1 // Auto-generated file. Do not edit!
2 // Template: src/f32-sigmoid/avx2-p5.c.in
3 // Generator: tools/xngen
4 //
5 // Copyright 2019 Google LLC
6 //
7 // This source code is licensed under the BSD-style license found in the
8 // LICENSE file in the root directory of this source tree.
9
10 #include <assert.h>
11
12 #include <immintrin.h>
13
14 #include <xnnpack/common.h>
15 #include <xnnpack/vunary.h>
16
17
18 static const int32_t mask_table[14] = {-1, -1, -1, -1, -1, -1, -1, 0, 0, 0, 0, 0, 0, 0};
19
xnn_f32_sigmoid_ukernel__avx2_rr1_p5_nr2fma_x64(size_t n,const float * x,float * y,const void * params)20 void xnn_f32_sigmoid_ukernel__avx2_rr1_p5_nr2fma_x64(
21 size_t n,
22 const float* x,
23 float* y,
24 const void* params)
25 {
26 assert(n % sizeof(float) == 0);
27
28 const __m256 vmagic_bias = _mm256_set1_ps(0x1.8000FEp23f);
29 // The smallest x for which sigmoidf(x) is normalized.
30 // This number is also the smallest x for which expf(x) is normalized.
31 const __m256 vdenorm_cutoff = _mm256_set1_ps(-0x1.5D589Ep+6f);
32 const __m256 vlog2e = _mm256_set1_ps(0x1.715476p+0f);
33 const __m256 vminus_ln2 = _mm256_set1_ps(-0x1.62E43p-1f);
34 const __m256 vone = _mm256_set1_ps(1.0f);
35 const __m256 vsign_mask = _mm256_set1_ps(-0.0f);
36
37 const __m256 vc1 = _mm256_set1_ps(0x1.FFFFF6p-1f);
38 const __m256 vc2 = _mm256_set1_ps(0x1.FFFDC6p-2f);
39 const __m256 vc3 = _mm256_set1_ps(0x1.555A80p-3f);
40 const __m256 vc4 = _mm256_set1_ps(0x1.573A1Ap-5f);
41 const __m256 vc5 = _mm256_set1_ps(0x1.0F9F9Cp-7f);
42
43 for (; n >= 64 * sizeof(float); n -= 64 * sizeof(float)) {
44 const __m256 vx0 = _mm256_loadu_ps(x);
45 const __m256 vx1 = _mm256_loadu_ps(x + 8);
46 const __m256 vx2 = _mm256_loadu_ps(x + 16);
47 const __m256 vx3 = _mm256_loadu_ps(x + 24);
48 const __m256 vx4 = _mm256_loadu_ps(x + 32);
49 const __m256 vx5 = _mm256_loadu_ps(x + 40);
50 const __m256 vx6 = _mm256_loadu_ps(x + 48);
51 const __m256 vx7 = _mm256_loadu_ps(x + 56);
52 x += 64;
53
54 // General structure of the algorithm:
55 // / exp(x) / (1 + exp(x)) if x <= 0
56 // f[x] :=
57 // \ 1 - f[-x] if x >= 0
58 //
59 // First we compute f[z] := exp(z) / (1 + exp(z)) where z = -abs(x),
60 // then replace result with 1 - f[z] if x >= 0.
61 const __m256 vz0 = _mm256_or_ps(vx0, vsign_mask);
62 const __m256 vz1 = _mm256_or_ps(vx1, vsign_mask);
63 const __m256 vz2 = _mm256_or_ps(vx2, vsign_mask);
64 const __m256 vz3 = _mm256_or_ps(vx3, vsign_mask);
65 const __m256 vz4 = _mm256_or_ps(vx4, vsign_mask);
66 const __m256 vz5 = _mm256_or_ps(vx5, vsign_mask);
67 const __m256 vz6 = _mm256_or_ps(vx6, vsign_mask);
68 const __m256 vz7 = _mm256_or_ps(vx7, vsign_mask);
69
70 // Compute reduced argument n := round(z / log(2)).
