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
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
|
.file "asinhf.s"
// Copyright (c) 2000 - 2003, Intel Corporation
// All rights reserved.
//
// Contributed 2000 by the Intel Numerics Group, Intel Corporation
//
// Redistribution and use in source and binary forms, with or without
// modification, are permitted provided that the following conditions are
// met:
//
// * Redistributions of source code must retain the above copyright
// notice, this list of conditions and the following disclaimer.
//
// * Redistributions in binary form must reproduce the above copyright
// notice, this list of conditions and the following disclaimer in the
// documentation and/or other materials provided with the distribution.
//
// * The name of Intel Corporation may not be used to endorse or promote
// products derived from this software without specific prior written
// permission.
// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
// "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
// LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
// A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL INTEL OR ITS
// CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
// EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
// PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
// PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY
// OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY OR TORT (INCLUDING
// NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
// SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
//
// Intel Corporation is the author of this code, and requests that all
// problem reports or change requests be submitted to it directly at
// http://www.intel.com/software/products/opensource/libraries/num.htm.
//
// ==============================================================
// History
// ==============================================================
// 04/02/01 Initial version
// 04/19/01 Improved speed of the paths #1,2,3,4,5
// 05/20/02 Cleaned up namespace and sf0 syntax
// 02/06/03 Reordered header: .section, .global, .proc, .align
// 05/21/03 Improved performance, fixed to handle unorms
//
// API
// ==============================================================
// float asinhf(float)
//
// Overview of operation
// ==============================================================
//
// There are 7 paths:
// 1. x = 0.0
// Return asinhf(x) = 0.0
// 2. 0.0 <|x| < 2^(-5)
// Return asinhf(x) = Pol5(x), where Pol5(x) = ((x^2)*C1 + C0)*x^3 + x
// 3. 2^(-5) <= |x| < 2^51
// Return asinhf(x) = sign(x)*(log(|x| + sqrt(x^2 + 1.0)))
// To compute x + sqrt(x^2 + 1.0) modified Newton Raphson method is used
// (2 iterations)
// Algorithm description for log function see below.
//
// 4. 2^51 <= |x| < +INF
// Return asinhf(x) = sign(x)*log(2*|x|)
// Algorithm description for log function see below.
//
// 5. x = INF
// Return asinhf(x) = INF
//
// 6. x = [S,Q]NaN
// Return asinhf(x) = QNaN
//
// 7. x = denormal
// Return asinhf(x) = x
//
//==============================================================
// Algorithm Description for log(x) function
// Below we are using the fact that inequality x - 1.0 > 2^(-6) is always
// true for this asinh implementation
//
// Consider x = 2^N 1.f1 f2 f3 f4...f63
// Log(x) = log(frcpa(x) x/frcpa(x))
// = log(1/frcpa(x)) + log(frcpa(x) x)
// = -log(frcpa(x)) + log(frcpa(x) x)
//
// frcpa(x) = 2^-N frcpa((1.f1 f2 ... f63)
//
// -log(frcpa(x)) = -log(C)
// = -log(2^-N) - log(frcpa(1.f1 f2 ... f63))
//
// -log(frcpa(x)) = -log(C)
// = +Nlog2 - log(frcpa(1.f1 f2 ... f63))
//
// -log(frcpa(x)) = -log(C)
// = +Nlog2 + log(frcpa(1.f1 f2 ... f63))
//
// Log(x) = log(1/frcpa(x)) + log(frcpa(x) x)
//
// Log(x) = +Nlog2 + log(1./frcpa(1.f1 f2 ... f63)) + log(frcpa(x) x)
// Log(x) = +Nlog2 - log(/frcpa(1.f1 f2 ... f63)) + log(frcpa(x) x)
// Log(x) = +Nlog2 + T + log(frcpa(x) x)
//
// Log(x) = +Nlog2 + T + log(C x)
//
// Cx = 1 + r
//
// Log(x) = +Nlog2 + T + log(1+r)
// Log(x) = +Nlog2 + T + Series( r - r^2/2 + r^3/3 - r^4/4 ....)
//
// 1.f1 f2 ... f8 has 256 entries.
// They are 1 + k/2^8, k = 0 ... 255
// These 256 values are the table entries.
