __m128i emm0 = _mm256_cvtpd_epi32(fx);
343 emm0 = _mm_add_epi32(emm0, p4i_1023);
344 emm0 = _mm_shuffle_epi32(emm0, _MM_SHUFFLE(3, 1, 2, 0));
345 __m128i lo = _mm_slli_epi64(emm0, 52);
346 __m128i hi = _mm_slli_epi64(_mm_srli_epi64(emm0, 32), 52);
347 __m256i e = _mm256_insertf128_si256(_mm256_setzero_si256(), lo, 0);
348 e = _mm256_insertf128_si256(e, hi, 1);
349
350 // Construct the result 2^n * exp(g) = e * x. The max is used to catch
351 // non-finite values in the input.
352 return pmax(pmul(x, _mm256_castsi256_pd(e)), _x);
353 }
354
355 // Functions for sqrt.
356 // The EIGEN_FAST_MATH version uses the _mm_rsqrt_ps approximation and one step
357 // of Newton's method, at a cost of 1-2 bits of precision as opposed to the
358 // exact solution. It does not handle +inf, or denormalized numbers correctly.
359 // The main advantage of this approach is not just speed, but also the fact that
360 // it can be inlined and pipelined with other computations, further reducing its
361 // effective latency. This is similar to Quake3's fast inverse square root.
362 // For detail see here: http://www.beyond3d.com/content/articles/8/
363 #if EIGEN_FAST_MATH
364 template <>
365 EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet8f
366 psqrt<Packet8f>(const Packet8f& _x) {
367 Packet8f half = pmul(_x, pset1<Packet8f>(.5f));
368 Packet8f denormal_mask = _mm256_and_ps(
369 _mm256_cmp_ps(_x, pset1<Packet8f>((std::numeric_limits<float>::min)()),
370 _CMP_LT_OQ),
371 _mm256_cmp_ps(_x, _mm256_setzero_ps(), _CMP_GE_OQ));
372
373 // Compute approximate reciprocal sqrt.
374 Packet8f x = _mm256_rsqrt_ps(_x);
375 // Do a single step of Newton's iteration.
376 x = pmul(x, psub(pset1<Packet8f>(1.5f), pmul(half, pmul(x,x))));
377 // Flush results for denormals to zero.
378 return _mm256_andnot_ps(denormal_mask, pmul(_x,x));
379 }
380 #else
381 template <> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED
382 Packet8f psqrt<Packet8f>(const Packet8f& x) {
383 return _mm256_sqrt_ps(x);
384 }
385 #endif
386 template <> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED
387 Packet4d psqrt<Packet4d>(const Packet4d& x) {
388 return _mm256_sqrt_pd(x);
389 }
390 #if EIGEN_FAST_MATH
391
392 template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED
393 Packet8f prsqrt<Packet8f>(const Packet8f& _x) {
394 _EIGEN_DECLARE_CONST_Packet8f_FROM_INT(inf, 0x7f800000);
395 _EIGEN_DECLARE_CONST_Packet8f_FROM_INT(nan, 0x7fc00000);
396 _EIGEN_DECLARE_CONST_Packet8f(one_point_five, 1.5f);
397 _EIGEN_DECLARE_CONST_Packet8f(minus_half, -0.5f);
398 _EIGEN_DECLARE_CONST_Packet8f_FROM_INT(flt_min, 0x00800000);
399
400 Packet8f neg_half = pmul(_x, p8f_minus_half);
401
402 // select only the inverse sqrt of positive normal inputs (denormals are
403 // flushed to zero and cause infs as well).
404 Packet8f le_zero_mask = _mm256_cmp_ps(_x, p8f_flt_min, _CMP_LT_OQ);
405 Packet8f x = _mm256_andnot_ps(le_zero_mask, _mm256_rsqrt_ps(_x));
406
407 // Fill in NaNs and Infs for the negative/zero entries.
408 Packet8f neg_mask = _mm256_cmp_ps(_x, _mm256_setzero_ps(), _CMP_LT_OQ);
409 Packet8f zero_mask = _mm256_andnot_ps(neg_mask, le_zero_mask);
410 Packet8f infs_and_nans = _mm256_or_ps(_mm256_and_ps(neg_mask, p8f_nan),
411 _mm256_and_ps(zero_mask, p8f_inf));
412
413 // Do a single step of Newton's iteration.
414 x = pmul(x, pmadd(neg_half, pmul(x, x), p8f_one_point_five));
415
416 // Insert NaNs and Infs in all the right places.
417 return _mm256_or_ps(x, infs_and_nans);
418 }
419
420 #else
421 template <> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED
422 Packet8f prsqrt<Packet8f>(const Packet8f& x) {
423 _EIGEN_DECLARE_CONST_Packet8f(one, 1.0f);
424 return _mm256_div_ps(p8f_one, _mm256_sqrt_ps(x));
425 }
426 #endif
427
428 template <> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED
429 Packet4d prsqrt<Packet4d>(const Packet4d& x) {
430 _EIGEN_DECLARE_CONST_Packet4d(one, 1.0);
431 return _mm256_div_pd(p4d_one, _mm256_sqrt_pd(x));
432 }
433
434
435 } // end namespace internal
436
437 } // end namespace Eigen
438
439 #endif // EIGEN_MATH_FUNCTIONS_AVX_H
440