Eigen  3.2.92
arch/AVX/MathFunctions.h
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 // For some reason, this function didn't make it into the avxintirn.h
14 // used by the compiler, so we'll just wrap it.
15 #define _mm256_setr_m128(lo, hi) \
16  _mm256_insertf128_si256(_mm256_castsi128_si256(lo), (hi), 1)
17 
18 /* The sin, cos, exp, and log functions of this file are loosely derived from
19  * Julien Pommier's sse math library: http://gruntthepeon.free.fr/ssemath/
20  */
21 
22 namespace Eigen {
23 
24 namespace internal {
25 
26 // Sine function
27 // Computes sin(x) by wrapping x to the interval [-Pi/4,3*Pi/4] and
28 // evaluating interpolants in [-Pi/4,Pi/4] or [Pi/4,3*Pi/4]. The interpolants
29 // are (anti-)symmetric and thus have only odd/even coefficients
30 template <>
31 EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet8f
32 psin<Packet8f>(const Packet8f& _x) {
33  Packet8f x = _x;
34 
35  // Some useful values.
36  _EIGEN_DECLARE_CONST_Packet8i(one, 1);
37  _EIGEN_DECLARE_CONST_Packet8f(one, 1.0f);
38  _EIGEN_DECLARE_CONST_Packet8f(two, 2.0f);
39  _EIGEN_DECLARE_CONST_Packet8f(one_over_four, 0.25f);
40  _EIGEN_DECLARE_CONST_Packet8f(one_over_pi, 3.183098861837907e-01f);
41  _EIGEN_DECLARE_CONST_Packet8f(neg_pi_first, -3.140625000000000e+00f);
42  _EIGEN_DECLARE_CONST_Packet8f(neg_pi_second, -9.670257568359375e-04f);
43  _EIGEN_DECLARE_CONST_Packet8f(neg_pi_third, -6.278329571784980e-07f);
44  _EIGEN_DECLARE_CONST_Packet8f(four_over_pi, 1.273239544735163e+00f);
45 
46  // Map x from [-Pi/4,3*Pi/4] to z in [-1,3] and subtract the shifted period.
47  Packet8f z = pmul(x, p8f_one_over_pi);
48  Packet8f shift = _mm256_floor_ps(padd(z, p8f_one_over_four));
49  x = pmadd(shift, p8f_neg_pi_first, x);
50  x = pmadd(shift, p8f_neg_pi_second, x);
51  x = pmadd(shift, p8f_neg_pi_third, x);
52  z = pmul(x, p8f_four_over_pi);
53 
54  // Make a mask for the entries that need flipping, i.e. wherever the shift
55  // is odd.
56  Packet8i shift_ints = _mm256_cvtps_epi32(shift);
57  Packet8i shift_isodd =
58  _mm256_castps_si256(_mm256_and_ps(_mm256_castsi256_ps(shift_ints), _mm256_castsi256_ps(p8i_one)));
59 #ifdef EIGEN_VECTORIZE_AVX2
60  Packet8i sign_flip_mask = _mm256_slli_epi32(shift_isodd, 31);
61 #else
62  __m128i lo =
63  _mm_slli_epi32(_mm256_extractf128_si256(shift_isodd, 0), 31);
64  __m128i hi =
65  _mm_slli_epi32(_mm256_extractf128_si256(shift_isodd, 1), 31);
66  Packet8i sign_flip_mask = _mm256_setr_m128(lo, hi);
67 #endif
68 
69  // Create a mask for which interpolant to use, i.e. if z > 1, then the mask
70  // is set to ones for that entry.
71  Packet8f ival_mask = _mm256_cmp_ps(z, p8f_one, _CMP_GT_OQ);
72 
73  // Evaluate the polynomial for the interval [1,3] in z.
