-
Notifications
You must be signed in to change notification settings - Fork 0
/
crlibm_private.h
1048 lines (877 loc) · 52.1 KB
/
crlibm_private.h
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
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
/*
* crlibm_private.h
*
* This file contains useful tools and data for the crlibm functions.
*
*/
#ifndef CRLIBM_PRIVATE_H
#define CRLIBM_PRIVATE_H 1
#include "scs_lib/scs.h"
#include "scs_lib/scs_private.h"
#ifdef HAVE_CONFIG_H
#include "crlibm_config.h"
#endif
/* otherwise CMake is used, and defines all the useful variables using -D switch */
#ifdef HAVE_INTTYPES_H
#include <inttypes.h>
#endif
#if (defined(CRLIBM_TYPECPU_X86) || defined(CRLIBM_TYPECPU_AMD64))
# ifdef HAVE_FPU_CONTROL_H
# include <fpu_control.h>
# endif
# ifndef _FPU_SETCW
# define _FPU_SETCW(cw) __asm__ ("fldcw %0" : : "m" (*&cw))
# endif
# ifndef _FPU_GETCW
# define _FPU_GETCW(cw) __asm__ ("fnstcw %0" : "=m" (*&cw))
# endif
#endif
/* 64 bit arithmetic may be standardised, but people still do what they want */
#ifdef HAVE_INTTYPES_H
#define ULL(bits) 0x##bits##uLL
#elif defined(_WIN32)
/* Windows garbage there */
typedef long long int int64_t;
typedef unsigned long long int uint64_t;
#define ULL(bits) 0x##bits##i64
/* Default, hoping it works, hopefully less and less relevant */
#else
typedef long long int int64_t;
typedef unsigned long long int uint64_t;
#define ULL(bits) 0x##bits##uLL
#endif
#ifndef SCS_DEF_INT64
#define SCS_DEF_INT64
#ifdef CRLIBM_TYPEOS_HPUX
#ifndef __LP64__ /* To solve the problem with 64 bits integer on HPPA */
typedef long long int64_t;
typedef unsigned long long uint64_t;
#define ULL(bits) 0x##bits##uLL
#endif
#endif
#endif
/* The Add22 and Add22 functions, as well as double-double
multiplications of the Dekker family may be either defined as
functions, or as #defines. Which one is better depends on the
processor/compiler/OS. As #define has to be used with more care (not
type-safe), the two following variables should be set to 1 in the
development/debugging phase, until no type warning remains.
*/
#define ADD22_AS_FUNCTIONS 0
#define DEKKER_AS_FUNCTIONS 0
#define SQRT_AS_FUNCTIONS 0
/* The conditional version of the Add12 can be implemented either
using 3 floating point additions, a absolute value test and
a branch or using 6 floating point additions but no branch.
The Add22 sequence is similar.
The branchless versions might be faster on some systems.
The function versions of Add12Cond and Add22Cond are not
implemented in branchless versions.
*/
#define AVOID_BRANCHES 1
/* setting the following variable adds variables and code for
monitoring the performance.
Note that sometimes only round to nearest is instrumented */
#define EVAL_PERF 1
#if EVAL_PERF==1
/* counter of calls to the second step (accurate step) */
extern int crlibm_second_step_taken;
#endif
/* The prototypes of the second steps */
/* extern void exp_SC(scs_ptr res_scs, double x);*/
/*
* i = d in rounding to nearest
The constant added is 2^52 + 2^51
*/
#define DOUBLE2INT(_i, _d) \
{db_number _t; \
_t.d = (_d+6755399441055744.0); \
_i = _t.i[LO];}
/* Same idea but beware: works only for |_i| < 2^51 -1 */
#define DOUBLE2LONGINT(_i, _d) \
{ \
db_number _t; \