71 // We do it by adding a large number (magic bias) to the product z * (1/log(2)), which cause rounding of the result
72 // to an integer, then subtracing the large number back. The trick with adding large number is valid only within
73 // certain bounds (|x| <= 2**22), but thats ok, because inputs x outside of [-87.336544, 17.328678] (i.e. z outsize
74 // [0, 87.336544]) underflow or saturate sigmoidf(x) anyway. We fixup the result for such inputs at the very end of
75 // the algorithm.
76 __m256 vn0 = _mm256_fmadd_ps(vz0, vlog2e, vmagic_bias);
77 __m256 vn1 = _mm256_fmadd_ps(vz1, vlog2e, vmagic_bias);
78 __m256 vn2 = _mm256_fmadd_ps(vz2, vlog2e, vmagic_bias);
79 __m256 vn3 = _mm256_fmadd_ps(vz3, vlog2e, vmagic_bias);
80 __m256 vn4 = _mm256_fmadd_ps(vz4, vlog2e, vmagic_bias);
81 __m256 vn5 = _mm256_fmadd_ps(vz5, vlog2e, vmagic_bias);
82 __m256 vn6 = _mm256_fmadd_ps(vz6, vlog2e, vmagic_bias);
83 __m256 vn7 = _mm256_fmadd_ps(vz7, vlog2e, vmagic_bias);
84
85 // Create a floating-point number s (scale) such that s == 2**n for inputs which don't cause underflow, i.e.
86 // -87.33642 <= z <= 0.0, and -126 <= n <= 0 accordingly.
87 const __m256 vs0 = _mm256_castsi256_ps(_mm256_slli_epi32(_mm256_castps_si256(vn0), 23));
88 const __m256 vs1 = _mm256_castsi256_ps(_mm256_slli_epi32(_mm256_castps_si256(vn1), 23));
89 const __m256 vs2 = _mm256_castsi256_ps(_mm256_slli_epi32(_mm256_castps_si256(vn2), 23));
90 const __m256 vs3 = _mm256_castsi256_ps(_mm256_slli_epi32(_mm256_castps_si256(vn3), 23));
91 const __m256 vs4 = _mm256_castsi256_ps(_mm256_slli_epi32(_mm256_castps_si256(vn4), 23));
92 const __m256 vs5 = _mm256_castsi256_ps(_mm256_slli_epi32(_mm256_castps_si256(vn5), 23));
93 const __m256 vs6 = _mm256_castsi256_ps(_mm256_slli_epi32(_mm256_castps_si256(vn6), 23));
94 const __m256 vs7 = _mm256_castsi256_ps(_mm256_slli_epi32(_mm256_castps_si256(vn7), 23));
95
96 // Subtract the large number back to get final n := round(z / log(2)).
97 vn0 = _mm256_sub_ps(vn0, vmagic_bias);
98 vn1 = _mm256_sub_ps(vn1, vmagic_bias);
99 vn2 = _mm256_sub_ps(vn2, vmagic_bias);
100 vn3 = _mm256_sub_ps(vn3, vmagic_bias);
101 vn4 = _mm256_sub_ps(vn4, vmagic_bias);
102 vn5 = _mm256_sub_ps(vn5, vmagic_bias);
103 vn6 = _mm256_sub_ps(vn6, vmagic_bias);
104 vn7 = _mm256_sub_ps(vn7, vmagic_bias);
105
106 // Compute reduced argument t := z - n * log(2).
107 __m256 vt0 = _mm256_fmadd_ps(vn0, vminus_ln2, vz0);
108 __m256 vt1 = _mm256_fmadd_ps(vn1, vminus_ln2, vz1);
109 __m256 vt2 = _mm256_fmadd_ps(vn2, vminus_ln2, vz2);
110 __m256 vt3 = _mm256_fmadd_ps(vn3, vminus_ln2, vz3);
111 __m256 vt4 = _mm256_fmadd_ps(vn4, vminus_ln2, vz4);
112 __m256 vt5 = _mm256_fmadd_ps(vn5, vminus_ln2, vz5);
113 __m256 vt6 = _mm256_fmadd_ps(vn6, vminus_ln2, vz6);
114 __m256 vt7 = _mm256_fmadd_ps(vn7, vminus_ln2, vz7);
115
116 // Compute degree-5 polynomial approxiatmion for exp(t) on [-log(2)/2, log(2)/2].