74  _EIGEN_DECLARE_CONST_Packet8f(coeff_right_0, 9.999999724233232e-01f);
75  _EIGEN_DECLARE_CONST_Packet8f(coeff_right_2, -3.084242535619928e-01f);
76  _EIGEN_DECLARE_CONST_Packet8f(coeff_right_4, 1.584991525700324e-02f);
77  _EIGEN_DECLARE_CONST_Packet8f(coeff_right_6, -3.188805084631342e-04f);
78  Packet8f z_minus_two = psub(z, p8f_two);
79  Packet8f z_minus_two2 = pmul(z_minus_two, z_minus_two);
80  Packet8f right = pmadd(p8f_coeff_right_6, z_minus_two2, p8f_coeff_right_4);
81  right = pmadd(right, z_minus_two2, p8f_coeff_right_2);
82  right = pmadd(right, z_minus_two2, p8f_coeff_right_0);
83 
84  // Evaluate the polynomial for the interval [-1,1] in z.
85  _EIGEN_DECLARE_CONST_Packet8f(coeff_left_1, 7.853981525427295e-01f);
86  _EIGEN_DECLARE_CONST_Packet8f(coeff_left_3, -8.074536727092352e-02f);
87  _EIGEN_DECLARE_CONST_Packet8f(coeff_left_5, 2.489871967827018e-03f);
88  _EIGEN_DECLARE_CONST_Packet8f(coeff_left_7, -3.587725841214251e-05f);
89  Packet8f z2 = pmul(z, z);
90  Packet8f left = pmadd(p8f_coeff_left_7, z2, p8f_coeff_left_5);
91  left = pmadd(left, z2, p8f_coeff_left_3);
92  left = pmadd(left, z2, p8f_coeff_left_1);
93  left = pmul(left, z);
94 
95  // Assemble the results, i.e. select the left and right polynomials.
96  left = _mm256_andnot_ps(ival_mask, left);
97  right = _mm256_and_ps(ival_mask, right);
98  Packet8f res = _mm256_or_ps(left, right);
99 
100  // Flip the sign on the odd intervals and return the result.
101  res = _mm256_xor_ps(res, _mm256_castsi256_ps(sign_flip_mask));
102  return res;
103 }
104 
105 // Natural logarithm
106 // Computes log(x) as log(2^e * m) = C*e + log(m), where the constant C =log(2)
107 // and m is in the range [sqrt(1/2),sqrt(2)). In this range, the logarithm can
108 // be easily approximated by a polynomial centered on m=1 for stability.
109 // TODO(gonnet): Further reduce the interval allowing for lower-degree
110 // polynomial interpolants -> ... -> profit!
111 template <>
112 EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet8f
113 plog<Packet8f>(const Packet8f& _x) {
114  Packet8f x = _x;
115  _EIGEN_DECLARE_CONST_Packet8f(1, 1.0f);
116  _EIGEN_DECLARE_CONST_Packet8f(half, 0.5f);
117  _EIGEN_DECLARE_CONST_Packet8f(126f, 126.0f);
118 
119  _EIGEN_DECLARE_CONST_Packet8f_FROM_INT(inv_mant_mask, ~0x7f800000);
120 
121  // The smallest non denormalized float number.
122  _EIGEN_DECLARE_CONST_Packet8f_FROM_INT(min_norm_pos, 0x00800000);
123  _EIGEN_DECLARE_CONST_Packet8f_FROM_INT(minus_inf, 0xff800000);
124 
125  // Polynomial coefficients.
