newlib/libm/ld/math_private.h

   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  * __double_t and __float_t are defined elsewhere in
  25  * freebsd and used to define double_t and float_t.
  26  * Newlib has already went through the process of
  27  * defining double_t and float_t so we should be able
  28  * to use them to define __double_t and __float_t for
  29  * this file.
  30  */
  31 typedef double_t          __double_t;
  32 typedef float_t           __float_t;
  33
  34 /*
  35  * Necessary to disable Protection Enabled specific
  36  * Free-bsd source.
  37  */
  38 #define NO_FPSETPREC
  39
  40 /*
  41  * The original fdlibm code used statements like:
  42  *      n0 = ((*(int*)&one)>>29)^1;             * index of high word *
  43  *      ix0 = *(n0+(int*)&x);                   * high word of x *
  44  *      ix1 = *((1-n0)+(int*)&x);               * low word of x *
  45  * to dig two 32 bit words out of the 64 bit IEEE floating point
  46  * value.  That is non-ANSI, and, moreover, the gcc instruction
  47  * scheduler gets it wrong.  We instead use the following macros.
  48  * Unlike the original code, we determine the endianness at compile
  49  * time, not at run time; I don't see much benefit to selecting
  50  * endianness at run time.
  51  */
  52
  53 /*
  54  * A union which permits us to convert between a double and two 32 bit
  55  * ints.
  56  */
  57
  58 #ifdef __arm__
  59 #if defined(__VFP_FP__) || defined(__ARM_EABI__)
  60 #define IEEE_WORD_ORDER BYTE_ORDER
  61 #else
  62 #define IEEE_WORD_ORDER BIG_ENDIAN
  63 #endif
  64 #else /* __arm__ */
  65 #define IEEE_WORD_ORDER BYTE_ORDER
  66 #endif
  67
  68 /* A union which permits us to convert between a long double and
  69    four 32 bit ints.  */
  70
  71 #if IEEE_WORD_ORDER == BIG_ENDIAN
  72
  73 typedef union
  74 {
  75   long double value;
  76   struct {
  77     u_int32_t mswhi;
  78     u_int32_t mswlo;
  79     u_int32_t lswhi;
  80     u_int32_t lswlo;
  81   } parts32;
  82   struct {
  83     u_int64_t msw;
  84     u_int64_t lsw;
  85   } parts64;
  86 } ieee_quad_shape_type;
  87
  88 #endif
  89
  90 #if IEEE_WORD_ORDER == LITTLE_ENDIAN
  91
  92 typedef union
  93 {
  94   long double value;
  95   struct {
  96     u_int32_t lswlo;
  97     u_int32_t lswhi;
  98     u_int32_t mswlo;
  99     u_int32_t mswhi;
 100   } parts32;
 101   struct {
 102     u_int64_t lsw;
 103     u_int64_t msw;
 104   } parts64;
 105 } ieee_quad_shape_type;
 106
 107 #endif
 108
 109 #if IEEE_WORD_ORDER == BIG_ENDIAN
 110
 111 typedef union
 112 {
 113   double value;
 114   struct
 115   {
 116     u_int32_t msw;
 117     u_int32_t lsw;
 118   } parts;
 119   struct
 120   {
 121     u_int64_t w;
 122   } xparts;
 123 } ieee_double_shape_type;
 124
 125 #endif
 126
 127 #if IEEE_WORD_ORDER == LITTLE_ENDIAN
 128
 129 typedef union
 130 {
 131   double value;
 132   struct
 133   {
 134     u_int32_t lsw;
 135     u_int32_t msw;
 136   } parts;
 137   struct
 138   {
 139     u_int64_t w;
 140   } xparts;
 141 } ieee_double_shape_type;
 142
 143 #endif
 144
 145 /* Get two 32 bit ints from a double.  */
 146
 147 #define EXTRACT_WORDS(ix0,ix1,d)                                \
 148 do {                                                            \
 149   ieee_double_shape_type ew_u;                                  \
 150   ew_u.value = (d);                                             \
 151   (ix0) = ew_u.parts.msw;                                       \
 152   (ix1) = ew_u.parts.lsw;                                       \
 153 } while (0)
 154
 155 /* Get a 64-bit int from a double. */
 156 #define EXTRACT_WORD64(ix,d)                                    \
 157 do {                                                            \
 158   ieee_double_shape_type ew_u;                                  \
 159   ew_u.value = (d);                                             \
 160   (ix) = ew_u.xparts.w;                                         \
 161 } while (0)
 162
 163 /* Get the more significant 32 bit int from a double.  */
 164
 165 #define GET_HIGH_WORD(i,d)                                      \
 166 do {                                                            \
 167   ieee_double_shape_type gh_u;                                  \
 168   gh_u.value = (d);                                             \
 169   (i) = gh_u.parts.msw;                                         \
 170 } while (0)
 171
 172 /* Get the less significant 32 bit int from a double.  */
 173
 174 #define GET_LOW_WORD(i,d)                                       \
 175 do {                                                            \
 176   ieee_double_shape_type gl_u;                                  \
 177   gl_u.value = (d);                                             \
 178   (i) = gl_u.parts.lsw;                                         \
 179 } while (0)
 180
 181 /* Set a double from two 32 bit ints.  */
 182
 183 #define INSERT_WORDS(d,ix0,ix1)                                 \
 184 do {                                                            \
 185   ieee_double_shape_type iw_u;                                  \
