summaryrefslogtreecommitdiffstats
path: root/modules/fdlibm/src/math_private.h
blob: d9ec448176ef194b911532a0dfa7593f670ddef1 (plain)
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
/*
 * ====================================================
 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
 *
 * Developed at SunPro, a Sun Microsystems, Inc. business.
 * Permission to use, copy, modify, and distribute this
 * software is freely granted, provided that this notice
 * is preserved.
 * ====================================================
 */

/*
 * from: @(#)fdlibm.h 5.1 93/09/24
 * $FreeBSD$
 */

#ifndef _MATH_PRIVATE_H_
#define	_MATH_PRIVATE_H_

#include <cfloat>
#include <stdint.h>
#include <sys/types.h>

#include "fdlibm.h"

#include "mozilla/EndianUtils.h"

/*
 * The original fdlibm code used statements like:
 *	n0 = ((*(int*)&one)>>29)^1;		* index of high word *
 *	ix0 = *(n0+(int*)&x);			* high word of x *
 *	ix1 = *((1-n0)+(int*)&x);		* low word of x *
 * to dig two 32 bit words out of the 64 bit IEEE floating point
 * value.  That is non-ANSI, and, moreover, the gcc instruction
 * scheduler gets it wrong.  We instead use the following macros.
 * Unlike the original code, we determine the endianness at compile
 * time, not at run time; I don't see much benefit to selecting
 * endianness at run time.
 */

#ifndef u_int32_t
#define u_int32_t uint32_t
#endif
#ifndef u_int64_t
#define u_int64_t uint64_t
#endif

/* A union which permits us to convert between a long double and
   four 32 bit ints.  */

#if MOZ_BIG_ENDIAN

typedef union
{
  long double value;
  struct {
    u_int32_t mswhi;
    u_int32_t mswlo;
    u_int32_t lswhi;
    u_int32_t lswlo;
  } parts32;
  struct {
    u_int64_t msw;
    u_int64_t lsw;
  } parts64;
} ieee_quad_shape_type;

#endif

#if MOZ_LITTLE_ENDIAN

typedef union
{
  long double value;
  struct {
    u_int32_t lswlo;
    u_int32_t lswhi;
    u_int32_t mswlo;
    u_int32_t mswhi;
  } parts32;
  struct {
    u_int64_t lsw;
    u_int64_t msw;
  } parts64;
} ieee_quad_shape_type;

#endif

/*
 * A union which permits us to convert between a double and two 32 bit
 * ints.
 */

#if MOZ_BIG_ENDIAN

typedef union
{
  double value;
  struct
  {
    u_int32_t msw;
    u_int32_t lsw;
  } parts;
  struct
  {
    u_int64_t w;
  } xparts;
} ieee_double_shape_type;

#endif

#if MOZ_LITTLE_ENDIAN

typedef union
{
  double value;
  struct
  {
    u_int32_t lsw;
    u_int32_t msw;
  } parts;
  struct
  {
    u_int64_t w;
  } xparts;
} ieee_double_shape_type;

#endif

/* Get two 32 bit ints from a double.  */

#define EXTRACT_WORDS(ix0,ix1,d)				\
do {								\
  ieee_double_shape_type ew_u;					\
  ew_u.value = (d);						\
  (ix0) = ew_u.parts.msw;					\
  (ix1) = ew_u.parts.lsw;					\
} while (0)

/* Get a 64-bit int from a double. */
#define EXTRACT_WORD64(ix,d)					\
do {								\
  ieee_double_shape_type ew_u;					\
  ew_u.value = (d);						\
  (ix) = ew_u.xparts.w;						\
} while (0)

/* Get the more significant 32 bit int from a double.  */

#define GET_HIGH_WORD(i,d)					\
do {								\
  ieee_double_shape_type gh_u;					\
  gh_u.value = (d);						\
  (i) = gh_u.parts.msw;						\
} while (0)

/* Get the less significant 32 bit int from a double.  */

#define GET_LOW_WORD(i,d)					\
do {								\
  ieee_double_shape_type gl_u;					\
  gl_u.value = (d);						\
  (i) = gl_u.parts.lsw;						\
} while (0)

