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 <immintrin.h>
10
11 #include <xnnpack/common.h>
12 #include <xnnpack/math-stubs.h>
13
14
xnn_math_f32_expm1minus__avx2_rr1_lut8_p4_perm(size_t n,const float * input,float * output)15 void xnn_math_f32_expm1minus__avx2_rr1_lut8_p4_perm(
16 size_t n,
17 const float* input,
18 float* output)
19 {
20 assert(n % (8 * sizeof(float)) == 0);
21
22 // The largest x for which expm1f(x) is saturated at -1.0f.
23 const __m256 vsat_cutoff = _mm256_set1_ps(-0x1.154246p+4f);
24 // Large number such that ulp(magic bias) == exp2(-3)
25 const __m256 vmagic_bias = _mm256_set1_ps(0x1.800000p20f);
26 const __m256 vlog2e = _mm256_set1_ps(0x1.715476p+0f);
27 // Table of exp2(k / 8) values decremented (as integer) by (k << 20), k = 0..7
28 const __m256i vtable = _mm256_set_epi32(
29 0x3F7AC0C7, 0x3F7744FD, 0x3F75672A, 0x3F7504F3, 0x3F75FED7, 0x3F7837F0, 0x3F7B95C2, 0x3F800000);
30 const __m256 vminus_ln2 = _mm256_set1_ps(-0x1.62E43p-1f);
31 // Coefficient of polynomial approximation
32 // exp(t) - 1 ~ t * (1 + t * (c2 + t * (c3 + t * c4)))
33 // on [-log(2)/16, log(2)/16]
34 const __m256 vc4 = _mm256_set1_ps(0x1.5558ECp-5f);
35 const __m256 vc3 = _mm256_set1_ps(0x1.555C20p-3f);
36 const __m256 vc2 = _mm256_set1_ps(0x1.000000p-1f);
37 const __m256 vone = _mm256_set1_ps(1.0f);
38
39 for (; n != 0; n -= 8 * sizeof(float)) {
40 __m256 vx = _mm256_loadu_ps(input);
41
42 // The function saturates at -1 for large negative inputs: expm1f(x) == -1.0f for x <= sat_cutoff ~= -17.328680.
43 // To guarantee this behaviour, we clip input at sat_cutoff, and leverage the fact that for our implementation
44 // expm1f(sat_cutoff) == -1.0f. The order of operands in the VMAXPS instruction matters: it ensures that NaN
45 // inputs are passed unchanged.
46 vx = _mm256_max_ps(vsat_cutoff, vx);
47
48 // Compute reduced argument n := round(x / log(2), 3).
49 // We do it by adding a large number (magic bias), which cause rounding of the result to 3 fractional bits, then
50 // subtracing the large number back. The first addition is combined with multiplication by log2e into a single FMA
51 // instruction. The trick with adding large number is valid only within certain bounds (|x / log(2)| <= 2**19,
52 // i.e. |x| <= 0x1.62E43p+18 = 363408.75), but that is acceptable, because inputs x are restricted to
53 // [-17.328680, 0].
54 // Note that addition-subtraction of the large number doesn't cause overflow for inputs in this range.
55 __m256 vn = _mm256_fmadd_ps(vx, vlog2e, vmagic_bias);
56
57 // Create a floating-point number s (scale) such that s := 2**n for valid inputs, i.e. -17.328680 <= x <= 0.0. As n
58 // has 4 fractional bits, we split s == 2**n = 2**int(n) * 2**frac(n). We create s in two steps:
59 // 1. Fetch 2**frac(n) from the table using the 4 low bits of n, as integer. Note that the fetched values are in
60 // the [1.0, 2.0) range, i.e. their floating-point exponent is 0.
61 // 2. Adjust fecthed value by addition of int(n) to its floating-point exponent. The result is always a normalized
62 // number, because for -17.328680 <= x <= 0.0 we have -25 <= int(n) <= 0, and thus the adjusted exponent is not
63 // lower than -25.
64 //
65 // Shift bits 3:11 into 23:31 (position of floating-point exponent).
66 const __m256i ven = _mm256_slli_epi32(_mm256_castps_si256(vn), 20);
67
68 // Use bits 0:3 bits of n, as integer, as an index for table lookup of l := 2**frac(n).
69 const __m256i vl = _mm256_permutevar8x32_epi32(vtable, _mm256_castps_si256(vn));
70
71 // Adjust exponent of the value l fetched from the table to get the final s value.
72 const __m256 vs = _mm256_castsi256_ps(_mm256_add_epi32(vl, ven));
73
74 // Subtract the large number back to get final n := round(x / log(2), 3).
75 vn = _mm256_sub_ps(vn, vmagic_bias);
76
77 // Compute reduced argument t := x - n * log(2).
78 __m256 vt = _mm256_fmadd_ps(vn, vminus_ln2, vx);
79
80 // Compute degree-4 polynomial approximation for exp(t) - 1 on [-log(2)/16, log(2)/16].
81 // P(t) = t * (1 + t * (c2 + t * (c3 + t * c4))) = t + t * (t * (c2 + t * (c3 + t * c4))) = t + t * p
82 __m256 vp = _mm256_fmadd_ps(vc4, vt, vc3);
83 vp = _mm256_fmadd_ps(vp, vt, vc2);
84 vp = _mm256_mul_ps(vp, vt);
85
86 // Reconstruct the exp(x) - 1 value:
87 // exp(x) - 1 = s * (1 + t * (1 + t * (c2 + t * (c3 + t * c4)))) - 1
88 // = (s - 1) + s * (t + t * p)
89 // = ((t * s) + (t * s) * p) + (s - 1)
90 vt = _mm256_mul_ps(vt, vs);
91 const __m256 vsm1 = _mm256_sub_ps(vs, vone);
92 vp = _mm256_fmadd_ps(vp, vt, vt);
93 const __m256 vf = _mm256_add_ps(vp, vsm1);
94
95 _mm256_storeu_ps(output, vf);
96
97 input += 8;
98 output += 8;
99 }
100 }
101