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
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
|
/* Function acoshf vectorized with SSE4.
Copyright (C) 2021-2023 Free Software Foundation, Inc.
This file is part of the GNU C Library.
The GNU C Library is free software; you can redistribute it and/or
modify it under the terms of the GNU Lesser General Public
License as published by the Free Software Foundation; either
version 2.1 of the License, or (at your option) any later version.
The GNU C Library is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
Lesser General Public License for more details.
You should have received a copy of the GNU Lesser General Public
License along with the GNU C Library; if not, see
https://www.gnu.org/licenses/. */
/*
* ALGORITHM DESCRIPTION:
*
* Compute acosh(x) as log(x + sqrt(x*x - 1))
*
* Special cases:
*
* acosh(NaN) = quiet NaN, and raise invalid exception
* acosh(-INF) = NaN
* acosh(+INF) = +INF
* acosh(x) = NaN if x < 1
* acosh(1) = +0
*
*/
/* Offsets for data table __svml_sacosh_data_internal
*/
#define sOne 0
#define sPoly 16
#define iBrkValue 144
#define iOffExpoMask 160
#define sBigThreshold 176
#define sC2 192
#define sC3 208
#define sHalf 224
#define sLargestFinite 240
#define sThirtyOne 256
#define sTopMask8 272
#define XScale 288
#define sLn2 304
#include <sysdep.h>
.section .text.sse4, "ax", @progbits
ENTRY(_ZGVbN4v_acoshf_sse4)
subq $72, %rsp
cfi_def_cfa_offset(80)
/* Compute U = X - 1 and V = X + 1, naively first. */
movaps %xmm0, %xmm12
/* Load constants, always including One = 1 */
movups sOne+__svml_sacosh_data_internal(%rip), %xmm2
/*
* Check that 1 < X < +inf; otherwise go to the callout function.
* We need the callout for X = 1 to avoid division by zero below.
* This test ensures that callout handles NaN and either infinity.
*/
movaps %xmm0, %xmm4
movaps %xmm2, %xmm9
/*
* Compute e = -(2 * d + d^2)
* The first FMR is exact, and the rounding error in the other is acceptable
* since d and e are ~ 2^-8
*/
movaps %xmm2, %xmm10
/* Finally, express Y + W = U * V accurately where Y has <= 8 bits */
movups sTopMask8+__svml_sacosh_data_internal(%rip), %xmm5
/*
* Now we feed into the log1p code, using H in place of _VARG1 and
* also adding L into Xl.
* compute 1+x as high, low parts
*/
movaps %xmm2, %xmm13
movaps %xmm5, %xmm11
movaps %xmm2, %xmm3
/*
* Now 1 / (1 + d)
* = 1 / (1 + (sqrt(1 - e) - 1))
* = 1 / sqrt(1 - e)
* = 1 + 1/2 * e + 3/8 * e^2 + 5/16 * e^3 + 35/128 * e^4 + ...
* So compute the first three nonconstant terms of that, so that
* we have a relative correction (1 + Corr) to apply to S etc.
* C1 = 1/2
* C2 = 3/8
* C3 = 5/16
*/
movups sC3+__svml_sacosh_data_internal(%rip), %xmm8
/*
* The following computation can go wrong for very large X, e.g.
* the X^2 - 1 = U * V can overflow. But for large X we have
* acosh(X) / log(2 X) - 1 =~= 1/(4 * X^2), so for X >= 2^30
* we can just later stick X back into the log and tweak up the exponent.
* Actually we scale X by 2^-30 and tweak the exponent up by 31,
* to stay in the safe range for the later log computation.
* Compute a flag now telling us when to do this.
*/
movaps %xmm0, %xmm1
cmpnleps sLargestFinite+__svml_sacosh_data_internal(%rip), %xmm4
cmpltps sBigThreshold+__svml_sacosh_data_internal(%rip), %xmm1
cmpnltps %xmm0, %xmm3
subps %xmm2, %xmm12
addps %xmm0, %xmm9
/* For low-accuracy versions, naivety is harmless */
mulps %xmm12, %xmm9
orps %xmm3, %xmm4
movmskps %xmm4, %edx
andps %xmm9, %xmm11
movaps %xmm1, %xmm3
/*
* Compute R = 1/sqrt(Y + W) * (1 + d)
* Force R to <= 8 significant bits.
* This means that R * Y and R^2 * Y are exactly representable.
*/
rsqrtps %xmm11, %xmm7
subps %xmm11, %xmm9
andps %xmm5, %xmm7
movaps %xmm2, %xmm4
/*
* Compute S = (Y/sqrt(Y + W)) * (1 + d)
* and T = (W/sqrt(Y + W)) * (1 + d)
* so that S + T = sqrt(Y + W) * (1 + d)
* S is exact, and the rounding error in T is OK.
