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/math-stubs.h>
12
13
xnn_math_f32_sigmoid__neonfma_rr2_p5_nr1recps1fma(size_t n,const float * input,float * output)14 void xnn_math_f32_sigmoid__neonfma_rr2_p5_nr1recps1fma(
15 size_t n,
16 const float* input,
17 float* output)
18 {
19 assert(n % (4 * sizeof(float)) == 0);
20
21 // Large number such that ulp(magic bias) == 1 and magic bias === 127 mod 2**22.
22 const float32x4_t vmagic_bias = vmovq_n_f32(0x1.8000FEp23f);
23 const float32x4_t vminus_log2e = vmovq_n_f32(-0x1.715476p+0f);
24 const float32x4_t vln2_hi = vmovq_n_f32(0x1.62E43p-1f);
25 const float32x4_t vln2_lo = vmovq_n_f32(-0x1.05C61p-29f);
26 // Coefficient of polynomial approximation of
27 // exp(-t) ~ 1 + t * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5)))) on [-log(2)/2, log(2)/2]
28 const float32x4_t vc5 = vmovq_n_f32(-0x1.0F9F9Cp-7f);
29 const float32x4_t vc4 = vmovq_n_f32(0x1.573A1Ap-5f);
30 const float32x4_t vc3 = vmovq_n_f32(-0x1.555A80p-3f);
31 const float32x4_t vc2 = vmovq_n_f32(0x1.FFFDC6p-2f);
32 const float32x4_t vc1 = vmovq_n_f32(-0x1.FFFFF6p-1f);
33 const float32x4_t vone = vmovq_n_f32(1.0f);
34 // The largest z for which sigmoidf(-z) is normalized.
35 // This number is also the largest z for which expf(-z) is normalized.
36 const float32x4_t vdenorm_cutoff = vmovq_n_f32(-0x1.5D589Ep+6f);
37
38 for (; n != 0; n -= 4 * sizeof(float)) {
39 const float32x4_t vx = vld1q_f32(input); input += 4;
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 float32x4_t vz = vabsq_f32(vx);
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 trick with adding large number is valid only within certain bounds
54 // (|-z / log(2)| <= 2**22, i.e. |z| <= 0x1.62E43p+22 = 5814540.0), but that is acceptable, because inputs x
55 // outside of [-87.336544, 17.328678] (i.e. z outsize [0, 87.336544]) underflow or saturate sigmoidf(x). We fixup
56 // the result for such inputs at the very end of the algorithm.
57 float32x4_t vn = vfmaq_f32(vmagic_bias, vz, vminus_log2e);
58
59 // Create a floating-point number s (scale) such that s == 2**n for inputs which don't cause underflow, i.e.
60 // -87.336544 <= -z <= 0.0, and -126 <= n <= 0 accordingly.
61 const float32x4_t vs = vreinterpretq_f32_s32(vshlq_n_s32(vreinterpretq_s32_f32(vn), 23));
62
63 // Subtract the large number back to get the final n := round(-z / log(2)) as a floating-point number.
64 vn = vsubq_f32(vn, vmagic_bias);
65
66 // Compute reduced argument t := z + n * log(2). Note that -t = -z - n * log(2).
67 // Use Cody-Waite range reduction method (note two constants to represent log(2)) to improve accuracy.
68 float32x4_t vt = vfmaq_f32(vz, vn, vln2_hi);
69 vt = vfmaq_f32(vt, vn, vln2_lo);
70
71 // Compute degree-5 polynomial approximation for exp(-t) on [-log(2)/2, log(2)/2]:
72 // P(t) = 1 + t * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5)))) = 1 + t * p
73 float32x4_t vp = vfmaq_f32(vc4, vc5, vt);
74 vp = vfmaq_f32(vc3, vp, vt);
75 vp = vfmaq_f32(vc2, vp, vt);
76 vp = vfmaq_f32(vc1, vp, vt);
77
78 // Reconstruct the exp(-z) value:
79 // e = s * (1 + t * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5)))))
80 // = s * (1 + t * p)
81 // = s + (t * s) * p
82 vt = vmulq_f32(vt, vs);
83 float32x4_t ve = vfmaq_f32(vs, vp, vt);
84
85 // Denominator of the sigmoid fraction: 1.0 + exp(-z)
86 float32x4_t vd = vaddq_f32(ve, vone);
87
88 // Use Newton-Raphson method (2 iterations) to compute reciprocal of denominator.
89 // Note: 1 < d <= 2, because z >= 0.0 and 0 < exp(-z) <= 1.0.
90 // Thus the reciprocal of the denominator never overflows.
91 float32x4_t vr = vrecpeq_f32(vd);
92 vr = vmulq_f32(vr, vrecpsq_f32(vr, vd));
93 vr = vfmaq_f32(vr, vr, vfmsq_f32(vone, vr, vd));
94
95 // Reconstruct sigmoid(-z) = exp(-z) / (1.0 + exp(-z))
96 float32x4_t vf = vmulq_f32(ve, vr);
97
98 // For inputs below denormal cutoff, replace output with +0.0f.
99 // Note that for NaN inputs, comparison result is false, and outputs are left unchanged.
100 vf = vreinterpretq_f32_u32(vbicq_u32(vreinterpretq_u32_f32(vf), vcagtq_f32(vx, vdenorm_cutoff)));
101
102 // Reconstruct sigmoid(x) = x < 0 ? sigmoid(-z) : 1.0 - sigmoid(-z)
103 const uint32x4_t vm = vcltq_f32(vx, vmovq_n_f32(0.0f));
104 vf = vbslq_f32(vm, vf, vsubq_f32(vone, vf));
105
106 vst1q_f32(output, vf); output += 4;
107 }
108 }
109