 186   iw_u.parts.msw = (ix0);                                       \
 187   iw_u.parts.lsw = (ix1);                                       \
 188   (d) = iw_u.value;                                             \
 189 } while (0)
 190
 191 /* Set a double from a 64-bit int. */
 192 #define INSERT_WORD64(d,ix)                                     \
 193 do {                                                            \
 194   ieee_double_shape_type iw_u;                                  \
 195   iw_u.xparts.w = (ix);                                         \
 196   (d) = iw_u.value;                                             \
 197 } while (0)
 198
 199 /* Set the more significant 32 bits of a double from an int.  */
 200
 201 #define SET_HIGH_WORD(d,v)                                      \
 202 do {                                                            \
 203   ieee_double_shape_type sh_u;                                  \
 204   sh_u.value = (d);                                             \
 205   sh_u.parts.msw = (v);                                         \
 206   (d) = sh_u.value;                                             \
 207 } while (0)
 208
 209 /* Set the less significant 32 bits of a double from an int.  */
 210
 211 #define SET_LOW_WORD(d,v)                                       \
 212 do {                                                            \
 213   ieee_double_shape_type sl_u;                                  \
 214   sl_u.value = (d);                                             \
 215   sl_u.parts.lsw = (v);                                         \
 216   (d) = sl_u.value;                                             \
 217 } while (0)
 218
 219 /*
 220  * A union which permits us to convert between a float and a 32 bit
 221  * int.
 222  */
 223
 224 typedef union
 225 {
 226   float value;
 227   /* FIXME: Assumes 32 bit int.  */
 228   unsigned int word;
 229 } ieee_float_shape_type;
 230
 231 /* Get a 32 bit int from a float.  */
 232
 233 #define GET_FLOAT_WORD(i,d)                                     \
 234 do {                                                            \
 235   ieee_float_shape_type gf_u;                                   \
 236   gf_u.value = (d);                                             \
 237   (i) = gf_u.word;                                              \
 238 } while (0)
 239
 240 /* Set a float from a 32 bit int.  */
 241
 242 #define SET_FLOAT_WORD(d,i)                                     \
 243 do {                                                            \
 244   ieee_float_shape_type sf_u;                                   \
 245   sf_u.word = (i);                                              \
 246   (d) = sf_u.value;                                             \
 247 } while (0)
 248
 249 /*
 250  * Get expsign and mantissa as 16 bit and 64 bit ints from an 80 bit long
 251  * double.
 252  */
 253
 254 #define EXTRACT_LDBL80_WORDS(ix0,ix1,d)                         \
 255 do {                                                            \
 256   union IEEEl2bits ew_u;                                        \
 257   ew_u.e = (d);                                                 \
 258   (ix0) = ew_u.xbits.expsign;                                   \
 259   (ix1) = ew_u.xbits.man;                                       \
 260 } while (0)
 261
 262 /*
 263  * Get expsign and mantissa as one 16 bit and two 64 bit ints from a 128 bit
 264  * long double.
 265  */
 266
 267 #define EXTRACT_LDBL128_WORDS(ix0,ix1,ix2,d)                    \
 268 do {                                                            \
 269   union IEEEl2bits ew_u;                                        \
 270   ew_u.e = (d);                                                 \
 271   (ix0) = ew_u.xbits.expsign;                                   \
 272   (ix1) = ew_u.xbits.manh;                                      \
 273   (ix2) = ew_u.xbits.manl;                                      \
 274 } while (0)
 275
 276 /* Get expsign as a 16 bit int from a long double.  */
 277
 278 #define GET_LDBL_EXPSIGN(i,d)                                   \
 279 do {                                                            \
 280   union IEEEl2bits ge_u;                                        \
 281   ge_u.e = (d);                                                 \
 282   (i) = ge_u.xbits.expsign;                                     \
 283 } while (0)
 284
 285 /*
 286  * Set an 80 bit long double from a 16 bit int expsign and a 64 bit int
 287  * mantissa.
 288  */
 289
 290 #define INSERT_LDBL80_WORDS(d,ix0,ix1)                          \
 291 do {                                                            \
 292   union IEEEl2bits iw_u;                                        \
 293   iw_u.xbits.expsign = (ix0);                                   \
 294   iw_u.xbits.man = (ix1);                                       \
 295   (d) = iw_u.e;                                                 \
 296 } while (0)
 297
 298 /*
 299  * Set a 128 bit long double from a 16 bit int expsign and two 64 bit ints
 300  * comprising the mantissa.