_t.d = (_d+6755399441055744.0); \
if (_d >= 0) /* sign extend */ \
_i = _t.l & ULL(0007FFFFFFFFFFFF); \
else \
_i = (_t.l & ULL(0007FFFFFFFFFFFF)) | (ULL(FFF8000000000000)); \
}
/* Macros for the rounding tests in directed modes */
/* After Evgeny Gvozdev pointed out a bug in the rounding procedures I
decided to centralize them here
Note that these tests launch the accurate phase when yl=0, in
particular in the exceptional cases when the image of a double is a
double. See the chapter about the log for an example
All this does not work for denormals, of course
*/
#define TEST_AND_RETURN_RU(__yh__, __yl__, __eps__) \
{ \
db_number __yhdb__, __yldb__, u53; int yh_neg, yl_neg; \
__yhdb__.d = __yh__; __yldb__.d = __yl__; \
yh_neg = (__yhdb__.i[HI] & 0x80000000); \
yl_neg = (__yldb__.i[HI] & 0x80000000); \
__yhdb__.l = __yhdb__.l & 0x7fffffffffffffffLL; /* compute the absolute value*/ \
__yldb__.l = __yldb__.l & 0x7fffffffffffffffLL; /* compute the absolute value*/ \
u53.l = (__yhdb__.l & ULL(7ff0000000000000)) + ULL(0010000000000000); \
if(__yldb__.d > __eps__ * u53.d){ \
if(!yl_neg) { /* The case yl==0 is filtered by the above test*/ \
/* return next up */ \
__yhdb__.d = __yh__; \
if(yh_neg) __yhdb__.l--; else __yhdb__.l++; /* Beware: fails for zero */ \
return __yhdb__.d ; \
} \
else return __yh__; \
} \
}
#define TEST_AND_RETURN_RD(__yh__, __yl__, __eps__) \
{ \
db_number __yhdb__, __yldb__, u53; int yh_neg, yl_neg; \
__yhdb__.d = __yh__; __yldb__.d = __yl__; \
yh_neg = (__yhdb__.i[HI] & 0x80000000); \
yl_neg = (__yldb__.i[HI] & 0x80000000); \
__yhdb__.l = __yhdb__.l & 0x7fffffffffffffffLL; /* compute the absolute value*/ \
__yldb__.l = __yldb__.l & 0x7fffffffffffffffLL; /* compute the absolute value*/ \
u53.l = (__yhdb__.l & ULL(7ff0000000000000)) + ULL(0010000000000000); \
if(__yldb__.d > __eps__ * u53.d){ \
if(yl_neg) { /* The case yl==0 is filtered by the above test*/ \
/* return next down */ \
__yhdb__.d = __yh__; \
if(yh_neg) __yhdb__.l++; else __yhdb__.l--; /* Beware: fails for zero */ \
return __yhdb__.d ; \
} \
else return __yh__; \
} \
}
#define TEST_AND_RETURN_RZ(__yh__, __yl__, __eps__) \
{ \
db_number __yhdb__, __yldb__, u53; int yh_neg, yl_neg; \
__yhdb__.d = __yh__; __yldb__.d = __yl__; \
yh_neg = (__yhdb__.i[HI] & 0x80000000); \
yl_neg = (__yldb__.i[HI] & 0x80000000); \
__yhdb__.l = __yhdb__.l & ULL(7fffffffffffffff); /* compute the absolute value*/\
__yldb__.l = __yldb__.l & ULL(7fffffffffffffff); /* compute the absolute value*/\
u53.l = (__yhdb__.l & ULL(7ff0000000000000)) + ULL(0010000000000000); \
if(__yldb__.d > __eps__ * u53.d){ \
if(yl_neg!=yh_neg) { \
__yhdb__.d = __yh__; \
__yhdb__.l--; /* Beware: fails for zero */ \
return __yhdb__.d ; \
} \
else return __yh__; \
} \
}
#define TEST_AND_COPY_RU(__cond__, __res__, __yh__, __yl__, __eps__) \
{ \
db_number __yhdb__, __yldb__, u53; int yh_neg, yl_neg; \
__yhdb__.d = __yh__; __yldb__.d = __yl__; \
yh_neg = (__yhdb__.i[HI] & 0x80000000); \
yl_neg = (__yldb__.i[HI] & 0x80000000); \
__yhdb__.l = __yhdb__.l & 0x7fffffffffffffffLL; /* compute the absolute value*/ \
__yldb__.l = __yldb__.l & 0x7fffffffffffffffLL; /* compute the absolute value*/ \
u53.l = (__yhdb__.l & ULL(7ff0000000000000)) + ULL(0010000000000000); \
