1 // Auto-generated file. Do not edit!
2 // Template: src/f32-sigmoid/neon-lut2048-p1.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 <arm_neon.h>
13
14 #include <xnnpack/common.h>
15 #include <xnnpack/vunary.h>
16
17
18 extern XNN_INTERNAL const float xnn_table_exp2_k_over_2048[2048];
19
xnn_f32_sigmoid_ukernel__neonfma_rr1_lut2048_p1_nr1recps1fma_x8(size_t n,const float * x,float * y,const void * params)20 void xnn_f32_sigmoid_ukernel__neonfma_rr1_lut2048_p1_nr1recps1fma_x8(
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 float32x4_t vmagic_bias = vmovq_n_f32(0x1.800000p23f);
29 // The largest z for which sigmoidf(-z) is normalized.
30 // This number is also the largest z for which expf(-z) is normalized.
31 const float32x4_t vdenorm_cutoff = vmovq_n_f32(0x1.5D589Ep+6f);
32 const float32x4_t vminus_log2e_x2048 = vmovq_n_f32(-0x1.715476p11f);
33 const float32x4_t vln2_o2048 = vmovq_n_f32(0x1.62E43p-12f);
34 const float32x4_t vone = vmovq_n_f32(1.0f);
35
36 const float32x4_t vc1 = vmovq_n_f32(-0x1.FFFFFEp-1f);
37
38 const int32x4_t vindex_mask = vmovq_n_s32(INT32_C(0x7FF));
39
40 for (; n >= 8 * sizeof(float); n -= 8 * sizeof(float)) {
41 const float32x4_t vx0123 = vld1q_f32(x); x += 4;
42 const float32x4_t vx4567 = vld1q_f32(x); x += 4;
43
44 // General structure of the algorithm:
45 // / exp(x) / (1 + exp(x)) if x <= 0
46 // f[x] :=
47 // \ 1 - f[-x] if x >= 0
48 //
49 // First we compute f[-z] := exp(-z) / (1 + exp(-z)) where z = abs(x),
50 // then replace result with 1 - f[-z] if x >= 0.
51 const float32x4_t vz0123 = vabsq_f32(vx0123);
52 const float32x4_t vz4567 = vabsq_f32(vx4567);
53
54 // Compute reduced argument n := round(-z * 2048 / log(2)).
55 // We do it by adding a large number (magic bias), which cause rounding of the result to an integer, then subtracing
56 // the large number back. The first addition is combined with multiplication by log2e into a single FMA instruction.
57 // The trick with adding large number is valid only within certain bounds (|z * 2048 / log(2)| <= 2**22, i.e.
58 // |z| <= 0x1.62E43p+10 = 1419.5654296875), but that is acceptable, because inputs x outside of
59 // [-87.336544, 17.328678] (i.e. z outsize [0, 87.336544]) underflow or saturate sigmoidf(x). We fixup the result
60 // for such inputs at the very end of the algorithm.
61 float32x4_t vn0123 = vfmaq_f32(vmagic_bias, vz0123, vminus_log2e_x2048);
62 float32x4_t vn4567 = vfmaq_f32(vmagic_bias, vz4567, vminus_log2e_x2048);
63
64 // Create a floating-point number s (scale) such that s := 2**(n / 2048) for such inputs that sigmoidf(-z) is
65 // normalized, i.e. 0 <= z <= 87.33642. As n has 11 fractional bits, we split s == 2**(n / 2048) =
66 // = 2**e * 2**(n / 2048 - e), where e := int(n / 2048). We create s in two steps:
67 // 1. Fetch 2**(n / 2048 - e) = 2**(n % 2048) from the table using the 6 low bits of n, as integer. Note that the
68 // fetched values are in the [1.0, 2.0) range, i.e. their floating-point exponent is 0.
69 // 2. Adjust fecthed value by addition of e to its floating-point exponent. The result is always a normalized
70 // number, because for 0 <= z <= 87.33642 (inputs for which sigmoidf(-z) is normalized) we have -126 <= e <= 0,
71 // and thus the adjusted exponent is not lower than -126.
