Eigen  3.3.0
 
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arch/AVX512/MathFunctions.h
1// This file is part of Eigen, a lightweight C++ template library
2// for linear algebra.
3//
4// Copyright (C) 2016 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 THIRD_PARTY_EIGEN3_EIGEN_SRC_CORE_ARCH_AVX512_MATHFUNCTIONS_H_
11#define THIRD_PARTY_EIGEN3_EIGEN_SRC_CORE_ARCH_AVX512_MATHFUNCTIONS_H_
12
13namespace Eigen {
14
15namespace internal {
16
17// Disable the code for older versions of gcc that don't support many of the required avx512 instrinsics.
18#if EIGEN_GNUC_AT_LEAST(5, 3)
19
20#define _EIGEN_DECLARE_CONST_Packet16f(NAME, X) \
21 const Packet16f p16f_##NAME = pset1<Packet16f>(X)
22
23#define _EIGEN_DECLARE_CONST_Packet16f_FROM_INT(NAME, X) \
24 const Packet16f p16f_##NAME = (__m512)pset1<Packet16i>(X)
25
26#define _EIGEN_DECLARE_CONST_Packet8d(NAME, X) \
27 const Packet8d p8d_##NAME = pset1<Packet8d>(X)
28
29#define _EIGEN_DECLARE_CONST_Packet8d_FROM_INT64(NAME, X) \
30 const Packet8d p8d_##NAME = _mm512_castsi512_pd(_mm512_set1_epi64(X))
31
32// Natural logarithm
33// Computes log(x) as log(2^e * m) = C*e + log(m), where the constant C =log(2)
34// and m is in the range [sqrt(1/2),sqrt(2)). In this range, the logarithm can
35// be easily approximated by a polynomial centered on m=1 for stability.
36#if defined(EIGEN_VECTORIZE_AVX512DQ)
37template <>
38EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet16f
39plog<Packet16f>(const Packet16f& _x) {
40 Packet16f x = _x;
41 _EIGEN_DECLARE_CONST_Packet16f(1, 1.0f);
42 _EIGEN_DECLARE_CONST_Packet16f(half, 0.5f);
43 _EIGEN_DECLARE_CONST_Packet16f(126f, 126.0f);
44
45 _EIGEN_DECLARE_CONST_Packet16f_FROM_INT(inv_mant_mask, ~0x7f800000);
46
47 // The smallest non denormalized float number.
48 _EIGEN_DECLARE_CONST_Packet16f_FROM_INT(min_norm_pos, 0x00800000);
49 _EIGEN_DECLARE_CONST_Packet16f_FROM_INT(minus_inf, 0xff800000);
50 _EIGEN_DECLARE_CONST_Packet16f_FROM_INT(nan, 0x7fc00000);
51
52 // Polynomial coefficients.
53 _EIGEN_DECLARE_CONST_Packet16f(cephes_SQRTHF, 0.707106781186547524f);
54 _EIGEN_DECLARE_CONST_Packet16f(cephes_log_p0, 7.0376836292E-2f);
55 _EIGEN_DECLARE_CONST_Packet16f(cephes_log_p1, -1.1514610310E-1f);
56 _EIGEN_DECLARE_CONST_Packet16f(cephes_log_p2, 1.1676998740E-1f);
57 _EIGEN_DECLARE_CONST_Packet16f(cephes_log_p3, -1.2420140846E-1f);
58 _EIGEN_DECLARE_CONST_Packet16f(cephes_log_p4, +1.4249322787E-1f);
59 _EIGEN_DECLARE_CONST_Packet16f(cephes_log_p5, -1.6668057665E-1f);
60 _EIGEN_DECLARE_CONST_Packet16f(cephes_log_p6, +2.0000714765E-1f);
61 _EIGEN_DECLARE_CONST_Packet16f(cephes_log_p7, -2.4999993993E-1f);
62 _EIGEN_DECLARE_CONST_Packet16f(cephes_log_p8, +3.3333331174E-1f);
63 _EIGEN_DECLARE_CONST_Packet16f(cephes_log_q1, -2.12194440e-4f);
64 _EIGEN_DECLARE_CONST_Packet16f(cephes_log_q2, 0.693359375f);
65
66 // invalid_mask is set to true when x is NaN
67 __mmask16 invalid_mask =
68 _mm512_cmp_ps_mask(x, _mm512_setzero_ps(), _CMP_NGE_UQ);
69 __mmask16 iszero_mask =
70 _mm512_cmp_ps_mask(x, _mm512_setzero_ps(), _CMP_EQ_UQ);
71
72 // Truncate input values to the minimum positive normal.