 301  */
 302
 303 #define INSERT_LDBL128_WORDS(d,ix0,ix1,ix2)                     \
 304 do {                                                            \
 305   union IEEEl2bits iw_u;                                        \
 306   iw_u.xbits.expsign = (ix0);                                   \
 307   iw_u.xbits.manh = (ix1);                                      \
 308   iw_u.xbits.manl = (ix2);                                      \
 309   (d) = iw_u.e;                                                 \
 310 } while (0)
 311
 312 /* Set expsign of a long double from a 16 bit int.  */
 313
 314 #define SET_LDBL_EXPSIGN(d,v)                                   \
 315 do {                                                            \
 316   union IEEEl2bits se_u;                                        \
 317   se_u.e = (d);                                                 \
 318   se_u.xbits.expsign = (v);                                     \
 319   (d) = se_u.e;                                                 \
 320 } while (0)
 321
 322 #ifdef __i386__
 323 /* Long double constants are broken on i386. */
 324 #define LD80C(m, ex, v) {                                               \
 325         .xbits.man = __CONCAT(m, ULL),                                  \
 326         .xbits.expsign = (0x3fff + (ex)) | ((v) < 0 ? 0x8000 : 0),      \
 327 }
 328 #else
 329 /* The above works on non-i386 too, but we use this to check v. */
 330 #define LD80C(m, ex, v) { .e = (v), }
 331 #endif
 332
 333 #ifdef FLT_EVAL_METHOD
 334 /*
 335  * Attempt to get strict C99 semantics for assignment with non-C99 compilers.
 336  */
 337 #if FLT_EVAL_METHOD == 0 || __GNUC__ == 0
 338 #define STRICT_ASSIGN(type, lval, rval) ((lval) = (rval))
 339 #else
 340 #define STRICT_ASSIGN(type, lval, rval) do {    \
 341         volatile type __lval;                   \
 342                                                 \
 343         if (sizeof(type) >= sizeof(long double))        \
 344                 (lval) = (rval);                \
 345         else {                                  \
 346                 __lval = (rval);                \
 347                 (lval) = __lval;                \
 348         }                                       \
 349 } while (0)
 350 #endif
 351 #endif /* FLT_EVAL_METHOD */
 352
 353 /* Support switching the mode to FP_PE if necessary. */
 354 #if defined(__i386__) && !defined(NO_FPSETPREC)
 355 #define ENTERI() ENTERIT(long double)
 356 #define ENTERIT(returntype)                     \
 357         returntype __retval;                    \
 358         fp_prec_t __oprec;                      \
 359                                                 \
 360         if ((__oprec = fpgetprec()) != FP_PE)   \
 361                 fpsetprec(FP_PE)
 362 #define RETURNI(x) do {                         \
 363         __retval = (x);                         \
 364         if (__oprec != FP_PE)                   \
 365                 fpsetprec(__oprec);             \
 366         RETURNF(__retval);                      \
 367 } while (0)
 368 #define ENTERV()                                \
 369         fp_prec_t __oprec;                      \
 370                                                 \
 371         if ((__oprec = fpgetprec()) != FP_PE)   \
 372                 fpsetprec(FP_PE)
 373 #define RETURNV() do {                          \
 374         if (__oprec != FP_PE)                   \
 375                 fpsetprec(__oprec);             \
 376         return;                 \
 377 } while (0)
 378 #else
 379 #define ENTERI()
 380 #define ENTERIT(x)
 381 #define RETURNI(x)      RETURNF(x)
 382 #define ENTERV()
 383 #define RETURNV()       return
 384 #endif
 385
 386 /* Default return statement if hack*_t() is not used. */
 387 #define      RETURNF(v)      return (v)
 388
 389 /*
 390  * 2sum gives the same result as 2sumF without requiring |a| >= |b| or
 391  * a == 0, but is slower.
 392  */
 393 #define _2sum(a, b) do {        \
 394         __typeof(a) __s, __w;   \
 395                                 \
 396         __w = (a) + (b);        \
 397         __s = __w - (a);        \
 398         (b) = ((a) - (__w - __s)) + ((b) - __s); \
 399         (a) = __w;              \
 400 } while (0)
 401
 402 /*
 403  * 2sumF algorithm.
 404  *
 405  * "Normalize" the terms in the infinite-precision expression a + b for
 406  * the sum of 2 floating point values so that b is as small as possible
 407  * relative to 'a'.  (The resulting 'a' is the value of the expression in
 408  * the same precision as 'a' and the resulting b is the rounding error.)
 409  * |a| must be >= |b| or 0, b's type must be no larger than 'a's type, and
 410  * exponent overflow or underflow must not occur.  This uses a Theorem of
 411  * Dekker (1971).  See Knuth (1981) 4.2.2 Theorem C.  The name "TwoSum"
 412  * is apparently due to Skewchuk (1997).
 413  *
 414  * For this to always work, assignment of a + b to 'a' must not retain any
 415  * extra precision in a + b.  This is required by C standards but broken
 416  * in many compilers.  The brokenness cannot be worked around using
 417  * STRICT_ASSIGN() like we do elsewhere, since the efficiency of this
 418  * algorithm would be destroyed by non-null strict assignments.  (The
 419  * compilers are correct to be broken -- the efficiency of all floating
 420  * point code calculations would be destroyed similarly if they forced the
 421  * conversions.)
