xref: /freebsd/lib/msun/src/math_private.h (revision 3468ddce672350a6d974b4f0fdf3f4a56eaab0a0)
1 /*
2  * ====================================================
3  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
4  *
5  * Developed at SunPro, a Sun Microsystems, Inc. business.
6  * Permission to use, copy, modify, and distribute this
7  * software is freely granted, provided that this notice
8  * is preserved.
9  * ====================================================
10  */
11 
12 /*
13  * from: @(#)fdlibm.h 5.1 93/09/24
14  * $FreeBSD$
15  */
16 
17 #ifndef _MATH_PRIVATE_H_
18 #define	_MATH_PRIVATE_H_
19 
20 #include <sys/types.h>
21 #include <machine/endian.h>
22 
23 /*
24  * The original fdlibm code used statements like:
25  *	n0 = ((*(int*)&one)>>29)^1;		* index of high word *
26  *	ix0 = *(n0+(int*)&x);			* high word of x *
27  *	ix1 = *((1-n0)+(int*)&x);		* low word of x *
28  * to dig two 32 bit words out of the 64 bit IEEE floating point
29  * value.  That is non-ANSI, and, moreover, the gcc instruction
30  * scheduler gets it wrong.  We instead use the following macros.
31  * Unlike the original code, we determine the endianness at compile
32  * time, not at run time; I don't see much benefit to selecting
33  * endianness at run time.
34  */
35 
36 /*
37  * A union which permits us to convert between a double and two 32 bit
38  * ints.
39  */
40 
41 #ifdef __arm__
42 #if defined(__VFP_FP__) || defined(__ARM_EABI__)
43 #define	IEEE_WORD_ORDER	BYTE_ORDER
44 #else
45 #define	IEEE_WORD_ORDER	BIG_ENDIAN
46 #endif
47 #else /* __arm__ */
48 #define	IEEE_WORD_ORDER	BYTE_ORDER
49 #endif
50 
51 /* A union which permits us to convert between a long double and
52    four 32 bit ints.  */
53 
54 #if IEEE_WORD_ORDER == BIG_ENDIAN
55 
56 typedef union
57 {
58   long double value;
59   struct {
60     u_int32_t mswhi;
61     u_int32_t mswlo;
62     u_int32_t lswhi;
63     u_int32_t lswlo;
64   } parts32;
65   struct {
66     u_int64_t msw;
67     u_int64_t lsw;
68   } parts64;
69 } ieee_quad_shape_type;
70 
71 #endif
72 
73 #if IEEE_WORD_ORDER == LITTLE_ENDIAN
74 
75 typedef union
76 {
77   long double value;
78   struct {
79     u_int32_t lswlo;
80     u_int32_t lswhi;
81     u_int32_t mswlo;
82     u_int32_t mswhi;
83   } parts32;
84   struct {
85     u_int64_t lsw;
86     u_int64_t msw;
87   } parts64;
88 } ieee_quad_shape_type;
89 
90 #endif
91 
92 #if IEEE_WORD_ORDER == BIG_ENDIAN
93 
94 typedef union
95 {
96   double value;
97   struct
98   {
99     u_int32_t msw;
100     u_int32_t lsw;
101   } parts;
102   struct
103   {
104     u_int64_t w;
105   } xparts;
106 } ieee_double_shape_type;
107 
108 #endif
109 
110 #if IEEE_WORD_ORDER == LITTLE_ENDIAN
111 
112 typedef union
113 {
114   double value;
115   struct
116   {
117     u_int32_t lsw;
118     u_int32_t msw;
119   } parts;
120   struct
121   {
122     u_int64_t w;
123   } xparts;
124 } ieee_double_shape_type;
125 
126 #endif
127 
128 /* Get two 32 bit ints from a double.  */
129 
130 #define EXTRACT_WORDS(ix0,ix1,d)				\
131 do {								\
132   ieee_double_shape_type ew_u;					\
