1 // Copyright 2020 Google LLC
2 //
3 // This source code is licensed under the BSD-style license found in the
4 // LICENSE file in the root directory of this source tree.
5
6 #include <assert.h>
7 #include <stddef.h>
8
9 #include <immintrin.h>
10
11 #include <xnnpack/common.h>
12 #include <xnnpack/math-stubs.h>
13
14
15 // Table of exp2(k / 64) values decremented (as integer) by (k << 17), k = 0..63
16 extern XNN_INTERNAL const float xnn_table_exp2minus_k_over_64[64];
17
xnn_math_f32_sigmoid__avx_rr2_lut64_p2_div(size_t n,const float * input,float * output)18 void xnn_math_f32_sigmoid__avx_rr2_lut64_p2_div(
19 size_t n,
20 const float* input,
21 float* output)
22 {
23 assert(n % (8 * sizeof(float)) == 0);
24
25 // Floating-point mask with only the sign bit set
26 const __m256 vsign_mask = _mm256_set1_ps(-0.0f);
27 // Large number such that ulp(magic bias) == exp2(-6)
28 const __m256 vmagic_bias = _mm256_set1_ps(0x1.800000p17f);
29 const __m256 vlog2e = _mm256_set1_ps(0x1.715476p0f);
30 // Mask for the lowest 6 bits
31 const __m256 vindex_mask = _mm256_castsi256_ps(_mm256_set1_epi32(INT32_C(0x3F)));
32 // Last 13 bits are zeroes
33 const __m256 vminus_ln2_hi = _mm256_set1_ps(-0x1.630000p-1f);
34 const __m256 vminus_ln2_lo = _mm256_set1_ps(0x1.BD0106p-13f);
35 // Coefficient of polynomial approximation of exp(t) ~ 1 + t * (1 + t * c2) on [-log(2)/128, log(2)/128]
36 const __m256 vc2 = _mm256_set1_ps(0x1.FFFF0Ap-2f);
37 const __m256 vone = _mm256_set1_ps(1.0f);
38 // The smallest x for which sigmoidf(x) is normalized.
39 // This number is also the smallest x for which expf(x) is normalized.
40 const __m256 vdenorm_cutoff = _mm256_set1_ps(-0x1.5D589Ep+6f);
41
42 for (; n != 0; n -= 8 * sizeof(float)) {
43 const __m256 vx = _mm256_loadu_ps(input);
44
45 // General structure of the algorithm:
46 //
47 // / exp(x) / (1 + exp(x)) if x <= 0
48 // f[x] :=
49 // \ 1 - f[-x] if x >= 0
50 //
51 // First we compute f[z] := exp(z) / (1 + exp(z)) where z = -abs(x), then replace result with 1 - f[z] if x >= 0.
52 const __m256 vz = _mm256_or_ps(vx, vsign_mask);
53
54 // Compute reduced argument n := round(z / log(2), 6).
55 // We do it by adding a large number (magic bias), which cause rounding of the result to 6 fractional bits, then
56 // subtracing the large number back. The trick with adding large number is valid only within certain bounds
57 // (|z / log(2)| <= 2**16, i.e. |z| <= 0x1.62E43p+15 = 45426.09375), but that is acceptable, because inputs x
58 // outside of [-87.336544, 17.328678] (i.e. z outsize [87.336544, 0]) underflow or saturate sigmoidf(x). We fixup
59 // the result for such inputs at the very end of the algorithm.
60 __m256 vn = _mm256_add_ps(_mm256_mul_ps(vz, vlog2e), vmagic_bias);
61
62 // Create a floating-point number s (scale) such that s := 2**n for such inputs that sigmoidf(z) is normalized,
63 // i.e. -87.33642 <= z <= 0. As n has 6 fractional bits, we split s == 2**n = 2**int(n) * 2**frac(n). We create s
64 // in two steps:
65 // 1. Fetch 2**frac(n) from the table using the 6 low bits of n, as integer. Note that the fetched values are in
66 // the [1.0, 2.0) range, i.e. their floating-point exponent is 0.