 422  *
 423  * Fortunately, a case that works well can usually be arranged by building
 424  * any extra precision into the type of 'a' -- 'a' should have type float_t,
 425  * double_t or long double.  b's type should be no larger than 'a's type.
 426  * Callers should use these types with scopes as large as possible, to
 427  * reduce their own extra-precision and efficiciency problems.  In
 428  * particular, they shouldn't convert back and forth just to call here.
 429  */
 430 #ifdef DEBUG
 431 #define _2sumF(a, b) do {                               \
 432         __typeof(a) __w;                                \
 433         volatile __typeof(a) __ia, __ib, __r, __vw;     \
 434                                                         \
 435         __ia = (a);                                     \
 436         __ib = (b);                                     \
 437         assert(__ia == 0 || fabsl(__ia) >= fabsl(__ib));        \
 438                                                         \
 439         __w = (a) + (b);                                \
 440         (b) = ((a) - __w) + (b);                        \
 441         (a) = __w;                                      \
 442                                                         \
 443         /* The next 2 assertions are weak if (a) is already long double. */ \
 444         assert((long double)__ia + __ib == (long double)(a) + (b));     \
 445         __vw = __ia + __ib;                             \
 446         __r = __ia - __vw;                              \
 447         __r += __ib;                                    \
 448         assert(__vw == (a) && __r == (b));              \
 449 } while (0)
 450 #else /* !DEBUG */
 451 #define _2sumF(a, b) do {       \
 452         __typeof(a) __w;        \
 453                                 \
 454         __w = (a) + (b);        \
 455         (b) = ((a) - __w) + (b); \
 456         (a) = __w;              \
 457 } while (0)
 458 #endif /* DEBUG */
 459
 460 /*
 461  * Set x += c, where x is represented in extra precision as a + b.
 462  * x must be sufficiently normalized and sufficiently larger than c,
 463  * and the result is then sufficiently normalized.
 464  *
 465  * The details of ordering are that |a| must be >= |c| (so that (a, c)
 466  * can be normalized without extra work to swap 'a' with c).  The details of
 467  * the normalization are that b must be small relative to the normalized 'a'.
 468  * Normalization of (a, c) makes the normalized c tiny relative to the
 469  * normalized a, so b remains small relative to 'a' in the result.  However,
 470  * b need not ever be tiny relative to 'a'.  For example, b might be about
 471  * 2**20 times smaller than 'a' to give about 20 extra bits of precision.
 472  * That is usually enough, and adding c (which by normalization is about
 473  * 2**53 times smaller than a) cannot change b significantly.  However,
 474  * cancellation of 'a' with c in normalization of (a, c) may reduce 'a'
 475  * significantly relative to b.  The caller must ensure that significant
 476  * cancellation doesn't occur, either by having c of the same sign as 'a',
 477  * or by having |c| a few percent smaller than |a|.  Pre-normalization of
 478  * (a, b) may help.
 479  *
 480  * This is a variant of an algorithm of Kahan (see Knuth (1981) 4.2.2
 481  * exercise 19).  We gain considerable efficiency by requiring the terms to
 482  * be sufficiently normalized and sufficiently increasing.
 483  */
 484 #define _3sumF(a, b, c) do {    \
 485         __typeof(a) __tmp;      \
 486                                 \
 487         __tmp = (c);            \
 488         _2sumF(__tmp, (a));     \
 489         (b) += (a);             \
 490         (a) = __tmp;            \
 491 } while (0)
 492
 493 /*
 494  * Common routine to process the arguments to nan(), nanf(), and nanl().
 495  */
 496 void _scan_nan(uint32_t *__words, int __num_words, const char *__s);
 497
 498 /*
 499  * Mix 0, 1 or 2 NaNs.  First add 0 to each arg.  This normally just turns
 500  * signaling NaNs into quiet NaNs by setting a quiet bit.  We do this
 501  * because we want to never return a signaling NaN, and also because we
 502  * don't want the quiet bit to affect the result.  Then mix the converted
 503  * args using the specified operation.
 504  *
 505  * When one arg is NaN, the result is typically that arg quieted.  When both
 506  * args are NaNs, the result is typically the quietening of the arg whose
 507  * mantissa is largest after quietening.  When neither arg is NaN, the
 508  * result may be NaN because it is indeterminate, or finite for subsequent
 509  * construction of a NaN as the indeterminate 0.0L/0.0L.
 510  *
 511  * Technical complications: the result in bits after rounding to the final
 512  * precision might depend on the runtime precision and/or on compiler
 513  * optimizations, especially when different register sets are used for
 514  * different precisions.  Try to make the result not depend on at least the
 515  * runtime precision by always doing the main mixing step in long double
 516  * precision.  Try to reduce dependencies on optimizations by adding the
 517  * the 0's in different precisions (unless everything is in long double
 518  * precision).