//
// Implementation
//==============================================================
// C = frcpa(x)
// r = C * x - 1
//
// Form rseries = r + P1*r^2 + P2*r^3 + P3*r^4
//
// x = f * 2*n where f is 1.f_1f_2f_3....f_63
// Nfloat = float(n) where n is the true unbiased exponent
// pre-index = f_1f_2....f_8
// index = pre_index * 8
// get the dxt table entry at index + offset = T
//
// result = (T + Nfloat * log(2)) + rseries
//
// The T table is calculated as follows
// Form x_k = 1 + k/2^8 where k goes from 0... 255
// y_k = frcpa(x_k)
// log(1/y_k) in quad and round to double-extended
//
//
// Registers used
//==============================================================
// Floating Point registers used:
// f8, input
// f9 -> f15, f32 -> f55
// General registers used:
// r14 -> r27
// Predicate registers used:
// p6 -> p14
// p6 to filter out case when x = [Q,S]NaN or INF or zero
// p7 to filter out case when x < 0.0
// p8 to select path #2
// p11 to filter out case when x >= 0
// p12 to filter out case when x = + denormal
// p13 to select path #4
// p14 to filtef out case when x = - denormal
// Assembly macros
//==============================================================
log_GR_exp_17_ones = r14
log_GR_signexp_f8 = r15
log_table_address2 = r16
log_GR_exp_16_ones = r17
log_GR_exp_f8 = r18
log_GR_true_exp_f8 = r19
log_GR_significand_f8 = r20
log_GR_index = r21
log_GR_comp2 = r22
asinh_GR_f8 = r23
asinh_GR_comp = r24
asinh_GR_f8 = r25
log_table_address3 = r26
NR_table_address = r27
//==============================================================
log_y = f9
NR1 = f10
NR2 = f11
log_y_rs = f12
log_y_rs_iter = f13
log_y_rs_iter1 = f14
fNormX = f15
asinh_w_sq = f32
log_arg_early = f33
log_y_rs_iter2 = f34
log_P3 = f35
log_P2 = f36
log_P1 = f37
log2 = f38
log_C0 = f39
log_C1 = f40
asinh_f8 = f41
log_C = f42
log_arg = f43
asinh_w_cube = f44
log_int_Nfloat = f45
log_r = f46
log_rsq = f47
asinh_w_1 = f48
log_rp_p32 = f49
log_rcube = f50
log_rp_p10 = f51
log_rp_p2 = f52
log_Nfloat = f53
log_T = f54
log_T_plus_Nlog2 = f55
// Data tables
//==============================================================
RODATA
.align 16
LOCAL_OBJECT_START(log_table_1)
data8 0xbfd0001008f39d59 // p3
data8 0x3fd5556073e0c45a // p2
data8 0xbfdffffffffaea15 // p1
data8 0x3fe62e42fefa39ef // log(2)
LOCAL_OBJECT_END(log_table_1)
LOCAL_OBJECT_START(log_table_2)
data8 0x3FE0000000000000 // 0.5
data8 0x4008000000000000 // 3.0
data8 0x9979C79685A5EB16, 0x00003FFB // C1 3FFB9979C79685A5EB16
data8 0xAAAAA96F80786D62, 0x0000BFFC // C0 BFFCAAAAA96F80786D62
LOCAL_OBJECT_END(log_table_2)
LOCAL_OBJECT_START(log_table_3)
data8 0x3F60040155D5889E //log(1/frcpa(1+ 0/256)
data8 0x3F78121214586B54 //log(1/frcpa(1+ 1/256)
data8 0x3F841929F96832F0 //log(1/frcpa(1+ 2/256)
data8 0x3F8C317384C75F06 //log(1/frcpa(1+ 3/256)
data8 0x3F91A6B91AC73386 //log(1/frcpa(1+ 4/256)
data8 0x3F95BA9A5D9AC039 //log(1/frcpa(1+ 5/256)
data8 0x3F99D2A8074325F4 //log(1/frcpa(1+ 6/256)
data8 0x3F9D6B2725979802 //log(1/frcpa(1+ 7/256)
data8 0x3FA0C58FA19DFAAA //log(1/frcpa(1+ 8/256)
data8 0x3FA2954C78CBCE1B //log(1/frcpa(1+ 9/256)
data8 0x3FA4A94D2DA96C56 //log(1/frcpa(1+ 10/256)
data8 0x3FA67C94F2D4BB58 //log(1/frcpa(1+ 11/256)
data8 0x3FA85188B630F068 //log(1/frcpa(1+ 12/256)
