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.h>
13 #include <xnnpack/math-stubs.h>
14
15
16 // Table of exp2(k / 64) values decremented (as integer) by (k << 17), k = 0..63
17 extern XNN_INTERNAL const uint32_t xnn_table_exp2minus_k_over_64[64];
18
xnn_math_f32_sigmoid__scalar_rr2_lut64_p2_div(size_t n,const float * input,float * output)19 void xnn_math_f32_sigmoid__scalar_rr2_lut64_p2_div(
20 size_t n,
21 const float* input,
22 float* output)
23 {
24 assert(n % sizeof(float) == 0);
25
26 // Large number such that ulp(magic bias) == exp2(-6)
27 const float vmagic_bias = 0x1.800000p17f;
28 const float vminus_log2e = -0x1.715476p0f;
29 // Mask for the lowest 6 bits
30 const uint32_t vindex_mask = UINT32_C(0x3F);
31 // Last 13 bits are zeroes
32 const float vln2_hi = 0x1.630000p-1f;
33 const float vln2_lo = -0x1.BD0106p-13f;
34 // Coefficient of polynomial approximation of exp(-t) ~ 1 + t * (1 + t * c2) on [-log(2)/128, log(2)/128]
35 const float vc2 = 0x1.FFFF0Ap-2f;
36 const float vone = 1.0f;
37 // The largest z for which sigmoidf(-z) is normalized.
38 // This number is also the largest z for which expf(-z) is normalized.
39 const float vdenorm_cutoff = 0x1.5D589Ep+6f;
40
41 for (; n != 0; n -= sizeof(float)) {
42 const float vx = *input++;
43
44 // General structure of the algorithm:
45 //
46 // / exp(x) / (1 + exp(x)) if x <= 0
47 // f[x] :=
48 // \ 1 - f[-x] if x >= 0
49 //
50 // First we compute f[-z] := exp(-z) / (1 + exp(-z)) where z = abs(x),
51 // then replace result with 1 - f[-z] if x >= 0.
52 const float vz = fabsf(vx);
53
54 // Compute reduced argument n := round(-z / log(2), 6).
55 // We do it by adding a large number (magic bias), which cause rounding of the result to integer, then subtracing
56 // the large number back. The trick with adding large number is valid only within certain bounds
57 // (|-z / log(2)| <= 2**16, i.e. |z| <= 0x1.62E43p+15 = 5814540.0), but that is acceptable, because inputs x
58 // outside of [-87.336544, 17.328678] (i.e. z outsize [0, 87.336544]) underflow or saturate sigmoidf(x). We fixup
59 // the result for such inputs at the very end of the algorithm.
60 float vn = vz * vminus_log2e + vmagic_bias;
61
62 // Create a floating-point number s (scale) such that s := 2**n for such inputs that sigmoidf(-z) is normalized,
63 // 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
64 // in two steps:
65 // 1. Fetch 2**frac(n) from the table using the 6 low bits of n, as integer. Note that the fetched values are in
66 // the [1.0, 2.0) range, i.e. their floating-point exponent is 0.
67 // 2. Adjust fecthed value by addition of int(n) to its floating-point exponent. The result is always a normalized
68 // number, because for 0 <= z <= 87.33642 (inputs for which sigmoidf(z) is normalized) we have
69 // -126 <= int(n) <= 0, and thus the adjusted exponent is not lower than -126.
70 //
71 // Shift bits 6:14 into 23:31 (position of floating-point exponent).
72 const uint32_t ve = float_as_uint32(vn) << 17;
73
74 // Use bits 0:6 of n, as integer, as an index for table lookup of l := 2**frac(n).
75 const uint32_t vidx = float_as_uint32(vn) & vindex_mask;
76 // Adjust exponent of the value l fetched from the table to get the final s value.
77 const float vs = uint32_as_float(xnn_table_exp2minus_k_over_64[vidx] + ve);
78
79 // Subtract the large number back to get the final n := round(-z / log(2), 6) as a floating-point number.
80 vn -= vmagic_bias;
81
82 // Compute reduced argument t := (z + n * log(2)). Note that -t = -z - n * log(2).
83 // Use Cody-Waite range reduction method (note two constants to represent log(2)) to improve accuracy.
84 float vt = vn * vln2_hi + vz;
85 vt = vn * vln2_lo + vt;
86
87 // Compute degree-2 polynomial approximation for exp(-t) on [-log(2)/128, log(2)/128].
88 // P(t) = 1 + t * (-1 + t * c2) = 1 - (t - t * (t * c2)) = 1 - p
89 float vp = vt * vc2;
90 vp = vt - vp * vt;
91
92 // Reconstruct the exp(-z) value:
93 // e = s * (1 + t * (-1 + t * c2))
94 // = s * (1 - p)
95 // = s - s * p
96 const float vy = vs - vs * vp;
97
98 // Reconstruct sigmoid(-z) = exp(-z) / (1.0 + exp(-z))
99 float vf = vy / (vy + vone);
100
101 // For inputs below denormal cutoff, replace output with +0.0f.
102 // Note that for NaN inputs, comparison result is false, and outputs are left unchanged.
103 if XNN_UNPREDICTABLE(vz > vdenorm_cutoff) {
104 vf = 0.0f;
105 }
106
107 // Reconstruct sigmoid(x) = x < 0 ? sigmoid(-z) : 1.0 - sigmoid(-z)
108 if XNN_UNPREDICTABLE(vx > 0.0f) {
109 vf = vone - vf;
110 }
111
112 *output++ = vf;
113 }
114 }
115