 519  */
 520 #define nan_mix(x, y)           (nan_mix_op((x), (y), +))
 521 #define nan_mix_op(x, y, op)    (((x) + 0.0L) op ((y) + 0))
 522
 523 #ifdef _COMPLEX_H
 524
 525 /*
 526  * C99 specifies that complex numbers have the same representation as
 527  * an array of two elements, where the first element is the real part
 528  * and the second element is the imaginary part.
 529  */
 530 typedef union {
 531         float complex f;
 532         float a[2];
 533 } float_complex;
 534 typedef union {
 535         double complex f;
 536         double a[2];
 537 } double_complex;
 538 typedef union {
 539         long double complex f;
 540         long double a[2];
 541 } long_double_complex;
 542 #define REALPART(z)     ((z).a[0])
 543 #define IMAGPART(z)     ((z).a[1])
 544
 545 /*
 546  * Inline functions that can be used to construct complex values.
 547  *
 548  * The C99 standard intends x+I*y to be used for this, but x+I*y is
 549  * currently unusable in general since gcc introduces many overflow,
 550  * underflow, sign and efficiency bugs by rewriting I*y as
 551  * (0.0+I)*(y+0.0*I) and laboriously computing the full complex product.
 552  * In particular, I*Inf is corrupted to NaN+I*Inf, and I*-0 is corrupted
 553  * to -0.0+I*0.0.
 554  *
 555  * The C11 standard introduced the macros CMPLX(), CMPLXF() and CMPLXL()
 556  * to construct complex values.  Compilers that conform to the C99
 557  * standard require the following functions to avoid the above issues.
 558  */
 559
 560 #ifndef CMPLXF
 561 static __inline float complex
 562 CMPLXF(float x, float y)
 563 {
 564         float_complex z;
 565
 566         REALPART(z) = x;
 567         IMAGPART(z) = y;
 568         return (z.f);
 569 }
 570 #endif
 571
 572 #ifndef CMPLX
 573 static __inline double complex
 574 CMPLX(double x, double y)
 575 {
 576         double_complex z;
 577
 578         REALPART(z) = x;
 579         IMAGPART(z) = y;
 580         return (z.f);
 581 }
 582 #endif
 583
 584 #ifndef CMPLXL
 585 static __inline long double complex
 586 CMPLXL(long double x, long double y)
 587 {
 588         long_double_complex z;
 589
 590         REALPART(z) = x;
 591         IMAGPART(z) = y;
 592         return (z.f);
 593 }
 594 #endif
 595
 596 #endif /* _COMPLEX_H */
 597
 598 /*
 599  * The rnint() family rounds to the nearest integer for a restricted range
 600  * range of args (up to about 2**MANT_DIG).  We assume that the current
 601  * rounding mode is FE_TONEAREST so that this can be done efficiently.
 602  * Extra precision causes more problems in practice, and we only centralize
 603  * this here to reduce those problems, and have not solved the efficiency
 604  * problems.  The exp2() family uses a more delicate version of this that
 605  * requires extracting bits from the intermediate value, so it is not
 606  * centralized here and should copy any solution of the efficiency problems.
 607  */
 608
 609 static inline double
 610 rnint(__double_t x)
 611 {
 612         /*
 613          * This casts to double to kill any extra precision.  This depends
 614          * on the cast being applied to a double_t to avoid compiler bugs
 615          * (this is a cleaner version of STRICT_ASSIGN()).  This is
 616          * inefficient if there actually is extra precision, but is hard
 617          * to improve on.  We use double_t in the API to minimise conversions
 618          * for just calling here.  Note that we cannot easily change the
 619          * magic number to the one that works directly with double_t, since
 620          * the rounding precision is variable at runtime on x86 so the
 621          * magic number would need to be variable.  Assuming that the
 622          * rounding precision is always the default is too fragile.  This
 623          * and many other complications will move when the default is
 624          * changed to FP_PE.
 625          */
 626         return ((double)(x + 0x1.8p52) - 0x1.8p52);
 627 }
 628
 629 static inline float
 630 rnintf(__float_t x)
 631 {
 632         /*
 633          * As for rnint(), except we could just call that to handle the
 634          * extra precision case, usually without losing efficiency.
 635          */
 636         return ((float)(x + 0x1.8p23F) - 0x1.8p23F);
 637 }
 638
 639 #ifdef LDBL_MANT_DIG
 640 /*
 641  * The complications for extra precision are smaller for rnintl() since it
 642  * can safely assume that the rounding precision has been increased from
 643  * its default to FP_PE on x86.  We don't exploit that here to get small
 644  * optimizations from limiting the rangle to double.  We just need it for
 645  * the magic number to work with long doubles.  ld128 callers should use
 646  * rnint() instead of this if possible.  ld80 callers should prefer
 647  * rnintl() since for amd64 this avoids swapping the register set, while
 648  * for i386 it makes no difference (assuming FP_PE), and for other arches
 649  * it makes little difference.