133   ew_u.value = (d);						\
134   (ix0) = ew_u.parts.msw;					\
135   (ix1) = ew_u.parts.lsw;					\
136 } while (0)
137 
138 /* Get a 64-bit int from a double. */
139 #define EXTRACT_WORD64(ix,d)					\
140 do {								\
141   ieee_double_shape_type ew_u;					\
142   ew_u.value = (d);						\
143   (ix) = ew_u.xparts.w;						\
144 } while (0)
145 
146 /* Get the more significant 32 bit int from a double.  */
147 
148 #define GET_HIGH_WORD(i,d)					\
149 do {								\
150   ieee_double_shape_type gh_u;					\
151   gh_u.value = (d);						\
152   (i) = gh_u.parts.msw;						\
153 } while (0)
154 
155 /* Get the less significant 32 bit int from a double.  */
156 
157 #define GET_LOW_WORD(i,d)					\
158 do {								\
159   ieee_double_shape_type gl_u;					\
160   gl_u.value = (d);						\
161   (i) = gl_u.parts.lsw;						\
162 } while (0)
163 
164 /* Set a double from two 32 bit ints.  */
165 
166 #define INSERT_WORDS(d,ix0,ix1)					\
167 do {								\
168   ieee_double_shape_type iw_u;					\
169   iw_u.parts.msw = (ix0);					\
170   iw_u.parts.lsw = (ix1);					\
171   (d) = iw_u.value;						\
172 } while (0)
173 
174 /* Set a double from a 64-bit int. */
175 #define INSERT_WORD64(d,ix)					\
176 do {								\
177   ieee_double_shape_type iw_u;					\
178   iw_u.xparts.w = (ix);						\
179   (d) = iw_u.value;						\
180 } while (0)
181 
182 /* Set the more significant 32 bits of a double from an int.  */
183 
184 #define SET_HIGH_WORD(d,v)					\
185 do {								\
186   ieee_double_shape_type sh_u;					\
187   sh_u.value = (d);						\
188   sh_u.parts.msw = (v);						\
189   (d) = sh_u.value;						\
190 } while (0)
191 
192 /* Set the less significant 32 bits of a double from an int.  */
193 
194 #define SET_LOW_WORD(d,v)					\
195 do {								\
196   ieee_double_shape_type sl_u;					\
197   sl_u.value = (d);						\
198   sl_u.parts.lsw = (v);						\
199   (d) = sl_u.value;						\
200 } while (0)
201 
202 /*
203  * A union which permits us to convert between a float and a 32 bit
204  * int.
205  */
206 
207 typedef union
208 {
209   float value;
210   /* FIXME: Assumes 32 bit int.  */
211   unsigned int word;
212 } ieee_float_shape_type;
213 
214 /* Get a 32 bit int from a float.  */
215 
216 #define GET_FLOAT_WORD(i,d)					\
217 do {								\
218   ieee_float_shape_type gf_u;					\
219   gf_u.value = (d);						\
220   (i) = gf_u.word;						\
221 } while (0)
222 
223 /* Set a float from a 32 bit int.  */
224 
225 #define SET_FLOAT_WORD(d,i)					\
226 do {								\
227   ieee_float_shape_type sf_u;					\
228   sf_u.word = (i);						\
229   (d) = sf_u.value;						\
230 } while (0)
231 
232 /*
233  * Get expsign and mantissa as 16 bit and 64 bit ints from an 80 bit long
234  * double.
235  */
236 
237 #define	EXTRACT_LDBL80_WORDS(ix0,ix1,d)				\
238 do {								\
239   union IEEEl2bits ew_u;					\
240   ew_u.e = (d);							\
241   (ix0) = ew_u.xbits.expsign;					\
242   (ix1) = ew_u.xbits.man;					\
243 } while (0)
244 
245 /*
246  * Get expsign and mantissa as one 16 bit and two 64 bit ints from a 128 bit
247  * long double.
