1 // Copyright 2019 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__avx2_rr2_p5_div(size_t n,const float * input,float * output)14 void xnn_math_f32_sigmoid__avx2_rr2_p5_div(
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 const __m256 vminus_ln2_hi = _mm256_set1_ps(-0x1.62E43p-1f);
27 const __m256 vminus_ln2_lo = _mm256_set1_ps(0x1.05C61p-29f);
28 // Coefficient of polynomial approximation of
29 // exp(t) ~ 1 + t * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5)))) on [-log(2)/2, log(2)/2]
30 const __m256 vc5 = _mm256_set1_ps(0x1.0F9F9Cp-7f);
31 const __m256 vc4 = _mm256_set1_ps(0x1.573A1Ap-5f);
32 const __m256 vc3 = _mm256_set1_ps(0x1.555A80p-3f);
33 const __m256 vc2 = _mm256_set1_ps(0x1.FFFDC6p-2f);
34 const __m256 vc1 = _mm256_set1_ps(0x1.FFFFF6p-1f);
35 const __m256 vone = _mm256_set1_ps(1.0f);
36 // The smallest x for which sigmoidf(x) is normalized.
37 // This number is also the smallest x for which expf(x) is normalized.
38 const __m256 vdenorm_cutoff = _mm256_set1_ps(-0x1.5D589Ep+6f);
39
40 for (; n != 0; n -= 8 * sizeof(float)) {
41 const __m256 vx = _mm256_loadu_ps(input);
42
43 // General structure of the algorithm:
44 //
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), then replace result with 1 - f[z] if x >= 0.
50 const __m256 vz = _mm256_or_ps(vx, vsign_mask);
51
52 // Compute reduced argument n := round(z / log(2)).
53 // We do it by adding a large number (magic bias), which cause rounding of the result to integer, then subtracing
54 // the large number back. The addition is combined with multiplication by log2e into a single FMA instruction. The
55 // trick with adding large number is valid only within certain bounds (|z / log(2)| <= 2**22, i.e.
56 // |z| <= 0x1.62E43p+21 = 2907270.0), but that is acceptable, because inputs x outside of [-87.336544, 17.328678]
57 // (i.e. z outsize [87.336544, 0]) underflow or saturate sigmoidf(x). We fixup the result for such inputs at the
58 // very end of the algorithm.
59 __m256 vn = _mm256_fmadd_ps(vz, vlog2e, vmagic_bias);
60
61 // Create a floating-point number s (scale) such that s == 2**n for inputs which don't cause underflow, i.e.
62 // -87.33642 <= z <= 0.0, and -126 <= n <= 0 accordingly.
63 const __m256 vs = _mm256_castsi256_ps(_mm256_slli_epi32(_mm256_castps_si256(vn), 23));
64
65 // Subtract the large number back to get the final n := round(z / log(2)) as a floating-point number.
66 vn = _mm256_sub_ps(vn, vmagic_bias);
67
68 // Compute reduced argument t := z - n * log(2).
69 // Use Cody-Waite range reduction method (note two constants to represent log(2)) to improve accuracy.
70 __m256 vt = _mm256_fmadd_ps(vn, vminus_ln2_hi, vz);
71 vt = _mm256_fmadd_ps(vn, vminus_ln2_lo, vt);
72
73 // Compute degree-5 polynomial approximation for exp(t) on [-log(2)/2, log(2)/2].
74 // P(t) = 1 + t * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5)))) = 1 + t * p
75 __m256 vp = _mm256_fmadd_ps(vc5, vt, vc4);
76 vp = _mm256_fmadd_ps(vp, vt, vc3);
77 vp = _mm256_fmadd_ps(vp, vt, vc2);
78 vp = _mm256_fmadd_ps(vp, vt, vc1);
79
80 // Reconstruct the exp(z) value:
81 // e = s * (1 + t * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5)))))
82 // = s + (t * s) * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5))))
83 // = s + (t * s) * p
84 vt = _mm256_mul_ps(vt, vs);
85 const __m256 ve = _mm256_fmadd_ps(vt, vp, vs);
86
87 // Denominator of the sigmoid fraction: 1.0 + exp(z)
88 const __m256 vd = _mm256_add_ps(ve, vone);
89
90 // Reconstruct sigmoid(z) = exp(z) / (1.0 + exp(z))
91 __m256 vf = _mm256_div_ps(ve, vd);
92
93 // For inputs below denormal cutoff, replace output with +0.0f.
94 // Note that for NaN inputs, comparison result is false, and outputs are left unchanged.
95 vf = _mm256_andnot_ps(_mm256_cmp_ps(vz, vdenorm_cutoff, _CMP_LT_OS), vf);
96
97 // Reconstruct sigmoid(x) = x < 0 ? sigmoid(z) : 1.0 - sigmoid(z)
98 vf = _mm256_blendv_ps(_mm256_sub_ps(vone, vf), vf, vx);
99
100 _mm256_storeu_ps(output, vf);
101
102 input += 8;
103 output += 8;
104 }
105 }
106