 650  */
 651 static inline long double
 652 rnintl(long double x)
 653 {
 654         return (x + __CONCAT(0x1.8p, LDBL_MANT_DIG) / 2 -
 655             __CONCAT(0x1.8p, LDBL_MANT_DIG) / 2);
 656 }
 657 #endif /* LDBL_MANT_DIG */
 658
 659 /*
 660  * irint() and i64rint() give the same result as casting to their integer
 661  * return type provided their arg is a floating point integer.  They can
 662  * sometimes be more efficient because no rounding is required.
 663  */
 664 #if defined(amd64) || (defined(__i386__) && (!defined _SOFT_FLOAT))
 665 #define irint(x)                                                \
 666     (sizeof(x) == sizeof(float) &&                              \
 667     sizeof(__float_t) == sizeof(long double) ? irintf(x) :      \
 668     sizeof(x) == sizeof(double) &&                              \
 669     sizeof(__double_t) == sizeof(long double) ? irintd(x) :     \
 670     sizeof(x) == sizeof(long double) ? irintl(x) : (int)(x))
 671 #else
 672 #define irint(x)        ((int)(x))
 673 #endif
 674
 675 #define i64rint(x)      ((int64_t)(x))  /* only needed for ld128 so not opt. */
 676
 677 #if defined(__i386__)
 678 static __inline int
 679 irintf(float x)
 680 {
 681         int n;
 682
 683         __asm("fistl %0" : "=m" (n) : "t" (x));
 684         return (n);
 685 }
 686
 687 static __inline int
 688 irintd(double x)
 689 {
 690         int n;
 691
 692         __asm("fistl %0" : "=m" (n) : "t" (x));
 693         return (n);
 694 }
 695 #endif
 696
 697 #if defined(__amd64__) || (defined(__i386__) && (!defined _SOFT_FLOAT))
 698 static __inline int
 699 irintl(long double x)
 700 {
 701         int n;
 702
 703         __asm("fistl %0" : "=m" (n) : "t" (x));
 704         return (n);
 705 }
 706 #endif
 707
 708 /*
 709  * The following are fast floor macros for 0 <= |x| < 0x1p(N-1), where
 710  * N is the precision of the type of x. These macros are used in the
 711  * half-cycle trignometric functions (e.g., sinpi(x)).
 712  */
 713 #define FFLOORF(x, j0, ix) do {                 \
 714         (j0) = (((ix) >> 23) & 0xff) - 0x7f;    \
 715         (ix) &= ~(0x007fffff >> (j0));          \
 716         SET_FLOAT_WORD((x), (ix));              \
 717 } while (0)
 718
 719 #define FFLOOR(x, j0, ix, lx) do {                              \
 720         (j0) = (((ix) >> 20) & 0x7ff) - 0x3ff;                  \
 721         if ((j0) < 20) {                                        \
 722                 (ix) &= ~(0x000fffff >> (j0));                  \
 723                 (lx) = 0;                                       \
 724         } else {                                                \
 725                 (lx) &= ~((uint32_t)0xffffffff >> ((j0) - 20)); \
 726         }                                                       \
 727         INSERT_WORDS((x), (ix), (lx));                          \
 728 } while (0)
 729
 730 #define FFLOORL80(x, j0, ix, lx) do {                   \
 731         j0 = ix - 0x3fff + 1;                           \
 732         if ((j0) < 32) {                                \
 733                 (lx) = ((lx) >> 32) << 32;              \
 734                 (lx) &= ~((((lx) << 32)-1) >> (j0));    \
 735         } else {                                        \
 736                 uint64_t _m;                            \
 737                 _m = (uint64_t)-1 >> (j0);              \
 738                 if ((lx) & _m) (lx) &= ~_m;             \
 739         }                                               \
 740         INSERT_LDBL80_WORDS((x), (ix), (lx));           \
 741 } while (0)
 742
 743 #define FFLOORL128(x, ai, ar) do {                      \
 744         union IEEEl2bits u;                             \
 745         uint64_t m;                                     \
 746         int e;                                          \
 747         u.e = (x);                                      \
 748         e = u.bits.exp - 16383;                         \
 749         if (e < 48) {                                   \
 750                 m = ((1llu << 49) - 1) >> (e + 1);      \
 751                 u.bits.manh &= ~m;                      \
 752                 u.bits.manl = 0;                        \
 753         } else {                                        \
 754                 m = (uint64_t)-1 >> (e - 48);           \
 755                 u.bits.manl &= ~m;                      \
 756         }                                               \
 757         (ai) = u.e;                                     \
 758         (ar) = (x) - (ai);                              \
 759 } while (0)
 760
 761 #ifdef DEBUG
 762 #if defined(__amd64__) || defined(__i386__)
 763 #define breakpoint()    asm("int $3")
 764 #else
 765 #include <signal.h>
 766
 767 #define breakpoint()    raise(SIGTRAP)
 768 #endif
 769 #endif
 770
 771 /* Write a pari script to test things externally. */
 772 #ifdef DOPRINT
 773 #include <stdio.h>
 774
 775 #ifndef DOPRINT_SWIZZLE
 776 #define DOPRINT_SWIZZLE         0
 777 #endif
 778
 779 #ifdef DOPRINT_LD80
 780
 781 #define DOPRINT_START(xp) do {                                          \
 782         uint64_t __lx;                                                  \
 783         uint16_t __hx;                                                  \
 784                                                                         \
 785         /* Hack to give more-problematic args. */                       \
 786         EXTRACT_LDBL80_WORDS(__hx, __lx, *xp);                          \
