modules/fdlibm/src/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 <cfloat>
  21 #include <stdint.h>
  22 #include <sys/types.h>
  23
  24 #include "mozilla/EndianUtils.h"
  25
  26 #include "fdlibm.h"
  27
  28 /*
  29  * Emulate FreeBSD internal double types.
  30  * Adapted from https://github.com/freebsd/freebsd-src/search?q=__double_t
  31  */
  32
  33 typedef double      __double_t;
  34 typedef __double_t  double_t;
  35 typedef float       __float_t;
  36
  37 /*
  38  * The original fdlibm code used statements like:
  39  *      n0 = ((*(int*)&one)>>29)^1;             * index of high word *
  40  *      ix0 = *(n0+(int*)&x);                   * high word of x *
  41  *      ix1 = *((1-n0)+(int*)&x);               * low word of x *
  42  * to dig two 32 bit words out of the 64 bit IEEE floating point
  43  * value.  That is non-ANSI, and, moreover, the gcc instruction
  44  * scheduler gets it wrong.  We instead use the following macros.
  45  * Unlike the original code, we determine the endianness at compile
  46  * time, not at run time; I don't see much benefit to selecting
  47  * endianness at run time.
  48  */
  49
  50 #ifndef u_int32_t
  51 #define u_int32_t uint32_t
  52 #endif
  53 #ifndef u_int64_t
  54 #define u_int64_t uint64_t
  55 #endif
  56
  57 /* A union which permits us to convert between a long double and
  58    four 32 bit ints.  */
  59
  60 #if MOZ_BIG_ENDIAN()
  61
  62 typedef union
  63 {
  64   long double value;
  65   struct {
  66     u_int32_t mswhi;
  67     u_int32_t mswlo;
  68     u_int32_t lswhi;
  69     u_int32_t lswlo;
  70   } parts32;
  71   struct {
  72     u_int64_t msw;
  73     u_int64_t lsw;
  74   } parts64;
  75 } ieee_quad_shape_type;
  76
  77 #endif
  78
  79 #if MOZ_LITTLE_ENDIAN()
  80
  81 typedef union
  82 {
  83   long double value;
  84   struct {
  85     u_int32_t lswlo;
  86     u_int32_t lswhi;
  87     u_int32_t mswlo;
  88     u_int32_t mswhi;
  89   } parts32;
  90   struct {
  91     u_int64_t lsw;
  92     u_int64_t msw;
  93   } parts64;
  94 } ieee_quad_shape_type;
  95
  96 #endif
  97
  98 #if MOZ_BIG_ENDIAN()
  99
 100 typedef union
 101 {
 102   double value;
 103   struct
 104   {
 105     u_int32_t msw;
 106     u_int32_t lsw;
 107   } parts;
 108   struct
 109   {
 110     u_int64_t w;
 111   } xparts;
 112 } ieee_double_shape_type;
 113
 114 #endif
 115
 116 #if MOZ_LITTLE_ENDIAN()
 117
 118 typedef union
 119 {
 120   double value;
 121   struct
 122   {
 123     u_int32_t lsw;
 124     u_int32_t msw;
 125   } parts;
 126   struct
 127   {
 128     u_int64_t w;
 129   } xparts;
 130 } ieee_double_shape_type;
 131
 132 #endif
 133
 134 /* Get two 32 bit ints from a double.  */
 135
 136 #define EXTRACT_WORDS(ix0,ix1,d)                                \
 137 do {                                                            \
 138   ieee_double_shape_type ew_u;                                  \
 139   ew_u.value = (d);                                             \
 140   (ix0) = ew_u.parts.msw;                                       \
 141   (ix1) = ew_u.parts.lsw;                                       \
 142 } while (0)
 143
 144 /* Get a 64-bit int from a double. */
 145 #define EXTRACT_WORD64(ix,d)                                    \
 146 do {                                                            \
 147   ieee_double_shape_type ew_u;                                  \
 148   ew_u.value = (d);                                             \
 149   (ix) = ew_u.xparts.w;                                         \
 150 } while (0)
 151
 152 /* Get the more significant 32 bit int from a double.  */
 153
 154 #define GET_HIGH_WORD(i,d)                                      \
 155 do {                                                            \
 156   ieee_double_shape_type gh_u;                                  \
 157   gh_u.value = (d);                                             \
 158   (i) = gh_u.parts.msw;                                         \
 159 } while (0)
 160
 161 /* Get the less significant 32 bit int from a double.  */
 162
 163 #define GET_LOW_WORD(i,d)                                       \
 164 do {                                                            \
 165   ieee_double_shape_type gl_u;                                  \
 166   gl_u.value = (d);                                             \
 167   (i) = gl_u.parts.lsw;                                         \
 168 } while (0)
 169
 170 /* Set a double from two 32 bit ints.  */
 171
 172 #define INSERT_WORDS(d,ix0,ix1)                                 \
 173 do {                                                            \
