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_rr1_p5_nr2fma(size_t n,const float * input,float * output)14 void xnn_math_f32_sigmoid__avx2_rr1_p5_nr2fma(
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 = _mm256_set1_ps(-0x1.62E43p-1f);
27 // Coefficient of polynomial approximation of
28 // exp(t) ~ 1 + t * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5)))) on [-log(2)/2, log(2)/2]
29 const __m256 vc5 = _mm256_set1_ps(0x1.0F9F9Cp-7f);
30 const __m256 vc4 = _mm256_set1_ps(0x1.573A1Ap-5f);
31 const __m256 vc3 = _mm256_set1_ps(0x1.555A80p-3f);
32 const __m256 vc2 = _mm256_set1_ps(0x1.FFFDC6p-2f);
33 const __m256 vc1 = _mm256_set1_ps(0x1.FFFFF6p-1f);
34 const __m256 vone = _mm256_set1_ps(1.0f);
35 // The smallest x for which sigmoidf(x) is normalized.
36 // This number is also the smallest x for which expf(x) is normalized.
37 const __m256 vdenorm_cutoff = _mm256_set1_ps(-0x1.5D589Ep+6f);
38
39 for (; n != 0; n -= 8 * sizeof(float)) {
40 const __m256 vx = _mm256_loadu_ps(input);
41
42 // General structure of the algorithm:
43 //
44 // / exp(x) / (1 + exp(x)) if x <= 0
45 // f[x] :=
46 // \ 1 - f[-x] if x >= 0
47 //
48 // First we compute f[z] := exp(z) / (1 + exp(z)) where z = -abs(x), then replace result with 1 - f[z] if x >= 0.
49 const __m256 vz = _mm256_or_ps(vx, vsign_mask);
50
51 // Compute reduced argument n := round(z / log(2)).
52 // We do it by adding a large number (magic bias), which cause rounding of the result to integer, then subtracing
53 // the large number back. The addition is combined with multiplication by log2e into a single FMA instruction. The
54 // trick with adding large number is valid only within certain bounds (|z / log(2)| <= 2**22, i.e.
55 // |z| <= 0x1.62E43p+21 = 2907270.0), but that is acceptable, because inputs x outside of [-87.336544, 17.328678]
56 // (i.e. z outsize [87.336544, 0]) underflow or saturate sigmoidf(x). We fixup the result for such inputs at the
57 // very end of the algorithm.
58 __m256 vn = _mm256_fmadd_ps(vz, vlog2e, vmagic_bias);
59
60 // Create a floating-point number s (scale) such that s == 2**n for inputs which don't cause underflow, i.e.
61 // -87.33642 <= z <= 0.0, and -126 <= n <= 0 accordingly.
62 const __m256 vs = _mm256_castsi256_ps(_mm256_slli_epi32(_mm256_castps_si256(vn), 23));
63
64 // Subtract the large number back to get the final n := round(z / log(2)) as a floating-point number.
65 vn = _mm256_sub_ps(vn, vmagic_bias);
66
67 // Compute reduced argument t := z - n * log(2).
68 __m256 vt = _mm256_fmadd_ps(vn, vminus_ln2, vz);
69
70 // Compute degree-5 polynomial approximation for exp(t) on [-log(2)/2, log(2)/2].
71 // P(t) = 1 + t * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5)))) = 1 + t * p
72 __m256 vp = _mm256_fmadd_ps(vc5, vt, vc4);
73 vp = _mm256_fmadd_ps(vp, vt, vc3);
74 vp = _mm256_fmadd_ps(vp, vt, vc2);
75 vp = _mm256_fmadd_ps(vp, vt, vc1);
76
77 // Reconstruct the exp(z) value:
78 // e = s * (1 + t * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5)))))
79 // = s + (t * s) * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5))))
80 // = s + (t * s) * p
81 vt = _mm256_mul_ps(vt, vs);
82 const __m256 ve = _mm256_fmadd_ps(vt, vp, vs);
83
84 // Denominator of the sigmoid fraction: 1.0 + exp(z)
85 const __m256 vd = _mm256_add_ps(ve, vone);
86
87 // Use Newton-Raphson method (2 iterations) to compute reciprocal of denominator.
88 // Note: 1 < d <= 2, because z >= 0.0 and 0 < exp(-z) <= 1.0.
89 // Thus the reciprocal of the denominator never overflows.
90 __m256 vr = _mm256_rcp_ps(vd);
91 vr = _mm256_fmadd_ps(_mm256_fnmadd_ps(vr, vd, vone), vr, vr);
92 vr = _mm256_fmadd_ps(_mm256_fnmadd_ps(vr, vd, vone), vr, vr);
93
94 // Reconstruct sigmoid(z) = exp(z) / (1.0 + exp(z))
95 __m256 vf = _mm256_mul_ps(ve, vr);
96
97 // For inputs below denormal cutoff, replace output with +0.0f.
98 // Note that for NaN inputs, comparison result is false, and outputs are left unchanged.
99 vf = _mm256_andnot_ps(_mm256_cmp_ps(vz, vdenorm_cutoff, _CMP_LT_OS), vf);
100
101 // Reconstruct sigmoid(x) = x < 0 ? sigmoid(z) : 1.0 - sigmoid(z)
102 vf = _mm256_blendv_ps(_mm256_sub_ps(vone, vf), vf, vx);
103
104 _mm256_storeu_ps(output, vf);
105
106 input += 8;
107 output += 8;
108 }
109 }
110