1 // Copyright 2022 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/math-stubs.h>
12
13
xnn_math_f16_sigmoid__neonfp16arith_rr2_p3_recpe(size_t n,const void * input,void * output)14 void xnn_math_f16_sigmoid__neonfp16arith_rr2_p3_recpe(
15 size_t n,
16 const void* input,
17 void* output)
18 {
19 assert(n % (8 * sizeof(__fp16)) == 0);
20
21 // Large number such that ulp(magic bias) == 1 and magic bias === 15 mod 2**9.
22 const float16x8_t vmagic_bias = vmovq_n_f16(0x1.83Cp+10f);
23 const float16x8_t vminus_log2e = vmovq_n_f16(-0x1.714p+0f);
24 const float16x8_t vln2_hi = vmovq_n_f16(0x1.630p-1f);
25 const float16x8_t vln2_lo = vmovq_n_f16(-0x1.BD0p-13f);
26 // Coefficient of polynomial approximation
27 // exp(-t) ~ 1 + t * (-1 + t * (c2 + t * c3))
28 // on [-log(2)/2, log(2)/2]
29 const float16x8_t vc3 = vmovq_n_f16(-0x1.558p-3f);
30 const float16x8_t vc2 = vmovq_n_f16(0x1.020p-1f);
31 const float16x8_t vone = vmovq_n_f16(1.0f);
32 // The largest z for which sigmoidh(-z) is normalized.
33 // This number is also the largest z for which exph(-z) is normalized.
34 const float16x8_t vdenorm_cutoff = vmovq_n_f16(-0x1.368p+3f);
35
36 const __fp16* i = (const __fp16*) input;
37 __fp16* o = (__fp16*) output;
38 for (; n != 0; n -= 8 * sizeof(__fp16)) {
39 const float16x8_t vx = vld1q_f16(i); i += 8;
40
41 // General structure of the algorithm:
42 //
43 // / exp(x) / (1 + exp(x)) if x <= 0
44 // f[x] :=
45 // \ 1 - f[-x] if x >= 0
46 //
47 // First we compute f[-z] := exp(-z) / (1 + exp(-z)) where z = abs(x),
48 // then replace result with 1 - f[-z] if x >= 0.
49 const float16x8_t vz = vabsq_f16(vx);
50
51 // Compute reduced argument n := round(-z / log(2)).
52 // We do it by adding a large number (magic bias) to the product z * (-1/log(2)), which cause rounding of the
53 // result to an integer, then subtracing the large number back. The first addition is combined with multiplication
54 // by -log2e into a single FMA instruction. The trick with adding large number is valid only within certain bounds
55 // (|-x / log(2)| <= 2**9, i.e. |z| <= 0x1.630p+8 = 355.0), but that is acceptable, because inputs outside
56 // of [-9.703125, 8.3125] (i.e. z outside [0, 9.703125]) underflow or saturate sigmoidh(x). We fixup the result for
57 // such inputs at the very end of the algorithm.
58 float16x8_t vn = vfmaq_f16(vmagic_bias, vz, vminus_log2e);
59
60 // Create a floating-point number s (scale) such that s == 2**n for inputs which don't cause underflow, i.e.
61 // -9.703125 <= -z <= 0.0, and -14 <= n <= 0 accordingly.
62 const float16x8_t vs = vreinterpretq_f16_s16(vshlq_n_s16(vreinterpretq_s16_f16(vn), 10));
63
64 // Subtract the large number back to get the final n := round(-z / log(2)) as a floating-point number.
65 vn = vsubq_f16(vn, vmagic_bias);
66
67 // Compute reduced argument t := z - n * log(2). Note that -t = -z - n * log(2).
68 // Use Cody-Waite range reduction method (note two constants to represent -log(2)) to improve accuracy.
69 float16x8_t vt = vfmaq_f16(vz, vn, vln2_hi);
70 vt = vfmaq_f16(vt, vn, vln2_lo);
71
72 // Compute degree-3 polynomial approximation for exp(-t) on [-log(2)/2, log(2)/2]:
73 // P(t) = 1 + t * (-1 + t * (c2 + t * c3)) = -(1 - t * p)
74 float16x8_t vp = vfmaq_f16(vc2, vc3, vt);
75 vp = vfmsq_f16(vone, vp, vt);
76
77 // Reconstruct the exp(-z) value:
78 // e = s * (1 + t * (-1 + t * (c2 + t * c3))
79 // = s * (1 - t * (-p))
80 // = s - (t * s) * (-p)
81 vt = vmulq_f16(vt, vs);
82 float16x8_t ve = vfmsq_f16(vs, vp, vt);
83
84 // Denominator of the sigmoid fraction: 1.0 + exp(-z)
85 float16x8_t vd = vaddq_f16(ve, vone);
86
87 // Compute approximate 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 const float16x8_t vr = vrecpeq_f16(vd);
91
92 // Reconstruct sigmoid(-z) = exp(-z) / (1.0 + exp(-z))
93 float16x8_t vf = vmulq_f16(ve, vr);
94
95 // For inputs below denormal cutoff, replace output with +0.0f.
96 // Note that for NaN inputs, comparison result is false, and outputs are left unchanged.
97 vf = vreinterpretq_f16_u16(vbicq_u16(vreinterpretq_u16_f16(vf), vcagtq_f16(vx, vdenorm_cutoff)));
98
99 // Reconstruct sigmoid(x) = x < 0 ? sigmoid(-z) : 1.0 - sigmoid(-z)
100 const uint16x8_t vm = vcltq_f16(vx, vmovq_n_f16(0.0f));
101 vf = vbslq_f16(vm, vf, vsubq_f16(vone, vf));
102
103 vst1q_f16(o, vf); o += 8;
104 }
105 }
106