data8 0x3FAA6B8ABE73AF4C //log(1/frcpa(1+ 13/256)
data8 0x3FAC441E06F72A9E //log(1/frcpa(1+ 14/256)
data8 0x3FAE1E6713606D07 //log(1/frcpa(1+ 15/256)
data8 0x3FAFFA6911AB9301 //log(1/frcpa(1+ 16/256)
data8 0x3FB0EC139C5DA601 //log(1/frcpa(1+ 17/256)
data8 0x3FB1DBD2643D190B //log(1/frcpa(1+ 18/256)
data8 0x3FB2CC7284FE5F1C //log(1/frcpa(1+ 19/256)
data8 0x3FB3BDF5A7D1EE64 //log(1/frcpa(1+ 20/256)
data8 0x3FB4B05D7AA012E0 //log(1/frcpa(1+ 21/256)
data8 0x3FB580DB7CEB5702 //log(1/frcpa(1+ 22/256)
data8 0x3FB674F089365A7A //log(1/frcpa(1+ 23/256)
data8 0x3FB769EF2C6B568D //log(1/frcpa(1+ 24/256)
data8 0x3FB85FD927506A48 //log(1/frcpa(1+ 25/256)
data8 0x3FB9335E5D594989 //log(1/frcpa(1+ 26/256)
data8 0x3FBA2B0220C8E5F5 //log(1/frcpa(1+ 27/256)
data8 0x3FBB0004AC1A86AC //log(1/frcpa(1+ 28/256)
data8 0x3FBBF968769FCA11 //log(1/frcpa(1+ 29/256)
data8 0x3FBCCFEDBFEE13A8 //log(1/frcpa(1+ 30/256)
data8 0x3FBDA727638446A2 //log(1/frcpa(1+ 31/256)
data8 0x3FBEA3257FE10F7A //log(1/frcpa(1+ 32/256)
data8 0x3FBF7BE9FEDBFDE6 //log(1/frcpa(1+ 33/256)
data8 0x3FC02AB352FF25F4 //log(1/frcpa(1+ 34/256)
data8 0x3FC097CE579D204D //log(1/frcpa(1+ 35/256)
data8 0x3FC1178E8227E47C //log(1/frcpa(1+ 36/256)
data8 0x3FC185747DBECF34 //log(1/frcpa(1+ 37/256)
data8 0x3FC1F3B925F25D41 //log(1/frcpa(1+ 38/256)
data8 0x3FC2625D1E6DDF57 //log(1/frcpa(1+ 39/256)
data8 0x3FC2D1610C86813A //log(1/frcpa(1+ 40/256)
data8 0x3FC340C59741142E //log(1/frcpa(1+ 41/256)
data8 0x3FC3B08B6757F2A9 //log(1/frcpa(1+ 42/256)
data8 0x3FC40DFB08378003 //log(1/frcpa(1+ 43/256)
data8 0x3FC47E74E8CA5F7C //log(1/frcpa(1+ 44/256)
data8 0x3FC4EF51F6466DE4 //log(1/frcpa(1+ 45/256)
data8 0x3FC56092E02BA516 //log(1/frcpa(1+ 46/256)
data8 0x3FC5D23857CD74D5 //log(1/frcpa(1+ 47/256)
data8 0x3FC6313A37335D76 //log(1/frcpa(1+ 48/256)
data8 0x3FC6A399DABBD383 //log(1/frcpa(1+ 49/256)
data8 0x3FC70337DD3CE41B //log(1/frcpa(1+ 50/256)
data8 0x3FC77654128F6127 //log(1/frcpa(1+ 51/256)
data8 0x3FC7E9D82A0B022D //log(1/frcpa(1+ 52/256)
data8 0x3FC84A6B759F512F //log(1/frcpa(1+ 53/256)
data8 0x3FC8AB47D5F5A310 //log(1/frcpa(1+ 54/256)
data8 0x3FC91FE49096581B //log(1/frcpa(1+ 55/256)
data8 0x3FC981634011AA75 //log(1/frcpa(1+ 56/256)
data8 0x3FC9F6C407089664 //log(1/frcpa(1+ 57/256)
data8 0x3FCA58E729348F43 //log(1/frcpa(1+ 58/256)
data8 0x3FCABB55C31693AD //log(1/frcpa(1+ 59/256)
data8 0x3FCB1E104919EFD0 //log(1/frcpa(1+ 60/256)
data8 0x3FCB94EE93E367CB //log(1/frcpa(1+ 61/256)
data8 0x3FCBF851C067555F //log(1/frcpa(1+ 62/256)
data8 0x3FCC5C0254BF23A6 //log(1/frcpa(1+ 63/256)
data8 0x3FCCC000C9DB3C52 //log(1/frcpa(1+ 64/256)
data8 0x3FCD244D99C85674 //log(1/frcpa(1+ 65/256)
data8 0x3FCD88E93FB2F450 //log(1/frcpa(1+ 66/256)
data8 0x3FCDEDD437EAEF01 //log(1/frcpa(1+ 67/256)
data8 0x3FCE530EFFE71012 //log(1/frcpa(1+ 68/256)
data8 0x3FCEB89A1648B971 //log(1/frcpa(1+ 69/256)
data8 0x3FCF1E75FADF9BDE //log(1/frcpa(1+ 70/256)
data8 0x3FCF84A32EAD7C35 //log(1/frcpa(1+ 71/256)
data8 0x3FCFEB2233EA07CD //log(1/frcpa(1+ 72/256)
data8 0x3FD028F9C7035C1C //log(1/frcpa(1+ 73/256)
data8 0x3FD05C8BE0D9635A //log(1/frcpa(1+ 74/256)
data8 0x3FD085EB8F8AE797 //log(1/frcpa(1+ 75/256)
data8 0x3FD0B9C8E32D1911 //log(1/frcpa(1+ 76/256)
data8 0x3FD0EDD060B78081 //log(1/frcpa(1+ 77/256)