126  _EIGEN_DECLARE_CONST_Packet8f(cephes_SQRTHF, 0.707106781186547524f);
127  _EIGEN_DECLARE_CONST_Packet8f(cephes_log_p0, 7.0376836292E-2f);
128  _EIGEN_DECLARE_CONST_Packet8f(cephes_log_p1, -1.1514610310E-1f);
129  _EIGEN_DECLARE_CONST_Packet8f(cephes_log_p2, 1.1676998740E-1f);
130  _EIGEN_DECLARE_CONST_Packet8f(cephes_log_p3, -1.2420140846E-1f);
131  _EIGEN_DECLARE_CONST_Packet8f(cephes_log_p4, +1.4249322787E-1f);
132  _EIGEN_DECLARE_CONST_Packet8f(cephes_log_p5, -1.6668057665E-1f);
133  _EIGEN_DECLARE_CONST_Packet8f(cephes_log_p6, +2.0000714765E-1f);
134  _EIGEN_DECLARE_CONST_Packet8f(cephes_log_p7, -2.4999993993E-1f);
135  _EIGEN_DECLARE_CONST_Packet8f(cephes_log_p8, +3.3333331174E-1f);
136  _EIGEN_DECLARE_CONST_Packet8f(cephes_log_q1, -2.12194440e-4f);
137  _EIGEN_DECLARE_CONST_Packet8f(cephes_log_q2, 0.693359375f);
138 
139  Packet8f invalid_mask = _mm256_cmp_ps(x, _mm256_setzero_ps(), _CMP_NGE_UQ); // not greater equal is true if x is NaN
140  Packet8f iszero_mask = _mm256_cmp_ps(x, _mm256_setzero_ps(), _CMP_EQ_OQ);
141 
142  // Truncate input values to the minimum positive normal.
143  x = pmax(x, p8f_min_norm_pos);
144 
145 // Extract the shifted exponents (No bitwise shifting in regular AVX, so
146 // convert to SSE and do it there).
147 #ifdef EIGEN_VECTORIZE_AVX2
148  Packet8f emm0 = _mm256_cvtepi32_ps(_mm256_srli_epi32(_mm256_castps_si256(x), 23));
149 #else
150  __m128i lo = _mm_srli_epi32(_mm256_extractf128_si256(_mm256_castps_si256(x), 0), 23);
151  __m128i hi = _mm_srli_epi32(_mm256_extractf128_si256(_mm256_castps_si256(x), 1), 23);
152  Packet8f emm0 = _mm256_cvtepi32_ps(_mm256_setr_m128(lo, hi));
153 #endif
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 #ifdef EIGEN_VECTORIZE_AVX2
263  emm0 = _mm256_slli_epi32(emm0, 23);
264 #else
265  __m128i lo = _mm_slli_epi32(_mm256_extractf128_si256(emm0, 0), 23);
266  __m128i hi = _mm_slli_epi32(_mm256_extractf128_si256(emm0, 1), 23);
267  emm0 = _mm256_setr_m128(lo, hi);
268 #endif
269 
270  // Return 2^m * exp(r).
271  return pmax(pmul(y, _mm256_castsi256_ps(emm0)), _x);
272 }
273 
274 template <>
275 EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet4d
276 pexp<Packet4d>(const Packet4d& _x) {
277  Packet4d x = _x;
278 
279  _EIGEN_DECLARE_CONST_Packet4d(1, 1.0);
280  _EIGEN_DECLARE_CONST_Packet4d(2, 2.0);
281  _EIGEN_DECLARE_CONST_Packet4d(half, 0.5);
282 
283  _EIGEN_DECLARE_CONST_Packet4d(exp_hi, 709.437);
284  _EIGEN_DECLARE_CONST_Packet4d(exp_lo, -709.436139303);
285 
286  _EIGEN_DECLARE_CONST_Packet4d(cephes_LOG2EF, 1.4426950408889634073599);
287 
288  _EIGEN_DECLARE_CONST_Packet4d(cephes_exp_p0, 1.26177193074810590878e-4);
289  _EIGEN_DECLARE_CONST_Packet4d(cephes_exp_p1, 3.02994407707441961300e-2);
290  _EIGEN_DECLARE_CONST_Packet4d(cephes_exp_p2, 9.99999999999999999910e-1);
291 
292  _EIGEN_DECLARE_CONST_Packet4d(cephes_exp_q0, 3.00198505138664455042e-6);
293  _EIGEN_DECLARE_CONST_Packet4d(cephes_exp_q1, 2.52448340349684104192e-3);
294  _EIGEN_DECLARE_CONST_Packet4d(cephes_exp_q2, 2.27265548208155028766e-1);
295  _EIGEN_DECLARE_CONST_Packet4d(cephes_exp_q3, 2.00000000000000000009e0);
296 
297  _EIGEN_DECLARE_CONST_Packet4d(cephes_exp_C1, 0.693145751953125);
298  _EIGEN_DECLARE_CONST_Packet4d(cephes_exp_C2, 1.42860682030941723212e-6);
299  _EIGEN_DECLARE_CONST_Packet4i(1023, 1023);
300 
301  Packet4d tmp, fx;
302 
303  // clamp x
304  x = pmax(pmin(x, p4d_exp_hi), p4d_exp_lo);
305  // Express exp(x) as exp(g + n*log(2)).