117 __m256 vp0 = _mm256_fmadd_ps(vc5, vt0, vc4);
118 __m256 vp1 = _mm256_fmadd_ps(vc5, vt1, vc4);
119 __m256 vp2 = _mm256_fmadd_ps(vc5, vt2, vc4);
120 __m256 vp3 = _mm256_fmadd_ps(vc5, vt3, vc4);
121 __m256 vp4 = _mm256_fmadd_ps(vc5, vt4, vc4);
122 __m256 vp5 = _mm256_fmadd_ps(vc5, vt5, vc4);
123 __m256 vp6 = _mm256_fmadd_ps(vc5, vt6, vc4);
124 __m256 vp7 = _mm256_fmadd_ps(vc5, vt7, vc4);
125
126 vp0 = _mm256_fmadd_ps(vp0, vt0, vc3);
127 vp1 = _mm256_fmadd_ps(vp1, vt1, vc3);
128 vp2 = _mm256_fmadd_ps(vp2, vt2, vc3);
129 vp3 = _mm256_fmadd_ps(vp3, vt3, vc3);
130 vp4 = _mm256_fmadd_ps(vp4, vt4, vc3);
131 vp5 = _mm256_fmadd_ps(vp5, vt5, vc3);
132 vp6 = _mm256_fmadd_ps(vp6, vt6, vc3);
133 vp7 = _mm256_fmadd_ps(vp7, vt7, vc3);
134
135 vp0 = _mm256_fmadd_ps(vp0, vt0, vc2);
136 vp1 = _mm256_fmadd_ps(vp1, vt1, vc2);
137 vp2 = _mm256_fmadd_ps(vp2, vt2, vc2);
138 vp3 = _mm256_fmadd_ps(vp3, vt3, vc2);
139 vp4 = _mm256_fmadd_ps(vp4, vt4, vc2);
140 vp5 = _mm256_fmadd_ps(vp5, vt5, vc2);
141 vp6 = _mm256_fmadd_ps(vp6, vt6, vc2);
142 vp7 = _mm256_fmadd_ps(vp7, vt7, vc2);
143
144 vp0 = _mm256_fmadd_ps(vp0, vt0, vc1);
145 vp1 = _mm256_fmadd_ps(vp1, vt1, vc1);
146 vp2 = _mm256_fmadd_ps(vp2, vt2, vc1);
147 vp3 = _mm256_fmadd_ps(vp3, vt3, vc1);
148 vp4 = _mm256_fmadd_ps(vp4, vt4, vc1);
149 vp5 = _mm256_fmadd_ps(vp5, vt5, vc1);
150 vp6 = _mm256_fmadd_ps(vp6, vt6, vc1);
151 vp7 = _mm256_fmadd_ps(vp7, vt7, vc1);
152
153 // Reconstruct the exp(z) value:
154 // e = s * (1 + t * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5)))))
155 // = s + (t * s) * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5))))
156 // = s + (t * s) * p
157 vt0 = _mm256_mul_ps(vt0, vs0);
158 vt1 = _mm256_mul_ps(vt1, vs1);
159 vt2 = _mm256_mul_ps(vt2, vs2);
160 vt3 = _mm256_mul_ps(vt3, vs3);
161 vt4 = _mm256_mul_ps(vt4, vs4);
162 vt5 = _mm256_mul_ps(vt5, vs5);
163 vt6 = _mm256_mul_ps(vt6, vs6);
164 vt7 = _mm256_mul_ps(vt7, vs7);
165
166 const __m256 ve0 = _mm256_fmadd_ps(vt0, vp0, vs0);
167 const __m256 ve1 = _mm256_fmadd_ps(vt1, vp1, vs1);
168 const __m256 ve2 = _mm256_fmadd_ps(vt2, vp2, vs2);
169 const __m256 ve3 = _mm256_fmadd_ps(vt3, vp3, vs3);
170 const __m256 ve4 = _mm256_fmadd_ps(vt4, vp4, vs4);
171 const __m256 ve5 = _mm256_fmadd_ps(vt5, vp5, vs5);
172 const __m256 ve6 = _mm256_fmadd_ps(vt6, vp6, vs6);
173 const __m256 ve7 = _mm256_fmadd_ps(vt7, vp7, vs7);
174
175 // Denominator of the sigmoid fraction: 1.0 + exp(z)
176 const __m256 vd0 = _mm256_add_ps(ve0, vone);
177 const __m256 vd1 = _mm256_add_ps(ve1, vone);
178 const __m256 vd2 = _mm256_add_ps(ve2, vone);
179 const __m256 vd3 = _mm256_add_ps(ve3, vone);
180 const __m256 vd4 = _mm256_add_ps(ve4, vone);
181 const __m256 vd5 = _mm256_add_ps(ve5, vone);
182 const __m256 vd6 = _mm256_add_ps(ve6, vone);
183 const __m256 vd7 = _mm256_add_ps(ve7, vone);
184
185 // Use Newton-Raphson method to compute reciprocal of denominator.