 787         __lx ^= DOPRINT_SWIZZLE;                                        \
 788         INSERT_LDBL80_WORDS(*xp, __hx, __lx);                           \
 789         printf("x = %.21Lg; ", (long double)*xp);                       \
 790 } while (0)
 791 #define DOPRINT_END1(v)                                                 \
 792         printf("y = %.21Lg; z = 0; show(x, y, z);\n", (long double)(v))
 793 #define DOPRINT_END2(hi, lo)                                            \
 794         printf("y = %.21Lg; z = %.21Lg; show(x, y, z);\n",              \
 795             (long double)(hi), (long double)(lo))
 796
 797 #elif defined(DOPRINT_D64)
 798
 799 #define DOPRINT_START(xp) do {                                          \
 800         uint32_t __hx, __lx;                                            \
 801                                                                         \
 802         EXTRACT_WORDS(__hx, __lx, *xp);                                 \
 803         __lx ^= DOPRINT_SWIZZLE;                                        \
 804         INSERT_WORDS(*xp, __hx, __lx);                                  \
 805         printf("x = %.21Lg; ", (long double)*xp);                       \
 806 } while (0)
 807 #define DOPRINT_END1(v)                                                 \
 808         printf("y = %.21Lg; z = 0; show(x, y, z);\n", (long double)(v))
 809 #define DOPRINT_END2(hi, lo)                                            \
 810         printf("y = %.21Lg; z = %.21Lg; show(x, y, z);\n",              \
 811             (long double)(hi), (long double)(lo))
 812
 813 #elif defined(DOPRINT_F32)
 814
 815 #define DOPRINT_START(xp) do {                                          \
 816         uint32_t __hx;                                                  \
 817                                                                         \
 818         GET_FLOAT_WORD(__hx, *xp);                                      \
 819         __hx ^= DOPRINT_SWIZZLE;                                        \
 820         SET_FLOAT_WORD(*xp, __hx);                                      \
 821         printf("x = %.21Lg; ", (long double)*xp);                       \
 822 } while (0)
 823 #define DOPRINT_END1(v)                                                 \
 824         printf("y = %.21Lg; z = 0; show(x, y, z);\n", (long double)(v))
 825 #define DOPRINT_END2(hi, lo)                                            \
 826         printf("y = %.21Lg; z = %.21Lg; show(x, y, z);\n",              \
 827             (long double)(hi), (long double)(lo))
 828
 829 #else /* !DOPRINT_LD80 && !DOPRINT_D64 (LD128 only) */
 830
 831 #ifndef DOPRINT_SWIZZLE_HIGH
 832 #define DOPRINT_SWIZZLE_HIGH    0
 833 #endif
 834
 835 #define DOPRINT_START(xp) do {                                          \
 836         uint64_t __lx, __llx;                                           \
 837         uint16_t __hx;                                                  \
 838                                                                         \
 839         EXTRACT_LDBL128_WORDS(__hx, __lx, __llx, *xp);                  \
 840         __llx ^= DOPRINT_SWIZZLE;                                       \
 841         __lx ^= DOPRINT_SWIZZLE_HIGH;                                   \
 842         INSERT_LDBL128_WORDS(*xp, __hx, __lx, __llx);                   \
 843         printf("x = %.36Lg; ", (long double)*xp);                                       \
 844 } while (0)
 845 #define DOPRINT_END1(v)                                                 \
 846         printf("y = %.36Lg; z = 0; show(x, y, z);\n", (long double)(v))
 847 #define DOPRINT_END2(hi, lo)                                            \
 848         printf("y = %.36Lg; z = %.36Lg; show(x, y, z);\n",              \
 849             (long double)(hi), (long double)(lo))
 850
 851 #endif /* DOPRINT_LD80 */
 852
 853 #else /* !DOPRINT */
 854 #define DOPRINT_START(xp)
 855 #define DOPRINT_END1(v)
 856 #define DOPRINT_END2(hi, lo)
 857 #endif /* DOPRINT */
 858
 859 #define RETURNP(x) do {                 \
 860         DOPRINT_END1(x);                \
 861         RETURNF(x);                     \
 862 } while (0)
 863 #define RETURNPI(x) do {                \
 864         DOPRINT_END1(x);                \
 865         RETURNI(x);                     \
 866 } while (0)
 867 #define RETURN2P(x, y) do {             \
 868         DOPRINT_END2((x), (y));         \
 869         RETURNF((x) + (y));             \
 870 } while (0)
 871 #define RETURN2PI(x, y) do {            \
 872         DOPRINT_END2((x), (y));         \
 873         RETURNI((x) + (y));             \
 874 } while (0)
 875 #ifdef STRUCT_RETURN
 876 #define RETURNSP(rp) do {               \
 877         if (!(rp)->lo_set)              \
 878                 RETURNP((rp)->hi);      \
 879         RETURN2P((rp)->hi, (rp)->lo);   \
 880 } while (0)
 881 #define RETURNSPI(rp) do {              \
 882         if (!(rp)->lo_set)              \
 883                 RETURNPI((rp)->hi);     \
 884         RETURN2PI((rp)->hi, (rp)->lo);  \
 885 } while (0)
 886 #endif
 887 #define SUM2P(x, y) ({                  \
 888         const __typeof (x) __x = (x);   \
 889         const __typeof (y) __y = (y);   \
 890                                         \
 891         DOPRINT_END2(__x, __y);         \
 892         __x + __y;                      \
 893 })
 894
 895 /*
 896  * ieee style elementary functions
 897  *
 898  * We rename functions here to improve other sources' diffability
 899  * against fdlibm.