data8 0x3FD122024CF0063F //log(1/frcpa(1+ 78/256)
data8 0x3FD14BE2927AECD4 //log(1/frcpa(1+ 79/256)
data8 0x3FD180618EF18ADF //log(1/frcpa(1+ 80/256)
data8 0x3FD1B50BBE2FC63B //log(1/frcpa(1+ 81/256)
data8 0x3FD1DF4CC7CF242D //log(1/frcpa(1+ 82/256)
data8 0x3FD214456D0EB8D4 //log(1/frcpa(1+ 83/256)
data8 0x3FD23EC5991EBA49 //log(1/frcpa(1+ 84/256)
data8 0x3FD2740D9F870AFB //log(1/frcpa(1+ 85/256)
data8 0x3FD29ECDABCDFA04 //log(1/frcpa(1+ 86/256)
data8 0x3FD2D46602ADCCEE //log(1/frcpa(1+ 87/256)
data8 0x3FD2FF66B04EA9D4 //log(1/frcpa(1+ 88/256)
data8 0x3FD335504B355A37 //log(1/frcpa(1+ 89/256)
data8 0x3FD360925EC44F5D //log(1/frcpa(1+ 90/256)
data8 0x3FD38BF1C3337E75 //log(1/frcpa(1+ 91/256)
data8 0x3FD3C25277333184 //log(1/frcpa(1+ 92/256)
data8 0x3FD3EDF463C1683E //log(1/frcpa(1+ 93/256)
data8 0x3FD419B423D5E8C7 //log(1/frcpa(1+ 94/256)
data8 0x3FD44591E0539F49 //log(1/frcpa(1+ 95/256)
data8 0x3FD47C9175B6F0AD //log(1/frcpa(1+ 96/256)
data8 0x3FD4A8B341552B09 //log(1/frcpa(1+ 97/256)
data8 0x3FD4D4F3908901A0 //log(1/frcpa(1+ 98/256)
data8 0x3FD501528DA1F968 //log(1/frcpa(1+ 99/256)
data8 0x3FD52DD06347D4F6 //log(1/frcpa(1+ 100/256)
data8 0x3FD55A6D3C7B8A8A //log(1/frcpa(1+ 101/256)
data8 0x3FD5925D2B112A59 //log(1/frcpa(1+ 102/256)
data8 0x3FD5BF406B543DB2 //log(1/frcpa(1+ 103/256)
data8 0x3FD5EC433D5C35AE //log(1/frcpa(1+ 104/256)
data8 0x3FD61965CDB02C1F //log(1/frcpa(1+ 105/256)
data8 0x3FD646A84935B2A2 //log(1/frcpa(1+ 106/256)
data8 0x3FD6740ADD31DE94 //log(1/frcpa(1+ 107/256)
data8 0x3FD6A18DB74A58C5 //log(1/frcpa(1+ 108/256)
data8 0x3FD6CF31058670EC //log(1/frcpa(1+ 109/256)
data8 0x3FD6F180E852F0BA //log(1/frcpa(1+ 110/256)
data8 0x3FD71F5D71B894F0 //log(1/frcpa(1+ 111/256)
data8 0x3FD74D5AEFD66D5C //log(1/frcpa(1+ 112/256)
data8 0x3FD77B79922BD37E //log(1/frcpa(1+ 113/256)
data8 0x3FD7A9B9889F19E2 //log(1/frcpa(1+ 114/256)
data8 0x3FD7D81B037EB6A6 //log(1/frcpa(1+ 115/256)
data8 0x3FD8069E33827231 //log(1/frcpa(1+ 116/256)
data8 0x3FD82996D3EF8BCB //log(1/frcpa(1+ 117/256)
data8 0x3FD85855776DCBFB //log(1/frcpa(1+ 118/256)
data8 0x3FD8873658327CCF //log(1/frcpa(1+ 119/256)
data8 0x3FD8AA75973AB8CF //log(1/frcpa(1+ 120/256)
data8 0x3FD8D992DC8824E5 //log(1/frcpa(1+ 121/256)
data8 0x3FD908D2EA7D9512 //log(1/frcpa(1+ 122/256)
data8 0x3FD92C59E79C0E56 //log(1/frcpa(1+ 123/256)
data8 0x3FD95BD750EE3ED3 //log(1/frcpa(1+ 124/256)
data8 0x3FD98B7811A3EE5B //log(1/frcpa(1+ 125/256)
data8 0x3FD9AF47F33D406C //log(1/frcpa(1+ 126/256)
data8 0x3FD9DF270C1914A8 //log(1/frcpa(1+ 127/256)
data8 0x3FDA0325ED14FDA4 //log(1/frcpa(1+ 128/256)
data8 0x3FDA33440224FA79 //log(1/frcpa(1+ 129/256)
data8 0x3FDA57725E80C383 //log(1/frcpa(1+ 130/256)
data8 0x3FDA87D0165DD199 //log(1/frcpa(1+ 131/256)
data8 0x3FDAAC2E6C03F896 //log(1/frcpa(1+ 132/256)
data8 0x3FDADCCC6FDF6A81 //log(1/frcpa(1+ 133/256)
data8 0x3FDB015B3EB1E790 //log(1/frcpa(1+ 134/256)
data8 0x3FDB323A3A635948 //log(1/frcpa(1+ 135/256)
data8 0x3FDB56FA04462909 //log(1/frcpa(1+ 136/256)
data8 0x3FDB881AA659BC93 //log(1/frcpa(1+ 137/256)
data8 0x3FDBAD0BEF3DB165 //log(1/frcpa(1+ 138/256)
data8 0x3FDBD21297781C2F //log(1/frcpa(1+ 139/256)
data8 0x3FDC039236F08819 //log(1/frcpa(1+ 140/256)
data8 0x3FDC28CB1E4D32FD //log(1/frcpa(1+ 141/256)
data8 0x3FDC4E19B84723C2 //log(1/frcpa(1+ 142/256)