186 // Note: 1 < d <= 2, because z >= 0.0 and 0 < exp(-z) <= 1.0.
187 // Thus the reciprocal of the denominator never overflows.
188 __m256 vr0 = _mm256_rcp_ps(vd0);
189 __m256 vr1 = _mm256_rcp_ps(vd1);
190 __m256 vr2 = _mm256_rcp_ps(vd2);
191 __m256 vr3 = _mm256_rcp_ps(vd3);
192 __m256 vr4 = _mm256_rcp_ps(vd4);
193 __m256 vr5 = _mm256_rcp_ps(vd5);
194 __m256 vr6 = _mm256_rcp_ps(vd6);
195 __m256 vr7 = _mm256_rcp_ps(vd7);
196
197 vr0 = _mm256_fmadd_ps(_mm256_fnmadd_ps(vr0, vd0, vone), vr0, vr0);
198 vr1 = _mm256_fmadd_ps(_mm256_fnmadd_ps(vr1, vd1, vone), vr1, vr1);
199 vr2 = _mm256_fmadd_ps(_mm256_fnmadd_ps(vr2, vd2, vone), vr2, vr2);
200 vr3 = _mm256_fmadd_ps(_mm256_fnmadd_ps(vr3, vd3, vone), vr3, vr3);
201 vr4 = _mm256_fmadd_ps(_mm256_fnmadd_ps(vr4, vd4, vone), vr4, vr4);
202 vr5 = _mm256_fmadd_ps(_mm256_fnmadd_ps(vr5, vd5, vone), vr5, vr5);
203 vr6 = _mm256_fmadd_ps(_mm256_fnmadd_ps(vr6, vd6, vone), vr6, vr6);
204 vr7 = _mm256_fmadd_ps(_mm256_fnmadd_ps(vr7, vd7, vone), vr7, vr7);
205
206 vr0 = _mm256_fmadd_ps(_mm256_fnmadd_ps(vr0, vd0, vone), vr0, vr0);
207 vr1 = _mm256_fmadd_ps(_mm256_fnmadd_ps(vr1, vd1, vone), vr1, vr1);
208 vr2 = _mm256_fmadd_ps(_mm256_fnmadd_ps(vr2, vd2, vone), vr2, vr2);
209 vr3 = _mm256_fmadd_ps(_mm256_fnmadd_ps(vr3, vd3, vone), vr3, vr3);
210 vr4 = _mm256_fmadd_ps(_mm256_fnmadd_ps(vr4, vd4, vone), vr4, vr4);
211 vr5 = _mm256_fmadd_ps(_mm256_fnmadd_ps(vr5, vd5, vone), vr5, vr5);
212 vr6 = _mm256_fmadd_ps(_mm256_fnmadd_ps(vr6, vd6, vone), vr6, vr6);
213 vr7 = _mm256_fmadd_ps(_mm256_fnmadd_ps(vr7, vd7, vone), vr7, vr7);
214
215 // Reconstruct sigmoid(z) = exp(z) * recip(1.0 + exp(z))
216 __m256 vf0 = _mm256_mul_ps(ve0, vr0);
217 __m256 vf1 = _mm256_mul_ps(ve1, vr1);
218 __m256 vf2 = _mm256_mul_ps(ve2, vr2);
219 __m256 vf3 = _mm256_mul_ps(ve3, vr3);
220 __m256 vf4 = _mm256_mul_ps(ve4, vr4);
221 __m256 vf5 = _mm256_mul_ps(ve5, vr5);
222 __m256 vf6 = _mm256_mul_ps(ve6, vr6);
223 __m256 vf7 = _mm256_mul_ps(ve7, vr7);
224
225 // For inputs below denormal cutoff, replace output with +0.0f.