 174   ieee_double_shape_type iw_u;                                  \
 175   iw_u.parts.msw = (ix0);                                       \
 176   iw_u.parts.lsw = (ix1);                                       \
 177   (d) = iw_u.value;                                             \
 178 } while (0)
 179
 180 /* Set a double from a 64-bit int. */
 181 #define INSERT_WORD64(d,ix)                                     \
 182 do {                                                            \
 183   ieee_double_shape_type iw_u;                                  \
 184   iw_u.xparts.w = (ix);                                         \
 185   (d) = iw_u.value;                                             \
 186 } while (0)
 187
 188 /* Set the more significant 32 bits of a double from an int.  */
 189
 190 #define SET_HIGH_WORD(d,v)                                      \
 191 do {                                                            \
 192   ieee_double_shape_type sh_u;                                  \
 193   sh_u.value = (d);                                             \
 194   sh_u.parts.msw = (v);                                         \
 195   (d) = sh_u.value;                                             \
 196 } while (0)
 197
 198 /* Set the less significant 32 bits of a double from an int.  */
 199
 200 #define SET_LOW_WORD(d,v)                                       \
 201 do {                                                            \
 202   ieee_double_shape_type sl_u;                                  \
 203   sl_u.value = (d);                                             \
 204   sl_u.parts.lsw = (v);                                         \
 205   (d) = sl_u.value;                                             \
 206 } while (0)
 207
 208 /*
 209  * A union which permits us to convert between a float and a 32 bit
 210  * int.
 211  */
 212
 213 typedef union
 214 {
 215   float value;
 216   /* FIXME: Assumes 32 bit int.  */
 217   unsigned int word;
 218 } ieee_float_shape_type;
 219
 220 /* Get a 32 bit int from a float.  */
 221
 222 #define GET_FLOAT_WORD(i,d)                                     \
 223 do {                                                            \
 224   ieee_float_shape_type gf_u;                                   \
 225   gf_u.value = (d);                                             \
 226   (i) = gf_u.word;                                              \
 227 } while (0)
 228
 229 /* Set a float from a 32 bit int.  */
 230
 231 #define SET_FLOAT_WORD(d,i)                                     \
 232 do {                                                            \
 233   ieee_float_shape_type sf_u;                                   \
 234   sf_u.word = (i);                                              \
 235   (d) = sf_u.value;                                             \
 236 } while (0)
 237
 238 /*
 239  * Get expsign and mantissa as 16 bit and 64 bit ints from an 80 bit long
 240  * double.
 241  */
 242
 243 #define EXTRACT_LDBL80_WORDS(ix0,ix1,d)                         \
 244 do {                                                            \
 245   union IEEEl2bits ew_u;                                        \
 246   ew_u.e = (d);                                                 \
 247   (ix0) = ew_u.xbits.expsign;                                   \
 248   (ix1) = ew_u.xbits.man;                                       \
 249 } while (0)
 250
 251 /*
 252  * Get expsign and mantissa as one 16 bit and two 64 bit ints from a 128 bit
 253  * long double.
 254  */
 255
 256 #define EXTRACT_LDBL128_WORDS(ix0,ix1,ix2,d)                    \
 257 do {                                                            \
 258   union IEEEl2bits ew_u;                                        \
 259   ew_u.e = (d);                                                 \
 260   (ix0) = ew_u.xbits.expsign;                                   \
 261   (ix1) = ew_u.xbits.manh;                                      \
 262   (ix2) = ew_u.xbits.manl;                                      \
 263 } while (0)
 264
 265 /* Get expsign as a 16 bit int from a long double.  */
 266
 267 #define GET_LDBL_EXPSIGN(i,d)                                   \
 268 do {                                                            \
 269   union IEEEl2bits ge_u;                                        \
 270   ge_u.e = (d);                                                 \
 271   (i) = ge_u.xbits.expsign;                                     \
 272 } while (0)
 273
 274 /*
 275  * Set an 80 bit long double from a 16 bit int expsign and a 64 bit int
 276  * mantissa.