248  */
249 
250 #define	EXTRACT_LDBL128_WORDS(ix0,ix1,ix2,d)			\
251 do {								\
252   union IEEEl2bits ew_u;					\
253   ew_u.e = (d);							\
254   (ix0) = ew_u.xbits.expsign;					\
255   (ix1) = ew_u.xbits.manh;					\
256   (ix2) = ew_u.xbits.manl;					\
257 } while (0)
258 
259 /* Get expsign as a 16 bit int from a long double.  */
260 
261 #define	GET_LDBL_EXPSIGN(i,d)					\
262 do {								\
263   union IEEEl2bits ge_u;					\
264   ge_u.e = (d);							\
265   (i) = ge_u.xbits.expsign;					\
266 } while (0)
267 
268 /*
269  * Set an 80 bit long double from a 16 bit int expsign and a 64 bit int
270  * mantissa.
271  */
272 
273 #define	INSERT_LDBL80_WORDS(d,ix0,ix1)				\
274 do {								\
275   union IEEEl2bits iw_u;					\
276   iw_u.xbits.expsign = (ix0);					\
277   iw_u.xbits.man = (ix1);					\
278   (d) = iw_u.e;							\
279 } while (0)
280 
281 /*
282  * Set a 128 bit long double from a 16 bit int expsign and two 64 bit ints
283  * comprising the mantissa.
284  */
285 
286 #define	INSERT_LDBL128_WORDS(d,ix0,ix1,ix2)			\
287 do {								\
288   union IEEEl2bits iw_u;					\
289   iw_u.xbits.expsign = (ix0);					\
290   iw_u.xbits.manh = (ix1);					\
291   iw_u.xbits.manl = (ix2);					\
292   (d) = iw_u.e;							\
293 } while (0)
294 
295 /* Set expsign of a long double from a 16 bit int.  */
296 
297 #define	SET_LDBL_EXPSIGN(d,v)					\
298 do {								\
299   union IEEEl2bits se_u;					\
300   se_u.e = (d);							\
301   se_u.xbits.expsign = (v);					\
302   (d) = se_u.e;							\
303 } while (0)
304 
305 #ifdef __i386__
306 /* Long double constants are broken on i386. */
307 #define	LD80C(m, ex, v) {						\
308 	.xbits.man = __CONCAT(m, ULL),					\
309 	.xbits.expsign = (0x3fff + (ex)) | ((v) < 0 ? 0x8000 : 0),	\
310 }
311 #else
312 /* The above works on non-i386 too, but we use this to check v. */
313 #define	LD80C(m, ex, v)	{ .e = (v), }
314 #endif
315 
316 #ifdef FLT_EVAL_METHOD
317 /*
318  * Attempt to get strict C99 semantics for assignment with non-C99 compilers.
319  */
320 #if FLT_EVAL_METHOD == 0 || __GNUC__ == 0
321 #define	STRICT_ASSIGN(type, lval, rval)	((lval) = (rval))
322 #else
323 #define	STRICT_ASSIGN(type, lval, rval) do {	\
324 	volatile type __lval;			\
325 						\
326 	if (sizeof(type) >= sizeof(long double))	\
327 		(lval) = (rval);		\
328 	else {					\
329 		__lval = (rval);		\
330 		(lval) = __lval;		\
331 	}					\
332 } while (0)
333 #endif
334 #endif /* FLT_EVAL_METHOD */
335 
336 /* Support switching the mode to FP_PE if necessary. */
337 #if defined(__i386__) && !defined(NO_FPSETPREC)
338 #define	ENTERI() ENTERIT(long double)
339 #define	ENTERIT(returntype)			\
340 	returntype __retval;			\
341 	fp_prec_t __oprec;			\
342 						\
343 	if ((__oprec = fpgetprec()) != FP_PE)	\
344 		fpsetprec(FP_PE)
345 #define	RETURNI(x) do {				\
346 	__retval = (x);				\
347 	if (__oprec != FP_PE)			\
348 		fpsetprec(__oprec);		\
349 	RETURNF(__retval);			\
350 } while (0)
351 #define	ENTERV()				\
352 	fp_prec_t __oprec;			\
353 						\
354 	if ((__oprec = fpgetprec()) != FP_PE)	\
355 		fpsetprec(FP_PE)
356 #define	RETURNV() do {				\
357 	if (__oprec != FP_PE)			\
358 		fpsetprec(__oprec);		\
359 	return;			\
360 } while (0)
361 #else
362 #define	ENTERI()
363 #define	ENTERIT(x)
364 #define	RETURNI(x)	RETURNF(x)
365 #define	ENTERV()
366 #define	RETURNV()	return
367 #endif
368 
369 /* Default return statement if hack*_t() is not used. */
370 #define      RETURNF(v)      return (v)
371 
372 /*
373  * 2sum gives the same result as 2sumF without requiring |a| >= |b| or
374  * a == 0, but is slower.
