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
15 // Table of exp2(k / 2048) values decremented (as integer) by (k << 12), k = 0..2048
16 extern XNN_INTERNAL const uint32_t xnn_table_exp2minus_k_over_2048[2048];
17
xnn_math_f32_sigmoid__scalar_rr2_lut2048_p1_div(size_t n,const float * input,float * output)18 void xnn_math_f32_sigmoid__scalar_rr2_lut2048_p1_div(
19 size_t n,
20 const float* input,
21 float* output)
22 {
23 assert(n % sizeof(float) == 0);
24
25 // Large number such that ulp(magic bias) == exp2(-11)
26 const float vmagic_bias = 0x1.800000p12f;
27 const float vminus_log2e = -0x1.715476p0f;
28 // Mask for the lowest 11 bits
29 const uint32_t vindex_mask = UINT32_C(0x7FF);
30 // Last 13 bits are zeroes
31 const float vln2_hi = 0x1.600000p-1f;
32 const float vln2_lo = 0x1.7217F8p-8f;
33 // Coefficient of polynomial approximation of exp(-t) ~ 1 + t * c1 on [-log(2)/2048, log(2)/2048]
34 const float vc1 = -0x1.FFFFFEp-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), 11).
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**11, i.e. |z| <= 0x1.62E43p+10 = 1419.5654296875), 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 such inputs that sigmoidf(-z) is normalized,
62 // i.e. 0 <= z <= 87.33642. As n has 11 fractional bits, we split s == 2**n = 2**int(n) * 2**frac(n). We create s
63 // in two steps:
64 // 1. Fetch 2**frac(n) from the table using the 11 low bits of n, as integer. Note that the fetched values are in
65 // the [1.0, 2.0) range, i.e. their floating-point exponent is 0.
66 // 2. Adjust fecthed value by addition of int(n) to its floating-point exponent. The result is always a normalized
67 // number, because for 0 <= z <= 87.33642 (inputs for which sigmoidf(z) is normalized) we have
68 // -126 <= int(n) <= 0, and thus the adjusted exponent is not lower than -126.
69 //
70 // Shift bits 11:19 into 23:31 (position of floating-point exponent).
71 const uint32_t ve = float_as_uint32(vn) << 12;
72
73 // Use bits 0:11 of n, as integer, as an index for table lookup of l := 2**frac(n).
74 const uint32_t vidx = float_as_uint32(vn) & vindex_mask;
75 // Adjust exponent of the value l fetched from the table to get the final s value.
76 const float vs = uint32_as_float(xnn_table_exp2minus_k_over_2048[vidx] + ve);
77
78 // Subtract the large number back to get the final n := round(-z / log(2), 11) as a floating-point number.
79 vn -= vmagic_bias;
80
81 // Compute reduced argument t := (z + n * log(2)). Note that -t = -z - n * log(2).
82 // Use Cody-Waite range reduction method (note two constants to represent log(2)) to improve accuracy.
83 float vt = vn * vln2_hi + vz;
84 vt = vn * vln2_lo + vt;
85
86 // Compute degree-1 polynomial approximation for exp(-t) on [-log(2)/2048, log(2)/2048]:
87 // P(t) = 1 + t * c1 = 1 + p
88 const float vp = vt * vc1;
89
90 // Reconstruct the exp(-z) value:
91 // e = s * (1 + t * c1)
92 // = s * (1 + p)
93 // = s + s * p
94 const float vy = vp * vs + vs;
95
96 // Reconstruct sigmoid(-z) = exp(-z) / (1.0 + exp(-z))
97 float vf = vy / (vy + vone);
98
99 // For inputs below denormal cutoff, replace output with +0.0f.
100 // Note that for NaN inputs, comparison result is false, and outputs are left unchanged.
101 if XNN_UNPREDICTABLE(vz > vdenorm_cutoff) {
102 vf = 0.0f;
103 }
104
105 // Reconstruct sigmoid(x) = x < 0 ? sigmoid(-z) : 1.0 - sigmoid(-z)
106 if XNN_UNPREDICTABLE(vx > 0.0f) {
107 vf = vone - vf;
108 }
109
110 *output++ = vf;
111 }
112 }
113