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 <emmintrin.h>
10
11 #include <xnnpack/math-stubs.h>
12
13
xnn_math_f32_sigmoid__sse2_rr2_p5_nr1(size_t n,const float * input,float * output)14 void xnn_math_f32_sigmoid__sse2_rr2_p5_nr1(
15 size_t n,
16 const float* input,
17 float* output)
18 {
19 assert(n % (4 * sizeof(float)) == 0);
20
21 // Floating-point mask with only the sign bit set
22 const __m128 vsign_mask = _mm_set1_ps(-0.0f);
23 // Large number such that ulp(magic bias) == 1 and magic bias === 127 mod 2**22.
24 const __m128 vmagic_bias = _mm_set1_ps(0x1.8000FEp23f);
25 const __m128 vlog2e = _mm_set1_ps(0x1.715476p0f);
26 // Last 7 bits are zeroes
27 const __m128 vminus_ln2_hi = _mm_set1_ps(-0x1.62E400p-1f);
28 const __m128 vminus_ln2_lo = _mm_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 __m128 vc5 = _mm_set1_ps(0x1.0F9F9Cp-7f);
32 const __m128 vc4 = _mm_set1_ps(0x1.573A1Ap-5f);
33 const __m128 vc3 = _mm_set1_ps(0x1.555A80p-3f);
34 const __m128 vc2 = _mm_set1_ps(0x1.FFFDC6p-2f);
35 const __m128 vc1 = _mm_set1_ps(0x1.FFFFF6p-1f);
36 const __m128 vone = _mm_set1_ps(1.0f);
37 const __m128 vtwo = _mm_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 __m128 vdenorm_cutoff = _mm_set1_ps(-0x1.5D589Ep+6f);
41
42 for (; n != 0; n -= 4 * sizeof(float)) {
43 const __m128 vx = _mm_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 __m128 vz = _mm_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 __m128 vn = _mm_add_ps(_mm_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 = _mm_castsi128_ps(_mm_slli_epi32(_mm_castps_si128(vn), 23));
65
66 // Subtract the large number back to get the final n := round(z / log(2)) as a floating-point number.
67 vn = _mm_sub_ps(vn, vmagic_bias);
68
69 // Compute reduced argument t := z - n * log(2).
70 // Use Cody-Waite range reduction method (note two constants to represent log(2)) to improve accuracy.
71 __m128 vt = _mm_add_ps(_mm_mul_ps(vn, vminus_ln2_hi), vz);
72 vt = _mm_add_ps(_mm_mul_ps(vn, vminus_ln2_lo), vt);
73
74 // Compute degree-5 polynomial approximation for exp(t) on [-log(2)/2, log(2)/2].
75 // P(t) = 1 + t * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5)))) = 1 + t * p
76 __m128 vp = _mm_add_ps(_mm_mul_ps(vc5, vt), vc4);
77 vp = _mm_add_ps(_mm_mul_ps(vp, vt), vc3);
78 vp = _mm_add_ps(_mm_mul_ps(vp, vt), vc2);
79 vp = _mm_add_ps(_mm_mul_ps(vp, vt), vc1);
80
81 // Reconstruct the exp(z) value:
82 // e = s * (1 + t * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5)))))
83 // = s + (t * s) * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5))))
84 // = s + (t * s) * p
85 vt = _mm_mul_ps(vt, vs);
86 __m128 ve = _mm_add_ps(_mm_mul_ps(vt, vp), vs);
87
88 // Denominator of the sigmoid fraction: 1.0 + exp(z)
89 __m128 vd = _mm_add_ps(ve, vone);
90
91 // Use Newton-Raphson method (1 iteration) to compute reciprocal of denominator.
92 // Note: 1 < d <= 2, because z >= 0.0 and 0 < exp(-z) <= 1.0.
93 // Thus the reciprocal of the denominator never overflows.
94 __m128 vr = _mm_rcp_ps(vd);
95 vr = _mm_mul_ps(vr, _mm_sub_ps(vtwo, _mm_mul_ps(vr, vd)));
96
97 // Reconstruct sigmoid(z) = exp(z) / (1.0 + exp(z))
98 __m128 vf = _mm_mul_ps(ve, vr);
99
100 // For inputs below denormal cutoff, replace output with +0.0f.
101 // Note that for NaN inputs, comparison result is false, and outputs are left unchanged.
102 vf = _mm_andnot_ps(_mm_cmplt_ps(vz, vdenorm_cutoff), vf);
103
104 // Reconstruct sigmoid(x) = x < 0 ? sigmoid(z) : 1.0 - sigmoid(z)
105 __m128 vm = _mm_castsi128_ps(_mm_cmpgt_epi32(_mm_setzero_si128(), _mm_castps_si128(vx)));
106 vf = _mm_or_ps(_mm_and_ps(vf, vm), _mm_andnot_ps(vm, _mm_sub_ps(vone, vf)));
107
108 _mm_storeu_ps(output, vf);
109
110 input += 4;
111 output += 4;
112 }
113 }
114