375  */
376 #define	_2sum(a, b) do {	\
377 	__typeof(a) __s, __w;	\
378 				\
379 	__w = (a) + (b);	\
380 	__s = __w - (a);	\
381 	(b) = ((a) - (__w - __s)) + ((b) - __s); \
382 	(a) = __w;		\
383 } while (0)
384 
385 /*
386  * 2sumF algorithm.
387  *
388  * "Normalize" the terms in the infinite-precision expression a + b for
389  * the sum of 2 floating point values so that b is as small as possible
390  * relative to 'a'.  (The resulting 'a' is the value of the expression in
391  * the same precision as 'a' and the resulting b is the rounding error.)
392  * |a| must be >= |b| or 0, b's type must be no larger than 'a's type, and
393  * exponent overflow or underflow must not occur.  This uses a Theorem of
394  * Dekker (1971).  See Knuth (1981) 4.2.2 Theorem C.  The name "TwoSum"
395  * is apparently due to Skewchuk (1997).
396  *
397  * For this to always work, assignment of a + b to 'a' must not retain any
398  * extra precision in a + b.  This is required by C standards but broken
399  * in many compilers.  The brokenness cannot be worked around using
400  * STRICT_ASSIGN() like we do elsewhere, since the efficiency of this
401  * algorithm would be destroyed by non-null strict assignments.  (The
402  * compilers are correct to be broken -- the efficiency of all floating
403  * point code calculations would be destroyed similarly if they forced the
404  * conversions.)
405  *
406  * Fortunately, a case that works well can usually be arranged by building
407  * any extra precision into the type of 'a' -- 'a' should have type float_t,
408  * double_t or long double.  b's type should be no larger than 'a's type.
409  * Callers should use these types with scopes as large as possible, to
410  * reduce their own extra-precision and efficiciency problems.  In
411  * particular, they shouldn't convert back and forth just to call here.
412  */
413 #ifdef DEBUG
414 #define	_2sumF(a, b) do {				\
415 	__typeof(a) __w;				\
416 	volatile __typeof(a) __ia, __ib, __r, __vw;	\
417 							\
418 	__ia = (a);					\
419 	__ib = (b);					\
420 	assert(__ia == 0 || fabsl(__ia) >= fabsl(__ib));	\
421 							\
422 	__w = (a) + (b);				\
423 	(b) = ((a) - __w) + (b);			\
424 	(a) = __w;					\
425 							\
426 	/* The next 2 assertions are weak if (a) is already long double. */ \
427 	assert((long double)__ia + __ib == (long double)(a) + (b));	\
428 	__vw = __ia + __ib;				\
429 	__r = __ia - __vw;				\
430 	__r += __ib;					\
431 	assert(__vw == (a) && __r == (b));		\
432 } while (0)
433 #else /* !DEBUG */
434 #define	_2sumF(a, b) do {	\
435 	__typeof(a) __w;	\
436 				\
437 	__w = (a) + (b);	\
438 	(b) = ((a) - __w) + (b); \
439 	(a) = __w;		\
440 } while (0)
441 #endif /* DEBUG */
442 
443 /*
444  * Set x += c, where x is represented in extra precision as a + b.
445  * x must be sufficiently normalized and sufficiently larger than c,
446  * and the result is then sufficiently normalized.
447  *
448  * The details of ordering are that |a| must be >= |c| (so that (a, c)
449  * can be normalized without extra work to swap 'a' with c).  The details of
450  * the normalization are that b must be small relative to the normalized 'a'.