73 x = pmax(x, p16f_min_norm_pos);
74
75 // Extract the shifted exponents.
76 Packet16f emm0 = _mm512_cvtepi32_ps(_mm512_srli_epi32((__m512i)x, 23));
77 Packet16f e = _mm512_sub_ps(emm0, p16f_126f);
78
79 // Set the exponents to -1, i.e. x are in the range [0.5,1).
80 x = _mm512_and_ps(x, p16f_inv_mant_mask);
81 x = _mm512_or_ps(x, p16f_half);
82
83 // part2: Shift the inputs from the range [0.5,1) to [sqrt(1/2),sqrt(2))
84 // and shift by -1. The values are then centered around 0, which improves
85 // the stability of the polynomial evaluation.
86 // if( x < SQRTHF ) {
87 // e -= 1;
88 // x = x + x - 1.0;
89 // } else { x = x - 1.0; }
90 __mmask16 mask = _mm512_cmp_ps_mask(x, p16f_cephes_SQRTHF, _CMP_LT_OQ);
91 Packet16f tmp = _mm512_mask_blend_ps(mask, x, _mm512_setzero_ps());
92 x = psub(x, p16f_1);
93 e = psub(e, _mm512_mask_blend_ps(mask, p16f_1, _mm512_setzero_ps()));
94 x = padd(x, tmp);
95
96 Packet16f x2 = pmul(x, x);
97 Packet16f x3 = pmul(x2, x);
98
99 // Evaluate the polynomial approximant of degree 8 in three parts, probably
100 // to improve instruction-level parallelism.
101 Packet16f y, y1, y2;
102 y = pmadd(p16f_cephes_log_p0, x, p16f_cephes_log_p1);
103 y1 = pmadd(p16f_cephes_log_p3, x, p16f_cephes_log_p4);
104 y2 = pmadd(p16f_cephes_log_p6, x, p16f_cephes_log_p7);
105 y = pmadd(y, x, p16f_cephes_log_p2);
106 y1 = pmadd(y1, x, p16f_cephes_log_p5);
107 y2 = pmadd(y2, x, p16f_cephes_log_p8);
108 y = pmadd(y, x3, y1);
109 y = pmadd(y, x3, y2);
110 y = pmul(y, x3);
111
112 // Add the logarithm of the exponent back to the result of the interpolation.
113 y1 = pmul(e, p16f_cephes_log_q1);
114 tmp = pmul(x2, p16f_half);
115 y = padd(y, y1);
116 x = psub(x, tmp);
117 y2 = pmul(e, p16f_cephes_log_q2);
118 x = padd(x, y);
119 x = padd(x, y2);
120
121 // Filter out invalid inputs, i.e. negative arg will be NAN, 0 will be -INF.
122 return _mm512_mask_blend_ps(iszero_mask, p16f_minus_inf,
123 _mm512_mask_blend_ps(invalid_mask, p16f_nan, x));
124}
125#endif
126
127// Exponential function. Works by writing "x = m*log(2) + r" where
128// "m = floor(x/log(2)+1/2)" and "r" is the remainder. The result is then
129// "exp(x) = 2^m*exp(r)" where exp(r) is in the range [-1,1).
130template <>
131EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet16f
132pexp<Packet16f>(const Packet16f& _x) {
133 _EIGEN_DECLARE_CONST_Packet16f(1, 1.0f);
134 _EIGEN_DECLARE_CONST_Packet16f(half, 0.5f);
135 _EIGEN_DECLARE_CONST_Packet16f(127, 127.0f);
136
137 _EIGEN_DECLARE_CONST_Packet16f(exp_hi, 88.3762626647950f);
138 _EIGEN_DECLARE_CONST_Packet16f(exp_lo, -88.3762626647949f);
139
140 _EIGEN_DECLARE_CONST_Packet16f(cephes_LOG2EF, 1.44269504088896341f);
141
142 _EIGEN_DECLARE_CONST_Packet16f(cephes_exp_p0, 1.9875691500E-4f);
143 _EIGEN_DECLARE_CONST_Packet16f(cephes_exp_p1, 1.3981999507E-3f);
144 _EIGEN_DECLARE_CONST_Packet16f(cephes_exp_p2, 8.3334519073E-3f);
145 _EIGEN_DECLARE_CONST_Packet16f(cephes_exp_p3, 4.1665795894E-2f);
146 _EIGEN_DECLARE_CONST_Packet16f(cephes_exp_p4, 1.6666665459E-1f);
147 _EIGEN_DECLARE_CONST_Packet16f(cephes_exp_p5, 5.0000001201E-1f);
148
149 // Clamp x.