67 // 2. Adjust fecthed value by addition of int(n) to its floating-point exponent. The result is always a normalized
68 // number, because for -87.33642 <= z <= 0 (inputs for which sigmoidf(z) is normalized) we have
69 // -126 <= int(n) <= 0, and thus the adjusted exponent is not lower than -126.
70 //
71 // Shift bits 6:14 into 23:31 (position of floating-point exponent).
72 __m128i ve_lo = _mm_slli_epi32(_mm_castps_si128(_mm256_castps256_ps128(vn)), 17);
73 __m128i ve_hi = _mm_slli_epi32(_mm_castps_si128(_mm256_extractf128_ps(vn, 1)), 17);
74
75 // Use bits 0:6 of n, as integer, as an index for table lookup of l := 2**frac(n).
76 const __m256 vidx = _mm256_and_ps(vn, vindex_mask);
77 const __m128i vidx_lo = _mm_slli_epi32(_mm_castps_si128(_mm256_castps256_ps128(vidx)), 2);
78 const __m128i vidx_hi = _mm_slli_epi32(_mm_castps_si128(_mm256_extractf128_ps(vidx, 1)), 2);
79 #if XNN_ARCH_X86_64
80 const uint64_t vidx_ll = (uint64_t) _mm_cvtsi128_si64(vidx_lo);
81 const uint64_t vidx_lh = (uint64_t) _mm_extract_epi64(vidx_lo, 1);
82 const uint64_t vidx_hl = (uint64_t) _mm_cvtsi128_si64(vidx_hi);
83 const uint64_t vidx_hh = (uint64_t) _mm_extract_epi64(vidx_hi, 1);
84 __m128i vl_ll = _mm_cvtsi32_si128(*((const int*) ((uintptr_t) xnn_table_exp2minus_k_over_64 + (uint32_t) vidx_ll)));
85 __m128i vl_lh = _mm_cvtsi32_si128(*((const int*) ((uintptr_t) xnn_table_exp2minus_k_over_64 + (uint32_t) vidx_lh)));
86 __m128i vl_hl = _mm_cvtsi32_si128(*((const int*) ((uintptr_t) xnn_table_exp2minus_k_over_64 + (uint32_t) vidx_hl)));
87 __m128i vl_hh = _mm_cvtsi32_si128(*((const int*) ((uintptr_t) xnn_table_exp2minus_k_over_64 + (uint32_t) vidx_hh)));
88 vl_ll = _mm_insert_epi32(vl_ll, *((const int*) ((uintptr_t) xnn_table_exp2minus_k_over_64 + (uint32_t) (vidx_ll >> 32))), 1);
89 vl_lh = _mm_insert_epi32(vl_lh, *((const int*) ((uintptr_t) xnn_table_exp2minus_k_over_64 + (uint32_t) (vidx_lh >> 32))), 1);
90 vl_hl = _mm_insert_epi32(vl_hl, *((const int*) ((uintptr_t) xnn_table_exp2minus_k_over_64 + (uint32_t) (vidx_hl >> 32))), 1);
91 vl_hh = _mm_insert_epi32(vl_hh, *((const int*) ((uintptr_t) xnn_table_exp2minus_k_over_64 + (uint32_t) (vidx_hh >> 32))), 1);
92 #else
93 __m128i vl_ll = _mm_cvtsi32_si128(*((const int*) ((uintptr_t) xnn_table_exp2minus_k_over_64 + (uint32_t) _mm_cvtsi128_si32(vidx_lo))));
94 __m128i vl_lh = _mm_cvtsi32_si128(*((const int*) ((uintptr_t) xnn_table_exp2minus_k_over_64 + (uint32_t) _mm_extract_epi32(vidx_lo, 2))));
95 __m128i vl_hl = _mm_cvtsi32_si128(*((const int*) ((uintptr_t) xnn_table_exp2minus_k_over_64 + (uint32_t) _mm_cvtsi128_si32(vidx_hi))));