/* Set a double from two 32 bit ints.  */

#define INSERT_WORDS(d,ix0,ix1)					\
do {								\
  ieee_double_shape_type iw_u;					\
  iw_u.parts.msw = (ix0);					\
  iw_u.parts.lsw = (ix1);					\
  (d) = iw_u.value;						\
} while (0)

/* Set a double from a 64-bit int. */
#define INSERT_WORD64(d,ix)					\
do {								\
  ieee_double_shape_type iw_u;					\
  iw_u.xparts.w = (ix);						\
  (d) = iw_u.value;						\
} while (0)

/* Set the more significant 32 bits of a double from an int.  */

#define SET_HIGH_WORD(d,v)					\
do {								\
  ieee_double_shape_type sh_u;					\
  sh_u.value = (d);						\
  sh_u.parts.msw = (v);						\
  (d) = sh_u.value;						\
} while (0)

/* Set the less significant 32 bits of a double from an int.  */

#define SET_LOW_WORD(d,v)					\
do {								\
  ieee_double_shape_type sl_u;					\
  sl_u.value = (d);						\
  sl_u.parts.lsw = (v);						\
  (d) = sl_u.value;						\
} while (0)

/*
 * A union which permits us to convert between a float and a 32 bit
 * int.
 */

typedef union
{
  float value;
  /* FIXME: Assumes 32 bit int.  */
  unsigned int word;
} ieee_float_shape_type;

/* Get a 32 bit int from a float.  */

#define GET_FLOAT_WORD(i,d)					\
do {								\
  ieee_float_shape_type gf_u;					\
  gf_u.value = (d);						\
  (i) = gf_u.word;						\
} while (0)

/* Set a float from a 32 bit int.  */

#define SET_FLOAT_WORD(d,i)					\
do {								\
  ieee_float_shape_type sf_u;					\
  sf_u.word = (i);						\
  (d) = sf_u.value;						\
} while (0)

/*
 * Get expsign and mantissa as 16 bit and 64 bit ints from an 80 bit long
 * double.
 */

#define	EXTRACT_LDBL80_WORDS(ix0,ix1,d)				\
do {								\
  union IEEEl2bits ew_u;					\
  ew_u.e = (d);							\
  (ix0) = ew_u.xbits.expsign;					\
  (ix1) = ew_u.xbits.man;					\
} while (0)

/*
 * Get expsign and mantissa as one 16 bit and two 64 bit ints from a 128 bit
 * long double.
 */

#define	EXTRACT_LDBL128_WORDS(ix0,ix1,ix2,d)			\
do {								\
  union IEEEl2bits ew_u;					\
  ew_u.e = (d);							\
  (ix0) = ew_u.xbits.expsign;					\
  (ix1) = ew_u.xbits.manh;					\
  (ix2) = ew_u.xbits.manl;					\
} while (0)

/* Get expsign as a 16 bit int from a long double.  */

#define	GET_LDBL_EXPSIGN(i,d)					\
do {								\
  union IEEEl2bits ge_u;					\
  ge_u.e = (d);							\
  (i) = ge_u.xbits.expsign;					\
} while (0)

/*
 * Set an 80 bit long double from a 16 bit int expsign and a 64 bit int
 * mantissa.
 */

#define	INSERT_LDBL80_WORDS(d,ix0,ix1)				\
do {								\
  union IEEEl2bits iw_u;					\
  iw_u.xbits.expsign = (ix0);					\
  iw_u.xbits.man = (ix1);					\
  (d) = iw_u.e;							\
} while (0)

/*
 * Set a 128 bit long double from a 16 bit int expsign and two 64 bit ints
 * comprising the mantissa.
 */

#define	INSERT_LDBL128_WORDS(d,ix0,ix1,ix2)			\
do {								\
  union IEEEl2bits iw_u;					\
  iw_u.xbits.expsign = (ix0);					\
  iw_u.xbits.manh = (ix1);					\
  iw_u.xbits.manl = (ix2);					\
  (d) = iw_u.e;							\
} while (0)

/* Set expsign of a long double from a 16 bit int.  */

#define	SET_LDBL_EXPSIGN(d,v)					\
do {								\
  union IEEEl2bits se_u;					\
  se_u.e = (d);							\
  se_u.xbits.expsign = (v);					\
  (d) = se_u.e;							\
} while (0)

