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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