226 // Note that for NaN inputs, comparison result is false, and outputs are left unchanged.
227 vf0 = _mm256_andnot_ps(_mm256_cmp_ps(vz0, vdenorm_cutoff, _CMP_LT_OS), vf0);
228 vf1 = _mm256_andnot_ps(_mm256_cmp_ps(vz1, vdenorm_cutoff, _CMP_LT_OS), vf1);
229 vf2 = _mm256_andnot_ps(_mm256_cmp_ps(vz2, vdenorm_cutoff, _CMP_LT_OS), vf2);
230 vf3 = _mm256_andnot_ps(_mm256_cmp_ps(vz3, vdenorm_cutoff, _CMP_LT_OS), vf3);
231 vf4 = _mm256_andnot_ps(_mm256_cmp_ps(vz4, vdenorm_cutoff, _CMP_LT_OS), vf4);
232 vf5 = _mm256_andnot_ps(_mm256_cmp_ps(vz5, vdenorm_cutoff, _CMP_LT_OS), vf5);
233 vf6 = _mm256_andnot_ps(_mm256_cmp_ps(vz6, vdenorm_cutoff, _CMP_LT_OS), vf6);
234 vf7 = _mm256_andnot_ps(_mm256_cmp_ps(vz7, vdenorm_cutoff, _CMP_LT_OS), vf7);
235
236 // Reconstruct sigmoid(x) = x < 0 ? sigmoid(z) : 1.0 - sigmoid(z)
237 vf0 = _mm256_blendv_ps(_mm256_sub_ps(vone, vf0), vf0, vx0);
238 vf1 = _mm256_blendv_ps(_mm256_sub_ps(vone, vf1), vf1, vx1);
239 vf2 = _mm256_blendv_ps(_mm256_sub_ps(vone, vf2), vf2, vx2);
240 vf3 = _mm256_blendv_ps(_mm256_sub_ps(vone, vf3), vf3, vx3);
241 vf4 = _mm256_blendv_ps(_mm256_sub_ps(vone, vf4), vf4, vx4);
242 vf5 = _mm256_blendv_ps(_mm256_sub_ps(vone, vf5), vf5, vx5);
243 vf6 = _mm256_blendv_ps(_mm256_sub_ps(vone, vf6), vf6, vx6);
244 vf7 = _mm256_blendv_ps(_mm256_sub_ps(vone, vf7), vf7, vx7);
245
246 _mm256_storeu_ps(y, vf0);
247 _mm256_storeu_ps(y + 8, vf1);
248 _mm256_storeu_ps(y + 16, vf2);
249 _mm256_storeu_ps(y + 24, vf3);
250 _mm256_storeu_ps(y + 32, vf4);
251 _mm256_storeu_ps(y + 40, vf5);
252 _mm256_storeu_ps(y + 48, vf6);
253 _mm256_storeu_ps(y + 56, vf7);
254 y += 64;
255 }
256 for (; n >= 8 * sizeof(float); n -= 8 * sizeof(float)) {
257 const __m256 vx = _mm256_loadu_ps(x);
258 x += 8;
259
260 // General structure of the algorithm:
261 // / exp(x) / (1 + exp(x)) if x <= 0
262 // f[x] :=
263 // \ 1 - f[-x] if x >= 0
264 //
265 // First we compute f[z] := exp(z) / (1 + exp(z)) where z = -abs(x),
266 // then replace result with 1 - f[z] if x >= 0.
267 const __m256 vz = _mm256_or_ps(vx, vsign_mask);
268
269 // Compute reduced argument n := round(z / log(2)).
