1 /* Double-precision vector (SVE) exp function.
2
3 Copyright (C) 2023 Free Software Foundation, Inc.
4 This file is part of the GNU C Library.
5
6 The GNU C Library is free software; you can redistribute it and/or
7 modify it under the terms of the GNU Lesser General Public
8 License as published by the Free Software Foundation; either
9 version 2.1 of the License, or (at your option) any later version.
10
11 The GNU C Library is distributed in the hope that it will be useful,
12 but WITHOUT ANY WARRANTY; without even the implied warranty of
13 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
14 Lesser General Public License for more details.
15
16 You should have received a copy of the GNU Lesser General Public
17 License along with the GNU C Library; if not, see
18 <https://www.gnu.org/licenses/>. */
19
20 #include "sv_math.h"
21
22 static const struct data
23 {
24 double poly[4];
25 double ln2_hi, ln2_lo, inv_ln2, shift, thres;
26 } data = {
27 .poly = { /* ulp error: 0.53. */
28 0x1.fffffffffdbcdp-2, 0x1.555555555444cp-3, 0x1.555573c6a9f7dp-5,
29 0x1.1111266d28935p-7 },
30 .ln2_hi = 0x1.62e42fefa3800p-1,
31 .ln2_lo = 0x1.ef35793c76730p-45,
32 /* 1/ln2. */
33 .inv_ln2 = 0x1.71547652b82fep+0,
34 /* 1.5*2^46+1023. This value is further explained below. */
35 .shift = 0x1.800000000ffc0p+46,
36 .thres = 704.0,
37 };
38
39 #define C(i) sv_f64 (d->poly[i])
40 #define SpecialOffset 0x6000000000000000 /* 0x1p513. */
41 /* SpecialBias1 + SpecialBias1 = asuint(1.0). */
42 #define SpecialBias1 0x7000000000000000 /* 0x1p769. */
43 #define SpecialBias2 0x3010000000000000 /* 0x1p-254. */
44
45 /* Update of both special and non-special cases, if any special case is
46 detected. */
47 static inline svfloat64_t
48 special_case (svbool_t pg, svfloat64_t s, svfloat64_t y, svfloat64_t n)
49 {
50 /* s=2^n may overflow, break it up into s=s1*s2,
51 such that exp = s + s*y can be computed as s1*(s2+s2*y)
52 and s1*s1 overflows only if n>0. */
53
54 /* If n<=0 then set b to 0x6, 0 otherwise. */
55 svbool_t p_sign = svcmple_n_f64 (pg, n, 0.0); /* n <= 0. */
56 svuint64_t b
57 = svdup_n_u64_z (p_sign, SpecialOffset); /* Inactive lanes set to 0. */
58
59 /* Set s1 to generate overflow depending on sign of exponent n. */
60 svfloat64_t s1 = svreinterpret_f64_u64 (
61 svsubr_n_u64_x (pg, b, SpecialBias1)); /* 0x70...0 - b. */
62 /* Offset s to avoid overflow in final result if n is below threshold. */
63 svfloat64_t s2 = svreinterpret_f64_u64 (svadd_u64_x (
64 pg, svsub_n_u64_x (pg, svreinterpret_u64_f64 (s), SpecialBias2),
65 b)); /* as_u64 (s) - 0x3010...0 + b. */
66
67 /* |n| > 1280 => 2^(n) overflows. */
68 svbool_t p_cmp = svacgt_n_f64 (pg, n, 1280.0);
69
70 svfloat64_t r1 = svmul_f64_x (pg, s1, s1);
71 svfloat64_t r2 = svmla_f64_x (pg, s2, s2, y);
72 svfloat64_t r0 = svmul_f64_x (pg, r2, s1);
73
74 return svsel_f64 (p_cmp, r1, r0);
75 }
76
77 /* SVE exp algorithm. Maximum measured error is 1.01ulps:
78 SV_NAME_D1 (exp)(0x1.4619d7b04da41p+6) got 0x1.885d9acc41da7p+117
79 want 0x1.885d9acc41da6p+117. */
80 svfloat64_t SV_NAME_D1 (exp) (svfloat64_t x, const svbool_t pg)
81 {
82 const struct data *d = ptr_barrier (&data);
83
84 svbool_t special = svacgt_n_f64 (pg, x, d->thres);
85
86 /* Use a modifed version of the shift used for flooring, such that x/ln2 is
87 rounded to a multiple of 2^-6=1/64, shift = 1.5 * 2^52 * 2^-6 = 1.5 *
88 2^46.
89
90 n is not an integer but can be written as n = m + i/64, with i and m
91 integer, 0 <= i < 64 and m <= n.
92
93 Bits 5:0 of z will be null every time x/ln2 reaches a new integer value
94 (n=m, i=0), and is incremented every time z (or n) is incremented by 1/64.
95 FEXPA expects i in bits 5:0 of the input so it can be used as index into
96 FEXPA hardwired table T[i] = 2^(i/64) for i = 0:63, that will in turn
97 populate the mantissa of the output. Therefore, we use u=asuint(z) as
98 input to FEXPA.
99
100 We add 1023 to the modified shift value in order to set bits 16:6 of u to
101 1, such that once these bits are moved to the exponent of the output of
102 FEXPA, we get the exponent of 2^n right, i.e. we get 2^m. */
103 svfloat64_t z = svmla_n_f64_x (pg, sv_f64 (d->shift), x, d->inv_ln2);
104 svuint64_t u = svreinterpret_u64_f64 (z);
105 svfloat64_t n = svsub_n_f64_x (pg, z, d->shift);
106
107 /* r = x - n * ln2, r is in [-ln2/(2N), ln2/(2N)]. */
108 svfloat64_t ln2 = svld1rq_f64 (svptrue_b64 (), &d->ln2_hi);
109 svfloat64_t r = svmls_lane_f64 (x, n, ln2, 0);
110 r = svmls_lane_f64 (r, n, ln2, 1);
111
112 /* y = exp(r) - 1 ~= r + C0 r^2 + C1 r^3 + C2 r^4 + C3 r^5. */
113 svfloat64_t r2 = svmul_f64_x (pg, r, r);
114 svfloat64_t p01 = svmla_f64_x (pg, C (0), C (1), r);
115 svfloat64_t p23 = svmla_f64_x (pg, C (2), C (3), r);
116 svfloat64_t p04 = svmla_f64_x (pg, p01, p23, r2);
117 svfloat64_t y = svmla_f64_x (pg, r, p04, r2);
118
119 /* s = 2^n, computed using FEXPA. FEXPA does not propagate NaNs, so for
120 consistent NaN handling we have to manually propagate them. This comes at
121 significant performance cost. */
122 svfloat64_t s = svexpa_f64 (u);
123
124 /* Assemble result as exp(x) = 2^n * exp(r). If |x| > Thresh the
125 multiplication may overflow, so use special case routine. */
126
127 if (__glibc_unlikely (svptest_any (pg, special)))
128 {
129 /* FEXPA zeroes the sign bit, however the sign is meaningful to the
130 special case function so needs to be copied.
131 e = sign bit of u << 46. */
132 svuint64_t e
133 = svand_n_u64_x (pg, svlsl_n_u64_x (pg, u, 46), 0x8000000000000000);
134 /* Copy sign to s. */
135 s = svreinterpret_f64_u64 (
136 svadd_u64_x (pg, e, svreinterpret_u64_f64 (s)));
137 return special_case (pg, s, y, n);
138 }
139
140 /* No special case. */
141 return svmla_f64_x (pg, s, s, y);
142 }