__cond__ = 0; \
if(__yldb__.d > __eps__ * u53.d){ \
__cond__ = 1; \
if(!yl_neg) { /* The case yl==0 is filtered by the above test*/ \
/* return next up */ \
__yhdb__.d = __yh__; \
if(yh_neg) __yhdb__.l--; else __yhdb__.l++; /* Beware: fails for zero */ \
__res__ = __yhdb__.d ; \
} \
else { \
__res__ = __yh__; \
} \
} \
}
#define TEST_AND_COPY_RD(__cond__, __res__, __yh__, __yl__, __eps__) \
{ \
db_number __yhdb__, __yldb__, u53; int yh_neg, yl_neg; \
__yhdb__.d = __yh__; __yldb__.d = __yl__; \
yh_neg = (__yhdb__.i[HI] & 0x80000000); \
yl_neg = (__yldb__.i[HI] & 0x80000000); \
__yhdb__.l = __yhdb__.l & 0x7fffffffffffffffLL; /* compute the absolute value*/ \
__yldb__.l = __yldb__.l & 0x7fffffffffffffffLL; /* compute the absolute value*/ \
u53.l = (__yhdb__.l & ULL(7ff0000000000000)) + ULL(0010000000000000); \
__cond__ = 0; \
if(__yldb__.d > __eps__ * u53.d){ \
__cond__ = 1; \
if(yl_neg) { /* The case yl==0 is filtered by the above test*/ \
/* return next down */ \
__yhdb__.d = __yh__; \
if(yh_neg) __yhdb__.l++; else __yhdb__.l--; /* Beware: fails for zero */ \
__res__ = __yhdb__.d ; \
} \
else { \
__res__ = __yh__; \
} \
} \
}
#define TEST_AND_COPY_RZ(__cond__, __res__, __yh__, __yl__, __eps__) \
{ \
db_number __yhdb__, __yldb__, u53; int yh_neg, yl_neg; \
__yhdb__.d = __yh__; __yldb__.d = __yl__; \
yh_neg = (__yhdb__.i[HI] & 0x80000000); \
yl_neg = (__yldb__.i[HI] & 0x80000000); \
__yhdb__.l = __yhdb__.l & ULL(7fffffffffffffff); /* compute the absolute value*/\
__yldb__.l = __yldb__.l & ULL(7fffffffffffffff); /* compute the absolute value*/\
u53.l = (__yhdb__.l & ULL(7ff0000000000000)) + ULL(0010000000000000); \
__cond__ = 0; \
if(__yldb__.d > __eps__ * u53.d){ \
if(yl_neg!=yh_neg) { \
__yhdb__.d = __yh__; \
__yhdb__.l--; /* Beware: fails for zero */ \
__res__ = __yhdb__.d ; \
__cond__ = 1; \
} \
else { \
__res__ = __yh__; \
__cond__ = 1; \
} \
}
/* If the processor has a FMA, use it ! **/
/* All this probably works only with gcc.
See Markstein book for the case of HP's compiler */
#if defined(CRLIBM_TYPECPU_POWERPC) && defined(__GNUC__)
#define PROCESSOR_HAS_FMA 1
#define FMA(a,b,c) /* r = a*b + c*/ \
({ \
double _a, _b,_c,_r; \
_a=a; _b=b;_c=c; \
__asm__ ("fmadd %0, %1, %2, %3\n ;;\n" \
: "=f"(_r) \
: "f"(_a), "f"(_b), "f"(_c) \
); \
_r; \
})
#define FMS(a,b,c) /* r = a*b - c*/ \
({ \
double _a, _b,_c,_r; \
_a=a; _b=b;_c=c; \
__asm__ ("fmsub %0, %1, %2, %3\n ;;\n" \
: "=f"(_r) \
: "f"(_a), "f"(_b), "f"(_c) \
); \
_r; \
})
#endif /* defined(CRLIBM_TYPECPU_POWERPC) && defined(__GCC__) */
/* On the Itanium 1 / gcc3.2 we lose 10 cycles when using the FMA !?!
It probably breaks the scheduling algorithms somehow...
To test again with higher gcc versions
*/
#if defined(CRLIBM_TYPECPU_ITANIUM) && defined(__GNUC__) && !defined(__INTEL_COMPILER) && 0
#define PROCESSOR_HAS_FMA 1
#define FMA(a,b,c) /* r = a*b + c*/ \
({ \
double _a, _b,_c,_r; \
_a=a; _b=b;_c=c; \
__asm__ ("fma %0 = %1, %2, %3\n ;;\n" \
: "=f"(_r) \
: "f"(_a), "f"(_b), "f"(_c) \
); \
_r; \
})
#define FMS(a,b,c) /* r = a*b - c*/ \
({ \
double _a, _b, _c, _r; \
_a=a; _b=b;_c=c; \
__asm__ ("fms %0 = %1, %2, %3\n ;;\n" \
: "=f"(_r) \
: "f"(_a), "f"(_b), "f"(_c) \
); \
_r; \
})
#endif /* defined(CRLIBM_TYPECPU_ITANIUM) && defined(__GCC__) && !defined(__INTEL_COMPILER) */