270 // We do it by adding a large number (magic bias) to the product z * (1/log(2)), which cause rounding of the result
271 // to an integer, then subtracing the large number back. The trick with adding large number is valid only within
272 // certain bounds (|x| <= 2**22), but thats ok, because inputs x outside of [-87.336544, 17.328678] (i.e. z outsize
273 // [0, 87.336544]) underflow or saturate sigmoidf(x) anyway. We fixup the result for such inputs at the very end of
274 // the algorithm.
275 __m256 vn = _mm256_fmadd_ps(vz, vlog2e, vmagic_bias);
276
277 // Create a floating-point number s (scale) such that s == 2**n for inputs which don't cause underflow, i.e.
278 // -87.33642 <= z <= 0.0, and -126 <= n <= 0 accordingly.
279 const __m256 vs = _mm256_castsi256_ps(_mm256_slli_epi32(_mm256_castps_si256(vn), 23));
280
281 // Subtract the large number back to get final n := round(z / log(2)).
282 vn = _mm256_sub_ps(vn, vmagic_bias);
283
284 // Compute reduced argument t := z - n * log(2).
285 __m256 vt = _mm256_fmadd_ps(vn, vminus_ln2, vz);
286
287 // Compute degree-5 polynomial approxiatmion for exp(t) on [-log(2)/2, log(2)/2].
288 __m256 vp = _mm256_fmadd_ps(vc5, vt, vc4);
289 vp = _mm256_fmadd_ps(vp, vt, vc3);
290 vp = _mm256_fmadd_ps(vp, vt, vc2);
291 vp = _mm256_fmadd_ps(vp, vt, vc1);
292
293 // Reconstruct the exp(z) value:
294 // e = s * (1 + t * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5)))))
295 // = s + (t * s) * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5))))
296 // = s + (t * s) * p
297 vt = _mm256_mul_ps(vt, vs);
298 const __m256 ve = _mm256_fmadd_ps(vt, vp, vs);
299
300 // Denominator of the sigmoid fraction: 1.0 + exp(z)
301 const __m256 vd = _mm256_add_ps(ve, vone);
302
303 // Use Newton-Raphson method to compute reciprocal of denominator.
304 // Note: 1 < d <= 2, because z >= 0.0 and 0 < exp(-z) <= 1.0.
305 // Thus the reciprocal of the denominator never overflows.
306 __m256 vr = _mm256_rcp_ps(vd);
307 vr = _mm256_fmadd_ps(_mm256_fnmadd_ps(vr, vd, vone), vr, vr);
308 vr = _mm256_fmadd_ps(_mm256_fnmadd_ps(vr, vd, vone), vr, vr);
309
310 // Reconstruct sigmoid(z) = exp(z) * recip(1.0 + exp(z))
311 __m256 vf = _mm256_mul_ps(ve, vr);
312
313 // For inputs below denormal cutoff, replace output with +0.0f.
314 // Note that for NaN inputs, comparison result is false, and outputs are left unchanged.
315 vf = _mm256_andnot_ps(_mm256_cmp_ps(vz, vdenorm_cutoff, _CMP_LT_OS), vf);
316
317 // Reconstruct sigmoid(x) = x < 0 ? sigmoid(z) : 1.0 - sigmoid(z)
318 vf = _mm256_blendv_ps(_mm256_sub_ps(vone, vf), vf, vx);
319
320 _mm256_storeu_ps(y, vf);
321 y += 8;
322 }
323 if XNN_UNLIKELY(n != 0) {
324 assert(n >= 1 * sizeof(float));
325 assert(n <= 7 * sizeof(float));
326 __m256i vmask = _mm256_loadu_si256((const __m256i*) ((uintptr_t) &mask_table[7] - n));
327
328 const __m256 vx = _mm256_maskload_ps(x, vmask);
329
330 // General structure of the algorithm:
331 // / exp(x) / (1 + exp(x)) if x <= 0
332 // f[x] :=
333 // \ 1 - f[-x] if x >= 0
334 //
335 // First we compute f[z] := exp(z) / (1 + exp(z)) where z = -abs(x),
336 // then replace result with 1 - f[z] if x >= 0.
337 const __m256 vz = _mm256_or_ps(vx, vsign_mask);
338
339 // Compute reduced argument n := round(z / log(2)).