306  fx = pmadd(p4d_cephes_LOG2EF, x, p4d_half);
307 
308  // Get the integer modulus of log(2), i.e. the "n" described above.
309  fx = _mm256_floor_pd(fx);
310 
311  // Get the remainder modulo log(2), i.e. the "g" described above. Subtract
312  // n*log(2) out in two steps, i.e. n*C1 + n*C2, C1+C2=log2 to get the last
313  // digits right.
314  tmp = pmul(fx, p4d_cephes_exp_C1);
315  Packet4d z = pmul(fx, p4d_cephes_exp_C2);
316  x = psub(x, tmp);
317  x = psub(x, z);
318 
319  Packet4d x2 = pmul(x, x);
320 
321  // Evaluate the numerator polynomial of the rational interpolant.
322  Packet4d px = p4d_cephes_exp_p0;
323  px = pmadd(px, x2, p4d_cephes_exp_p1);
324  px = pmadd(px, x2, p4d_cephes_exp_p2);
325  px = pmul(px, x);
326 
327  // Evaluate the denominator polynomial of the rational interpolant.
328  Packet4d qx = p4d_cephes_exp_q0;
329  qx = pmadd(qx, x2, p4d_cephes_exp_q1);
330  qx = pmadd(qx, x2, p4d_cephes_exp_q2);
331  qx = pmadd(qx, x2, p4d_cephes_exp_q3);
332 
333  // I don't really get this bit, copied from the SSE2 routines, so...
334  // TODO(gonnet): Figure out what is going on here, perhaps find a better
335  // rational interpolant?
336  x = _mm256_div_pd(px, psub(qx, px));
337  x = pmadd(p4d_2, x, p4d_1);
338 
339  // Build e=2^n by constructing the exponents in a 128-bit vector and
340  // shifting them to where they belong in double-precision values.
341  __m128i emm0 = _mm256_cvtpd_epi32(fx);
342  emm0 = _mm_add_epi32(emm0, p4i_1023);
343  emm0 = _mm_shuffle_epi32(emm0, _MM_SHUFFLE(3, 1, 2, 0));
344  __m128i lo = _mm_slli_epi64(emm0, 52);
345  __m128i hi = _mm_slli_epi64(_mm_srli_epi64(emm0, 32), 52);
346  __m256i e = _mm256_insertf128_si256(_mm256_setzero_si256(), lo, 0);
347  e = _mm256_insertf128_si256(e, hi, 1);
348 
349  // Construct the result 2^n * exp(g) = e * x. The max is used to catch
350  // non-finite values in the input.
351  return pmax(pmul(x, _mm256_castsi256_pd(e)), _x);
352 }
353 
354 // Functions for sqrt.
355 // The EIGEN_FAST_MATH version uses the _mm_rsqrt_ps approximation and one step
356 // of Newton's method, at a cost of 1-2 bits of precision as opposed to the
357 // exact solution. The main advantage of this approach is not just speed, but
358 // also the fact that it can be inlined and pipelined with other computations,
359 // further reducing its effective latency.