#if defined(CRLIBM_TYPECPU_ITANIUM) && defined(__INTEL_COMPILER)
#define PROCESSOR_HAS_FMA 1
#if 0 /* Commented out because it shouldn't be there: There should be
a standard #include doing all this, but as of april 2005
it doesn't exist, say intel people). Leave
it as documentation, though, until it is replaced by #include
*/
/* Table 1-17: legal floating-point precision completers (.pc) */
typedef enum {
_PC_S = 1 /* single .s */
,_PC_D = 2 /* double .d */
,_PC_NONE = 3 /* dynamic */
} _Asm_pc;
/* Table 1-22: legal getf/setf floating-point register access completers */
typedef enum {
_FR_S = 1 /* single form .s */
,_FR_D = 2 /* double form .d */
,_FR_EXP = 3 /* exponent form .exp */
,_FR_SIG = 4 /* significand form .sig */
} _Asm_fr_access;
/* Table 1-24: legal floating-point FPSR status field completers (.sf) */
typedef enum {
_SF0 = 0 /* FPSR status field 0 .s0 */
,_SF1 = 1 /* FPSR status field 1 .s1 */
,_SF2 = 2 /* FPSR status field 2 .s2 */
,_SF3 = 3 /* FPSR status field 3 .s3 */
} _Asm_sf;
#endif
#define FMA(a,b,c) /* r = a*b + c*/ \
_Asm_fma( 2/*_PC_D*/, a, b, c, 0/*_SF0*/ );
#define FMS(a,b,c) /* r = a*b - c*/ \
_Asm_fms( 2/*_PC_D*/, a, b, c, 0/*_SF0*/);
#endif /*defined(CRLIBM_TYPECPU_ITANIUM) && defined(__INTEL_COMPILER)*/
#ifdef WORDS_BIGENDIAN
#define DB_ONE {{0x3ff00000, 0x00000000}}
#else
#define DB_ONE {{0x00000000 ,0x3ff00000}}
#endif
extern const scs scs_zer, scs_half, scs_one, scs_two, scs_sixinv;
#define SCS_ZERO (scs_ptr)(&scs_zer)
#define SCS_HALF (scs_ptr)(&scs_half)
#define SCS_ONE (scs_ptr)(&scs_one)
#define SCS_TWO (scs_ptr)(&scs_two)
#define SCS_SIXINV (scs_ptr)(&scs_sixinv)
#if defined(__GNUC__)
#define ABS(x) (__builtin_fabs((x)))
#else
#define ABS(x) (((x)>0) ? (x) : (-(x)))
#endif
/*
* In the following, when an operator is preceded by a '@' it means that we
* are considering the IEEE-compliant machine operator, otherwise it
* is the mathematical operator.
*
*/
/*
* computes s and r such that s + r = a + b, with s = a @+ b exactly
*/
#if AVOID_BRANCHES
#define Add12Cond(s, r, a, b) \
{ \
double _u1, _u2, _u3, _u4; \
double _a=a, _b=b; \
\
s = _a + _b; \
_u1 = s - _a; \
_u2 = s - _u1; \
_u3 = _b - _u1; \
_u4 = _a - _u2; \
r = _u4 + _u3; \
}
#else
#define Add12Cond(s, r, a, b) \
{double _z, _a=a, _b=b; \
s = _a + _b; \
if (ABS(a) > ABS(b)){ \
_z = s - _a; \
r = _b - _z; \
}else { \
_z = s - _b; \
r = _a - _z;}}
#endif
/*
* computes s and r such that s + r = a + b, with s = a @+ b exactly
* under the condition a >= b
*/
#define Add12(s, r, a, b) \
{double _z, _a=a, _b=b; \
s = _a + _b; \
_z = s - _a; \
r = _b - _z; }
/*
* computes r1, r2, r3 such that r1 + r2 + r3 = a + b + c exactly
*/
#define Fast3Sum(r1, r2, r3, a, b, c) \
{double u, v, w; \
Fast2Sum(u, v, b, c); \
Fast2Sum(r1, w, a, u); \
Fast2Sum(r2, r3, w, v); }
/*
* Functions to computes double-double addition: zh+zl = xh+xl + yh+yl
* knowing that xh>yh
* relative error is smaller than 2^-103
*/
#if ADD22_AS_FUNCTIONS
extern void Add22(double *zh, double *zl, double xh, double xl, double yh, double yl);
extern void Add22Cond(double *zh, double *zl, double xh, double xl, double yh, double yl);
#else /* ADD22_AS_FUNCTIONS */
#if AVOID_BRANCHES
