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 <xnnpack/common.h>
10 #include <xnnpack/math.h>
11 #include <xnnpack/math-stubs.h>
12
13
14 // Table of exp2(k / 64) values decremented (as integer) by (k << 17), k = 0..63
15 extern XNN_INTERNAL const uint32_t xnn_table_exp2minus_k_over_64[64];
16
xnn_math_f32_expminus__scalar_rr2_lut64_p2(size_t n,const float * input,float * output)17 void xnn_math_f32_expminus__scalar_rr2_lut64_p2(
18 size_t n,
19 const float* input,
20 float* output)
21 {
22 assert(n % sizeof(float) == 0);
23
24 // Large number such that ulp(magic bias) == exp2(-6)
25 const float vmagic_bias = 0x1.800000p17f;
26 const float vlog2e = 0x1.715476p0f;
27 // Mask for the lowest 6 bits
28 const uint32_t vindex_mask = UINT32_C(0x3F);
29 // Last 13 bits are zeroes
30 const float vminus_ln2_hi = -0x1.630000p-1f;
31 const float vminus_ln2_lo = 0x1.BD0106p-13f;
32 // Coefficient of polynomial approximation
33 // exp(t) ~ 1 + t * (1 + t * c2)
34 // on [-log(2)/128, log(2)/128]
35 const float vc2 = 0x1.FFFF0Ap-2f;
36 // The smallest x for which expf(x) is normalized.
37 const float vdenorm_cutoff = -0x1.5D589Ep6f;
38
39 for (; n != 0; n -= sizeof(float)) {
40 const float vx = *input++;
41
42 // Compute reduced argument n := round(x / log(2), 6).
43 // We do it by adding a large number (magic bias), which cause rounding of the result to 6 fractional bits, then
44 // subtracing the large number back. The trick with adding large number is valid only within certain bounds
45 // (|x / log(2)| <= 2**16, i.e. |x| <= 0x1.62E43p+15 = 45426.09375), but that is acceptable, because inputs x
46 // outside of [-87.336544, 0] underflow expf(x). We fixup the result for such inputs at the very end of the
47 // algorithm.
48 float vn = vx * vlog2e + vmagic_bias;
49
50 // Create a floating-point number s (scale) such that s := 2**n for such inputs that expf(x) is normalized, i.e.
51 // -87.336544 <= x <= 0. As n has 6 fractional bits, we split s == 2**n = 2**int(n) * 2**frac(n). We create s in
52 // two steps:
53 // 1. Fetch 2**frac(n) from the table using the 6 low bits of n, as integer. Note that the fetched values are in
54 // the [1.0, 2.0) range, i.e. their floating-point exponent is 0.
55 // 2. Adjust fecthed value by addition of int(n) to its floating-point exponent. The result is always a normalized
56 // number, because for -87.33642 <= x <= 0 (inputs for which expf(x) is normalized) we have -126 <= int(n) <= 0,
57 // and thus the adjusted exponent is not lower than -126.
58 //
59 // Shift bits 6:14 into 23:31 (position of floating-point exponent).
60 const uint32_t ve = float_as_uint32(vn) << 17;
61
62 // Use bits 0:6 of n, as integer, as an index for table lookup of l := 2**frac(n).
63 const uint32_t vidx = float_as_uint32(vn) & vindex_mask;
64 // Adjust exponent of the value l fetched from the table to get the final s value.
65 const float vs = uint32_as_float(xnn_table_exp2minus_k_over_64[vidx] + ve);
66
67 // Subtract the large number back to get the final n := round(x / log(2), 6) as a floating-point number.
68 vn -= vmagic_bias;
69
70 // Compute reduced argument t := x - n * log(2)
71 // Use Cody-Waite range reduction method (note the two constants representing log(2)) to improve accuracy.
72 float vt = vn * vminus_ln2_hi + vx;
73 vt = vn * vminus_ln2_lo + vt;
74
75 // Compute degree-2 polynomial approximation for exp(t) on [-log(2)/128, log(2)/128].
76 // P(t) = 1 + t * (1 + t * c2) = 1 + (t + t * (t * c2)) = 1 + p
77 float vp = vt * vc2;
78 vp = vp * vt + vt;
79
80 // Reconstruct the exp(x) value:
81 // exp(x) = s * (1 + t * (1 + t * c2))
82 // = s * (1 + p)
83 // = s + s * p
84 float vf = vp * vs + vs;
85
86 // For inputs below denormal cutoff, replace output with +0.0f.
87 // Note that for NaN inputs, comparison result is false, and outputs are left unchanged.
88 if XNN_UNPREDICTABLE(vx < vdenorm_cutoff) {
89 vf = 0.0f;
90 }
91
92 *output++ = vf;
93 }
94 }
95