1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
|
/*
* Single-precision vector powf function.
*
* Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
* See https://llvm.org/LICENSE.txt for license information.
* SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
*/
#include "mathlib.h"
#include "v_math.h"
#if V_SUPPORTED
#define Min v_u32 (0x00800000)
#define Max v_u32 (0x7f800000)
#define SBITS 5
#define Tlog v__powf_log2_data.tab
#define Texp v__exp2f_data.tab
#define A v__powf_log2_data.poly
#define C v__exp2f_data.poly
#define LOGDEG 4
#if LOGDEG == 5
/* 1.01 ulp */
#define OFF v_u32 (0x3f330000)
#define TBITS 4
#elif LOGDEG == 4
/* 2.6 ulp ~ 0.5 + 2^24 (128*Ln2*relerr_log2 + relerr_exp2) */
#define OFF v_u32 (0x3f35d000)
#define TBITS 5
#endif
#define V_EXP2F_TABLE_BITS SBITS
#define V_EXP2F_POLY_ORDER 3
struct v_exp2f_data
{
uint64_t tab[1 << V_EXP2F_TABLE_BITS];
double poly[V_EXP2F_POLY_ORDER];
};
#define V_POWF_LOG2_TABLE_BITS TBITS
#define V_POWF_LOG2_POLY_ORDER LOGDEG
#define SCALE ((double) (1 << SBITS))
struct v_powf_log2_data
{
struct
{
double invc, logc;
} tab[1 << V_POWF_LOG2_TABLE_BITS];
double poly[V_POWF_LOG2_POLY_ORDER];
};
static const struct v_powf_log2_data v__powf_log2_data = {
#if LOGDEG == 5
.tab = {
{ 0x1.661ec79f8f3bep+0, -0x1.efec65b963019p-2 * SCALE },
{ 0x1.571ed4aaf883dp+0, -0x1.b0b6832d4fca4p-2 * SCALE },
{ 0x1.49539f0f010bp+0, -0x1.7418b0a1fb77bp-2 * SCALE },
{ 0x1.3c995b0b80385p+0, -0x1.39de91a6dcf7bp-2 * SCALE },
{ 0x1.30d190c8864a5p+0, -0x1.01d9bf3f2b631p-2 * SCALE },
{ 0x1.25e227b0b8eap+0, -0x1.97c1d1b3b7afp-3 * SCALE },
{ 0x1.1bb4a4a1a343fp+0, -0x1.2f9e393af3c9fp-3 * SCALE },
{ 0x1.12358f08ae5bap+0, -0x1.960cbbf788d5cp-4 * SCALE },
{ 0x1.0953f419900a7p+0, -0x1.a6f9db6475fcep-5 * SCALE },
{ 0x1p+0, 0x0p+0 * SCALE },
{ 0x1.e608cfd9a47acp-1, 0x1.338ca9f24f53dp-4 * SCALE },
{ 0x1.ca4b31f026aap-1, 0x1.476a9543891bap-3 * SCALE },
{ 0x1.b2036576afce6p-1, 0x1.e840b4ac4e4d2p-3 * SCALE },
{ 0x1.9c2d163a1aa2dp-1, 0x1.40645f0c6651cp-2 * SCALE },
{ 0x1.886e6037841edp-1, 0x1.88e9c2c1b9ff8p-2 * SCALE },
{ 0x1.767dcf5534862p-1, 0x1.ce0a44eb17bccp-2 * SCALE },
},
/* rel err: 1.46 * 2^-32 */
.poly = {
0x1.27616c9496e0bp-2 * SCALE, -0x1.71969a075c67ap-2 * SCALE,
0x1.ec70a6ca7baddp-2 * SCALE, -0x1.7154748bef6c8p-1 * SCALE,
0x1.71547652ab82bp0 * SCALE,
}
#elif LOGDEG == 4
.tab = {
{0x1.6489890582816p+0, -0x1.e960f97b22702p-2 * SCALE},