#define Add22Cond(zh,zl,xh,xl,yh,yl) \
do { \
double _v1, _v2, _v3, _v4; \
\
Add12Cond(_v1, _v2, (xh), (yh)); \
_v3 = (xl) + (yl); \
_v4 = _v2 + _v3; \
Add12((*(zh)),(*(zl)),_v1,_v4); \
} while (2+2==5)
#else
#define Add22Cond(zh,zl,xh,xl,yh,yl) \
do { \
double _r,_s; \
_r = (xh)+(yh); \
_s = ((ABS(xh)) > (ABS(yh)))? ((xh)-_r+(yh)+(yl)+(xl)) : ((yh)-_r+(xh)+(xl)+(yl)); \
*zh = _r+_s; \
*zl = (_r - (*zh)) + _s; \
} while(2+2==5)
#endif
#define Add22(zh,zl,xh,xl,yh,yl) \
do { \
double _r,_s; \
_r = (xh)+(yh); \
_s = ((((xh)-_r) +(yh)) + (yl)) + (xl); \
*zh = _r+_s; \
*zl = (_r - (*zh)) + _s; \
} while(0)
#endif /* ADD22_AS_FUNCTIONS */
#ifdef PROCESSOR_HAS_FMA
/* One of the nice things with the fused multiply-and-add is that it
greatly simplifies the double-double multiplications : */
#define Mul12(rh,rl,u,v) \
{ \
*rh = u*v; \
*rl = FMS(u,v, *rh); \
}
#define Mul22(pzh,pzl, xh,xl, yh,yl) \
{ \
double ph, pl; \
ph = xh*yh; \
pl = FMS(xh, yh, ph); \
pl = FMA(xh,yl, pl); \
pl = FMA(xl,yh,pl); \
*pzh = ph+pl; \
*pzl = ph - (*pzh); \
*pzl += pl; \
}
/* besides we don't care anymore about overflows in the mult */
#define Mul12Cond Mul12
#define Mul22cond Mul22
#else /* ! PROCESSOR_HAS_FMA */
#if DEKKER_AS_FUNCTIONS
extern void Mul12(double *rh, double *rl, double u, double v);
extern void Mul12Cond(double *rh, double *rl, double a, double b);
extern void Mul22(double *zh, double *zl, double xh, double xl, double yh, double yl);
#else /* if DEKKER_AS_FUNCTIONS */
/*
* computes rh and rl such that rh + rl = a * b with rh = a @* b exactly
* under the conditions : a < 2^970 et b < 2^970
*/
#if 1
#define Mul12(rh,rl,u,v) \
{ \
const double c = 134217729.; /* 2^27 +1 */ \
double up, u1, u2, vp, v1, v2; \
double _u=u, _v=v; \
up = _u*c; vp = _v*c; \
u1 = (_u-up)+up; v1 = (_v-vp)+vp; \
u2 = _u-u1; v2 = _v-v1; \
\
*rh = _u*_v; \
*rl = (((u1*v1-*rh)+(u1*v2))+(u2*v1))+(u2*v2);\
}
#else
/* This works but is much slower. Problem:
SSE2 instructions are two-address, and intrinsincs are 3-address */
#include<emmintrin.h>
#define Mul12(rh,rl,u,v) \
{ \
const double c = 134217729.; /* 2^27 +1 */ \
__m128d _u_v = _mm_set_pd (u,v); \
__m128d c2=_mm_set1_pd(c); \
c2 = _mm_mul_pd(c2, _u_v); \
__m128d u1v1 = _mm_sub_pd(_u_v, c2); \
u1v1 = _mm_add_pd(u1v1, c2); \
__m128d u2v2 = _mm_sub_pd(_u_v, u1v1); \
__m128d _v_u = _mm_shuffle_pd(_u_v, _u_v, _MM_SHUFFLE2 (0,1)); \
__m128d rhrh = _mm_mul_pd(_v_u, _u_v); \
_mm_store_sd (rh, rhrh); \
__m128d v2u2 = _mm_shuffle_pd(u2v2, u2v2, _MM_SHUFFLE2 (0,1)); \
__m128d u1v2u2v1 = _mm_mul_pd(u1v1, v2u2); \
__m128d u2v1u1v2 = _mm_shuffle_pd(u1v2u2v1, u1v2u2v1, _MM_SHUFFLE2 (0,1)); \
__m128d uvmed = _mm_add_pd(u1v2u2v1, u2v1u1v2); \
__m128d u1u2 = _mm_shuffle_pd(u1v1, u2v2, _MM_SHUFFLE2 (1,1)); \
__m128d v1v2 = _mm_shuffle_pd(u1v1, u2v2, _MM_SHUFFLE2 (0,0)); \
__m128d u1v1u2v2 = _mm_mul_pd(u1u2, v1v2); \
__m128d tmp = _mm_sub_pd(u1v1u2v2, rhrh); \
tmp = _mm_add_pd(tmp, uvmed); \
__m128d u2v2u2v2 = _mm_mul_pd(u2v2, v2u2); \
tmp = _mm_add_pd(tmp, u2v2u2v2); \
_mm_store_sd (rl, tmp); \
}
#endif
/*
double _u =u, _v=v; \
__m128d _u_v = _mm_set_pd(_u, _v); \
*/ \
/*
* Computes rh and rl such that rh + rl = a * b and rh = a @* b exactly
*/
#define Mul12Cond(rh, rl, a, b) \
{\