#ifdef __i386__
/* Long double constants are broken on i386. */
#define	LD80C(m, ex, v) {						\
	.xbits.man = __CONCAT(m, ULL),					\
	.xbits.expsign = (0x3fff + (ex)) | ((v) < 0 ? 0x8000 : 0),	\
}
#else
/* The above works on non-i386 too, but we use this to check v. */
#define	LD80C(m, ex, v)	{ .e = (v), }
#endif

#ifdef FLT_EVAL_METHOD
/*
 * Attempt to get strict C99 semantics for assignment with non-C99 compilers.
 */
#if !defined(_MSC_VER) && (FLT_EVAL_METHOD == 0 || __GNUC__ == 0)
#define	STRICT_ASSIGN(type, lval, rval)	((lval) = (rval))
#else
#define	STRICT_ASSIGN(type, lval, rval) do {	\
	volatile type __lval;			\
						\
	if (sizeof(type) >= sizeof(long double))	\
		(lval) = (rval);		\
	else {					\
		__lval = (rval);		\
		(lval) = __lval;		\
	}					\
} while (0)
#endif
#else
#define	STRICT_ASSIGN(type, lval, rval) do {	\
	volatile type __lval;			\
						\
	if (sizeof(type) >= sizeof(long double))	\
		(lval) = (rval);		\
	else {					\
		__lval = (rval);		\
		(lval) = __lval;		\
	}					\
} while (0)
#endif /* FLT_EVAL_METHOD */

/* Support switching the mode to FP_PE if necessary. */
#if defined(__i386__) && !defined(NO_FPSETPREC)
#define	ENTERI() ENTERIT(long double)
#define	ENTERIT(returntype)			\
	returntype __retval;			\
	fp_prec_t __oprec;			\
						\
	if ((__oprec = fpgetprec()) != FP_PE)	\
		fpsetprec(FP_PE)
#define	RETURNI(x) do {				\
	__retval = (x);				\
	if (__oprec != FP_PE)			\
		fpsetprec(__oprec);		\
	RETURNF(__retval);			\
} while (0)
#define	ENTERV()				\
	fp_prec_t __oprec;			\
						\
	if ((__oprec = fpgetprec()) != FP_PE)	\
		fpsetprec(FP_PE)
#define	RETURNV() do {				\
	if (__oprec != FP_PE)			\
		fpsetprec(__oprec);		\
	return;			\
} while (0)
#else
#define	ENTERI()
#define	ENTERIT(x)
#define	RETURNI(x)	RETURNF(x)
#define	ENTERV()
#define	RETURNV()	return
#endif

/* Default return statement if hack*_t() is not used. */
#define      RETURNF(v)      return (v)

/*
 * 2sum gives the same result as 2sumF without requiring |a| >= |b| or
 * a == 0, but is slower.
 */
#define	_2sum(a, b) do {	\
	__typeof(a) __s, __w;	\
				\
	__w = (a) + (b);	\
	__s = __w - (a);	\
	(b) = ((a) - (__w - __s)) + ((b) - __s); \
	(a) = __w;		\
} while (0)