72 //
73 // Extract e from bits 11:19 of n and shift it into bits 23:31 (position of floating-point exponent).
74 const int32x4_t ve0123 = vshlq_n_s32(vbicq_s32(vreinterpretq_s32_f32(vn0123), vmovq_n_s32(INT32_C(0x7FF))), 12);
75 const int32x4_t ve4567 = vshlq_n_s32(vbicq_s32(vreinterpretq_s32_f32(vn4567), vmovq_n_s32(INT32_C(0x7FF))), 12);
76
77 // Use bits 0:11 bits of n, as integer, as an index for table lookup of l := 2**(n % 2048).
78 const uint64x2_t vidx0123 = vreinterpretq_u64_s32(vandq_s32(vreinterpretq_s32_f32(vn0123), vindex_mask));
79 const uint64x2_t vidx4567 = vreinterpretq_u64_s32(vandq_s32(vreinterpretq_s32_f32(vn4567), vindex_mask));
80
81 const uint64_t vidx01 = vgetq_lane_u64(vidx0123, 0);
82 const uint64_t vidx23 = vgetq_lane_u64(vidx0123, 1);
83 float32x2_t vl01 = vld1_dup_f32(&xnn_table_exp2_k_over_2048[(uint32_t) vidx01]);
84 float32x2_t vl23 = vld1_dup_f32(&xnn_table_exp2_k_over_2048[(uint32_t) vidx23]);
85 const uint64_t vidx45 = vgetq_lane_u64(vidx4567, 0);
86 const uint64_t vidx67 = vgetq_lane_u64(vidx4567, 1);
87 float32x2_t vl45 = vld1_dup_f32(&xnn_table_exp2_k_over_2048[(uint32_t) vidx45]);
88 float32x2_t vl67 = vld1_dup_f32(&xnn_table_exp2_k_over_2048[(uint32_t) vidx67]);
89
90 vl01 = vld1_lane_f32(&xnn_table_exp2_k_over_2048[(uint32_t) (vidx01 >> 32)], vl01, 1);
91 vl23 = vld1_lane_f32(&xnn_table_exp2_k_over_2048[(uint32_t) (vidx23 >> 32)], vl23, 1);
92 const float32x4_t vl0123 = vcombine_f32(vl01, vl23);
93 vl45 = vld1_lane_f32(&xnn_table_exp2_k_over_2048[(uint32_t) (vidx45 >> 32)], vl45, 1);
94 vl67 = vld1_lane_f32(&xnn_table_exp2_k_over_2048[(uint32_t) (vidx67 >> 32)], vl67, 1);
95 const float32x4_t vl4567 = vcombine_f32(vl45, vl67);
96
97 // Adjust exponent of the value l fetched from the table to get the final s value.
98 const float32x4_t vs0123 = vreinterpretq_f32_s32(vaddq_s32(vreinterpretq_s32_f32(vl0123), ve0123));
99 const float32x4_t vs4567 = vreinterpretq_f32_s32(vaddq_s32(vreinterpretq_s32_f32(vl4567), ve4567));
100
101 // Subtract the large number back to get the final n := round(-z * 2048 / log(2)) as a floating-point number.
102 vn0123 = vsubq_f32(vn0123, vmagic_bias);
103 vn4567 = vsubq_f32(vn4567, vmagic_bias);
104
105 // Compute reduced argument t := (z + n * log(2) / 2048). Note that -t = -z - n * log(2) / 2048.