96 __m128i vl_hh = _mm_cvtsi32_si128(*((const int*) ((uintptr_t) xnn_table_exp2minus_k_over_64 + (uint32_t) _mm_extract_epi32(vidx_hi, 2))));
97 vl_ll = _mm_insert_epi32(vl_ll, *((const int*) ((uintptr_t) xnn_table_exp2minus_k_over_64 + (uint32_t) _mm_extract_epi32(vidx_lo, 1))), 1);
98 vl_lh = _mm_insert_epi32(vl_lh, *((const int*) ((uintptr_t) xnn_table_exp2minus_k_over_64 + (uint32_t) _mm_extract_epi32(vidx_lo, 3))), 1);
99 vl_hl = _mm_insert_epi32(vl_hl, *((const int*) ((uintptr_t) xnn_table_exp2minus_k_over_64 + (uint32_t) _mm_extract_epi32(vidx_hi, 1))), 1);
100 vl_hh = _mm_insert_epi32(vl_hh, *((const int*) ((uintptr_t) xnn_table_exp2minus_k_over_64 + (uint32_t) _mm_extract_epi32(vidx_hi, 3))), 1);
101 #endif
102 const __m128i vl_lo = _mm_unpacklo_epi64(vl_ll, vl_lh);
103 const __m128i vl_hi = _mm_unpacklo_epi64(vl_hl, vl_hh);
104 // Adjust exponent of the value l fetched from the table to get the final s value.
105 const __m128 vs_lo = _mm_castsi128_ps(_mm_add_epi32(vl_lo, ve_lo));
106 const __m128 vs_hi = _mm_castsi128_ps(_mm_add_epi32(vl_hi, ve_hi));
107 const __m256 vs = _mm256_insertf128_ps(_mm256_castps128_ps256(vs_lo), vs_hi, 1);
108
109 // Subtract the large number back to get the final n := round(z / log(2), 6) as a floating-point number.
110 vn = _mm256_sub_ps(vn, vmagic_bias);
111
112 // Compute reduced argument t := z - n * log(2).
113 // Use Cody-Waite range reduction method (note two constants to represent log(2)) to improve accuracy.
114 __m256 vt = _mm256_add_ps(_mm256_mul_ps(vn, vminus_ln2_hi), vz);
115 vt = _mm256_add_ps(_mm256_mul_ps(vn, vminus_ln2_lo), vt);
116
117 // Compute degree-2 polynomial approximation for exp(t) on [-log(2)/128, log(2)/128].
118 // P(t) = 1 + t * (1 + t * c2) = 1 + (t + t * (t * c2)) = 1 + p
119 __m256 vp = _mm256_mul_ps(vt, vc2);
120 vp = _mm256_add_ps(vt, _mm256_mul_ps(vp, vt));
121
122 // Reconstruct the exp(z) value:
123 // e = s * (1 + t * (1 + t * c2))
124 // = s * (1 + p)
125 // = s + s * p
126 const __m256 ve = _mm256_add_ps(vs, _mm256_mul_ps(vs, vp));
127
128 // Denominator of the sigmoid fraction: 1.0 + exp(z)
129 const __m256 vd = _mm256_add_ps(ve, vone);
130
131 // Reconstruct sigmoid(z) = exp(z) / (1.0 + exp(z))
132 __m256 vf = _mm256_div_ps(ve, vd);
133
134 // For inputs below denormal cutoff, replace output with +0.0f.
135 // Note that for NaN inputs, comparison result is false, and outputs are left unchanged.
136 vf = _mm256_andnot_ps(_mm256_cmp_ps(vz, vdenorm_cutoff, _CMP_LT_OS), vf);
137
138 // Reconstruct sigmoid(x) = x < 0 ? sigmoid(z) : 1.0 - sigmoid(z)
139 vf = _mm256_blendv_ps(_mm256_sub_ps(vone, vf), vf, vx);
140
141 _mm256_storeu_ps(output, vf);
142
143 input += 8;
144 output += 8;
145 }
146 }
147