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 <arm_neon.h>
10
11 #include <xnnpack/common.h>
12 #include <xnnpack/math-stubs.h>
13
14
15 // Table of exp2(k / 64) values, k = 0..63
16 extern XNN_INTERNAL const float xnn_table_exp2_k_over_64[64];
17
xnn_math_f32_sigmoid__neonfma_rr1_lut64_p2_div(size_t n,const float * input,float * output)18 void xnn_math_f32_sigmoid__neonfma_rr1_lut64_p2_div(
19 size_t n,
20 const float* input,
21 float* output)
22 {
23 assert(n % (4 * sizeof(float)) == 0);
24
25 const float32x4_t vmagic_bias = vmovq_n_f32(0x1.800000p23f);
26 // The largest z for which sigmoidf(-z) is normalized.
27 // This number is also the largest z for which expf(-z) is normalized.
28 const float32x4_t vdenorm_cutoff = vmovq_n_f32(-0x1.5D589Ep+6f);
29 const float32x4_t vminus_log2e_x64 = vmovq_n_f32(-0x1.715476p6f);
30 const float32x4_t vln2_o64 = vmovq_n_f32(0x1.62E43p-7f);
31 const float32x4_t vone = vmovq_n_f32(1.0f);
32
33 const float32x4_t vc2 = vmovq_n_f32(0x1.FFFF0Ap-2f);
34
35 const int32x4_t vindex_mask = vmovq_n_s32(INT32_C(0x3F));
36
37 for (; n != 0; n -= 4 * sizeof(float)) {
38 const float32x4_t vx = vld1q_f32(input); input += 4;
39
40 // General structure of the algorithm:
41 // / exp(x) / (1 + exp(x)) if x <= 0
42 // f[x] :=
43 // \ 1 - f[-x] if x >= 0
44 //
45 // First we compute f[-z] := exp(-z) / (1 + exp(-z)) where z = abs(x),
46 // then replace result with 1 - f[-z] if x >= 0.
47 const float32x4_t vz = vabsq_f32(vx);
48
49 // Compute reduced argument n := round(-z * 64 / log(2)).
50 // We do it by adding a large number (magic bias), which cause rounding of the result to an integer, then subtracing
51 // the large number back. The first addition is combined with multiplication by log2e into a single FMA instruction.
52 // The trick with adding large number is valid only within certain bounds (|z * 64 / log(2)| <= 2**22, i.e.
53 // |z| <= 0x1.62E43p+15 = 45426.09375), but that is acceptable, because inputs x outside of [-87.336544, 17.328678]
54 // (i.e. z outsize [0, 87.336544]) underflow or saturate sigmoidf(x). We fixup the result for such inputs at the
55 // very end of the algorithm.
56 float32x4_t vn = vfmaq_f32(vmagic_bias, vz, vminus_log2e_x64);
57
58 // Create a floating-point number s (scale) such that s := 2**(n / 64) for such inputs that sigmoidf(-z) is
59 // normalized, i.e. 0 <= z <= 87.33642. As n has 6 fractional bits, we split s == 2**(n / 64) =
60 // = 2**e * 2**(n / 64 - e), where e := int(n / 64). We create s in two steps:
61 // 1. Fetch 2**(n / 64 - e) = 2**(n % 64) from the table using the 6 low bits of n, as integer. Note that the
62 // fetched values are in the [1.0, 2.0) range, i.e. their floating-point exponent is 0.
63 // 2. Adjust fecthed value by addition of e to its floating-point exponent. The result is always a normalized
64 // number, because for 0 <= z <= 87.33642 (inputs for which sigmoidf(-z) is normalized) we have -126 <= e <= 0,
65 // and thus the adjusted exponent is not lower than -126.
66 //
67 // Extract e from bits 6:14 of n and shift it into bits 23:31 (position of floating-point exponent).
68 const int32x4_t ve = vshlq_n_s32(vbicq_s32(vreinterpretq_s32_f32(vn), vmovq_n_s32(INT32_C(0x3F))), 17);
69
70 // Use bits 0:6 bits of n, as integer, as an index for table lookup of l := 2**(n % 64).
71 const uint64x2_t vidx = vreinterpretq_u64_s32(vandq_s32(vreinterpretq_s32_f32(vn), vindex_mask));
72 const uint64_t vidx_lo = vgetq_lane_u64(vidx, 0);
73 const uint64_t vidx_hi = vgetq_lane_u64(vidx, 1);
74 float32x2_t vl_lo = vld1_dup_f32(&xnn_table_exp2_k_over_64[(uint32_t) vidx_lo]);
75 float32x2_t vl_hi = vld1_dup_f32(&xnn_table_exp2_k_over_64[(uint32_t) vidx_hi]);
76 vl_lo = vld1_lane_f32(&xnn_table_exp2_k_over_64[(uint32_t) (vidx_lo >> 32)], vl_lo, 1);
77 vl_hi = vld1_lane_f32(&xnn_table_exp2_k_over_64[(uint32_t) (vidx_hi >> 32)], vl_hi, 1);
78 const float32x4_t vl = vcombine_f32(vl_lo, vl_hi);
79 // Adjust exponent of the value l fetched from the table to get the final s value.
80 const float32x4_t vs = vreinterpretq_f32_s32(vaddq_s32(vreinterpretq_s32_f32(vl), ve));
81
82 // Subtract the large number back to get the final n := round(-z * 64 / log(2)) as a floating-point number.
83 vn = vsubq_f32(vn, vmagic_bias);
84
85 // Compute reduced argument t := (z + n * log(2) / 64). Note that -t = -z - n * log(2) / 64.
86 float32x4_t vt = vfmaq_f32(vz, vn, vln2_o64);
87
88 // Compute degree-2 polynomial approxiatmion for exp(-t) on [-log(2)/128, log(2)/128].
89 // P1(t) = 1 + t * (-1 + t * c2)
90 float32x4_t vp = vmulq_f32(vt, vc2);
91 vp = vfmsq_f32(vt, vp, vt);
92
93 // Reconstruct the exp(-z) value:
94 // f = s * (1 + t * (-1 + t * c2))
95 // = s * (1 - t + t * (t * c2))
96 // = s - s * (t - t * (t * c2))
97 // = s - s * p
98 const float32x4_t vy = vfmsq_f32(vs, vs, vp);
99
100 // Denominator of the sigmoid fraction: 1.0 + exp(-z)
101 const float32x4_t vd = vaddq_f32(vy, vone);
102
103 // Reconstruct sigmoid(-z) = exp(-z) / (1.0 + exp(-z))
104 float32x4_t vf = vdivq_f32(vy, vd);
105
106 // For inputs below denormal cutoff, replace output with +0.0f.
107 // Note that for NaN inputs, comparison result is false, and outputs are left unchanged.
108 vf = vreinterpretq_f32_u32(vbicq_u32(vreinterpretq_u32_f32(vf), vcagtq_f32(vx, vdenorm_cutoff)));
109
110 // Reconstruct sigmoid(x) = x < 0 ? sigmoid(-z) : 1.0 - sigmoid(-z)
111 const uint32x4_t vm = vcltq_f32(vx, vmovq_n_f32(0.0f));
112 vf = vbslq_f32(vm, vf, vsubq_f32(vone, vf));
113
114 vst1q_f32(output, vf); output += 4;
115 }
116 }
117