1 // Copyright 2020 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 / 4) values decremented (as integer) by (k << 21), k = 0..3
16 extern XNN_INTERNAL const uint32_t xnn_table_exp2minus_k_over_4[4];
17
xnn_math_f32_expm1minus__scalar_rr2_lut4_p4(size_t n,const float * input,float * output)18 void xnn_math_f32_expm1minus__scalar_rr2_lut4_p4(
19 size_t n,
20 const float* input,
21 float* output)
22 {
23 assert(n % (4 * sizeof(float)) == 0);
24
25 // Large number such that ulp(magic bias) == exp2(-2)
26 const float vmagic_bias = 0x1.800000p21f;
27 const float vlog2e = 0x1.715476p+0f;
28 // Mask for the lowest 2 bits
29 const uint32_t vindex_mask = UINT32_C(0x3);
30 // The largest x for which expm1f(x) is saturated at -1.0f.
31 const float vsat_cutoff = -0x1.154246p+4f;
32 // Last 7 bits are zeroes
33 const float vminus_ln2_hi = -0x1.62E400p-1f;
34 const float vminus_ln2_lo = -0x1.7F7D1Cp-20f;
35 // Coefficient of polynomial approximation
36 // exp(t) - 1 ~ t * (1 + t * (c2 + t * (c3 + t * c4)))
37 // on [-log(2)/8, log(2)/8]
38 const float vc4 = 0x1.554F9Ap-5f;
39 const float vc3 = 0x1.557082p-3f;
40 const float vc2 = 0x1.000002p-1f;
41 const float vone = 1.0f;
42
43 for (; n != 0; n -= sizeof(float)) {
44 float vx = *input++;
45
46 // Compute reduced argument n := round(x / log(2), 2).
47 // We do it by adding a large number (magic bias), which cause rounding of the result to 2 fractional bits, then
48 // subtracing the large number back. The trick with adding large number is valid only within certain bounds
49 // (|x / log(2)| <= 2**20, i.e. |x| <= 0x1.62E43p+19 = 726817.5), but that is acceptable, because inputs x are
50 // restricted to [-17.328680, 0].
51 // Note that addition-subtraction of the large number doesn't cause overflow for inputs in this range.
52 float vn = vx * vlog2e + vmagic_bias;
53
54 // Create a floating-point number s (scale) such that s := 2**n for valid inputs, i.e. -17.328680 <= x <= 0.0. As n
55 // has 2 fractional bits, we split s == 2**n = 2**int(n) * 2**frac(n). We create s in two steps:
56 // 1. Fetch 2**frac(n) from the table using the 2 low bits of n, as integer. Note that the fetched values are in
57 // the [1.0, 2.0) range, i.e. their floating-point exponent is 0.
58 // 2. Adjust fecthed value by addition of int(n) to its floating-point exponent. The result is always a normalized
59 // number, because for -17.328680 <= x <= 0.0 we have -25 <= int(n) <= 0, and thus the adjusted exponent is not
60 // lower than -25.
61 //
62 // Shift bits 2:10 into 23:31 (position of floating-point exponent).
63 const uint32_t ven = fp32_to_bits(vn) << 21;
64
65 // Use bits 0:2 bits of n, as integer, as an index for table lookup of l := 2**frac(n).
66 const uint32_t vidx = fp32_to_bits(vn) & vindex_mask;
67 // Adjust exponent of the value l fetched from the table to get the final s value.
68 float vs = fp32_from_bits(xnn_table_exp2minus_k_over_4[vidx] + ven);
69
70 // Subtract the large number back to get final n := round(x / log(2), 2).
71 vn -= vmagic_bias;
72
73 // Compute reduced argument t := x - n * log(2).
74 // Use Cody-Waite range reduction method (note two constants to represent log(2)) to improve accuracy.
75 float vt = vn * vminus_ln2_hi + vx;
76 vt = vn * vminus_ln2_lo + vt;
77
78 // The function saturates at -1 for large negative inputs: expm1f(x) == -1.0f for x <= sat_cutoff ~= -17.328680.
79 // To guarantee this behaviour, we zero out s (scale) and t (reduced argument) for x <= sat_cutoff.
80 if XNN_UNPREDICTABLE(vx <= vsat_cutoff) {
81 vs = 0.0f;
82 vt = 0.0f;
83 }
84
85 // Compute degree-4 polynomial approximation for exp(t) - 1 on [-log(2)/8, log(2)/8].
86 // P(t) = t * (1 + t * (c2 + t * (c3 + t * c4))) = t + t * (t * (c2 + t * (c3 + t * c4))) = t + t * p
87 float vp = vc4 * vt + vc3;
88 vp = vp * vt + vc2;
89 vp *= vt;
90
91 // Reconstruct the exp(x) - 1 value:
92 // exp(x) - 1 = s * (1 + t * (1 + t * (c2 + t * (c3 + t * c4)))) - 1
93 // = (s - 1) + s * (t + t * p)
94 // = ((t * s) + (t * s) * p) + (s - 1)
95 vt *= vs;
96 const float vsm1 = vs - vone;
97 vp = vp * vt + vt;
98 const float vf = vp + vsm1;
99
100 *output++ = vf;
101 }
102 }
103