150 Packet16f x = pmax(pmin(_x, p16f_exp_hi), p16f_exp_lo);
151
152 // Express exp(x) as exp(m*ln(2) + r), start by extracting
153 // m = floor(x/ln(2) + 0.5).
154 Packet16f m = _mm512_floor_ps(pmadd(x, p16f_cephes_LOG2EF, p16f_half));
155
156 // Get r = x - m*ln(2). Note that we can do this without losing more than one
157 // ulp precision due to the FMA instruction.
158 _EIGEN_DECLARE_CONST_Packet16f(nln2, -0.6931471805599453f);
159 Packet16f r = _mm512_fmadd_ps(m, p16f_nln2, x);
160 Packet16f r2 = pmul(r, r);
161
162 // TODO(gonnet): Split into odd/even polynomials and try to exploit
163 // instruction-level parallelism.
164 Packet16f y = p16f_cephes_exp_p0;
165 y = pmadd(y, r, p16f_cephes_exp_p1);
166 y = pmadd(y, r, p16f_cephes_exp_p2);
167 y = pmadd(y, r, p16f_cephes_exp_p3);
168 y = pmadd(y, r, p16f_cephes_exp_p4);
169 y = pmadd(y, r, p16f_cephes_exp_p5);
170 y = pmadd(y, r2, r);
171 y = padd(y, p16f_1);
172
173 // Build emm0 = 2^m.
174 Packet16i emm0 = _mm512_cvttps_epi32(padd(m, p16f_127));
175 emm0 = _mm512_slli_epi32(emm0, 23);
176
177 // Return 2^m * exp(r).
178 return pmax(pmul(y, _mm512_castsi512_ps(emm0)), _x);
179}
180
181/*template <>
182EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet8d
183pexp<Packet8d>(const Packet8d& _x) {
184 Packet8d x = _x;
185
186 _EIGEN_DECLARE_CONST_Packet8d(1, 1.0);
187 _EIGEN_DECLARE_CONST_Packet8d(2, 2.0);
188
189 _EIGEN_DECLARE_CONST_Packet8d(exp_hi, 709.437);
190 _EIGEN_DECLARE_CONST_Packet8d(exp_lo, -709.436139303);
191
192 _EIGEN_DECLARE_CONST_Packet8d(cephes_LOG2EF, 1.4426950408889634073599);
193
194 _EIGEN_DECLARE_CONST_Packet8d(cephes_exp_p0, 1.26177193074810590878e-4);
195 _EIGEN_DECLARE_CONST_Packet8d(cephes_exp_p1, 3.02994407707441961300e-2);
196 _EIGEN_DECLARE_CONST_Packet8d(cephes_exp_p2, 9.99999999999999999910e-1);
197
198 _EIGEN_DECLARE_CONST_Packet8d(cephes_exp_q0, 3.00198505138664455042e-6);
199 _EIGEN_DECLARE_CONST_Packet8d(cephes_exp_q1, 2.52448340349684104192e-3);
200 _EIGEN_DECLARE_CONST_Packet8d(cephes_exp_q2, 2.27265548208155028766e-1);
201 _EIGEN_DECLARE_CONST_Packet8d(cephes_exp_q3, 2.00000000000000000009e0);
202
203 _EIGEN_DECLARE_CONST_Packet8d(cephes_exp_C1, 0.693145751953125);
204 _EIGEN_DECLARE_CONST_Packet8d(cephes_exp_C2, 1.42860682030941723212e-6);
205
206 // clamp x
207 x = pmax(pmin(x, p8d_exp_hi), p8d_exp_lo);
208
209 // Express exp(x) as exp(g + n*log(2)).
210 const Packet8d n =
211 _mm512_mul_round_pd(p8d_cephes_LOG2EF, x, _MM_FROUND_TO_NEAREST_INT);
212
213 // Get the remainder modulo log(2), i.e. the "g" described above. Subtract
214 // n*log(2) out in two steps, i.e. n*C1 + n*C2, C1+C2=log2 to get the last
215 // digits right.
216 const Packet8d nC1 = pmul(n, p8d_cephes_exp_C1);
217 const Packet8d nC2 = pmul(n, p8d_cephes_exp_C2);
218 x = psub(x, nC1);
219 x = psub(x, nC2);
220
221 const Packet8d x2 = pmul(x, x);
222
223 // Evaluate the numerator polynomial of the rational interpolant.
224 Packet8d px = p8d_cephes_exp_p0;
225 px = pmadd(px, x2, p8d_cephes_exp_p1);
226 px = pmadd(px, x2, p8d_cephes_exp_p2);
227 px = pmul(px, x);
228
229 // Evaluate the denominator polynomial of the rational interpolant.