451  * Normalization of (a, c) makes the normalized c tiny relative to the
452  * normalized a, so b remains small relative to 'a' in the result.  However,
453  * b need not ever be tiny relative to 'a'.  For example, b might be about
454  * 2**20 times smaller than 'a' to give about 20 extra bits of precision.
455  * That is usually enough, and adding c (which by normalization is about
456  * 2**53 times smaller than a) cannot change b significantly.  However,
457  * cancellation of 'a' with c in normalization of (a, c) may reduce 'a'
458  * significantly relative to b.  The caller must ensure that significant
459  * cancellation doesn't occur, either by having c of the same sign as 'a',
460  * or by having |c| a few percent smaller than |a|.  Pre-normalization of
461  * (a, b) may help.
462  *
463  * This is is a variant of an algorithm of Kahan (see Knuth (1981) 4.2.2
464  * exercise 19).  We gain considerable efficiency by requiring the terms to
465  * be sufficiently normalized and sufficiently increasing.
466  */
467 #define	_3sumF(a, b, c) do {	\
468 	__typeof(a) __tmp;	\
469 				\
470 	__tmp = (c);		\
471 	_2sumF(__tmp, (a));	\
472 	(b) += (a);		\
473 	(a) = __tmp;		\
474 } while (0)
475 
476 /*
477  * Common routine to process the arguments to nan(), nanf(), and nanl().
478  */
479 void _scan_nan(uint32_t *__words, int __num_words, const char *__s);
480 
481 /*
482  * Mix 1 or 2 NaNs.  First add 0 to each arg.  This normally just turns
483  * signaling NaNs into quiet NaNs by setting a quiet bit.  We do this
484  * because we want to never return a signaling NaN, and also because we
485  * don't want the quiet bit to affect the result.  Then mix the converted
486  * args using addition.  The result is typically the arg whose mantissa
487  * bits (considered as in integer) are largest.
488  *
489  * Technical complications: the result in bits might depend on the precision
490  * and/or on compiler optimizations, especially when different register sets
491  * are used for different precisions.  Try to make the result not depend on
492  * at least the precision by always doing the main mixing step in long double
493  * precision.  Try to reduce dependencies on optimizations by adding the
494  * the 0's in different precisions (unless everything is in long double
495  * precision).
496  */
497 #define	nan_mix(x, y)	(((x) + 0.0L) + ((y) + 0))
498 
499 #ifdef _COMPLEX_H
500 
501 /*
502  * C99 specifies that complex numbers have the same representation as
503  * an array of two elements, where the first element is the real part
504  * and the second element is the imaginary part.
505  */
506 typedef union {
507 	float complex f;
508 	float a[2];
509 } float_complex;
510 typedef union {
511 	double complex f;
512 	double a[2];
513 } double_complex;
514 typedef union {
515 	long double complex f;
516 	long double a[2];
517 } long_double_complex;
518 #define	REALPART(z)	((z).a[0])
519 #define	IMAGPART(z)	((z).a[1])
520 
521 /*
522  * Inline functions that can be used to construct complex values.
523  *
524  * The C99 standard intends x+I*y to be used for this, but x+I*y is
525  * currently unusable in general since gcc introduces many overflow,
526  * underflow, sign and efficiency bugs by rewriting I*y as
527  * (0.0+I)*(y+0.0*I) and laboriously computing the full complex product.
528  * In particular, I*Inf is corrupted to NaN+I*Inf, and I*-0 is corrupted
529  * to -0.0+I*0.0.
530  *
531  * The C11 standard introduced the macros CMPLX(), CMPLXF() and CMPLXL()
532  * to construct complex values.  Compilers that conform to the C99
533  * standard require the following functions to avoid the above issues.