360 #if EIGEN_FAST_MATH
361 template <>
362 EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet8f
363 psqrt<Packet8f>(const Packet8f& _x) {
364  _EIGEN_DECLARE_CONST_Packet8f(one_point_five, 1.5f);
365  _EIGEN_DECLARE_CONST_Packet8f(minus_half, -0.5f);
366  _EIGEN_DECLARE_CONST_Packet8f_FROM_INT(flt_min, 0x00800000);
367 
368  Packet8f neg_half = pmul(_x, p8f_minus_half);
369 
370  // select only the inverse sqrt of positive normal inputs (denormals are
371  // flushed to zero and cause infs as well).
372  Packet8f non_zero_mask = _mm256_cmp_ps(_x, p8f_flt_min, _CMP_GE_OQ);
373  Packet8f x = _mm256_and_ps(non_zero_mask, _mm256_rsqrt_ps(_x));
374 
375  // Do a single step of Newton's iteration.
376  x = pmul(x, pmadd(neg_half, pmul(x, x), p8f_one_point_five));
377 
378  // Multiply the original _x by it's reciprocal square root to extract the
379  // square root.
380  return pmul(_x, x);
381 }
382 #else
383 template <> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED
384 Packet8f psqrt<Packet8f>(const Packet8f& x) {
385  return _mm256_sqrt_ps(x);
386 }
387 #endif
388 template <> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED
389 Packet4d psqrt<Packet4d>(const Packet4d& x) {
390  return _mm256_sqrt_pd(x);
391 }
392 #if EIGEN_FAST_MATH
393 
394 template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED
395 Packet8f prsqrt<Packet8f>(const Packet8f& _x) {
396  _EIGEN_DECLARE_CONST_Packet8f_FROM_INT(inf, 0x7f800000);
397  _EIGEN_DECLARE_CONST_Packet8f_FROM_INT(nan, 0x7fc00000);
398  _EIGEN_DECLARE_CONST_Packet8f(one_point_five, 1.5f);
399  _EIGEN_DECLARE_CONST_Packet8f(minus_half, -0.5f);
400  _EIGEN_DECLARE_CONST_Packet8f_FROM_INT(flt_min, 0x00800000);
401 
402  Packet8f neg_half = pmul(_x, p8f_minus_half);
403 
404  // select only the inverse sqrt of positive normal inputs (denormals are
405  // flushed to zero and cause infs as well).
406  Packet8f le_zero_mask = _mm256_cmp_ps(_x, p8f_flt_min, _CMP_LT_OQ);
407  Packet8f x = _mm256_andnot_ps(le_zero_mask, _mm256_rsqrt_ps(_x));
408 
409  // Fill in NaNs and Infs for the negative/zero entries.
410  Packet8f neg_mask = _mm256_cmp_ps(_x, _mm256_setzero_ps(), _CMP_LT_OQ);
411  Packet8f zero_mask = _mm256_andnot_ps(neg_mask, le_zero_mask);
412  Packet8f infs_and_nans = _mm256_or_ps(_mm256_and_ps(neg_mask, p8f_nan),
413  _mm256_and_ps(zero_mask, p8f_inf));
414 
415  // Do a single step of Newton's iteration.
416  x = pmul(x, pmadd(neg_half, pmul(x, x), p8f_one_point_five));
417 
418  // Insert NaNs and Infs in all the right places.
419  return _mm256_or_ps(x, infs_and_nans);
420 }
421 
422 #else
423 template <> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED
424 Packet8f prsqrt<Packet8f>(const Packet8f& x) {
425  _EIGEN_DECLARE_CONST_Packet8f(one, 1.0f);
426  return _mm256_div_ps(p8f_one, _mm256_sqrt_ps(x));
427 }
428 #endif
429 
430 template <> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED
431 Packet4d prsqrt<Packet4d>(const Packet4d& x) {
432  _EIGEN_DECLARE_CONST_Packet4d(one, 1.0);
433  return _mm256_div_pd(p4d_one, _mm256_sqrt_pd(x));
434 }
435 
436 
437 } // end namespace internal
438 
439 } // end namespace Eigen
440 
441 #endif // EIGEN_MATH_FUNCTIONS_AVX_H
Definition: LDLT.h:16
Definition: Eigen_Colamd.h:54