 277  */
 278
 279 #define INSERT_LDBL80_WORDS(d,ix0,ix1)                          \
 280 do {                                                            \
 281   union IEEEl2bits iw_u;                                        \
 282   iw_u.xbits.expsign = (ix0);                                   \
 283   iw_u.xbits.man = (ix1);                                       \
 284   (d) = iw_u.e;                                                 \
 285 } while (0)
 286
 287 /*
 288  * Set a 128 bit long double from a 16 bit int expsign and two 64 bit ints
 289  * comprising the mantissa.
 290  */
 291
 292 #define INSERT_LDBL128_WORDS(d,ix0,ix1,ix2)                     \
 293 do {                                                            \
 294   union IEEEl2bits iw_u;                                        \
 295   iw_u.xbits.expsign = (ix0);                                   \
 296   iw_u.xbits.manh = (ix1);                                      \
 297   iw_u.xbits.manl = (ix2);                                      \
 298   (d) = iw_u.e;                                                 \
 299 } while (0)
 300
 301 /* Set expsign of a long double from a 16 bit int.  */
 302
 303 #define SET_LDBL_EXPSIGN(d,v)                                   \
 304 do {                                                            \
 305   union IEEEl2bits se_u;                                        \
 306   se_u.e = (d);                                                 \
 307   se_u.xbits.expsign = (v);                                     \
 308   (d) = se_u.e;                                                 \
 309 } while (0)
 310
 311 #ifdef __i386__
 312 /* Long double constants are broken on i386. */
 313 #define LD80C(m, ex, v) {                                               \
 314         .xbits.man = __CONCAT(m, ULL),                                  \
 315         .xbits.expsign = (0x3fff + (ex)) | ((v) < 0 ? 0x8000 : 0),      \
 316 }
 317 #else
 318 /* The above works on non-i386 too, but we use this to check v. */
 319 #define LD80C(m, ex, v) { .e = (v), }
 320 #endif
 321
 322 #ifdef FLT_EVAL_METHOD
 323 /*
 324  * Attempt to get strict C99 semantics for assignment with non-C99 compilers.
 325  */
 326 #if !defined(_MSC_VER) && (FLT_EVAL_METHOD == 0 || __GNUC__ == 0)
 327 #define STRICT_ASSIGN(type, lval, rval) ((lval) = (rval))
 328 #else
 329 #define STRICT_ASSIGN(type, lval, rval) do {    \
 330         volatile type __lval;                   \
 331                                                 \
 332         if (sizeof(type) >= sizeof(long double))        \
 333                 (lval) = (rval);                \
 334         else {                                  \
 335                 __lval = (rval);                \
 336                 (lval) = __lval;                \
 337         }                                       \
 338 } while (0)
 339 #endif
 340 #else
 341 #define STRICT_ASSIGN(type, lval, rval) do {    \
 342         volatile type __lval;                   \
 343                                                 \
 344         if (sizeof(type) >= sizeof(long double))        \
 345                 (lval) = (rval);                \
 346         else {                                  \
 347                 __lval = (rval);                \
 348                 (lval) = __lval;                \
 349         }                                       \
 350 } while (0)
 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 /*
 630  * irint() and i64rint() give the same result as casting to their integer
 631  * return type provided their arg is a floating point integer.  They can
 632  * sometimes be more efficient because no rounding is required.