*/
mulps %xmm7, %xmm11
movaps %xmm7, %xmm6
mulps %xmm7, %xmm9
mulps %xmm11, %xmm6
mulps %xmm9, %xmm7
/*
* For low-accuracy versions, the computation can be done
* just as U + ((S + T) + (S + T) * Corr)
*/
addps %xmm9, %xmm11
subps %xmm6, %xmm10
movaps %xmm2, %xmm9
subps %xmm7, %xmm10
mulps %xmm10, %xmm8
/* Now multiplex to the case X = 2^-30 * input, Xl = 0 in the "big" case. */
movups XScale+__svml_sacosh_data_internal(%rip), %xmm14
mulps %xmm0, %xmm14
addps sC2+__svml_sacosh_data_internal(%rip), %xmm8
mulps %xmm10, %xmm8
andnps %xmm14, %xmm3
/*
* Now resume the main code.
* reduction: compute r, n
*/
movdqu iBrkValue+__svml_sacosh_data_internal(%rip), %xmm14
movdqu iOffExpoMask+__svml_sacosh_data_internal(%rip), %xmm5
/* Add 31 to the exponent in the "large" case to get log(2 * input) */
movups sThirtyOne+__svml_sacosh_data_internal(%rip), %xmm6
addps sHalf+__svml_sacosh_data_internal(%rip), %xmm8
mulps %xmm8, %xmm10
movaps %xmm1, %xmm8
mulps %xmm11, %xmm10
addps %xmm10, %xmm11
addps %xmm11, %xmm12
maxps %xmm12, %xmm13
minps %xmm12, %xmm9
movaps %xmm13, %xmm15
addps %xmm9, %xmm15
subps %xmm15, %xmm13
andps %xmm1, %xmm15
orps %xmm15, %xmm3
addps %xmm13, %xmm9
psubd %xmm14, %xmm3
andps %xmm1, %xmm9
pand %xmm3, %xmm5
psrad $23, %xmm3
cvtdq2ps %xmm3, %xmm7
pslld $23, %xmm3
paddd %xmm14, %xmm5
psubd %xmm3, %xmm4
/* polynomial evaluation */
subps %xmm2, %xmm5
mulps %xmm4, %xmm9
addps %xmm7, %xmm6
movups sPoly+112+__svml_sacosh_data_internal(%rip), %xmm2
andnps %xmm6, %xmm8
andps %xmm1, %xmm7
addps %xmm5, %xmm9
mulps %xmm9, %xmm2
orps %xmm7, %xmm8
/* final reconstruction */
mulps sLn2+__svml_sacosh_data_internal(%rip), %xmm8
addps sPoly+96+__svml_sacosh_data_internal(%rip), %xmm2
mulps %xmm9, %xmm2
addps sPoly+80+__svml_sacosh_data_internal(%rip), %xmm2
mulps %xmm9, %xmm2
addps sPoly+64+__svml_sacosh_data_internal(%rip), %xmm2
mulps %xmm9, %xmm2
addps sPoly+48+__svml_sacosh_data_internal(%rip), %xmm2
mulps %xmm9, %xmm2
addps sPoly+32+__svml_sacosh_data_internal(%rip), %xmm2
mulps %xmm9, %xmm2
addps sPoly+16+__svml_sacosh_data_internal(%rip), %xmm2
mulps %xmm9, %xmm2
addps sPoly+__svml_sacosh_data_internal(%rip), %xmm2
mulps %xmm9, %xmm2
mulps %xmm9, %xmm2
addps %xmm2, %xmm9
addps %xmm8, %xmm9
testl %edx, %edx
/* Go to special inputs processing branch */
jne L(SPECIAL_VALUES_BRANCH)
# LOE rbx rbp r12 r13 r14 r15 edx xmm0 xmm9
/* Restore registers
* and exit the function
*/
L(EXIT):
movaps %xmm9, %xmm0
addq $72, %rsp
cfi_def_cfa_offset(8)
ret
cfi_def_cfa_offset(80)
/* Branch to process
* special inputs
*/
L(SPECIAL_VALUES_BRANCH):
movups %xmm0, 32(%rsp)
movups %xmm9, 48(%rsp)
# LOE rbx rbp r12 r13 r14 r15 edx
xorl %eax, %eax
movq %r12, 16(%rsp)
cfi_offset(12, -64)
movl %eax, %r12d
movq %r13, 8(%rsp)
cfi_offset(13, -72)
movl %edx, %r13d
movq %r14, (%rsp)
cfi_offset(14, -80)
# LOE rbx rbp r15 r12d r13d
/* Range mask
* bits check
*/
L(RANGEMASK_CHECK):
btl %r12d, %r13d
/* Call scalar math function */
jc L(SCALAR_MATH_CALL)
# LOE rbx rbp r15 r12d r13d
/* Special inputs
* processing loop
*/
L(SPECIAL_VALUES_LOOP):
incl %r12d
cmpl $4, %r12d
/* Check bits in range mask */
jl L(RANGEMASK_CHECK)
# LOE rbx rbp r15 r12d r13d
movq 16(%rsp), %r12
cfi_restore(12)
