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-stubs.h>
11
12 #include <fp16/bitcasts.h>
13
14
15 // Table of exp2(k / 64) values decremented (as integer) by (k << 17), k = 0..63
16 extern XNN_INTERNAL const uint32_t xnn_table_exp2minus_k_over_64[64];
17
xnn_math_f32_expminus__scalar_rr2_lut64_p2(size_t n,const float * input,float * output)18 void xnn_math_f32_expminus__scalar_rr2_lut64_p2(
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(-6)
26 const float vmagic_bias = 0x1.800000p17f;
27 const float vlog2e = 0x1.715476p0f;
28 // Mask for the lowest 6 bits
29 const uint32_t vindex_mask = UINT32_C(0x3F);
30 // Last 13 bits are zeroes
31 const float vminus_ln2_hi = -0x1.630000p-1f;
32 const float vminus_ln2_lo = 0x1.BD0106p-13f;
33 // Coefficient of polynomial approximation
34 // exp(t) ~ 1 + t * (1 + t * c2)
35 // on [-log(2)/128, log(2)/128]
36 const float vc2 = 0x1.FFFF0Ap-2f;
37 // The smallest x for which expf(x) is normalized.
38 const float vdenorm_cutoff = -0x1.5D589Ep6f;
39
40 for (; n != 0; n -= sizeof(float)) {
41 const float vx = *input++;
42
43 // Compute reduced argument n := round(x / log(2), 6).
44 // We do it by adding a large number (magic bias), which cause rounding of the result to 6 fractional bits, then
45 // subtracing the large number back. The trick with adding large number is valid only within certain bounds
46 // (|x / log(2)| <= 2**16, i.e. |x| <= 0x1.62E43p+15 = 45426.09375), but that is acceptable, because inputs x
47 // outside of [-87.336544, 0] underflow expf(x). We fixup the result for such inputs at the very end of the
48 // algorithm.
49 float vn = vx * vlog2e + vmagic_bias;
50
51 // Create a floating-point number s (scale) such that s := 2**n for such inputs that expf(x) is normalized, i.e.
52 // -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
53 // two steps:
54 // 1. Fetch 2**frac(n) from the table using the 6 low bits of n, as integer. Note that the fetched values are in
55 // the [1.0, 2.0) range, i.e. their floating-point exponent is 0.
56 // 2. Adjust fecthed value by addition of int(n) to its floating-point exponent. The result is always a normalized
57 // number, because for -87.33642 <= x <= 0 (inputs for which expf(x) is normalized) we have -126 <= int(n) <= 0,
58 // and thus the adjusted exponent is not lower than -126.
59 //
60 // Shift bits 6:14 into 23:31 (position of floating-point exponent).
61 const uint32_t ve = fp32_to_bits(vn) << 17;
62
63 // Use bits 0:6 of n, as integer, as an index for table lookup of l := 2**frac(n).
64 const uint32_t vidx = fp32_to_bits(vn) & vindex_mask;
65 // Adjust exponent of the value l fetched from the table to get the final s value.
66 const float vs = fp32_from_bits(xnn_table_exp2minus_k_over_64[vidx] + ve);
67
68 // Subtract the large number back to get the final n := round(x / log(2), 6) as a floating-point number.
69 vn -= vmagic_bias;
70
71 // Compute reduced argument t := x - n * log(2)
72 // Use Cody-Waite range reduction method (note the two constants representing log(2)) to improve accuracy.
73 float vt = vn * vminus_ln2_hi + vx;
74 vt = vn * vminus_ln2_lo + vt;
75
76 // Compute degree-2 polynomial approximation for exp(t) on [-log(2)/128, log(2)/128].
77 // P(t) = 1 + t * (1 + t * c2) = 1 + (t + t * (t * c2)) = 1 + p
78 float vp = vt * vc2;
79 vp = vp * vt + vt;
80
81 // Reconstruct the exp(x) value:
82 // exp(x) = s * (1 + t * (1 + t * c2))
83 // = s * (1 + p)
84 // = s + s * p
85 float vf = vp * vs + vs;
86
87 // For inputs below denormal cutoff, replace output with +0.0f.
88 // Note that for NaN inputs, comparison result is false, and outputs are left unchanged.
89 if XNN_UNPREDICTABLE(vx < vdenorm_cutoff) {
90 vf = 0.0f;
91 }
92
93 *output++ = vf;
94 }
95 }
96