/*
 * 2sumF algorithm.
 *
 * "Normalize" the terms in the infinite-precision expression a + b for
 * the sum of 2 floating point values so that b is as small as possible
 * relative to 'a'.  (The resulting 'a' is the value of the expression in
 * the same precision as 'a' and the resulting b is the rounding error.)
 * |a| must be >= |b| or 0, b's type must be no larger than 'a's type, and
 * exponent overflow or underflow must not occur.  This uses a Theorem of
 * Dekker (1971).  See Knuth (1981) 4.2.2 Theorem C.  The name "TwoSum"
 * is apparently due to Skewchuk (1997).
 *
 * For this to always work, assignment of a + b to 'a' must not retain any
 * extra precision in a + b.  This is required by C standards but broken
 * in many compilers.  The brokenness cannot be worked around using
 * STRICT_ASSIGN() like we do elsewhere, since the efficiency of this
 * algorithm would be destroyed by non-null strict assignments.  (The
 * compilers are correct to be broken -- the efficiency of all floating
 * point code calculations would be destroyed similarly if they forced the
 * conversions.)
 *
 * Fortunately, a case that works well can usually be arranged by building
 * any extra precision into the type of 'a' -- 'a' should have type float_t,
 * double_t or long double.  b's type should be no larger than 'a's type.
 * Callers should use these types with scopes as large as possible, to
 * reduce their own extra-precision and efficiciency problems.  In
 * particular, they shouldn't convert back and forth just to call here.
 */
#ifdef DEBUG
#define	_2sumF(a, b) do {				\
	__typeof(a) __w;				\
	volatile __typeof(a) __ia, __ib, __r, __vw;	\
							\
	__ia = (a);					\
	__ib = (b);					\
	assert(__ia == 0 || fabsl(__ia) >= fabsl(__ib));	\
							\
	__w = (a) + (b);				\
	(b) = ((a) - __w) + (b);			\
	(a) = __w;					\
							\
	/* The next 2 assertions are weak if (a) is already long double. */ \
	assert((long double)__ia + __ib == (long double)(a) + (b));	\
	__vw = __ia + __ib;				\
	__r = __ia - __vw;				\
	__r += __ib;					\
	assert(__vw == (a) && __r == (b));		\
} while (0)
#else /* !DEBUG */
#define	_2sumF(a, b) do {	\
	__typeof(a) __w;	\
				\
	__w = (a) + (b);	\
	(b) = ((a) - __w) + (b); \
	(a) = __w;		\
} while (0)
#endif /* DEBUG */

/*
 * Set x += c, where x is represented in extra precision as a + b.
 * x must be sufficiently normalized and sufficiently larger than c,
 * and the result is then sufficiently normalized.
 *
 * The details of ordering are that |a| must be >= |c| (so that (a, c)
 * can be normalized without extra work to swap 'a' with c).  The details of
 * the normalization are that b must be small relative to the normalized 'a'.
 * Normalization of (a, c) makes the normalized c tiny relative to the
 * normalized a, so b remains small relative to 'a' in the result.  However,
 * b need not ever be tiny relative to 'a'.  For example, b might be about
 * 2**20 times smaller than 'a' to give about 20 extra bits of precision.
 * That is usually enough, and adding c (which by normalization is about
 * 2**53 times smaller than a) cannot change b significantly.  However,
 * cancellation of 'a' with c in normalization of (a, c) may reduce 'a'
 * significantly relative to b.  The caller must ensure that significant
 * cancellation doesn't occur, either by having c of the same sign as 'a',
 * or by having |c| a few percent smaller than |a|.  Pre-normalization of
 * (a, b) may help.
 *
 * This is is a variant of an algorithm of Kahan (see Knuth (1981) 4.2.2
 * exercise 19).  We gain considerable efficiency by requiring the terms to
 * be sufficiently normalized and sufficiently increasing.
 */
#define	_3sumF(a, b, c) do {	\
	__typeof(a) __tmp;	\
				\
	__tmp = (c);		\
	_2sumF(__tmp, (a));	\
	(b) += (a);		\
	(a) = __tmp;		\
} while (0)

/*
 * Common routine to process the arguments to nan(), nanf(), and nanl().
 */
void _scan_nan(uint32_t *__words, int __num_words, const char *__s);

/*
 * Mix 0, 1 or 2 NaNs.  First add 0 to each arg.  This normally just turns
 * signaling NaNs into quiet NaNs by setting a quiet bit.  We do this
 * because we want to never return a signaling NaN, and also because we
 * don't want the quiet bit to affect the result.  Then mix the converted
 * args using the specified operation.
 *
 * When one arg is NaN, the result is typically that arg quieted.  When both
 * args are NaNs, the result is typically the quietening of the arg whose
 * mantissa is largest after quietening.  When neither arg is NaN, the
 * result may be NaN because it is indeterminate, or finite for subsequent
 * construction of a NaN as the indeterminate 0.0L/0.0L.
 *
 * Technical complications: the result in bits after rounding to the final
 * precision might depend on the runtime precision and/or on compiler
 * optimizations, especially when different register sets are used for
 * different precisions.  Try to make the result not depend on at least the
 * runtime precision by always doing the main mixing step in long double
 * precision.  Try to reduce dependencies on optimizations by adding the
 * the 0's in different precisions (unless everything is in long double
 * precision).
 */
#define	nan_mix(x, y)		(nan_mix_op((x), (y), +))
#define	nan_mix_op(x, y, op)	(((x) + 0.0L) op ((y) + 0))