230 Packet8d qx = p8d_cephes_exp_q0;
231 qx = pmadd(qx, x2, p8d_cephes_exp_q1);
232 qx = pmadd(qx, x2, p8d_cephes_exp_q2);
233 qx = pmadd(qx, x2, p8d_cephes_exp_q3);
234
235 // I don't really get this bit, copied from the SSE2 routines, so...
236 // TODO(gonnet): Figure out what is going on here, perhaps find a better
237 // rational interpolant?
238 x = _mm512_div_pd(px, psub(qx, px));
239 x = pmadd(p8d_2, x, p8d_1);
240
241 // Build e=2^n.
242 const Packet8d e = _mm512_castsi512_pd(_mm512_slli_epi64(
243 _mm512_add_epi64(_mm512_cvtpd_epi64(n), _mm512_set1_epi64(1023)), 52));
244
245 // Construct the result 2^n * exp(g) = e * x. The max is used to catch
246 // non-finite values in the input.
247 return pmax(pmul(x, e), _x);
248 }*/
249
250// Functions for sqrt.
251// The EIGEN_FAST_MATH version uses the _mm_rsqrt_ps approximation and one step
252// of Newton's method, at a cost of 1-2 bits of precision as opposed to the
253// exact solution. The main advantage of this approach is not just speed, but
254// also the fact that it can be inlined and pipelined with other computations,
255// further reducing its effective latency.
256#if EIGEN_FAST_MATH
257template <>
258EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet16f
259psqrt<Packet16f>(const Packet16f& _x) {
260 _EIGEN_DECLARE_CONST_Packet16f(one_point_five, 1.5f);
261 _EIGEN_DECLARE_CONST_Packet16f(minus_half, -0.5f);
262 _EIGEN_DECLARE_CONST_Packet16f_FROM_INT(flt_min, 0x00800000);
263
264 Packet16f neg_half = pmul(_x, p16f_minus_half);
265
266 // select only the inverse sqrt of positive normal inputs (denormals are
267 // flushed to zero and cause infs as well).
268 __mmask16 non_zero_mask = _mm512_cmp_ps_mask(_x, p16f_flt_min, _CMP_GE_OQ);
269 Packet16f x = _mm512_mask_blend_ps(non_zero_mask, _mm512_rsqrt14_ps(_x),
270 _mm512_setzero_ps());
271
272 // Do a single step of Newton's iteration.
273 x = pmul(x, pmadd(neg_half, pmul(x, x), p16f_one_point_five));
274
275 // Multiply the original _x by it's reciprocal square root to extract the
276 // square root.
277 return pmul(_x, x);
278}
279
280template <>
281EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet8d
282psqrt<Packet8d>(const Packet8d& _x) {
283 _EIGEN_DECLARE_CONST_Packet8d(one_point_five, 1.5);
284 _EIGEN_DECLARE_CONST_Packet8d(minus_half, -0.5);
285 _EIGEN_DECLARE_CONST_Packet8d_FROM_INT64(dbl_min, 0x0010000000000000LL);
286
287 Packet8d neg_half = pmul(_x, p8d_minus_half);
288
289 // select only the inverse sqrt of positive normal inputs (denormals are
290 // flushed to zero and cause infs as well).
291 __mmask8 non_zero_mask = _mm512_cmp_pd_mask(_x, p8d_dbl_min, _CMP_GE_OQ);
292 Packet8d x = _mm512_mask_blend_pd(non_zero_mask, _mm512_rsqrt14_pd(_x),
293 _mm512_setzero_pd());
294
295 // Do a first step of Newton's iteration.
296 x = pmul(x, pmadd(neg_half, pmul(x, x), p8d_one_point_five));
297
298 // Do a second step of Newton's iteration.
299 x = pmul(x, pmadd(neg_half, pmul(x, x), p8d_one_point_five));
300
301 // Multiply the original _x by it's reciprocal square root to extract the
302 // square root.
303 return pmul(_x, x);
304}
305#else
306template <>
307EIGEN_STRONG_INLINE Packet16f psqrt<Packet16f>(const Packet16f& x) {
308 return _mm512_sqrt_ps(x);
309}
310template <>
311EIGEN_STRONG_INLINE Packet8d psqrt<Packet8d>(const Packet8d& x) {
312 return _mm512_sqrt_pd(x);
313}
314#endif
315
316// Functions for rsqrt.
317// Almost identical to the sqrt routine, just leave out the last multiplication
318// and fill in NaN/Inf where needed. Note that this function only exists as an
319// iterative version for doubles since there is no instruction for diretly
320// computing the reciprocal square root in AVX-512.