{0x1.5cf19b35e3472p+0, -0x1.c993406cd4db6p-2 * SCALE},
{0x1.55aac0e956d65p+0, -0x1.aa711d9a7d0f3p-2 * SCALE},
{0x1.4eb0022977e01p+0, -0x1.8bf37bacdce9bp-2 * SCALE},
{0x1.47fcccda1dd1fp+0, -0x1.6e13b3519946ep-2 * SCALE},
{0x1.418ceabab68c1p+0, -0x1.50cb8281e4089p-2 * SCALE},
{0x1.3b5c788f1edb3p+0, -0x1.341504a237e2bp-2 * SCALE},
{0x1.3567de48e9c9ap+0, -0x1.17eaab624ffbbp-2 * SCALE},
{0x1.2fabc80fd19bap+0, -0x1.f88e708f8c853p-3 * SCALE},
{0x1.2a25200ce536bp+0, -0x1.c24b6da113914p-3 * SCALE},
{0x1.24d108e0152e3p+0, -0x1.8d02ee397cb1dp-3 * SCALE},
{0x1.1facd8ab2fbe1p+0, -0x1.58ac1223408b3p-3 * SCALE},
{0x1.1ab614a03efdfp+0, -0x1.253e6fd190e89p-3 * SCALE},
{0x1.15ea6d03af9ffp+0, -0x1.e5641882c12ffp-4 * SCALE},
{0x1.1147b994bb776p+0, -0x1.81fea712926f7p-4 * SCALE},
{0x1.0ccbf650593aap+0, -0x1.203e240de64a3p-4 * SCALE},
{0x1.0875408477302p+0, -0x1.8029b86a78281p-5 * SCALE},
{0x1.0441d42a93328p+0, -0x1.85d713190fb9p-6 * SCALE},
{0x1p+0, 0x0p+0 * SCALE},
{0x1.f1d006c855e86p-1, 0x1.4c1cc07312997p-5 * SCALE},
{0x1.e28c3341aa301p-1, 0x1.5e1848ccec948p-4 * SCALE},
{0x1.d4bdf9aa64747p-1, 0x1.04cfcb7f1196fp-3 * SCALE},
{0x1.c7b45a24e5803p-1, 0x1.582813d463c21p-3 * SCALE},
{0x1.bb5f5eb2ed60ap-1, 0x1.a936fa68760ccp-3 * SCALE},
{0x1.afb0bff8fe6b4p-1, 0x1.f81bc31d6cc4ep-3 * SCALE},
{0x1.a49badf7ab1f5p-1, 0x1.2279a09fae6b1p-2 * SCALE},
{0x1.9a14a111fc4c9p-1, 0x1.47ec0b6df5526p-2 * SCALE},
{0x1.901131f5b2fdcp-1, 0x1.6c71762280f1p-2 * SCALE},
{0x1.8687f73f6d865p-1, 0x1.90155070798dap-2 * SCALE},
{0x1.7d7067eb77986p-1, 0x1.b2e23b1d3068cp-2 * SCALE},
{0x1.74c2c1cf97b65p-1, 0x1.d4e21b0daa86ap-2 * SCALE},
{0x1.6c77f37cff2a1p-1, 0x1.f61e2a2f67f3fp-2 * SCALE},
},
/* rel err: 1.5 * 2^-30 */
.poly = {
-0x1.6ff5daa3b3d7cp-2 * SCALE,
0x1.ec81d03c01aebp-2 * SCALE,
-0x1.71547bb43f101p-1 * SCALE,
0x1.7154764a815cbp0 * SCALE,
}
#endif
};
static const struct v_exp2f_data v__exp2f_data = {
.tab = {
0x3ff0000000000000, 0x3fefd9b0d3158574, 0x3fefb5586cf9890f, 0x3fef9301d0125b51,
0x3fef72b83c7d517b, 0x3fef54873168b9aa, 0x3fef387a6e756238, 0x3fef1e9df51fdee1,
0x3fef06fe0a31b715, 0x3feef1a7373aa9cb, 0x3feedea64c123422, 0x3feece086061892d,
0x3feebfdad5362a27, 0x3feeb42b569d4f82, 0x3feeab07dd485429, 0x3feea47eb03a5585,
0x3feea09e667f3bcd, 0x3fee9f75e8ec5f74, 0x3feea11473eb0187, 0x3feea589994cce13,
0x3feeace5422aa0db, 0x3feeb737b0cdc5e5, 0x3feec49182a3f090, 0x3feed503b23e255d,