const double two_em53 = 1.1102230246251565404e-16; /* 0x3CA00000, 0x00000000 */\
const double two_e53 = 9007199254740992.; /* 0x43400000, 0x00000000 */\
double u, v; \
db_number _a=a, _b=b; \
\
if (_a.i[HI]>0x7C900000) u = _a*two_em53; \
else u = _a; \
if (_b.i[HI]>0x7C900000) v = _b*two_em53; \
else v = _b; \
\
Mul12(rh, rl, u, v); \
\
if (_a.i[HI]>0x7C900000) {*rh *= two_e53; *rl *= two_e53;} \
if (_b.i[HI]>0x7C900000) {*rh *= two_e53; *rl *= two_e53;} \
}
/*
* computes double-double multiplication: zh+zl = (xh+xl) * (yh+yl)
* relative error is smaller than 2^-102
*/
#define Mul22(zh,zl,xh,xl,yh,yl) \
{ \
double mh, ml; \
\
const double c = 134217729.; \
double up, u1, u2, vp, v1, v2; \
\
up = (xh)*c; vp = (yh)*c; \
u1 = ((xh)-up)+up; v1 = ((yh)-vp)+vp; \
u2 = (xh)-u1; v2 = (yh)-v1; \
\
mh = (xh)*(yh); \
ml = (((u1*v1-mh)+(u1*v2))+(u2*v1))+(u2*v2); \
\
ml += (xh)*(yl) + (xl)*(yh); \
*zh = mh+ml; \
*zl = mh - (*zh) + ml; \
}
#endif /* DEKKER_AS_FUNCTIONS */
#endif /* PROCESSOR_HAS_FMA */
/* Additional double-double operators */
/* Eps Mul122 <= 2^-102 */
#define Mul122(resh,resl,a,bh,bl) \
{ \
double _t1, _t2, _t3, _t4; \
\
Mul12(&_t1,&_t2,(a),(bh)); \
_t3 = (a) * (bl); \
_t4 = _t2 + _t3; \
Add12((*(resh)),(*(resl)),_t1,_t4); \
}
/* Eps MulAdd212 <= 2^-100 for |a * (bh + bl)| <= 1/4 * |ch + cl| */
#define MulAdd212(resh,resl,ch,cl,a,bh,bl) \
{ \
double _t1, _t2, _t3, _t4, _t5, _t6, _t7, _t8; \
\
Mul12(&_t1,&_t2,(a),(bh)); \
Add12(_t3,_t4,(ch),_t1); \
_t5 = (bl) * (a); \
_t6 = (cl) + _t2; \
_t7 = _t5 + _t6; \
_t8 = _t7 + _t4; \
Add12((*(resh)),(*(resl)),_t3,_t8); \
}
/* Eps MulAdd212 <= 2^-100
for |(ah + bh) * (bh + bl)| <= 1/4 * |ch + cl|
*/
#define MulAdd22(resh,resl,ch,cl,ah,al,bh,bl) \
{ \
double _t1, _t2, _t3, _t4, _t5, _t6, _t7, _t8; \
double _t9, _t10; \
\
Mul12(&_t1,&_t2,(ah),(bh)); \
Add12(_t3,_t4,(ch),_t1); \
_t5 = (ah) * (bl); \
_t6 = (al) * (bh); \
_t7 = _t2 + (cl); \
_t8 = _t4 + _t7; \
_t9 = _t5 + _t6; \
_t10 = _t8 + _t9; \
Add12((*(resh)),(*(resl)),_t3,_t10); \
}
#define Add122(resh,resl,a,bh,bl) \
{ \
double _t1, _t2, _t3; \
\
Add12(_t1,_t2,(a),(bh)); \
_t3 = _t2 + (bl); \
Add12((*(resh)),(*(resl)),_t1,_t3); \
}
#define Add122Cond(resh,resl,a,bh,bl) \
{ \
double _t1, _t2, _t3; \
\
Add12Cond(_t1,_t2,(a),(bh)); \
_t3 = _t2 + (bl); \
Add12((*(resh)),(*(resl)),_t1,_t3); \
}
#define Add212(resh,resl,ah,al,b) \
{ \
double _t1, _t2, _t3; \
\
Add12(_t1,_t2,(ah),b); \
_t3 = _t2 + (al); \
Add12((*(resh)),(*(resl)),_t1,_t3); \
}
/* In the following the one-line computation of _cl was split so that
icc(8.1) would compile it properly. It's a bug of icc */
#if DEKKER_AS_FUNCTIONS
extern void Div22(double *z, double *zz, double x, double xx, double y, double yy);
#else
#define Div22(pzh,pzl,xh,xl,yh,yl) { \
double _ch,_cl,_uh,_ul; \
_ch=(xh)/(yh); Mul12(&_uh,&_ul,_ch,(yh)); \
_cl=((xh)-_uh); \
_cl -= _ul; \
_cl += (xl); \
_cl -= _ch*(yl); \
_cl /= (yh); \
*pzh=_ch+_cl; *pzl=(_ch-(*pzh))+_cl; \
}
#endif /* DEKKER_AS_FUNCTIONS */
/*
Coefficients for 1/sqrt(m) with 1/2 < m < 2
The corresponding relative polynomial approximation error is less than
eps < 2^(-8.3127) (cf. Maple file)
The Itanium instruction frsqrta is slightly more accurate; it can
therefore easily replace the polynomial evaluation.