data8 0x3FDC7FF9C74554C9 //log(1/frcpa(1+ 143/256)
data8 0x3FDCA57B64E9DB05 //log(1/frcpa(1+ 144/256)
data8 0x3FDCCB130A5CEBB0 //log(1/frcpa(1+ 145/256)
data8 0x3FDCF0C0D18F326F //log(1/frcpa(1+ 146/256)
data8 0x3FDD232075B5A201 //log(1/frcpa(1+ 147/256)
data8 0x3FDD490246DEFA6B //log(1/frcpa(1+ 148/256)
data8 0x3FDD6EFA918D25CD //log(1/frcpa(1+ 149/256)
data8 0x3FDD9509707AE52F //log(1/frcpa(1+ 150/256)
data8 0x3FDDBB2EFE92C554 //log(1/frcpa(1+ 151/256)
data8 0x3FDDEE2F3445E4AF //log(1/frcpa(1+ 152/256)
data8 0x3FDE148A1A2726CE //log(1/frcpa(1+ 153/256)
data8 0x3FDE3AFC0A49FF40 //log(1/frcpa(1+ 154/256)
data8 0x3FDE6185206D516E //log(1/frcpa(1+ 155/256)
data8 0x3FDE882578823D52 //log(1/frcpa(1+ 156/256)
data8 0x3FDEAEDD2EAC990C //log(1/frcpa(1+ 157/256)
data8 0x3FDED5AC5F436BE3 //log(1/frcpa(1+ 158/256)
data8 0x3FDEFC9326D16AB9 //log(1/frcpa(1+ 159/256)
data8 0x3FDF2391A2157600 //log(1/frcpa(1+ 160/256)
data8 0x3FDF4AA7EE03192D //log(1/frcpa(1+ 161/256)
data8 0x3FDF71D627C30BB0 //log(1/frcpa(1+ 162/256)
data8 0x3FDF991C6CB3B379 //log(1/frcpa(1+ 163/256)
data8 0x3FDFC07ADA69A910 //log(1/frcpa(1+ 164/256)
data8 0x3FDFE7F18EB03D3E //log(1/frcpa(1+ 165/256)
data8 0x3FE007C053C5002E //log(1/frcpa(1+ 166/256)
data8 0x3FE01B942198A5A1 //log(1/frcpa(1+ 167/256)
data8 0x3FE02F74400C64EB //log(1/frcpa(1+ 168/256)
data8 0x3FE04360BE7603AD //log(1/frcpa(1+ 169/256)
data8 0x3FE05759AC47FE34 //log(1/frcpa(1+ 170/256)
data8 0x3FE06B5F1911CF52 //log(1/frcpa(1+ 171/256)
data8 0x3FE078BF0533C568 //log(1/frcpa(1+ 172/256)
data8 0x3FE08CD9687E7B0E //log(1/frcpa(1+ 173/256)
data8 0x3FE0A10074CF9019 //log(1/frcpa(1+ 174/256)
data8 0x3FE0B5343A234477 //log(1/frcpa(1+ 175/256)
data8 0x3FE0C974C89431CE //log(1/frcpa(1+ 176/256)
data8 0x3FE0DDC2305B9886 //log(1/frcpa(1+ 177/256)
data8 0x3FE0EB524BAFC918 //log(1/frcpa(1+ 178/256)
data8 0x3FE0FFB54213A476 //log(1/frcpa(1+ 179/256)
data8 0x3FE114253DA97D9F //log(1/frcpa(1+ 180/256)
data8 0x3FE128A24F1D9AFF //log(1/frcpa(1+ 181/256)
data8 0x3FE1365252BF0865 //log(1/frcpa(1+ 182/256)
data8 0x3FE14AE558B4A92D //log(1/frcpa(1+ 183/256)
data8 0x3FE15F85A19C765B //log(1/frcpa(1+ 184/256)
data8 0x3FE16D4D38C119FA //log(1/frcpa(1+ 185/256)
data8 0x3FE18203C20DD133 //log(1/frcpa(1+ 186/256)
data8 0x3FE196C7BC4B1F3B //log(1/frcpa(1+ 187/256)
data8 0x3FE1A4A738B7A33C //log(1/frcpa(1+ 188/256)
data8 0x3FE1B981C0C9653D //log(1/frcpa(1+ 189/256)
data8 0x3FE1CE69E8BB106B //log(1/frcpa(1+ 190/256)
data8 0x3FE1DC619DE06944 //log(1/frcpa(1+ 191/256)
data8 0x3FE1F160A2AD0DA4 //log(1/frcpa(1+ 192/256)
data8 0x3FE2066D7740737E //log(1/frcpa(1+ 193/256)
data8 0x3FE2147DBA47A394 //log(1/frcpa(1+ 194/256)
data8 0x3FE229A1BC5EBAC3 //log(1/frcpa(1+ 195/256)
data8 0x3FE237C1841A502E //log(1/frcpa(1+ 196/256)
data8 0x3FE24CFCE6F80D9A //log(1/frcpa(1+ 197/256)
data8 0x3FE25B2C55CD5762 //log(1/frcpa(1+ 198/256)
data8 0x3FE2707F4D5F7C41 //log(1/frcpa(1+ 199/256)
data8 0x3FE285E0842CA384 //log(1/frcpa(1+ 200/256)
data8 0x3FE294294708B773 //log(1/frcpa(1+ 201/256)
data8 0x3FE2A9A2670AFF0C //log(1/frcpa(1+ 202/256)
data8 0x3FE2B7FB2C8D1CC1 //log(1/frcpa(1+ 203/256)
data8 0x3FE2C65A6395F5F5 //log(1/frcpa(1+ 204/256)
data8 0x3FE2DBF557B0DF43 //log(1/frcpa(1+ 205/256)
data8 0x3FE2EA64C3F97655 //log(1/frcpa(1+ 206/256)
