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 <math.h>
10
11 #include <xnnpack/common.h>
12 #include <xnnpack/math-stubs.h>
13
14 #include <fp16/bitcasts.h>
15
16
xnn_math_f32_sigmoid__scalar_p5_div(size_t n,const float * input,float * output)17 void xnn_math_f32_sigmoid__scalar_p5_div(
18 size_t n,
19 const float* input,
20 float* output)
21 {
22 assert(n % sizeof(float) == 0);
23
24 const float vmagic_bias = 0x1.8000FEp23f;
25 // The largest z for which sigmoidf(-z) is normalized.
26 // This number is also the largest z for which expf(-z) is normalized.
27 const float vdenorm_cutoff = 0x1.5D589Ep+6f;
28 const float vminus_log2e = -0x1.715476p+0f;
29 // Last 7 bits are zeroes
30 const float vln2_hi = 0x1.62E400p-1f;
31 const float vln2_lo = 0x1.7F7D1Cp-20f;
32 const float vone = 1.0f;
33
34 const float vc1 = -0x1.FFFFF6p-1f;
35 const float vc2 = 0x1.FFFDC6p-2f;
36 const float vc3 = -0x1.555A80p-3f;
37 const float vc4 = 0x1.573A1Ap-5f;
38 const float vc5 = -0x1.0F9F9Cp-7f;
39
40 for (; n != 0; n -= sizeof(float)) {
41 const float vx = *input++;
42
43 // General structure of the algorithm:
44 // / exp(x) / (1 + exp(x)) if x <= 0
45 // f[x] :=
46 // \ 1 - f[-x] if x >= 0
47 //
48 // First we compute f[-z] := exp(-z) / (1 + exp(-z)) where z = abs(x),
49 // then replace result with 1 - f[-z] if x >= 0.
50 const float vz = fabsf(vx);
51
52 // Compute reduced argument n := round(-z / log(2)).
53 // We do it by adding a large number (magic bias), which cause rounding of result to an integer, then subtracing the
54 // large number back. The first addition is combined with multiplication by log2e into a single FMA instruction.
55 // The trick with adding large number is valid only within certain bounds (|x| <= 2**22), but thats ok, because
56 // inputs x outside of [-87.336544, 17.328678] (i.e. z outsize [0, 87.336544]) underflow or saturate sigmoidf(x)
57 // anyway. We fixup the result for such inputs at the very end of the algorithm.
58 float vn = vz * vminus_log2e + vmagic_bias;
59
60 // Create a floating-point number s (scale) such that s == 2**n for inputs which don't cause underflow, i.e.
61 // -87.336544 <= -z <= 0.0, and -126 <= n <= 0 accordingly.
62 const float vs = fp32_from_bits(fp32_to_bits(vn) << 23);
63
64 // Subtract the large number back to get the final n := round(-z / log(2)) as a floating-point number.
65 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 float vt = vn * vln2_hi + vz;
70 vt = vn * vln2_lo + vt;
71
72 // Compute degree-5 polynomial approximation for exp(-t) on [-log(2)/2, log(2)/2]:
73 // P5(t) = 1 + t * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5))))
74 float vp = vt * vc5 + vc4;
75 vp = vt * vp + vc3;
76 vp = vt * vp + vc2;
77 vp = vt * vp + vc1;
78
79 // Reconstruct the exp(-z) value:
80 // e = s * (1 + t * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5)))))
81 // = s + (t * s) * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5))))
82 // = s + (t * s) * p
83 vt *= vs;
84 const float ve = vt * vp + vs;
85
86 // Reconstruct sigmoid(-z) = exp(-z) / (1.0 + exp(-z))
87 float vf = ve / (ve + vone);
88
89 // For inputs below denormal cutoff, replace output with +0.0f.
90 // Note that for NaN inputs, comparison result is false, and outputs are left unchanged.
91 if XNN_UNPREDICTABLE(vz > vdenorm_cutoff) {
92 vf = 0.0f;
93 }
94
95 // Reconstruct sigmoid(x) = x < 0 ? sigmoid(-z) : 1.0 - sigmoid(-z)
96 if XNN_UNPREDICTABLE(vx > 0.0f) {
97 vf = vone - vf;
98 }
99
100 *output++ = vf;
101 }
102 }
103