321#ifdef EIGEN_FAST_MATH
322template <>
323EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet16f
324prsqrt<Packet16f>(const Packet16f& _x) {
325 _EIGEN_DECLARE_CONST_Packet16f_FROM_INT(inf, 0x7f800000);
326 _EIGEN_DECLARE_CONST_Packet16f_FROM_INT(nan, 0x7fc00000);
327 _EIGEN_DECLARE_CONST_Packet16f(one_point_five, 1.5f);
328 _EIGEN_DECLARE_CONST_Packet16f(minus_half, -0.5f);
329 _EIGEN_DECLARE_CONST_Packet16f_FROM_INT(flt_min, 0x00800000);
330
331 Packet16f neg_half = pmul(_x, p16f_minus_half);
332
333 // select only the inverse sqrt of positive normal inputs (denormals are
334 // flushed to zero and cause infs as well).
335 __mmask16 le_zero_mask = _mm512_cmp_ps_mask(_x, p16f_flt_min, _CMP_LT_OQ);
336 Packet16f x = _mm512_mask_blend_ps(le_zero_mask, _mm512_setzero_ps(),
337 _mm512_rsqrt14_ps(_x));
338
339 // Fill in NaNs and Infs for the negative/zero entries.
340 __mmask16 neg_mask = _mm512_cmp_ps_mask(_x, _mm512_setzero_ps(), _CMP_LT_OQ);
341 Packet16f infs_and_nans = _mm512_mask_blend_ps(
342 neg_mask, p16f_nan,
343 _mm512_mask_blend_ps(le_zero_mask, p16f_inf, _mm512_setzero_ps()));
344
345 // Do a single step of Newton's iteration.
346 x = pmul(x, pmadd(neg_half, pmul(x, x), p16f_one_point_five));
347
348 // Insert NaNs and Infs in all the right places.
349 return _mm512_mask_blend_ps(le_zero_mask, infs_and_nans, x);
350}
351
352template <>
353EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet8d
354prsqrt<Packet8d>(const Packet8d& _x) {
355 _EIGEN_DECLARE_CONST_Packet8d_FROM_INT64(inf, 0x7ff0000000000000LL);
356 _EIGEN_DECLARE_CONST_Packet8d_FROM_INT64(nan, 0x7ff1000000000000LL);
357 _EIGEN_DECLARE_CONST_Packet8d(one_point_five, 1.5);
358 _EIGEN_DECLARE_CONST_Packet8d(minus_half, -0.5);
359 _EIGEN_DECLARE_CONST_Packet8d_FROM_INT64(dbl_min, 0x0010000000000000LL);
360
361 Packet8d neg_half = pmul(_x, p8d_minus_half);
362
363 // select only the inverse sqrt of positive normal inputs (denormals are
364 // flushed to zero and cause infs as well).
365 __mmask8 le_zero_mask = _mm512_cmp_pd_mask(_x, p8d_dbl_min, _CMP_LT_OQ);
366 Packet8d x = _mm512_mask_blend_pd(le_zero_mask, _mm512_setzero_pd(),
367 _mm512_rsqrt14_pd(_x));
368
369 // Fill in NaNs and Infs for the negative/zero entries.
370 __mmask8 neg_mask = _mm512_cmp_pd_mask(_x, _mm512_setzero_pd(), _CMP_LT_OQ);
371 Packet8d infs_and_nans = _mm512_mask_blend_pd(
372 neg_mask, p8d_nan,
373 _mm512_mask_blend_pd(le_zero_mask, p8d_inf, _mm512_setzero_pd()));
374
375 // Do a first step of Newton's iteration.
376 x = pmul(x, pmadd(neg_half, pmul(x, x), p8d_one_point_five));
377
378 // Do a second step of Newton's iteration.
379 x = pmul(x, pmadd(neg_half, pmul(x, x), p8d_one_point_five));
380
381 // Insert NaNs and Infs in all the right places.
382 return _mm512_mask_blend_pd(le_zero_mask, infs_and_nans, x);
383}
384#else
385template <>
386EIGEN_STRONG_INLINE Packet16f prsqrt<Packet16f>(const Packet16f& x) {
387 return _mm512_rsqrt28_ps(x);
388}
389#endif
390#endif
391
392} // end namespace internal
393
394} // end namespace Eigen
395
396#endif // THIRD_PARTY_EIGEN3_EIGEN_SRC_CORE_ARCH_AVX512_MATHFUNCTIONS_H_
Namespace containing all symbols from the Eigen library.
Definition: Core:287