 633  */
 634 #if defined(amd64) || defined(__i386__)
 635 #define irint(x)                                                \
 636     (sizeof(x) == sizeof(float) &&                              \
 637     sizeof(__float_t) == sizeof(long double) ? irintf(x) :      \
 638     sizeof(x) == sizeof(double) &&                              \
 639     sizeof(__double_t) == sizeof(long double) ? irintd(x) :     \
 640     sizeof(x) == sizeof(long double) ? irintl(x) : (int)(x))
 641 #else
 642 #define irint(x)        ((int)(x))
 643 #endif
 644
 645 #define i64rint(x)      ((int64_t)(x))  /* only needed for ld128 so not opt. */
 646
 647 #if defined(__i386__)
 648 static __inline int
 649 irintf(float x)
 650 {
 651         int n;
 652
 653         __asm("fistl %0" : "=m" (n) : "t" (x));
 654         return (n);
 655 }
 656
 657 static __inline int
 658 irintd(double x)
 659 {
 660         int n;
 661
 662         __asm("fistl %0" : "=m" (n) : "t" (x));
 663         return (n);
 664 }
 665 #endif
 666
 667 #if defined(__amd64__) || defined(__i386__)
 668 static __inline int
 669 irintl(long double x)
 670 {
 671         int n;
 672
 673         __asm("fistl %0" : "=m" (n) : "t" (x));
 674         return (n);
 675 }
 676 #endif
 677
 678 #ifdef DEBUG
 679 #if defined(__amd64__) || defined(__i386__)
 680 #define breakpoint()    asm("int $3")
 681 #else
 682 #include <signal.h>
 683
 684 #define breakpoint()    raise(SIGTRAP)
 685 #endif
 686 #endif
 687
 688 /* Write a pari script to test things externally. */
 689 #ifdef DOPRINT
 690 #include <stdio.h>
 691
 692 #ifndef DOPRINT_SWIZZLE
 693 #define DOPRINT_SWIZZLE         0
 694 #endif
 695
 696 #ifdef DOPRINT_LD80
 697
 698 #define DOPRINT_START(xp) do {                                          \
 699         uint64_t __lx;                                                  \
 700         uint16_t __hx;                                                  \
 701                                                                         \
 702         /* Hack to give more-problematic args. */                       \
 703         EXTRACT_LDBL80_WORDS(__hx, __lx, *xp);                          \
 704         __lx ^= DOPRINT_SWIZZLE;                                        \
 705         INSERT_LDBL80_WORDS(*xp, __hx, __lx);                           \
 706         printf("x = %.21Lg; ", (long double)*xp);                       \
 707 } while (0)
 708 #define DOPRINT_END1(v)                                                 \
 709         printf("y = %.21Lg; z = 0; show(x, y, z);\n", (long double)(v))
 710 #define DOPRINT_END2(hi, lo)                                            \
 711         printf("y = %.21Lg; z = %.21Lg; show(x, y, z);\n",              \
 712             (long double)(hi), (long double)(lo))
 713
 714 #elif defined(DOPRINT_D64)
 715
 716 #define DOPRINT_START(xp) do {                                          \
 717         uint32_t __hx, __lx;                                            \
 718                                                                         \
 719         EXTRACT_WORDS(__hx, __lx, *xp);                                 \
 720         __lx ^= DOPRINT_SWIZZLE;                                        \
 721         INSERT_WORDS(*xp, __hx, __lx);                                  \
 722         printf("x = %.21Lg; ", (long double)*xp);                       \
 723 } while (0)
 724 #define DOPRINT_END1(v)                                                 \
 725         printf("y = %.21Lg; z = 0; show(x, y, z);\n", (long double)(v))
 726 #define DOPRINT_END2(hi, lo)                                            \
 727         printf("y = %.21Lg; z = %.21Lg; show(x, y, z);\n",              \
 728             (long double)(hi), (long double)(lo))
 729
 730 #elif defined(DOPRINT_F32)
 731
 732 #define DOPRINT_START(xp) do {                                          \
 733         uint32_t __hx;                                                  \
 734                                                                         \
 735         GET_FLOAT_WORD(__hx, *xp);                                      \
 736         __hx ^= DOPRINT_SWIZZLE;                                        \
 737         SET_FLOAT_WORD(*xp, __hx);                                      \
 738         printf("x = %.21Lg; ", (long double)*xp);                       \
 739 } while (0)
 740 #define DOPRINT_END1(v)                                                 \