movq 8(%rsp), %r13
cfi_restore(13)
movq (%rsp), %r14
cfi_restore(14)
movups 48(%rsp), %xmm9
/* Go to exit */
jmp L(EXIT)
cfi_offset(12, -64)
cfi_offset(13, -72)
cfi_offset(14, -80)
# LOE rbx rbp r12 r13 r14 r15 xmm9
/* Scalar math function call
* to process special input
*/
L(SCALAR_MATH_CALL):
movl %r12d, %r14d
movss 32(%rsp, %r14, 4), %xmm0
call acoshf@PLT
# LOE rbx rbp r14 r15 r12d r13d xmm0
movss %xmm0, 48(%rsp, %r14, 4)
/* Process special inputs in loop */
jmp L(SPECIAL_VALUES_LOOP)
# LOE rbx rbp r15 r12d r13d
END(_ZGVbN4v_acoshf_sse4)
.section .rodata, "a"
.align 16
#ifdef __svml_sacosh_data_internal_typedef
typedef unsigned int VUINT32;
typedef struct {
__declspec(align(16)) VUINT32 sOne[4][1];
__declspec(align(16)) VUINT32 sPoly[8][4][1];
__declspec(align(16)) VUINT32 iBrkValue[4][1];
__declspec(align(16)) VUINT32 iOffExpoMask[4][1];
__declspec(align(16)) VUINT32 sBigThreshold[4][1];
__declspec(align(16)) VUINT32 sC2[4][1];
__declspec(align(16)) VUINT32 sC3[4][1];
__declspec(align(16)) VUINT32 sHalf[4][1];
__declspec(align(16)) VUINT32 sLargestFinite[4][1];
__declspec(align(16)) VUINT32 sThirtyOne[4][1];
__declspec(align(16)) VUINT32 sTopMask8[4][1];
__declspec(align(16)) VUINT32 XScale[4][1];
__declspec(align(16)) VUINT32 sLn2[4][1];
} __svml_sacosh_data_internal;
#endif
__svml_sacosh_data_internal:
/* sOne = SP 1.0 */
.long 0x3f800000, 0x3f800000, 0x3f800000, 0x3f800000
/* sPoly[] = SP polynomial */
.align 16
.long 0xbf000000, 0xbf000000, 0xbf000000, 0xbf000000 /* -5.0000000000000000000000000e-01 P0 */
.long 0x3eaaaa94, 0x3eaaaa94, 0x3eaaaa94, 0x3eaaaa94 /* 3.3333265781402587890625000e-01 P1 */
.long 0xbe80058e, 0xbe80058e, 0xbe80058e, 0xbe80058e /* -2.5004237890243530273437500e-01 P2 */
.long 0x3e4ce190, 0x3e4ce190, 0x3e4ce190, 0x3e4ce190 /* 2.0007920265197753906250000e-01 P3 */
.long 0xbe28ad37, 0xbe28ad37, 0xbe28ad37, 0xbe28ad37 /* -1.6472326219081878662109375e-01 P4 */
.long 0x3e0fcb12, 0x3e0fcb12, 0x3e0fcb12, 0x3e0fcb12 /* 1.4042308926582336425781250e-01 P5 */
.long 0xbe1ad9e3, 0xbe1ad9e3, 0xbe1ad9e3, 0xbe1ad9e3 /* -1.5122179687023162841796875e-01 P6 */
.long 0x3e0d84ed, 0x3e0d84ed, 0x3e0d84ed, 0x3e0d84ed /* 1.3820238411426544189453125e-01 P7 */
/* iBrkValue = SP 2/3 */
.align 16
.long 0x3f2aaaab, 0x3f2aaaab, 0x3f2aaaab, 0x3f2aaaab
/* iOffExpoMask = SP significand mask */
.align 16
.long 0x007fffff, 0x007fffff, 0x007fffff, 0x007fffff
/* sBigThreshold */
.align 16
.long 0x4E800000, 0x4E800000, 0x4E800000, 0x4E800000
/* sC2 */
.align 16
.long 0x3EC00000, 0x3EC00000, 0x3EC00000, 0x3EC00000
/* sC3 */
.align 16
.long 0x3EA00000, 0x3EA00000, 0x3EA00000, 0x3EA00000
/* sHalf */
.align 16
.long 0x3F000000, 0x3F000000, 0x3F000000, 0x3F000000
/* sLargestFinite */
.align 16
.long 0x7F7FFFFF, 0x7F7FFFFF, 0x7F7FFFFF, 0x7F7FFFFF
/* sThirtyOne */
.align 16
.long 0x41F80000, 0x41F80000, 0x41F80000, 0x41F80000
/* sTopMask8 */
.align 16
.long 0xFFFF0000, 0xFFFF0000, 0xFFFF0000, 0xFFFF0000
/* XScale */
.align 16
.long 0x30800000, 0x30800000, 0x30800000, 0x30800000
/* sLn2 = SP ln(2) */
.align 16
.long 0x3f317218, 0x3f317218, 0x3f317218, 0x3f317218
.align 16
.type __svml_sacosh_data_internal, @object
.size __svml_sacosh_data_internal, .-__svml_sacosh_data_internal
|