106 float32x4_t vt0123 = vfmaq_f32(vz0123, vn0123, vln2_o2048);
107 float32x4_t vt4567 = vfmaq_f32(vz4567, vn4567, vln2_o2048);
108
109 // Compute degree-1 polynomial approximation for exp(-t) on [-log(2)/2048, log(2)/2048]:
110 // P1(t) = 1 + t * c1
111 const float32x4_t vp0123 = vmulq_f32(vt0123, vc1);
112 const float32x4_t vp4567 = vmulq_f32(vt4567, vc1);
113
114 // Reconstruct the exp(-z) value:
115 // y = s * (1 + t * c1)
116 // = s + s * (t * c1))
117 // = s + s * p
118 const float32x4_t vy0123 = vfmaq_f32(vs0123, vs0123, vp0123);
119 const float32x4_t vy4567 = vfmaq_f32(vs4567, vs4567, vp4567);
120
121 // Denominator of the sigmoid fraction: 1.0 + exp(-z)
122 const float32x4_t vd0123 = vaddq_f32(vy0123, vone);
123 const float32x4_t vd4567 = vaddq_f32(vy4567, vone);
124
125 // Use Newton-Raphson method (2 iterations) to compute reciprocal of denominator.
126 // Note: 1 < d <= 2, because z >= 0.0 and 0 < exp(-z) <= 1.0.
127 // Thus the reciprocal of the denominator never overflows.
128 float32x4_t vr0123 = vrecpeq_f32(vd0123);
129 float32x4_t vr4567 = vrecpeq_f32(vd4567);
130
131 vr0123 = vmulq_f32(vr0123, vrecpsq_f32(vr0123, vd0123));
132 vr4567 = vmulq_f32(vr4567, vrecpsq_f32(vr4567, vd4567));
133
134 vr0123 = vfmaq_f32(vr0123, vr0123, vfmsq_f32(vone, vr0123, vd0123));
135 vr4567 = vfmaq_f32(vr4567, vr4567, vfmsq_f32(vone, vr4567, vd4567));
136
137 // Reconstruct sigmoid(-z) = exp(-z) / (1.0 + exp(-z))
138 float32x4_t vf0123 = vmulq_f32(vy0123, vr0123);
139 float32x4_t vf4567 = vmulq_f32(vy4567, vr4567);
140
141 // For inputs below denormal cutoff, replace output with +0.0f.
142 // Note that for NaN inputs, comparison result is false, and outputs are left unchanged.
143 vf0123 = vreinterpretq_f32_u32(vbicq_u32(vreinterpretq_u32_f32(vf0123), vcagtq_f32(vx0123, vdenorm_cutoff)));
144 vf4567 = vreinterpretq_f32_u32(vbicq_u32(vreinterpretq_u32_f32(vf4567), vcagtq_f32(vx4567, vdenorm_cutoff)));
145
146 // Reconstruct sigmoid(x) = x < 0 ? sigmoid(-z) : 1.0 - sigmoid(-z)
147 const uint32x4_t vm0123 = vcltq_f32(vx0123, vmovq_n_f32(0.0f));
148 const uint32x4_t vm4567 = vcltq_f32(vx4567, vmovq_n_f32(0.0f));
149
150 vf0123 = vbslq_f32(vm0123, vf0123, vsubq_f32(vone, vf0123));
151 vf4567 = vbslq_f32(vm4567, vf4567, vsubq_f32(vone, vf4567));
152
153 vst1q_f32(y, vf0123); y += 4;
154 vst1q_f32(y, vf4567); y += 4;
155 }
156 for (; n >= 4 * sizeof(float); n -= 4 * sizeof(float)) {
157 const float32x4_t vx = vld1q_f32(x); x += 4;
158
159 // General structure of the algorithm:
160 // / exp(x) / (1 + exp(x)) if x <= 0
161 // f[x] :=
162 // \ 1 - f[-x] if x >= 0
163 //
164 // First we compute f[-z] := exp(-z) / (1 + exp(-z)) where z = abs(x),
165 // then replace result with 1 - f[-z] if x >= 0.
166 const float32x4_t vz = vabsq_f32(vx);
167
168 // Compute reduced argument n := round(-z * 2048 / log(2)).
169 // We do it by adding a large number (magic bias), which cause rounding of the result to an integer, then subtracing
170 // the large number back. The first addition is combined with multiplication by log2e into a single FMA instruction.
171 // The trick with adding large number is valid only within certain bounds (|z * 2048 / log(2)| <= 2**22, i.e.