*/
#define SQRTPOLYC0 2.50385236695888790947606139525305479764938354492188e+00
#define SQRTPOLYC1 -3.29763389114324168005509818613063544034957885742188e+00
#define SQRTPOLYC2 2.75726076139124520736345402838196605443954467773438e+00
#define SQRTPOLYC3 -1.15233725777933848632983426796272397041320800781250e+00
#define SQRTPOLYC4 1.86900066679800969104974228685023263096809387207031e-01
#define SQRTTWO52 4.50359962737049600000000000000000000000000000000000e+15
#if SQRT_AS_FUNCTIONS
extern void sqrt12(double *resh, double *resl, double x);
#else
/* Concerning special case handling see crlibm_private.h */
#define sqrt12(resh, resl, x) { \
db_number _xdb; \
int _E; \
double _m, _r0, _r1, _r2, _r3h, _r3l, _r4h, _r4l, _srtmh, _srtml; \
double _r2PHr2h, _r2PHr2l, _r2Sqh, _r2Sql; \
double _mMr2h, _mMr2l, _mMr2Ch, _mMr2Cl; \
double _MHmMr2Ch, _MHmMr2Cl; \
double _r3Sqh, _r3Sql, _mMr3Sqh, _mMr3Sql; \
double _half; \
\
/* Special case x = 0 */ \
if ((x) == 0.0) { \
(*(resh)) = (x); \
(*(resl)) = 0.0; \
} else { \
\
_E = 0; \
\
/* Convert to integer format */ \
_xdb.d = (x); \
\
/* Handle subnormal case */ \
if (_xdb.i[HI] < 0x00100000) { \
_E = -52; \
_xdb.d *= ((db_number) ((double) SQRTTWO52)).d; \
/* make x a normal number */ \
} \
\
/* Extract exponent E and mantissa m */ \
_E += (_xdb.i[HI]>>20)-1023; \
_xdb.i[HI] = (_xdb.i[HI] & 0x000fffff) | 0x3ff00000; \
_m = _xdb.d; \
\
_half = 0.5; \
/* Make exponent even */ \
if (_E & 0x00000001) { \
_E++; \
_m *= _half; /* Suppose now 1/2 <= m <= 2 */ \
} \
\
/* Construct sqrt(2^E) = 2^(E/2) */ \
_xdb.i[HI] = (_E/2 + 1023) << 20; \
_xdb.i[LO] = 0; \
\
/* Compute initial approximation to r = 1/sqrt(m) */ \
\
_r0 = SQRTPOLYC0 + \
_m * (SQRTPOLYC1 + _m * (SQRTPOLYC2 + _m * (SQRTPOLYC3 + _m * SQRTPOLYC4))); \
\
/* Iterate two times on double precision */ \
\
_r1 = _half * _r0 * (3.0 - _m * (_r0 * _r0)); \
_r2 = _half * _r1 * (3.0 - _m * (_r1 * _r1)); \
\
/* Iterate two times on double-double precision */ \
\
Mul12(&_r2Sqh, &_r2Sql, _r2, _r2); \
Add12(_r2PHr2h, _r2PHr2l, _r2, (_half * _r2)); \
Mul12(&_mMr2h, &_mMr2l, _m, _r2); \
Mul22(&_mMr2Ch, &_mMr2Cl, _mMr2h, _mMr2l, _r2Sqh, _r2Sql); \
\
_MHmMr2Ch = -_half * _mMr2Ch; \
_MHmMr2Cl = -_half * _mMr2Cl; \
\
Add22(&_r3h, &_r3l, _r2PHr2h, _r2PHr2l, _MHmMr2Ch, _MHmMr2Cl); \
\
Mul22(&_r3Sqh, &_r3Sql, _r3h, _r3l, _r3h, _r3l); \
Mul22(&_mMr3Sqh, &_mMr3Sql, _m, 0.0, _r3Sqh, _r3Sql); \
/* To prove: mMr3Sqh = 1.0 in each case */ \
\
Mul22(&_r4h, &_r4l, _r3h, _r3l, 1.0, (-_half * _mMr3Sql)); \
\
/* Multiply obtained reciprocal square root by m */ \
\
Mul22(&_srtmh,&_srtml,_m,0.0,_r4h,_r4l); \
\
/* Multiply componentwise by sqrt(2^E) */ \
/* which is an integer power of 2 that may not produce a subnormal */ \
\
(*(resh)) = _xdb.d * _srtmh; \