534  */
535 
536 #ifndef CMPLXF
537 static __inline float complex
538 CMPLXF(float x, float y)
539 {
540 	float_complex z;
541 
542 	REALPART(z) = x;
543 	IMAGPART(z) = y;
544 	return (z.f);
545 }
546 #endif
547 
548 #ifndef CMPLX
549 static __inline double complex
550 CMPLX(double x, double y)
551 {
552 	double_complex z;
553 
554 	REALPART(z) = x;
555 	IMAGPART(z) = y;
556 	return (z.f);
557 }
558 #endif
559 
560 #ifndef CMPLXL
561 static __inline long double complex
562 CMPLXL(long double x, long double y)
563 {
564 	long_double_complex z;
565 
566 	REALPART(z) = x;
567 	IMAGPART(z) = y;
568 	return (z.f);
569 }
570 #endif
571 
572 #endif /* _COMPLEX_H */
573 
574 #ifdef __GNUCLIKE_ASM
575 
576 /* Asm versions of some functions. */
577 
578 #ifdef __amd64__
579 static __inline int
580 irint(double x)
581 {
582 	int n;
583 
584 	asm("cvtsd2si %1,%0" : "=r" (n) : "x" (x));
585 	return (n);
586 }
587 #define	HAVE_EFFICIENT_IRINT
588 #endif
589 
590 #ifdef __i386__
591 static __inline int
592 irint(double x)
593 {
594 	int n;
595 
596 	asm("fistl %0" : "=m" (n) : "t" (x));
597 	return (n);
598 }
599 #define	HAVE_EFFICIENT_IRINT
600 #endif
601 
602 #if defined(__amd64__) || defined(__i386__)
603 static __inline int
604 irintl(long double x)
605 {
606 	int n;
607 
608 	asm("fistl %0" : "=m" (n) : "t" (x));
609 	return (n);
610 }
611 #define	HAVE_EFFICIENT_IRINTL
612 #endif
613 
614 #endif /* __GNUCLIKE_ASM */
615 
616 #ifdef DEBUG
617 #if defined(__amd64__) || defined(__i386__)
618 #define	breakpoint()	asm("int $3")
619 #else
620 #include <signal.h>
621 
622 #define	breakpoint()	raise(SIGTRAP)
623 #endif
624 #endif
625 
626 /* Write a pari script to test things externally. */
627 #ifdef DOPRINT
628 #include <stdio.h>
629 
630 #ifndef DOPRINT_SWIZZLE
631 #define	DOPRINT_SWIZZLE		0
632 #endif
633 
634 #ifdef DOPRINT_LD80
635 
636 #define	DOPRINT_START(xp) do {						\
637 	uint64_t __lx;							\
638 	uint16_t __hx;							\
639 									\
640 	/* Hack to give more-problematic args. */			\
641 	EXTRACT_LDBL80_WORDS(__hx, __lx, *xp);				\
642 	__lx ^= DOPRINT_SWIZZLE;					\
643 	INSERT_LDBL80_WORDS(*xp, __hx, __lx);				\
644 	printf("x = %.21Lg; ", (long double)*xp);			\
645 } while (0)
646 #define	DOPRINT_END1(v)							\
647 	printf("y = %.21Lg; z = 0; show(x, y, z);\n", (long double)(v))
648 #define	DOPRINT_END2(hi, lo)						\
649 	printf("y = %.21Lg; z = %.21Lg; show(x, y, z);\n",		\
650 	    (long double)(hi), (long double)(lo))
651 
652 #elif defined(DOPRINT_D64)
653 
654 #define	DOPRINT_START(xp) do {						\
655 	uint32_t __hx, __lx;						\
656 									\
657 	EXTRACT_WORDS(__hx, __lx, *xp);					\
658 	__lx ^= DOPRINT_SWIZZLE;					\
659 	INSERT_WORDS(*xp, __hx, __lx);					\
660 	printf("x = %.21Lg; ", (long double)*xp);			\
661 } while (0)
662 #define	DOPRINT_END1(v)							\