data8 0x3FE3001823684D73 //log(1/frcpa(1+ 207/256)
data8 0x3FE30E97E9A8B5CD //log(1/frcpa(1+ 208/256)
data8 0x3FE32463EBDD34EA //log(1/frcpa(1+ 209/256)
data8 0x3FE332F4314AD796 //log(1/frcpa(1+ 210/256)
data8 0x3FE348D90E7464D0 //log(1/frcpa(1+ 211/256)
data8 0x3FE35779F8C43D6E //log(1/frcpa(1+ 212/256)
data8 0x3FE36621961A6A99 //log(1/frcpa(1+ 213/256)
data8 0x3FE37C299F3C366A //log(1/frcpa(1+ 214/256)
data8 0x3FE38AE2171976E7 //log(1/frcpa(1+ 215/256)
data8 0x3FE399A157A603E7 //log(1/frcpa(1+ 216/256)
data8 0x3FE3AFCCFE77B9D1 //log(1/frcpa(1+ 217/256)
data8 0x3FE3BE9D503533B5 //log(1/frcpa(1+ 218/256)
data8 0x3FE3CD7480B4A8A3 //log(1/frcpa(1+ 219/256)
data8 0x3FE3E3C43918F76C //log(1/frcpa(1+ 220/256)
data8 0x3FE3F2ACB27ED6C7 //log(1/frcpa(1+ 221/256)
data8 0x3FE4019C2125CA93 //log(1/frcpa(1+ 222/256)
data8 0x3FE4181061389722 //log(1/frcpa(1+ 223/256)
data8 0x3FE42711518DF545 //log(1/frcpa(1+ 224/256)
data8 0x3FE436194E12B6BF //log(1/frcpa(1+ 225/256)
data8 0x3FE445285D68EA69 //log(1/frcpa(1+ 226/256)
data8 0x3FE45BCC464C893A //log(1/frcpa(1+ 227/256)
data8 0x3FE46AED21F117FC //log(1/frcpa(1+ 228/256)
data8 0x3FE47A1527E8A2D3 //log(1/frcpa(1+ 229/256)
data8 0x3FE489445EFFFCCC //log(1/frcpa(1+ 230/256)
data8 0x3FE4A018BCB69835 //log(1/frcpa(1+ 231/256)
data8 0x3FE4AF5A0C9D65D7 //log(1/frcpa(1+ 232/256)
data8 0x3FE4BEA2A5BDBE87 //log(1/frcpa(1+ 233/256)
data8 0x3FE4CDF28F10AC46 //log(1/frcpa(1+ 234/256)
data8 0x3FE4DD49CF994058 //log(1/frcpa(1+ 235/256)
data8 0x3FE4ECA86E64A684 //log(1/frcpa(1+ 236/256)
data8 0x3FE503C43CD8EB68 //log(1/frcpa(1+ 237/256)
data8 0x3FE513356667FC57 //log(1/frcpa(1+ 238/256)
data8 0x3FE522AE0738A3D8 //log(1/frcpa(1+ 239/256)
data8 0x3FE5322E26867857 //log(1/frcpa(1+ 240/256)
data8 0x3FE541B5CB979809 //log(1/frcpa(1+ 241/256)
data8 0x3FE55144FDBCBD62 //log(1/frcpa(1+ 242/256)
data8 0x3FE560DBC45153C7 //log(1/frcpa(1+ 243/256)
data8 0x3FE5707A26BB8C66 //log(1/frcpa(1+ 244/256)
data8 0x3FE587F60ED5B900 //log(1/frcpa(1+ 245/256)
data8 0x3FE597A7977C8F31 //log(1/frcpa(1+ 246/256)
data8 0x3FE5A760D634BB8B //log(1/frcpa(1+ 247/256)
data8 0x3FE5B721D295F10F //log(1/frcpa(1+ 248/256)
data8 0x3FE5C6EA94431EF9 //log(1/frcpa(1+ 249/256)
data8 0x3FE5D6BB22EA86F6 //log(1/frcpa(1+ 250/256)
data8 0x3FE5E6938645D390 //log(1/frcpa(1+ 251/256)
data8 0x3FE5F673C61A2ED2 //log(1/frcpa(1+ 252/256)
data8 0x3FE6065BEA385926 //log(1/frcpa(1+ 253/256)
data8 0x3FE6164BFA7CC06B //log(1/frcpa(1+ 254/256)
data8 0x3FE62643FECF9743 //log(1/frcpa(1+ 255/256)
LOCAL_OBJECT_END(log_table_3)
.section .text
GLOBAL_LIBM_ENTRY(asinhf)
{ .mfi
getf.exp asinh_GR_f8 = f8 // Must recompute later if x unorm
fclass.m p12,p0 = f8, 0x0b // Test x unorm
mov log_GR_exp_17_ones = 0x1ffff
}
{ .mfi
addl NR_table_address = @ltoff(log_table_1), gp
fma.s1 log_y = f8, f8, f1 // y = x^2 + 1
mov asinh_GR_comp = 0xfffa
}
;;
{ .mfi
mov log_GR_exp_16_ones = 0xffff //BIAS
fclass.m p6,p0 = f8, 0xe7 // Test for x = NaN and inf and zero
mov log_GR_comp2 = 0x10032
}
{ .mfi
ld8 NR_table_address = [NR_table_address]
fma.s1 asinh_w_sq = f8,f8,f0 // x^2
nop.i 0
}
;;
{ .mfi
nop.m 0
fcmp.lt.s1 p7,p11 = f8,f0 // if x<0
nop.i 0
}
{ .mfb
nop.m 0
fnorm.s1 fNormX = f8 // Normalize x
(p12) br.cond.spnt ASINH_UNORM // Branch if x=unorm
}
;;
ASINH_COMMON:
// Return here if x=unorm and not denorm
{ .mfi
//to get second table address
adds log_table_address2 = 0x20, NR_table_address
fma.s1 log_arg = f8,f1,f8
}
{ .mfb
nop.m 0
(p6) fma.s.s0 f8 = f8,f1,f8 // quietize nan result if x=nan
(p6) br.ret.spnt b0 // Exit for x=nan and inf and zero
}
;;
{ .mfi
ldfpd NR1,NR2 = [log_table_address2],16
frsqrta.s1 log_y_rs,p0 = log_y // z=1/sqrt(y)
nop.i 0
}
;;
{ .mfi
ldfe log_C1 = [log_table_address2],16
nop.f 0
and asinh_GR_f8 = asinh_GR_f8,log_GR_exp_17_ones
}
;;
{ .mib
ldfe log_C0 = [log_table_address2],16
cmp.le p13,p0 = log_GR_comp2,asinh_GR_f8
(p13) br.cond.spnt LOG_COMMON1 // Branch if path 4: |x| >= 2^51
}
;;
{ .mfi
nop.m 0
fma.s1 log_y_rs_iter = log_y_rs,log_y,f0 // y*z
nop.i 0
}
;;
.pred.rel "mutex",p7,p11
{ .mfi
nop.m 0
(p11) mov asinh_f8 = fNormX
nop.i 0
}
{ .mfb
cmp.gt p8,p0 = asinh_GR_comp,asinh_GR_f8
(p7) fnma.s1 asinh_f8 = fNormX,f1,f0
(p8) br.cond.spnt ASINH_NEAR_ZERO // Branch if path 2: 0 < |x| < 2^-5
}
;;
// Here if main path, 2^-5 <= |x| < 2^51
///////////////////////////////// The first iteration /////////////////////////
{ .mfi
ldfpd log_P3,log_P2 = [NR_table_address],16
fnma.s1 log_y_rs_iter2 = log_y_rs_iter,log_y_rs,NR2 // 3-(y*z)*z
nop.i 0
}
{ .mfi
nop.m 0
fma.s1 log_y_rs_iter1 = log_y_rs,NR1,f0 // 0.5*z
nop.i 0
}
;;
{ .mfi
ldfpd log_P1,log2 = [NR_table_address],16
// (0.5*z)*(3-(y*z)*z)
fma.s1 log_y_rs_iter = log_y_rs_iter1,log_y_rs_iter2,f0
nop.i 0
}
{ .mfi
nop.m 0
// (0.5*z)*(3-(y*z)*z)
fma.s1 log_arg_early = log_y_rs_iter1,log_y_rs_iter2,f0
nop.i 0
}
;;
////////////////////////////////// The second iteration ////////////////////////
{ .mfi
nop.m 0
fma.s1 log_y_rs = log_y_rs_iter,log_y,f0
nop.i 0
}
{ .mfi
nop.m 0
fma.s1 log_y_rs_iter1 = log_y_rs_iter,NR1,f0
nop.i 0
}
;;
{ .mfi
nop.m 0
fma.s1 log_arg_early = log_arg_early,log_y,asinh_f8
nop.i 0
}
;;
{ .mfi
nop.m 0
fnma.s1 log_y_rs = log_y_rs,log_y_rs_iter,NR2
nop.i 0
}
{ .mfi
nop.m 0
fma.s1 log_y_rs_iter1 = log_y_rs_iter1,log_y,f0
nop.i 0
}
;;
{ .mfi
nop.m 0
frcpa.s1 log_C,p0 = f1,log_arg_early
nop.i 0
}
;;
{ .mfi
getf.exp log_GR_signexp_f8 = log_arg_early
nop.f 0
nop.i 0
}
;;
{ .mfi
getf.sig log_GR_significand_f8 = log_arg_early
// (0.5*z)*(3-(y*z)*z)*y + |x|
fma.s1 log_arg = log_y_rs_iter1,log_y_rs,asinh_f8
//to get third table address
adds log_table_address3 = 0x30, NR_table_address
}
;;
/////////////////////////////////////////// The end NR iterations /////////////
{ .mfi
nop.m 0
nop.f 0
//significant bit destruction
and log_GR_exp_f8 = log_GR_signexp_f8, log_GR_exp_17_ones
}
;;
{ .mfi
//BIAS subtraction
sub log_GR_true_exp_f8 = log_GR_exp_f8, log_GR_exp_16_ones
(p7) fnma.s1 log2 = log2,f1,f0
nop.i 0
}
;;
{ .mfi
setf.sig log_int_Nfloat = log_GR_true_exp_f8
fms.s1 log_r = log_C,log_arg,f1 //C = frcpa(x); r = C * x - 1
extr.u log_GR_index = log_GR_significand_f8,55,8 //Extract 8 bits
}
;;
{ .mmi
//pre-index*16 + index
shladd log_table_address3 = log_GR_index,3,log_table_address3
;;
ldfd log_T = [log_table_address3]
nop.i 0
}
;;
{ .mfi
nop.m 0
fma.s1 log_rsq = log_r, log_r, f0 //r^2
nop.i 0
}
{ .mfi
nop.m 0
fma.s1 log_rp_p32 = log_P3, log_r, log_P2 //P3*r + P2
nop.i 0
}
;;
{ .mfi
nop.m 0
fma.s1 log_rp_p10 = log_P1, log_r, f1
nop.i 0
}
;;
{ .mfi
nop.m 0
//convert N to the floating-point format
