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 #include <math.h>
9
10 #include <xnnpack/common.h>
11 #include <xnnpack/math.h>
12 #include <xnnpack/math-stubs.h>
13
14
xnn_math_f32_sigmoid__scalar_rr2_p5_div(size_t n,const float * input,float * output)15 void xnn_math_f32_sigmoid__scalar_rr2_p5_div(
16 size_t n,
17 const float* input,
18 float* output)
19 {
20 assert(n % sizeof(float) == 0);
21
22 // Large number such that ulp(magic bias) == 1 and magic bias === 127 mod 2**22.
23 const float vmagic_bias = 0x1.8000FEp23f;
24 const float vminus_log2e = -0x1.715476p+0f;
25 // Last 7 bits are zeroes
26 const float vln2_hi = 0x1.62E400p-1f;
27 const float vln2_lo = 0x1.7F7D1Cp-20f;
28 // Coefficient of polynomial approximation of
29 // exp(-t) ~ 1 + t * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5)))) on [-log(2)/2, log(2)/2]
30 const float vc5 = -0x1.0F9F9Cp-7f;
31 const float vc4 = 0x1.573A1Ap-5f;
32 const float vc3 = -0x1.555A80p-3f;
33 const float vc2 = 0x1.FFFDC6p-2f;
34 const float vc1 = -0x1.FFFFF6p-1f;
35 const float vone = 1.0f;
36 // The largest z for which sigmoidf(-z) is normalized.
37 // This number is also the largest z for which expf(-z) is normalized.
38 const float vdenorm_cutoff = 0x1.5D589Ep+6f;
39
40 for (; n != 0; n -= sizeof(float)) {
41 const float vx = *input++;
42
43 // General structure of the algorithm:
44 //
45 // / exp(x) / (1 + exp(x)) if x <= 0
46 // f[x] :=
47 // \ 1 - f[-x] if x >= 0
48 //
49 // First we compute f[-z] := exp(-z) / (1 + exp(-z)) where z = abs(x),
50 // then replace result with 1 - f[-z] if x >= 0.
51 const float vz = fabsf(vx);
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
57 // outside of [-87.336544, 17.328678] (i.e. z outsize [0, 87.336544]) underflow or saturate sigmoidf(x). We fixup
58 // the result for such inputs at the very end of the algorithm.
59 float vn = vz * vminus_log2e + 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 // -87.336544 <= -z <= 0.0, and -126 <= n <= 0 accordingly.
63 const float vs = uint32_as_float(float_as_uint32(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 -= vmagic_bias;
67
68 // Compute reduced argument t := z + n * log(2). Note that -t = -z - n * log(2).
69 // Use Cody-Waite range reduction method (note two constants to represent log(2)) to improve accuracy.
70 float vt = vn * vln2_hi + vz;
71 vt = vn * vln2_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 float vp = vt * vc5 + vc4;
76 vp = vt * vp + vc3;
77 vp = vt * vp + vc2;
78 vp = vt * vp + vc1;
79
80 // Reconstruct the exp(-z) value:
81 // e = s * (1 + t * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5)))))
82 // = s * (1 + t * p)
83 // = s + (t * s) * p
84 vt *= vs;
85 const float ve = vt * vp + vs;
86
87 // Reconstruct sigmoid(-z) = exp(-z) / (1.0 + exp(-z))
88 float vf = ve / (ve + vone);
89
90 // For inputs below denormal cutoff, replace output with +0.0f.
91 // Note that for NaN inputs, comparison result is false, and outputs are left unchanged.
92 if XNN_UNPREDICTABLE(vz > vdenorm_cutoff) {
93 vf = 0.0f;
94 }
95
96 // Reconstruct sigmoid(x) = x < 0 ? sigmoid(-z) : 1.0 - sigmoid(-z)
97 if XNN_UNPREDICTABLE(vx > 0.0f) {
98 vf = vone - vf;
99 }
100
101 *output++ = vf;
102 }
103 }
104