172 // |z| <= 0x1.62E43p+10 = 1419.5654296875), but that is acceptable, because inputs x outside of
173 // [-87.336544, 17.328678] (i.e. z outsize [0, 87.336544]) underflow or saturate sigmoidf(x). We fixup the result
174 // for such inputs at the very end of the algorithm.
175 float32x4_t vn = vfmaq_f32(vmagic_bias, vz, vminus_log2e_x2048);
176
177 // Create a floating-point number s (scale) such that s := 2**(n / 2048) for such inputs that sigmoidf(-z) is
178 // normalized, i.e. 0 <= z <= 87.33642. As n has 11 fractional bits, we split s == 2**(n / 2048) =
179 // = 2**e * 2**(n / 2048 - e), where e := int(n / 2048). We create s in two steps:
180 // 1. Fetch 2**(n / 2048 - e) = 2**(n % 2048) from exp2_k_over_2048_table using the 6 low bits of n, as integer. Note that the
181 // fetched values are in the [1.0, 2.0) range, i.e. their floating-point exponent is 0.
182 // 2. Adjust fecthed value by addition of e to its floating-point exponent. The result is always a normalized
183 // number, because for 0 <= z <= 87.33642 (inputs for which sigmoidf(-z) is normalized) we have -126 <= e <= 0,
184 // and thus the adjusted exponent is not lower than -126.
185 //
186 // Extract e from bits 11:19 of n and shift it into bits 23:31 (position of floating-point exponent).
187 const int32x4_t ve = vshlq_n_s32(vbicq_s32(vreinterpretq_s32_f32(vn), vmovq_n_s32(INT32_C(0x7FF))), 12);
188
189 // Use bits 0:11 bits of n, as integer, as an index for table lookup of l := 2**(n % 2048).
190 const uint64x2_t vidx = vreinterpretq_u64_s32(vandq_s32(vreinterpretq_s32_f32(vn), vindex_mask));
191 const uint64_t vidx_lo = vgetq_lane_u64(vidx, 0);
192 const uint64_t vidx_hi = vgetq_lane_u64(vidx, 1);
193 float32x2_t vl_lo = vld1_dup_f32(&xnn_table_exp2_k_over_2048[(uint32_t) vidx_lo]);
194 float32x2_t vl_hi = vld1_dup_f32(&xnn_table_exp2_k_over_2048[(uint32_t) vidx_hi]);
195 vl_lo = vld1_lane_f32(&xnn_table_exp2_k_over_2048[(uint32_t) (vidx_lo >> 32)], vl_lo, 1);
196 vl_hi = vld1_lane_f32(&xnn_table_exp2_k_over_2048[(uint32_t) (vidx_hi >> 32)], vl_hi, 1);
197 const float32x4_t vl = vcombine_f32(vl_lo, vl_hi);
198 // Adjust exponent of the value l fetched from the exp2_k_over_2048_table to get the final s value.
199 const float32x4_t vs = vreinterpretq_f32_s32(vaddq_s32(vreinterpretq_s32_f32(vl), ve));
200
201 // Subtract the large number back to get the final n := round(-z * 2048 / log(2)) as a floating-point number.
202 vn = vsubq_f32(vn, vmagic_bias);
203
204 // Compute reduced argument t := (z + n * log(2) / 2048). Note that -t = -z - n * log(2) / 2048.
205 float32x4_t vt = vfmaq_f32(vz, vn, vln2_o2048);
206
207 // Compute degree-1 polynomial approximation for exp(-t) on [-log(2)/2048, log(2)/2048]:
208 // P1(t) = 1 + t * c1
209 const float32x4_t vp = vmulq_f32(vt, vc1);
210
211 // Reconstruct the exp(-z) value:
212 // y = s * (1 + t * c1)
213 // = s + s * (t * c1))
214 // = s + s * p
215 const float32x4_t vy = vfmaq_f32(vs, vs, vp);
216
217 // Denominator of the sigmoid fraction: 1.0 + exp(-z)
218 const float32x4_t vd = vaddq_f32(vy, vone);
219
220 // Use Newton-Raphson method (2 iterations) to compute reciprocal of denominator.