#ifdef _COMPLEX_H

/*
 * C99 specifies that complex numbers have the same representation as
 * an array of two elements, where the first element is the real part
 * and the second element is the imaginary part.
 */
typedef union {
	float complex f;
	float a[2];
} float_complex;
typedef union {
	double complex f;
	double a[2];
} double_complex;
typedef union {
	long double complex f;
	long double a[2];
} long_double_complex;
#define	REALPART(z)	((z).a[0])
#define	IMAGPART(z)	((z).a[1])

/*
 * Inline functions that can be used to construct complex values.
 *
 * The C99 standard intends x+I*y to be used for this, but x+I*y is
 * currently unusable in general since gcc introduces many overflow,
 * underflow, sign and efficiency bugs by rewriting I*y as
 * (0.0+I)*(y+0.0*I) and laboriously computing the full complex product.
 * In particular, I*Inf is corrupted to NaN+I*Inf, and I*-0 is corrupted
 * to -0.0+I*0.0.
 *
 * The C11 standard introduced the macros CMPLX(), CMPLXF() and CMPLXL()
 * to construct complex values.  Compilers that conform to the C99
 * standard require the following functions to avoid the above issues.
 */

#ifndef CMPLXF
static __inline float complex
CMPLXF(float x, float y)
{
	float_complex z;

	REALPART(z) = x;
	IMAGPART(z) = y;
	return (z.f);
}
#endif

#ifndef CMPLX
static __inline double complex
CMPLX(double x, double y)
{
	double_complex z;

	REALPART(z) = x;
	IMAGPART(z) = y;
	return (z.f);
}
#endif

#ifndef CMPLXL
static __inline long double complex
CMPLXL(long double x, long double y)
{
	long_double_complex z;

	REALPART(z) = x;
	IMAGPART(z) = y;
	return (z.f);
}
#endif

#endif /* _COMPLEX_H */
 
#ifdef DEBUG
#if defined(__amd64__) || defined(__i386__)
#define	breakpoint()	asm("int $3")
#else
#include <signal.h>

#define	breakpoint()	raise(SIGTRAP)
#endif
#endif

/* Write a pari script to test things externally. */
#ifdef DOPRINT
#include <stdio.h>

#ifndef DOPRINT_SWIZZLE
#define	DOPRINT_SWIZZLE		0
#endif

#ifdef DOPRINT_LD80

#define	DOPRINT_START(xp) do {						\
	uint64_t __lx;							\
	uint16_t __hx;							\
									\
	/* Hack to give more-problematic args. */			\
	EXTRACT_LDBL80_WORDS(__hx, __lx, *xp);				\
	__lx ^= DOPRINT_SWIZZLE;					\
	INSERT_LDBL80_WORDS(*xp, __hx, __lx);				\
	printf("x = %.21Lg; ", (long double)*xp);			\
} while (0)
#define	DOPRINT_END1(v)							\
	printf("y = %.21Lg; z = 0; show(x, y, z);\n", (long double)(v))
#define	DOPRINT_END2(hi, lo)						\
	printf("y = %.21Lg; z = %.21Lg; show(x, y, z);\n",		\
	    (long double)(hi), (long double)(lo))

#elif defined(DOPRINT_D64)

#define	DOPRINT_START(xp) do {						\
	uint32_t __hx, __lx;						\
									\
	EXTRACT_WORDS(__hx, __lx, *xp);					\
	__lx ^= DOPRINT_SWIZZLE;					\
	INSERT_WORDS(*xp, __hx, __lx);					\
	printf("x = %.21Lg; ", (long double)*xp);			\
} while (0)
#define	DOPRINT_END1(v)							\
	printf("y = %.21Lg; z = 0; show(x, y, z);\n", (long double)(v))
#define	DOPRINT_END2(hi, lo)						\
	printf("y = %.21Lg; z = %.21Lg; show(x, y, z);\n",		\
	    (long double)(hi), (long double)(lo))

