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 <emmintrin.h>
10
11 #include <xnnpack/math-stubs.h>
12
13
xnn_math_f32_sigmoid__sse2_rr2_p5_div(size_t n,const float * input,float * output)14 void xnn_math_f32_sigmoid__sse2_rr2_p5_div(
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 // The smallest x for which sigmoidf(x) is normalized.
38 // This number is also the smallest x for which expf(x) is normalized.
39 const __m128 vdenorm_cutoff = _mm_set1_ps(-0x1.5D589Ep+6f);
40
41 for (; n != 0; n -= 4 * sizeof(float)) {
42 const __m128 vx = _mm_loadu_ps(input);
43
44 // General structure of the algorithm:
45 //
46 // / exp(x) / (1 + exp(x)) if x <= 0
47 // f[x] :=
48 // \ 1 - f[-x] if x >= 0
49 //
50 // First we compute f[z] := exp(z) / (1 + exp(z)) where z = -abs(x), then replace result with 1 - f[z] if x >= 0.
51 const __m128 vz = _mm_or_ps(vx, vsign_mask);
52
53 // Compute reduced argument n := round(z / log(2)).
54 // We do it by adding a large number (magic bias), which cause rounding of the result to integer, then subtracing
55 // the large number back. The trick with adding large number is valid only within certain bounds
56 // (|z / log(2)| <= 2**22, i.e. |z| <= 0x1.62E43p+21 = 2907270.0), but that is acceptable, because inputs x outside
57 // of [-87.336544, 17.328678] (i.e. z outsize [87.336544, 0]) underflow or saturate sigmoidf(x). We fixup the
58 // result for such inputs at the very end of the algorithm.
59 __m128 vn = _mm_add_ps(_mm_mul_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 __m128 vs = _mm_castsi128_ps(_mm_slli_epi32(_mm_castps_si128(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 = _mm_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 __m128 vt = _mm_add_ps(_mm_mul_ps(vn, vminus_ln2_hi), vz);
71 vt = _mm_add_ps(_mm_mul_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 __m128 vp = _mm_add_ps(_mm_mul_ps(vc5, vt), vc4);
76 vp = _mm_add_ps(_mm_mul_ps(vp, vt), vc3);
77 vp = _mm_add_ps(_mm_mul_ps(vp, vt), vc2);
78 vp = _mm_add_ps(_mm_mul_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 = _mm_mul_ps(vt, vs);
85 __m128 ve = _mm_add_ps(_mm_mul_ps(vt, vp), vs);
86
87 // Denominator of the sigmoid fraction: 1.0 + exp(z)
88 __m128 vd = _mm_add_ps(ve, vone);
89
90 // Reconstruct sigmoid(z) = exp(z) / (1.0 + exp(z))
91 __m128 vf = _mm_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 = _mm_andnot_ps(_mm_cmplt_ps(vz, vdenorm_cutoff), vf);
96
97 // Reconstruct sigmoid(x) = x < 0 ? sigmoid(z) : 1.0 - sigmoid(z)
98 __m128 vm = _mm_castsi128_ps(_mm_cmpgt_epi32(_mm_setzero_si128(), _mm_castps_si128(vx)));
99 vf = _mm_or_ps(_mm_and_ps(vf, vm), _mm_andnot_ps(vm, _mm_sub_ps(vone, vf)));
100
101 _mm_storeu_ps(output, vf);
102
103 input += 4;
104 output += 4;
105 }
106 }
107