0x3feee89f995ad3ad, 0x3feeff76f2fb5e47, 0x3fef199bdd85529c, 0x3fef3720dcef9069,
0x3fef5818dcfba487, 0x3fef7c97337b9b5f, 0x3fefa4afa2a490da, 0x3fefd0765b6e4540,
},
/* rel err: 1.69 * 2^-34 */
.poly = {
0x1.c6af84b912394p-5/SCALE/SCALE/SCALE, 0x1.ebfce50fac4f3p-3/SCALE/SCALE, 0x1.62e42ff0c52d6p-1/SCALE
},
};
VPCS_ATTR
__attribute__ ((noinline)) static v_f32_t
specialcase (v_f32_t x, v_f32_t y, v_f32_t ret, v_u32_t cmp)
{
return v_call2_f32 (powf, x, y, ret, cmp);
}
VPCS_ATTR
v_f32_t
V_NAME(powf) (v_f32_t x, v_f32_t y)
{
v_u32_t u, tmp, cmp, i, top, iz;
v_s32_t k;
v_f32_t ret;
u = v_as_u32_f32 (x);
cmp = v_cond_u32 (u - Min >= Max - Min);
tmp = u - OFF;
i = (tmp >> (23 - TBITS)) % (1 << TBITS);
top = tmp & 0xff800000;
iz = u - top;
k = v_as_s32_u32 (top) >> (23 - SBITS); /* arithmetic shift */
for (int lane = 0; lane < v_lanes32 (); lane++)
{
uint32_t si, siz;
int32_t sk;
float sy;
/* Use double precision for each lane. */
double invc, logc, z, r, p, y0, logx, ylogx, kd, s;
uint64_t ki, t;
si = v_get_u32 (i, lane);
siz = v_get_u32 (iz, lane);
sk = v_get_s32 (k, lane);
sy = v_get_f32 (y, lane);
invc = Tlog[si].invc;
logc = Tlog[si].logc;
z = (double) as_f32_u32 (siz);
/* log2(x) = log1p(z/c-1)/ln2 + log2(c) + k */
r = __builtin_fma (z, invc, -1.0);
y0 = logc + (double) sk;
/* Polynomial to approximate log1p(r)/ln2. */
#if LOGDEG == 5
logx = A[0];
logx = r * logx + A[1];
logx = r * logx + A[2];
logx = r * logx + A[3];
logx = r * logx + A[4];
logx = r * logx + y0;
#elif LOGDEG == 4
logx = A[0];
logx = r * logx + A[1];
logx = r * logx + A[2];
logx = r * logx + A[3];
logx = r * logx + y0;
#endif
ylogx = sy * logx;
v_set_u32 (&cmp, lane,
(as_u64_f64 (ylogx) >> 47 & 0xffff)
>= as_u64_f64 (126.0 * (1 << SBITS)) >> 47
? 1
: v_get_u32 (cmp, lane));
/* N*x = k + r with r in [-1/2, 1/2] */
#if TOINT_INTRINSICS
kd = roundtoint (ylogx); /* k */
ki = converttoint (ylogx);
#else
# define SHIFT 0x1.8p52
kd = eval_as_double (ylogx + SHIFT);
ki = asuint64 (kd);
kd -= SHIFT;
#endif
r = ylogx - kd;
/* exp2(x) = 2^(k/N) * 2^r ~= s * (C0*r^3 + C1*r^2 + C2*r + 1) */
t = Texp[ki % (1 << SBITS)];
t += ki << (52 - SBITS);
s = as_f64_u64 (t);
p = C[0];
p = __builtin_fma (p, r, C[1]);
p = __builtin_fma (p, r, C[2]);
p = __builtin_fma (p, s * r, s);
v_set_f32 (&ret, lane, p);
}
if (unlikely (v_any_u32 (cmp)))
return specialcase (x, y, ret, cmp);
return ret;
}
VPCS_ALIAS
#endif
|