221 // Note: 1 < d <= 2, because z >= 0.0 and 0 < exp(-z) <= 1.0.
222 // Thus the reciprocal of the denominator never overflows.
223 float32x4_t vr = vrecpeq_f32(vd);
224
225 vr = vmulq_f32(vr, vrecpsq_f32(vr, vd));
226
227 vr = vfmaq_f32(vr, vr, vfmsq_f32(vone, vr, vd));
228
229 // Reconstruct sigmoid(-z) = exp(-z) / (1.0 + exp(-z))
230 float32x4_t vf = vmulq_f32(vy, vr);
231
232 // For inputs below denormal cutoff, replace output with +0.0f.
233 // Note that for NaN inputs, comparison result is false, and outputs are left unchanged.
234 vf = vreinterpretq_f32_u32(vbicq_u32(vreinterpretq_u32_f32(vf), vcagtq_f32(vx, vdenorm_cutoff)));
235
236 // Reconstruct sigmoid(x) = x < 0 ? sigmoid(-z) : 1.0 - sigmoid(-z)
237 const uint32x4_t vm = vcltq_f32(vx, vmovq_n_f32(0.0f));
238 vf = vbslq_f32(vm, vf, vsubq_f32(vone, vf));
239
240 vst1q_f32(y, vf); y += 4;
241 }
242 if XNN_UNLIKELY(n != 0) {
243 const float32x4_t vx = vld1q_f32(x);
244
245 // General structure of the algorithm:
246 // / exp(x) / (1 + exp(x)) if x <= 0
247 // f[x] :=
248 // \ 1 - f[-x] if x >= 0
249 //
250 // First we compute f[-z] := exp(-z) / (1 + exp(-z)) where z = abs(x),
251 // then replace result with 1 - f[-z] if x >= 0.
252 const float32x4_t vz = vabsq_f32(vx);
253
254 // Compute reduced argument n := round(-z * 2048 / log(2)).
255 // We do it by adding a large number (magic bias), which cause rounding of the result to an integer, then subtracing
256 // the large number back. The first addition is combined with multiplication by log2e into a single FMA instruction.
257 // The trick with adding large number is valid only within certain bounds (|z * 2048 / log(2)| <= 2**22, i.e.
258 // |z| <= 0x1.62E43p+10 = 1419.5654296875), but that is acceptable, because inputs x outside of
259 // [-87.336544, 17.328678] (i.e. z outsize [0, 87.336544]) underflow or saturate sigmoidf(x). We fixup the result
260 // for such inputs at the very end of the algorithm.
261 float32x4_t vn = vfmaq_f32(vmagic_bias, vz, vminus_log2e_x2048);
262
263 // Create a floating-point number s (scale) such that s := 2**(n / 2048) for such inputs that sigmoidf(-z) is
264 // normalized, i.e. 0 <= z <= 87.33642. As n has 11 fractional bits, we split s == 2**(n / 2048) =
265 // = 2**e * 2**(n / 2048 - e), where e := int(n / 2048). We create s in two steps:
266 // 1. Fetch 2**(n / 2048 - e) = 2**(n % 2048) from exp2_k_over_2048_table using the 6 low bits of n, as integer. Note that the
267 // fetched values are in the [1.0, 2.0) range, i.e. their floating-point exponent is 0.
268 // 2. Adjust fecthed value by addition of e to its floating-point exponent. The result is always a normalized
269 // number, because for 0 <= z <= 87.33642 (inputs for which sigmoidf(-z) is normalized) we have -126 <= e <= 0,
270 // and thus the adjusted exponent is not lower than -126.
271 //
272 // Extract e from bits 11:19 of n and shift it into bits 23:31 (position of floating-point exponent).
273 const int32x4_t ve = vshlq_n_s32(vbicq_s32(vreinterpretq_s32_f32(vn), vmovq_n_s32(INT32_C(0x7FF))), 12);
274
275 // Use bits 0:11 bits of n, as integer, as an index for table lookup of l := 2**(n % 2048).
