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/math-stubs.h>
12
13
xnn_math_f32_sigmoid__avx_rr2_p5_nr1(size_t n,const float * input,float * output)14 void xnn_math_f32_sigmoid__avx_rr2_p5_nr1(
15 size_t n,
16 const float* input,
17 float* output)
18 {
19 assert(n % (8 * sizeof(float)) == 0);
20
21 // Floating-point mask with only the sign bit set
22 const __m256 vsign_mask = _mm256_set1_ps(-0.0f);
23 // Large number such that ulp(magic bias) == 1 and magic bias === 127 mod 2**22.
24 const __m256 vmagic_bias = _mm256_set1_ps(0x1.8000FEp23f);
25 const __m256 vlog2e = _mm256_set1_ps(0x1.715476p0f);
26 // Last 7 bits are zeroes
27 const __m256 vminus_ln2_hi = _mm256_set1_ps(-0x1.62E400p-1f);
28 const __m256 vminus_ln2_lo = _mm256_set1_ps(-0x1.7F7D1Cp-20f);
29 // Coefficient of polynomial approximation of
30 // exp(t) ~ 1 + t * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5)))) on [-log(2)/2, log(2)/2]
31 const __m256 vc5 = _mm256_set1_ps(0x1.0F9F9Cp-7f);
32 const __m256 vc4 = _mm256_set1_ps(0x1.573A1Ap-5f);
33 const __m256 vc3 = _mm256_set1_ps(0x1.555A80p-3f);
34 const __m256 vc2 = _mm256_set1_ps(0x1.FFFDC6p-2f);
35 const __m256 vc1 = _mm256_set1_ps(0x1.FFFFF6p-1f);
36 const __m256 vone = _mm256_set1_ps(1.0f);
37 const __m256 vtwo = _mm256_set1_ps(2.0f);
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)).
55 // We do it by adding a large number (magic bias), which cause rounding of the result to integer, then subtracing
56 // the large number back. The trick with adding large number is valid only within certain bounds
57 // (|z / log(2)| <= 2**22, i.e. |z| <= 0x1.62E43p+21 = 2907270.0), but that is acceptable, because inputs x outside
58 // of [-87.336544, 17.328678] (i.e. z outsize [87.336544, 0]) underflow or saturate sigmoidf(x). We fixup the
59 // 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 inputs which don't cause underflow, i.e.
63 // -87.33642 <= z <= 0.0, and -126 <= n <= 0 accordingly.
64 const __m128 vs_lo = _mm_castsi128_ps(_mm_slli_epi32(_mm_castps_si128(_mm256_castps256_ps128(vn)), 23));
65 const __m128 vs_hi = _mm_castsi128_ps(_mm_slli_epi32(_mm_castps_si128(_mm256_extractf128_ps(vn, 1)), 23));
66 const __m256 vs = _mm256_insertf128_ps(_mm256_castps128_ps256(vs_lo), vs_hi, 1);
67
68 // Subtract the large number back to get the final n := round(z / log(2)) as a floating-point number.
69 vn = _mm256_sub_ps(vn, vmagic_bias);
70
71 // Compute reduced argument t := z - n * log(2).
72 // Use Cody-Waite range reduction method (note two constants to represent log(2)) to improve accuracy.
73 __m256 vt = _mm256_add_ps(_mm256_mul_ps(vn, vminus_ln2_hi), vz);
74 vt = _mm256_add_ps(_mm256_mul_ps(vn, vminus_ln2_lo), vt);
75
76 // Compute degree-5 polynomial approximation for exp(t) on [-log(2)/2, log(2)/2].
77 // P(t) = 1 + t * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5)))) = 1 + t * p
78 __m256 vp = _mm256_add_ps(_mm256_mul_ps(vc5, vt), vc4);
79 vp = _mm256_add_ps(_mm256_mul_ps(vp, vt), vc3);
80 vp = _mm256_add_ps(_mm256_mul_ps(vp, vt), vc2);
81 vp = _mm256_add_ps(_mm256_mul_ps(vp, vt), vc1);
82
83 // Reconstruct the exp(z) value:
84 // e = s * (1 + t * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5)))))
85 // = s + (t * s) * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5))))
86 // = s + (t * s) * p
87 vt = _mm256_mul_ps(vt, vs);
88 const __m256 ve = _mm256_add_ps(_mm256_mul_ps(vt, vp), vs);
89
90 // Denominator of the sigmoid fraction: 1.0 + exp(z)
91 const __m256 vd = _mm256_add_ps(ve, vone);
92
93 // Use Newton-Raphson method (1 iteration) to compute reciprocal of denominator.
94 // Note: 1 < d <= 2, because z >= 0.0 and 0 < exp(-z) <= 1.0.
95 // Thus the reciprocal of the denominator never overflows.
96 __m256 vr = _mm256_rcp_ps(vd);
97 vr = _mm256_mul_ps(vr, _mm256_sub_ps(vtwo, _mm256_mul_ps(vr, vd)));
98
99 // Reconstruct sigmoid(z) = exp(z) / (1.0 + exp(z))
100 __m256 vf = _mm256_mul_ps(ve, vr);
101
102 // For inputs below denormal cutoff, replace output with +0.0f.
103 // Note that for NaN inputs, comparison result is false, and outputs are left unchanged.
104 vf = _mm256_andnot_ps(_mm256_cmp_ps(vz, vdenorm_cutoff, _CMP_LT_OS), vf);
105
106 // Reconstruct sigmoid(x) = x < 0 ? sigmoid(z) : 1.0 - sigmoid(z)
107 vf = _mm256_blendv_ps(_mm256_sub_ps(vone, vf), vf, vx);
108
109 _mm256_storeu_ps(output, vf);
110
111 input += 8;
112 output += 8;
113 }
114 }
115