663 	printf("y = %.21Lg; z = 0; show(x, y, z);\n", (long double)(v))
664 #define	DOPRINT_END2(hi, lo)						\
665 	printf("y = %.21Lg; z = %.21Lg; show(x, y, z);\n",		\
666 	    (long double)(hi), (long double)(lo))
667 
668 #elif defined(DOPRINT_F32)
669 
670 #define	DOPRINT_START(xp) do {						\
671 	uint32_t __hx;							\
672 									\
673 	GET_FLOAT_WORD(__hx, *xp);					\
674 	__hx ^= DOPRINT_SWIZZLE;					\
675 	SET_FLOAT_WORD(*xp, __hx);					\
676 	printf("x = %.21Lg; ", (long double)*xp);			\
677 } while (0)
678 #define	DOPRINT_END1(v)							\
679 	printf("y = %.21Lg; z = 0; show(x, y, z);\n", (long double)(v))
680 #define	DOPRINT_END2(hi, lo)						\
681 	printf("y = %.21Lg; z = %.21Lg; show(x, y, z);\n",		\
682 	    (long double)(hi), (long double)(lo))
683 
684 #else /* !DOPRINT_LD80 && !DOPRINT_D64 (LD128 only) */
685 
686 #ifndef DOPRINT_SWIZZLE_HIGH
687 #define	DOPRINT_SWIZZLE_HIGH	0
688 #endif
689 
690 #define	DOPRINT_START(xp) do {						\
691 	uint64_t __lx, __llx;						\
692 	uint16_t __hx;							\
693 									\
694 	EXTRACT_LDBL128_WORDS(__hx, __lx, __llx, *xp);			\
695 	__llx ^= DOPRINT_SWIZZLE;					\
696 	__lx ^= DOPRINT_SWIZZLE_HIGH;					\
697 	INSERT_LDBL128_WORDS(*xp, __hx, __lx, __llx);			\
698 	printf("x = %.36Lg; ", (long double)*xp);					\
699 } while (0)
700 #define	DOPRINT_END1(v)							\
701 	printf("y = %.36Lg; z = 0; show(x, y, z);\n", (long double)(v))
702 #define	DOPRINT_END2(hi, lo)						\
703 	printf("y = %.36Lg; z = %.36Lg; show(x, y, z);\n",		\
704 	    (long double)(hi), (long double)(lo))
705 
706 #endif /* DOPRINT_LD80 */
707 
708 #else /* !DOPRINT */
709 #define	DOPRINT_START(xp)
710 #define	DOPRINT_END1(v)
711 #define	DOPRINT_END2(hi, lo)
712 #endif /* DOPRINT */
713 
714 #define	RETURNP(x) do {			\
715 	DOPRINT_END1(x);		\
716 	RETURNF(x);			\
717 } while (0)
718 #define	RETURNPI(x) do {		\
719 	DOPRINT_END1(x);		\
720 	RETURNI(x);			\
721 } while (0)
722 #define	RETURN2P(x, y) do {		\
723 	DOPRINT_END2((x), (y));		\
724 	RETURNF((x) + (y));		\
725 } while (0)
726 #define	RETURN2PI(x, y) do {		\
727 	DOPRINT_END2((x), (y));		\
728 	RETURNI((x) + (y));		\
729 } while (0)
730 #ifdef STRUCT_RETURN
731 #define	RETURNSP(rp) do {		\
732 	if (!(rp)->lo_set)		\
733 		RETURNP((rp)->hi);	\
734 	RETURN2P((rp)->hi, (rp)->lo);	\
735 } while (0)
736 #define	RETURNSPI(rp) do {		\
737 	if (!(rp)->lo_set)		\
738 		RETURNPI((rp)->hi);	\
739 	RETURN2PI((rp)->hi, (rp)->lo);	\
740 } while (0)
741 #endif
742 #define	SUM2P(x, y) ({			\
743 	const __typeof (x) __x = (x);	\
744 	const __typeof (y) __y = (y);	\
745 					\
746 	DOPRINT_END2(__x, __y);		\
747 	__x + __y;			\
748 })
749 
750 /*
751  * ieee style elementary functions
752  *
753  * We rename functions here to improve other sources' diffability
754  * against fdlibm.
