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
17 // Table of exp2(k / 64) values decremented (as integer) by (k << 17), k = 0..63
18 extern XNN_INTERNAL const uint32_t xnn_table_exp2minus_k_over_64[64];
19
xnn_math_f32_sigmoid__scalar_rr2_lut64_p2_div(size_t n,const float * input,float * output)20 void xnn_math_f32_sigmoid__scalar_rr2_lut64_p2_div(
21 size_t n,
22 const float* input,
23 float* output)
24 {
25 assert(n % sizeof(float) == 0);
26
27 // Large number such that ulp(magic bias) == exp2(-6)
28 const float vmagic_bias = 0x1.800000p17f;
29 const float vminus_log2e = -0x1.715476p0f;
30 // Mask for the lowest 6 bits
31 const uint32_t vindex_mask = UINT32_C(0x3F);
32 // Last 13 bits are zeroes
33 const float vln2_hi = 0x1.630000p-1f;
34 const float vln2_lo = -0x1.BD0106p-13f;
35 // Coefficient of polynomial approximation of exp(-t) ~ 1 + t * (1 + t * c2) on [-log(2)/128, log(2)/128]
36 const float vc2 = 0x1.FFFF0Ap-2f;
37 const float vone = 1.0f;
38 // The largest z for which sigmoidf(-z) is normalized.
39 // This number is also the largest z for which expf(-z) is normalized.
40 const float vdenorm_cutoff = 0x1.5D589Ep+6f;
41
42 for (; n != 0; n -= sizeof(float)) {
43 const float vx = *input++;
44
45 // General structure of the algorithm:
46 //
47 // / exp(x) / (1 + exp(x)) if x <= 0
48 // f[x] :=
49 // \ 1 - f[-x] if x >= 0
50 //
51 // First we compute f[-z] := exp(-z) / (1 + exp(-z)) where z = abs(x),
52 // then replace result with 1 - f[-z] if x >= 0.
53 const float vz = fabsf(vx);
54
55 // Compute reduced argument n := round(-z / log(2), 6).
56 // We do it by adding a large number (magic bias), which cause rounding of the result to integer, then subtracing
57 // the large number back. The trick with adding large number is valid only within certain bounds
58 // (|-z / log(2)| <= 2**16, i.e. |z| <= 0x1.62E43p+15 = 5814540.0), but that is acceptable, because inputs x
59 // outside of [-87.336544, 17.328678] (i.e. z outsize [0, 87.336544]) underflow or saturate sigmoidf(x). We fixup
60 // the result for such inputs at the very end of the algorithm.
61 float vn = vz * vminus_log2e + vmagic_bias;
62
63 // Create a floating-point number s (scale) such that s := 2**n for such inputs that sigmoidf(-z) is normalized,
64 // i.e. 0 <= z <= 87.33642. As n has 6 fractional bits, we split s == 2**n = 2**int(n) * 2**frac(n). We create s
65 // in two steps:
66 // 1. Fetch 2**frac(n) from the table using the 6 low bits of n, as integer. Note that the fetched values are in
67 // the [1.0, 2.0) range, i.e. their floating-point exponent is 0.
68 // 2. Adjust fecthed value by addition of int(n) to its floating-point exponent. The result is always a normalized
69 // number, because for 0 <= z <= 87.33642 (inputs for which sigmoidf(z) is normalized) we have
70 // -126 <= int(n) <= 0, and thus the adjusted exponent is not lower than -126.
71 //
72 // Shift bits 6:14 into 23:31 (position of floating-point exponent).
73 const uint32_t ve = fp32_to_bits(vn) << 17;
74
75 // Use bits 0:6 of n, as integer, as an index for table lookup of l := 2**frac(n).
76 const uint32_t vidx = fp32_to_bits(vn) & vindex_mask;
77 // Adjust exponent of the value l fetched from the table to get the final s value.
78 const float vs = fp32_from_bits(xnn_table_exp2minus_k_over_64[vidx] + ve);
79
80 // Subtract the large number back to get the final n := round(-z / log(2), 6) as a floating-point number.
81 vn -= vmagic_bias;
82
83 // Compute reduced argument t := (z + n * log(2)). Note that -t = -z - n * log(2).
84 // Use Cody-Waite range reduction method (note two constants to represent log(2)) to improve accuracy.
85 float vt = vn * vln2_hi + vz;
86 vt = vn * vln2_lo + vt;
87
88 // Compute degree-2 polynomial approximation for exp(-t) on [-log(2)/128, log(2)/128].
89 // P(t) = 1 + t * (-1 + t * c2) = 1 - (t - t * (t * c2)) = 1 - p
90 float vp = vt * vc2;
91 vp = vt - vp * vt;
92
93 // Reconstruct the exp(-z) value:
94 // e = s * (1 + t * (-1 + t * c2))
95 // = s * (1 - p)
96 // = s - s * p
97 const float vy = vs - vs * vp;
98
99 // Reconstruct sigmoid(-z) = exp(-z) / (1.0 + exp(-z))
100 float vf = vy / (vy + vone);
101
102 // For inputs below denormal cutoff, replace output with +0.0f.
103 // Note that for NaN inputs, comparison result is false, and outputs are left unchanged.
104 if XNN_UNPREDICTABLE(vz > vdenorm_cutoff) {
105 vf = 0.0f;
106 }
107
108 // Reconstruct sigmoid(x) = x < 0 ? sigmoid(-z) : 1.0 - sigmoid(-z)
109 if XNN_UNPREDICTABLE(vx > 0.0f) {
110 vf = vone - vf;
111 }
112
113 *output++ = vf;
114 }
115 }
116