#elif defined(DOPRINT_F32)

#define	DOPRINT_START(xp) do {						\
	uint32_t __hx;							\
									\
	GET_FLOAT_WORD(__hx, *xp);					\
	__hx ^= DOPRINT_SWIZZLE;					\
	SET_FLOAT_WORD(*xp, __hx);					\
	printf("x = %.21Lg; ", (long double)*xp);			\
} while (0)
#define	DOPRINT_END1(v)							\
	printf("y = %.21Lg; z = 0; show(x, y, z);\n", (long double)(v))
#define	DOPRINT_END2(hi, lo)						\
	printf("y = %.21Lg; z = %.21Lg; show(x, y, z);\n",		\
	    (long double)(hi), (long double)(lo))

#else /* !DOPRINT_LD80 && !DOPRINT_D64 (LD128 only) */

#ifndef DOPRINT_SWIZZLE_HIGH
#define	DOPRINT_SWIZZLE_HIGH	0
#endif

#define	DOPRINT_START(xp) do {						\
	uint64_t __lx, __llx;						\
	uint16_t __hx;							\
									\
	EXTRACT_LDBL128_WORDS(__hx, __lx, __llx, *xp);			\
	__llx ^= DOPRINT_SWIZZLE;					\
	__lx ^= DOPRINT_SWIZZLE_HIGH;					\
	INSERT_LDBL128_WORDS(*xp, __hx, __lx, __llx);			\
	printf("x = %.36Lg; ", (long double)*xp);					\
} while (0)
#define	DOPRINT_END1(v)							\
	printf("y = %.36Lg; z = 0; show(x, y, z);\n", (long double)(v))
#define	DOPRINT_END2(hi, lo)						\
	printf("y = %.36Lg; z = %.36Lg; show(x, y, z);\n",		\
	    (long double)(hi), (long double)(lo))

#endif /* DOPRINT_LD80 */

#else /* !DOPRINT */
#define	DOPRINT_START(xp)
#define	DOPRINT_END1(v)
#define	DOPRINT_END2(hi, lo)
#endif /* DOPRINT */

#define	RETURNP(x) do {			\
	DOPRINT_END1(x);		\
	RETURNF(x);			\
} while (0)
#define	RETURNPI(x) do {		\
	DOPRINT_END1(x);		\
	RETURNI(x);			\
} while (0)
#define	RETURN2P(x, y) do {		\
	DOPRINT_END2((x), (y));		\
	RETURNF((x) + (y));		\
} while (0)
#define	RETURN2PI(x, y) do {		\
	DOPRINT_END2((x), (y));		\
	RETURNI((x) + (y));		\
} while (0)
#ifdef STRUCT_RETURN
#define	RETURNSP(rp) do {		\
	if (!(rp)->lo_set)		\
		RETURNP((rp)->hi);	\
	RETURN2P((rp)->hi, (rp)->lo);	\
} while (0)
#define	RETURNSPI(rp) do {		\
	if (!(rp)->lo_set)		\
		RETURNPI((rp)->hi);	\
	RETURN2PI((rp)->hi, (rp)->lo);	\
} while (0)
#endif
#define	SUM2P(x, y) ({			\
	const __typeof (x) __x = (x);	\
	const __typeof (y) __y = (y);	\
					\
	DOPRINT_END2(__x, __y);		\
	__x + __y;			\
})