755  */
756 #define	__ieee754_sqrt	sqrt
757 #define	__ieee754_acos	acos
758 #define	__ieee754_acosh	acosh
759 #define	__ieee754_log	log
760 #define	__ieee754_log2	log2
761 #define	__ieee754_atanh	atanh
762 #define	__ieee754_asin	asin
763 #define	__ieee754_atan2	atan2
764 #define	__ieee754_exp	exp
765 #define	__ieee754_cosh	cosh
766 #define	__ieee754_fmod	fmod
767 #define	__ieee754_pow	pow
768 #define	__ieee754_lgamma lgamma
769 #define	__ieee754_gamma	gamma
770 #define	__ieee754_lgamma_r lgamma_r
771 #define	__ieee754_gamma_r gamma_r
772 #define	__ieee754_log10	log10
773 #define	__ieee754_sinh	sinh
774 #define	__ieee754_hypot	hypot
775 #define	__ieee754_j0	j0
776 #define	__ieee754_j1	j1
777 #define	__ieee754_y0	y0
778 #define	__ieee754_y1	y1
779 #define	__ieee754_jn	jn
780 #define	__ieee754_yn	yn
781 #define	__ieee754_remainder remainder
782 #define	__ieee754_scalb	scalb
783 #define	__ieee754_sqrtf	sqrtf
784 #define	__ieee754_acosf	acosf
785 #define	__ieee754_acoshf acoshf
786 #define	__ieee754_logf	logf
787 #define	__ieee754_atanhf atanhf
788 #define	__ieee754_asinf	asinf
789 #define	__ieee754_atan2f atan2f
790 #define	__ieee754_expf	expf
791 #define	__ieee754_coshf	coshf
792 #define	__ieee754_fmodf	fmodf
793 #define	__ieee754_powf	powf
794 #define	__ieee754_lgammaf lgammaf
795 #define	__ieee754_gammaf gammaf
796 #define	__ieee754_lgammaf_r lgammaf_r
797 #define	__ieee754_gammaf_r gammaf_r
798 #define	__ieee754_log10f log10f
799 #define	__ieee754_log2f log2f
800 #define	__ieee754_sinhf	sinhf
801 #define	__ieee754_hypotf hypotf
802 #define	__ieee754_j0f	j0f
803 #define	__ieee754_j1f	j1f
804 #define	__ieee754_y0f	y0f
805 #define	__ieee754_y1f	y1f
806 #define	__ieee754_jnf	jnf
807 #define	__ieee754_ynf	ynf
808 #define	__ieee754_remainderf remainderf
809 #define	__ieee754_scalbf scalbf
810 
811 /* fdlibm kernel function */
812 int	__kernel_rem_pio2(double*,double*,int,int,int);
813 
814 /* double precision kernel functions */
815 #ifndef INLINE_REM_PIO2
816 int	__ieee754_rem_pio2(double,double*);
817 #endif
818 double	__kernel_sin(double,double,int);
819 double	__kernel_cos(double,double);
820 double	__kernel_tan(double,double,int);
821 double	__ldexp_exp(double,int);
822 #ifdef _COMPLEX_H
823 double complex __ldexp_cexp(double complex,int);
824 #endif
825 
826 /* float precision kernel functions */
827 #ifndef INLINE_REM_PIO2F
828 int	__ieee754_rem_pio2f(float,double*);
829 #endif
830 #ifndef INLINE_KERNEL_SINDF
831 float	__kernel_sindf(double);
832 #endif
833 #ifndef INLINE_KERNEL_COSDF
834 float	__kernel_cosdf(double);
835 #endif
836 #ifndef INLINE_KERNEL_TANDF
837 float	__kernel_tandf(double,int);
838 #endif
839 float	__ldexp_expf(float,int);
840 #ifdef _COMPLEX_H
841 float complex __ldexp_cexpf(float complex,int);
842 #endif
843 
844 /* long double precision kernel functions */
845 long double __kernel_sinl(long double, long double, int);
846 long double __kernel_cosl(long double, long double);
847 long double __kernel_tanl(long double, long double, int);
848 
849 long double __p1evll(long double, void *, int);
850 long double __polevll(long double, void *, int);
851 
852 #endif /* !_MATH_PRIVATE_H_ */
853