340 // We do it by adding a large number (magic bias) to the product z * (1/log(2)), which cause rounding of the result
341 // to an integer, then subtracing the large number back. The trick with adding large number is valid only within
342 // certain bounds (|x| <= 2**22), but thats ok, because inputs x outside of [-87.336544, 17.328678] (i.e. z outsize
343 // [0, 87.336544]) underflow or saturate sigmoidf(x) anyway. We fixup the result for such inputs at the very end of
344 // the algorithm.
345 __m256 vn = _mm256_fmadd_ps(vz, vlog2e, vmagic_bias);
346
347 // Create a floating-point number s (scale) such that s == 2**n for inputs which don't cause underflow, i.e.
348 // -87.33642 <= z <= 0.0, and -126 <= n <= 0 accordingly.
349 const __m256 vs = _mm256_castsi256_ps(_mm256_slli_epi32(_mm256_castps_si256(vn), 23));
350
351 // Subtract the large number back to get final n := round(z / log(2)).
352 vn = _mm256_sub_ps(vn, vmagic_bias);
353
354 // Compute reduced argument t := z - n * log(2).
355 __m256 vt = _mm256_fmadd_ps(vn, vminus_ln2, vz);
356
357 // Compute degree-5 polynomial approxiatmion for exp(t) on [-log(2)/2, log(2)/2].
358 __m256 vp = _mm256_fmadd_ps(vc5, vt, vc4);
359 vp = _mm256_fmadd_ps(vp, vt, vc3);
360 vp = _mm256_fmadd_ps(vp, vt, vc2);
361 vp = _mm256_fmadd_ps(vp, vt, vc1);
362
363 // Reconstruct the exp(z) value:
364 // e = s * (1 + t * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5)))))
365 // = s + (t * s) * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5))))
366 // = s + (t * s) * p
367 vt = _mm256_mul_ps(vt, vs);
368 const __m256 ve = _mm256_fmadd_ps(vt, vp, vs);
369
370 // Denominator of the sigmoid fraction: 1.0 + exp(z)
371 const __m256 vd = _mm256_add_ps(ve, vone);
372
373 // Use Newton-Raphson method to compute reciprocal of denominator.
374 // Note: 1 < d <= 2, because z >= 0.0 and 0 < exp(-z) <= 1.0.
375 // Thus the reciprocal of the denominator never overflows.
376 __m256 vr = _mm256_rcp_ps(vd);
377 vr = _mm256_fmadd_ps(_mm256_fnmadd_ps(vr, vd, vone), vr, vr);
378 vr = _mm256_fmadd_ps(_mm256_fnmadd_ps(vr, vd, vone), vr, vr);
379
380 // Reconstruct sigmoid(z) = exp(z) * recip(1.0 + exp(z))
381 __m256 vf = _mm256_mul_ps(ve, vr);
382
383 // For inputs below denormal cutoff, replace output with +0.0f.
384 // Note that for NaN inputs, comparison result is false, and outputs are left unchanged.
385 vf = _mm256_andnot_ps(_mm256_cmp_ps(vz, vdenorm_cutoff, _CMP_LT_OS), vf);
386
387 // Reconstruct sigmoid(x) = x < 0 ? sigmoid(z) : 1.0 - sigmoid(z)
388 vf = _mm256_blendv_ps(_mm256_sub_ps(vone, vf), vf, vx);
389
390 // _mm256_maskstore_ps(y, vmask, vf) could be used here, but triggers msan failures (probably an msan bug).
391 __m128 vf_lo = _mm256_castps256_ps128(vf);
392 if (n & (4 * sizeof(float))) {
393 _mm_storeu_ps(y, vf_lo);
394 vf_lo = _mm256_extractf128_ps(vf, 1);
395 y += 4;
396 }
397 if (n & (2 * sizeof(float))) {
398 _mm_storel_pi((__m64*) y, vf_lo);
399 vf_lo = _mm_movehl_ps(vf_lo, vf_lo);
400 y += 2;
401 }
402 if (n & (1 * sizeof(float))) {
403 _mm_store_ss(y, vf_lo);
404 }
405 }
406 }
407