/*
 * ieee style elementary functions
 *
 * We rename functions here to improve other sources' diffability
 * against fdlibm.
 */
#define	__ieee754_sqrt	sqrt
#define	__ieee754_acos	acos
#define	__ieee754_acosh	acosh
#define	__ieee754_log	log
#define	__ieee754_log2	log2
#define	__ieee754_atanh	atanh
#define	__ieee754_asin	asin
#define	__ieee754_atan2	atan2
#define	__ieee754_exp	exp
#define	__ieee754_cosh	cosh
#define	__ieee754_fmod	fmod
#define	__ieee754_pow	pow
#define	__ieee754_lgamma lgamma
#define	__ieee754_gamma	gamma
#define	__ieee754_lgamma_r lgamma_r
#define	__ieee754_gamma_r gamma_r
#define	__ieee754_log10	log10
#define	__ieee754_sinh	sinh
#define	__ieee754_hypot	hypot
#define	__ieee754_j0	j0
#define	__ieee754_j1	j1
#define	__ieee754_y0	y0
#define	__ieee754_y1	y1
#define	__ieee754_jn	jn
#define	__ieee754_yn	yn
#define	__ieee754_remainder remainder
#define	__ieee754_scalb	scalb
#define	__ieee754_sqrtf	sqrtf
#define	__ieee754_acosf	acosf
#define	__ieee754_acoshf acoshf
#define	__ieee754_logf	logf
#define	__ieee754_atanhf atanhf
#define	__ieee754_asinf	asinf
#define	__ieee754_atan2f atan2f
#define	__ieee754_expf	expf
#define	__ieee754_coshf	coshf
#define	__ieee754_fmodf	fmodf
#define	__ieee754_powf	powf
#define	__ieee754_lgammaf lgammaf
#define	__ieee754_gammaf gammaf
#define	__ieee754_lgammaf_r lgammaf_r
#define	__ieee754_gammaf_r gammaf_r
#define	__ieee754_log10f log10f
#define	__ieee754_log2f log2f
#define	__ieee754_sinhf	sinhf
#define	__ieee754_hypotf hypotf
#define	__ieee754_j0f	j0f
#define	__ieee754_j1f	j1f
#define	__ieee754_y0f	y0f
#define	__ieee754_y1f	y1f
#define	__ieee754_jnf	jnf
#define	__ieee754_ynf	ynf
#define	__ieee754_remainderf remainderf
#define	__ieee754_scalbf scalbf

#define acos fdlibm::acos
#define asin fdlibm::asin
#define atan fdlibm::atan
#define atan2 fdlibm::atan2
#define cosh fdlibm::cosh
#define sinh fdlibm::sinh
#define tanh fdlibm::tanh
#define exp fdlibm::exp
#define log fdlibm::log
#define log10 fdlibm::log10
#define pow fdlibm::pow
#define ceil fdlibm::ceil
#define ceilf fdlibm::ceilf
#define fabs fdlibm::fabs
#define floor fdlibm::floor
#define acosh fdlibm::acosh
#define asinh fdlibm::asinh
#define atanh fdlibm::atanh
#define cbrt fdlibm::cbrt
#define expm1 fdlibm::expm1
#define hypot fdlibm::hypot
#define log1p fdlibm::log1p
#define log2 fdlibm::log2
#define scalb fdlibm::scalb
#define copysign fdlibm::copysign
#define scalbn fdlibm::scalbn
#define trunc fdlibm::trunc
#define truncf fdlibm::truncf
#define floorf fdlibm::floorf
#define nearbyint fdlibm::nearbyint
#define nearbyintf fdlibm::nearbyintf
#define rint fdlibm::rint
#define rintf fdlibm::rintf

/* fdlibm kernel function */
int	__kernel_rem_pio2(double*,double*,int,int,int);

/* double precision kernel functions */
#ifndef INLINE_REM_PIO2
int	__ieee754_rem_pio2(double,double*);
#endif
double	__kernel_sin(double,double,int);
double	__kernel_cos(double,double);
double	__kernel_tan(double,double,int);
double	__ldexp_exp(double,int);
#ifdef _COMPLEX_H
double complex __ldexp_cexp(double complex,int);
#endif

/* float precision kernel functions */
#ifndef INLINE_REM_PIO2F
int	__ieee754_rem_pio2f(float,double*);
#endif
#ifndef INLINE_KERNEL_SINDF
float	__kernel_sindf(double);
#endif
#ifndef INLINE_KERNEL_COSDF
float	__kernel_cosdf(double);
#endif
#ifndef INLINE_KERNEL_TANDF
float	__kernel_tandf(double,int);
#endif
float	__ldexp_expf(float,int);
#ifdef _COMPLEX_H
float complex __ldexp_cexpf(float complex,int);
#endif

/* long double precision kernel functions */
long double __kernel_sinl(long double, long double, int);
long double __kernel_cosl(long double, long double);
long double __kernel_tanl(long double, long double, int);

#endif /* !_MATH_PRIVATE_H_ */