(*(resl)) = _xdb.d * _srtml; \
\
} /* End: special case 0 */ \
}
#define sqrt12_64(resh, resl, x) { \
db_number _xdb; \
int _E; \
double _m, _r0, _r1, _r2, _r3h, _r3l, _r4h, _r4l, _srtmh, _srtml; \
double _r2PHr2h, _r2PHr2l, _r2Sqh, _r2Sql; \
double _mMr2h, _mMr2l, _mMr2Ch, _mMr2Cl; \
double _MHmMr2Ch, _MHmMr2Cl; \
double _r3Sqh, _r3Sql, _mMr3Sqh, _mMr3Sql; \
double _half; \
\
/* Special case x = 0 */ \
if ((x) == 0.0) { \
(*(resh)) = (x); \
(*(resl)) = 0.0; \
} else { \
\
_E = 0.0; \
\
/* Convert to integer format */ \
_xdb.d = (x); \
\
/* Handle subnormal case */ \
if (_xdb.i[HI] < 0x00100000) { \
_E = -52; \
_xdb.d *= ((db_number) ((double) SQRTTWO52)).d; \
/* make x a normal number */ \
} \
\
/* Extract exponent E and mantissa m */ \
_E += (_xdb.i[HI]>>20)-1023; \
_xdb.i[HI] = (_xdb.i[HI] & 0x000fffff) | 0x3ff00000; \
_m = _xdb.d; \
\
_half = 0.5; \
/* Make exponent even */ \
if (_E & 0x00000001) { \
_E++; \
_m *= _half; /* Suppose now 1/2 <= m <= 2 */ \
} \
\
/* Construct sqrt(2^E) = 2^(E/2) */ \
_xdb.i[HI] = (_E/2 + 1023) << 20; \
_xdb.i[LO] = 0; \
\
/* Compute initial approximation to r = 1/sqrt(m) */ \
\
_r0 = SQRTPOLYC0 + \
_m * (SQRTPOLYC1 + _m * (SQRTPOLYC2 + _m * (SQRTPOLYC3 + _m * SQRTPOLYC4))); \
\
/* Iterate two times on double precision */ \
\
_r1 = _half * _r0 * (3.0 - _m * (_r0 * _r0)); \
_r2 = _half * _r1 * (3.0 - _m * (_r1 * _r1)); \
\
/* Iterate once on double-double precision */ \
\
Mul12(&_r2Sqh, &_r2Sql, _r2, _r2); \
Add12(_r2PHr2h, _r2PHr2l, _r2, (_half * _r2)); \
Mul12(&_mMr2h, &_mMr2l, _m, _r2); \
Mul22(&_mMr2Ch, &_mMr2Cl, _mMr2h, _mMr2l, _r2Sqh, _r2Sql); \
\
_MHmMr2Ch = -_half * _mMr2Ch; \
_MHmMr2Cl = -_half * _mMr2Cl; \
\
Add22(&_r3h, &_r3l, _r2PHr2h, _r2PHr2l, _MHmMr2Ch, _MHmMr2Cl); \
\
/* Multiply obtained reciprocal square root by m */ \
\
Mul22(&_srtmh,&_srtml,_m,0.0,_r3h,_r3l); \
\
/* Multiply componentwise by sqrt(2^E) */ \
/* which is an integer power of 2 that may not produce a subnormal */ \
\
(*(resh)) = _xdb.d * _srtmh; \
(*(resl)) = _xdb.d * _srtml; \
\
} /* End: special case 0 */ \
}
/*
sqrt12_64_unfiltered = sqrt(x) * (1 + eps) where abs(eps) <= 2^(-64)
if x is neither subnormal nor 0
*/
#define sqrt12_64_unfiltered(resh, resl, x) { \
db_number _xdb; \
int _E; \
double _m, _r0, _r1, _r2, _r3h, _r3l, _srtmh, _srtml; \
double _r2PHr2h, _r2PHr2l, _r2Sqh, _r2Sql; \
double _mMr2h, _mMr2l, _mMr2Ch, _mMr2Cl; \
double _MHmMr2Ch, _MHmMr2Cl; \
double _half; \
\
\
\
/* Convert to integer format */ \
_xdb.d = (x); \
\
\
/* Extract exponent E and mantissa m */ \
_E = (_xdb.i[HI]>>20)-1023; \
_xdb.i[HI] = (_xdb.i[HI] & 0x000fffff) | 0x3ff00000; \
_m = _xdb.d; \
\
_half = 0.5; \
/* Make exponent even */ \
if (_E & 0x00000001) { \
_E++; \
_m *= _half; /* Suppose now 1/2 <= m <= 2 */ \
} \
\