276 const uint64x2_t vidx = vreinterpretq_u64_s32(vandq_s32(vreinterpretq_s32_f32(vn), vindex_mask));
277 const uint64_t vidx_lo = vgetq_lane_u64(vidx, 0);
278 const uint64_t vidx_hi = vgetq_lane_u64(vidx, 1);
279 float32x2_t vl_lo = vld1_dup_f32(&xnn_table_exp2_k_over_2048[(uint32_t) vidx_lo]);
280 float32x2_t vl_hi = vld1_dup_f32(&xnn_table_exp2_k_over_2048[(uint32_t) vidx_hi]);
281 vl_lo = vld1_lane_f32(&xnn_table_exp2_k_over_2048[(uint32_t) (vidx_lo >> 32)], vl_lo, 1);
282 vl_hi = vld1_lane_f32(&xnn_table_exp2_k_over_2048[(uint32_t) (vidx_hi >> 32)], vl_hi, 1);
283 const float32x4_t vl = vcombine_f32(vl_lo, vl_hi);
284 // Adjust exponent of the value l fetched from the exp2_k_over_2048_table to get the final s value.
285 const float32x4_t vs = vreinterpretq_f32_s32(vaddq_s32(vreinterpretq_s32_f32(vl), ve));
286
287 // Subtract the large number back to get the final n := round(-z * 2048 / log(2)) as a floating-point number.
288 vn = vsubq_f32(vn, vmagic_bias);
289
290 // Compute reduced argument t := (z + n * log(2) / 2048). Note that -t = -z - n * log(2) / 2048.
291 float32x4_t vt = vfmaq_f32(vz, vn, vln2_o2048);
292
293 // Compute degree-1 polynomial approximation for exp(-t) on [-log(2)/2048, log(2)/2048]:
294 // P1(t) = 1 + t * c1
295 const float32x4_t vp = vmulq_f32(vt, vc1);
296
297 // Reconstruct the exp(-z) value:
298 // y = s * (1 + t * c1)
299 // = s + s * (t * c1))
300 // = s + s * p
301 const float32x4_t vy = vfmaq_f32(vs, vs, vp);
302
303 // Denominator of the sigmoid fraction: 1.0 + exp(-z)
304 const float32x4_t vd = vaddq_f32(vy, vone);
305
306 // Use Newton-Raphson method (2 iterations) to compute reciprocal of denominator.
307 // Note: 1 < d <= 2, because z >= 0.0 and 0 < exp(-z) <= 1.0.
308 // Thus the reciprocal of the denominator never overflows.
309 float32x4_t vr = vrecpeq_f32(vd);
310
311 vr = vmulq_f32(vr, vrecpsq_f32(vr, vd));
312
313 vr = vfmaq_f32(vr, vr, vfmsq_f32(vone, vr, vd));
314
315 // Reconstruct sigmoid(-z) = exp(-z) / (1.0 + exp(-z))
316 float32x4_t vf = vmulq_f32(vy, vr);
317
318 // For inputs below denormal cutoff, replace output with +0.0f.
319 // Note that for NaN inputs, comparison result is false, and outputs are left unchanged.
320 vf = vreinterpretq_f32_u32(vbicq_u32(vreinterpretq_u32_f32(vf), vcagtq_f32(vx, vdenorm_cutoff)));
321
322 // Reconstruct sigmoid(x) = x < 0 ? sigmoid(-z) : 1.0 - sigmoid(-z)
323 const uint32x4_t vm = vcltq_f32(vx, vmovq_n_f32(0.0f));
324 vf = vbslq_f32(vm, vf, vsubq_f32(vone, vf));
325
326 float32x2_t vf_lo = vget_low_f32(vf);
327 if (n & (2 * sizeof(float))) {
328 vst1_f32(y, vf_lo); y += 2;
329 vf_lo = vget_high_f32(vf);
330 }
331 if (n & (1 * sizeof(float))) {
332 vst1_lane_f32(y, vf_lo, 0);
333 }
334 }
335 }
336