fcvt.xf log_Nfloat = log_int_Nfloat
nop.i 0
}
;;
{ .mfi
nop.m 0
fma.s1 log_rp_p2 = log_rp_p32, log_rsq, log_rp_p10
nop.i 0
}
;;
.pred.rel "mutex",p7,p11
{ .mfi
nop.m 0
(p11) fma.s1 log_T_plus_Nlog2 = log_Nfloat,log2,log_T //N*log2 + T if x>0
nop.i 0
}
{ .mfi
nop.m 0
(p7) fms.s1 log_T_plus_Nlog2 = log_Nfloat,log2,log_T //N*log2 - T if x<0
nop.i 0
}
;;
{ .mfi
nop.m 0
(p11) fma.s.s0 f8 = log_rp_p2,log_r,log_T_plus_Nlog2
nop.i 0
}
{ .mfb
nop.m 0
(p7) fnma.s.s0 f8 = log_rp_p2,log_r,log_T_plus_Nlog2
br.ret.sptk b0 // Exit main path, path 3: 2^-5 <= |x| < 2^51
}
;;
// Here if path 4, |x| >= 2^51
LOG_COMMON1:
{ .mfi
ldfpd log_P3,log_P2 = [NR_table_address],16
nop.f 0
nop.i 0
}
;;
{ .mfi
ldfpd log_P1,log2 = [NR_table_address],16
frcpa.s1 log_C,p0 = f1,log_arg
nop.i 0
}
;;
{ .mfi
getf.exp log_GR_signexp_f8 = log_arg
nop.f 0
//to get third table address
adds log_table_address3 = 0x30, NR_table_address
}
;;
{ .mfi
getf.sig log_GR_significand_f8 = log_arg
nop.f 0
nop.i 0
}
;;
{ .mfi
nop.m 0
nop.f 0
//to destroy the most bit in the significant area
and log_GR_exp_f8 = log_GR_signexp_f8, log_GR_exp_17_ones
}
;;
{ .mmf
nop.m 0
//BIAS subtraction
sub log_GR_true_exp_f8 = log_GR_exp_f8, log_GR_exp_16_ones
fms.s1 log_r = log_C,log_arg,f1 //C = frcpa(x); r = C * x - 1
}
;;
{ .mfi
setf.sig log_int_Nfloat = log_GR_true_exp_f8
nop.f 0
extr.u log_GR_index = log_GR_significand_f8,55,8 //Extract 8 bits
}
;;
{ .mmi
//pre-index*16 + index
shladd log_table_address3 = log_GR_index,3,log_table_address3
;;
ldfd log_T = [log_table_address3]
nop.i 0
}
;;
{ .mfi
nop.m 0
fma.s1 log_rsq = log_r, log_r, f0 //r^2
nop.i 0
}
{ .mfi
nop.m 0
fma.s1 log_rp_p32 = log_P3, log_r, log_P2 //P3*r + P2
nop.i 0
}
;;
{ .mfi
nop.m 0
fma.s1 log_rp_p10 = log_P1, log_r, f1
nop.i 0
}
{ .mfi
nop.m 0
(p7) fnma.s1 log2 = log2,f1,f0
nop.i 0
}
;;
{ .mfi
nop.m 0
//convert N to the floating-point format
fcvt.xf log_Nfloat = log_int_Nfloat
nop.i 0
}
{ .mfi
nop.m 0
fma.s1 log_rp_p2 = log_rp_p32, log_rsq, log_rp_p10
nop.i 0
}
;;
.pred.rel "mutex",p7,p11
{ .mfi
nop.m 0
(p11) fma.s1 log_T_plus_Nlog2 = log_Nfloat,log2,log_T //N*log2 + T if x>0
nop.i 0
}
{ .mfi
nop.m 0
(p7) fms.s1 log_T_plus_Nlog2 = log_Nfloat,log2,log_T //N*log2 - T if x<0
nop.i 0
}
;;
{ .mfi
nop.m 0
(p11) fma.s.s0 f8 = log_rp_p2,log_r,log_T_plus_Nlog2
nop.i 0
}
{ .mfb
nop.m 0
(p7) fnma.s.s0 f8 = log_rp_p2,log_r,log_T_plus_Nlog2
br.ret.sptk b0 // Exit path 4, |x| >= 2^51
}
;;
// Here if path 2, 0 < |x| < 2^-5
ASINH_NEAR_ZERO:
{ .mfi
nop.m 0
fma.s1 asinh_w_1 = asinh_w_sq,log_C1,log_C0
nop.i 0
}
{ .mfi
nop.m 0
fma.s1 asinh_w_cube = asinh_w_sq,fNormX,f0
nop.i 0
}
;;
{ .mfb
nop.m 0
fma.s.s0 f8 = asinh_w_1,asinh_w_cube,fNormX
br.ret.sptk b0 // Exit path 2, 0 < |x| < 2^-5
}
;;
ASINH_UNORM:
// Here if x=unorm
{ .mfi
getf.exp asinh_GR_f8 = fNormX // Recompute if x unorm
fclass.m p0,p13 = fNormX, 0x0b // Test x denorm
nop.i 0
}
;;
{ .mfb
nop.m 0
fcmp.eq.s0 p14,p0 = f8, f0 // Dummy to set denormal flag
(p13) br.cond.sptk ASINH_COMMON // Continue if x unorm and not denorm
}
;;
.pred.rel "mutex",p7,p11
{ .mfi
nop.m 0
(p7) fma.s.s0 f8 = f8,f8,f8 // Result x+x^2 if x=-denorm
nop.i 0
}
{ .mfb
nop.m 0
(p11) fnma.s.s0 f8 = f8,f8,f8 // Result x-x^2 if x=+denorm
br.ret.spnt b0 // Exit if denorm
}
;;
GLOBAL_LIBM_END(asinhf)
|