 900  */
 901 #define __ieee754_sqrt  sqrt
 902 #define __ieee754_acos  acos
 903 #define __ieee754_acosh acosh
 904 #define __ieee754_log   log
 905 #define __ieee754_log2  log2
 906 #define __ieee754_atanh atanh
 907 #define __ieee754_asin  asin
 908 #define __ieee754_atan2 atan2
 909 #define __ieee754_exp   exp
 910 #define __ieee754_cosh  cosh
 911 #define __ieee754_fmod  fmod
 912 #define __ieee754_pow   pow
 913 #define __ieee754_lgamma lgamma
 914 #define __ieee754_gamma gamma
 915 #define __ieee754_lgamma_r lgamma_r
 916 #define __ieee754_gamma_r gamma_r
 917 #define __ieee754_log10 log10
 918 #define __ieee754_sinh  sinh
 919 #define __ieee754_hypot hypot
 920 #define __ieee754_j0    j0
 921 #define __ieee754_j1    j1
 922 #define __ieee754_y0    y0
 923 #define __ieee754_y1    y1
 924 #define __ieee754_jn    jn
 925 #define __ieee754_yn    yn
 926 #define __ieee754_remainder remainder
 927 #define __ieee754_scalb scalb
 928 #define __ieee754_sqrtf sqrtf
 929 #define __ieee754_acosf acosf
 930 #define __ieee754_acoshf acoshf
 931 #define __ieee754_logf  logf
 932 #define __ieee754_atanhf atanhf
 933 #define __ieee754_asinf asinf
 934 #define __ieee754_atan2f atan2f
 935 #define __ieee754_expf  expf
 936 #define __ieee754_coshf coshf
 937 #define __ieee754_fmodf fmodf
 938 #define __ieee754_powf  powf
 939 #define __ieee754_lgammaf lgammaf
 940 #define __ieee754_gammaf gammaf
 941 #define __ieee754_lgammaf_r lgammaf_r
 942 #define __ieee754_gammaf_r gammaf_r
 943 #define __ieee754_log10f log10f
 944 #define __ieee754_log2f log2f
 945 #define __ieee754_sinhf sinhf
 946 #define __ieee754_hypotf hypotf
 947 #define __ieee754_j0f   j0f
 948 #define __ieee754_j1f   j1f
 949 #define __ieee754_y0f   y0f
 950 #define __ieee754_y1f   y1f
 951 #define __ieee754_jnf   jnf
 952 #define __ieee754_ynf   ynf
 953 #define __ieee754_remainderf remainderf
 954 #define __ieee754_scalbf scalbf
 955
 956 /* fdlibm kernel function */
 957 int     __kernel_rem_pio2(double*,double*,int,int,int);
 958
 959 /* double precision kernel functions */
 960 #ifndef INLINE_REM_PIO2
 961 int     __ieee754_rem_pio2(double,double*);
 962 #endif
 963 double  __kernel_sin(double,double,int);
 964 double  __kernel_cos(double,double);
 965 double  __kernel_tan(double,double,int);
 966 double  __ldexp_exp(double,int);
 967 #ifdef _COMPLEX_H
 968 double complex __ldexp_cexp(double complex,int);
 969 #endif
 970
 971 /* float precision kernel functions */
 972 #ifndef INLINE_REM_PIO2F
 973 int     __ieee754_rem_pio2f(float,double*);
 974 #endif
 975 #ifndef INLINE_KERNEL_SINDF
 976 float   __kernel_sindf(double);
 977 #endif
 978 #ifndef INLINE_KERNEL_COSDF
 979 float   __kernel_cosdf(double);
 980 #endif
 981 #ifndef INLINE_KERNEL_TANDF
 982 float   __kernel_tandf(double,int);
 983 #endif
 984 float   __ldexp_expf(float,int);
 985 #ifdef _COMPLEX_H
 986 float complex __ldexp_cexpf(float complex,int);
 987 #endif
 988
 989 /* long double precision kernel functions */
 990 long double __kernel_sinl(long double, long double, int);
 991 long double __kernel_cosl(long double, long double);
 992 long double __kernel_tanl(long double, long double, int);
 993
 994 #endif /* !_MATH_PRIVATE_H_ */