 741         printf("y = %.21Lg; z = 0; show(x, y, z);\n", (long double)(v))
 742 #define DOPRINT_END2(hi, lo)                                            \
 743         printf("y = %.21Lg; z = %.21Lg; show(x, y, z);\n",              \
 744             (long double)(hi), (long double)(lo))
 745
 746 #else /* !DOPRINT_LD80 && !DOPRINT_D64 (LD128 only) */
 747
 748 #ifndef DOPRINT_SWIZZLE_HIGH
 749 #define DOPRINT_SWIZZLE_HIGH    0
 750 #endif
 751
 752 #define DOPRINT_START(xp) do {                                          \
 753         uint64_t __lx, __llx;                                           \
 754         uint16_t __hx;                                                  \
 755                                                                         \
 756         EXTRACT_LDBL128_WORDS(__hx, __lx, __llx, *xp);                  \
 757         __llx ^= DOPRINT_SWIZZLE;                                       \
 758         __lx ^= DOPRINT_SWIZZLE_HIGH;                                   \
 759         INSERT_LDBL128_WORDS(*xp, __hx, __lx, __llx);                   \
 760         printf("x = %.36Lg; ", (long double)*xp);                                       \
 761 } while (0)
 762 #define DOPRINT_END1(v)                                                 \
 763         printf("y = %.36Lg; z = 0; show(x, y, z);\n", (long double)(v))
 764 #define DOPRINT_END2(hi, lo)                                            \
 765         printf("y = %.36Lg; z = %.36Lg; show(x, y, z);\n",              \
 766             (long double)(hi), (long double)(lo))
 767
 768 #endif /* DOPRINT_LD80 */
 769
 770 #else /* !DOPRINT */
 771 #define DOPRINT_START(xp)
 772 #define DOPRINT_END1(v)
 773 #define DOPRINT_END2(hi, lo)
 774 #endif /* DOPRINT */
 775
 776 #define RETURNP(x) do {                 \
 777         DOPRINT_END1(x);                \
 778         RETURNF(x);                     \
 779 } while (0)
 780 #define RETURNPI(x) do {                \
 781         DOPRINT_END1(x);                \
 782         RETURNI(x);                     \
 783 } while (0)
 784 #define RETURN2P(x, y) do {             \
 785         DOPRINT_END2((x), (y));         \
 786         RETURNF((x) + (y));             \
 787 } while (0)
 788 #define RETURN2PI(x, y) do {            \
 789         DOPRINT_END2((x), (y));         \
 790         RETURNI((x) + (y));             \
 791 } while (0)
 792 #ifdef STRUCT_RETURN
 793 #define RETURNSP(rp) do {               \
 794         if (!(rp)->lo_set)              \
 795                 RETURNP((rp)->hi);      \
 796         RETURN2P((rp)->hi, (rp)->lo);   \
 797 } while (0)
 798 #define RETURNSPI(rp) do {              \
 799         if (!(rp)->lo_set)              \
 800                 RETURNPI((rp)->hi);     \
 801         RETURN2PI((rp)->hi, (rp)->lo);  \
 802 } while (0)
 803 #endif
 804 #define SUM2P(x, y) ({                  \
 805         const __typeof (x) __x = (x);   \
 806         const __typeof (y) __y = (y);   \
 807                                         \
 808         DOPRINT_END2(__x, __y);         \
 809         __x + __y;                      \
 810 })
 811
 812 /*
 813  * ieee style elementary functions
 814  *
 815  * We rename functions here to improve other sources' diffability
 816  * against fdlibm.
 817  */
 818 #define __ieee754_sqrt  sqrt
 819 #define __ieee754_acos  acos
 820 #define __ieee754_acosh acosh
 821 #define __ieee754_log   log
 822 #define __ieee754_log2  log2
 823 #define __ieee754_atanh atanh
 824 #define __ieee754_asin  asin
 825 #define __ieee754_atan2 atan2
 826 #define __ieee754_exp   exp
 827 #define __ieee754_cosh  cosh
 828 #define __ieee754_fmod  fmod
 829 #define __ieee754_pow   pow
 830 #define __ieee754_lgamma lgamma
 831 #define __ieee754_gamma gamma
 832 #define __ieee754_lgamma_r lgamma_r
 833 #define __ieee754_gamma_r gamma_r
 834 #define __ieee754_log10 log10
 835 #define __ieee754_sinh  sinh
 836 #define __ieee754_hypot hypot
 837 #define __ieee754_j0    j0
 838 #define __ieee754_j1    j1
 839 #define __ieee754_y0    y0
 840 #define __ieee754_y1    y1
 841 #define __ieee754_jn    jn
 842 #define __ieee754_yn    yn
 843 #define __ieee754_remainder remainder
 844 #define __ieee754_scalb scalb
 845 #define __ieee754_sqrtf sqrtf
 846 #define __ieee754_acosf acosf
 847 #define __ieee754_acoshf acoshf
 848 #define __ieee754_logf  logf
 849 #define __ieee754_atanhf atanhf
 850 #define __ieee754_asinf asinf
 851 #define __ieee754_atan2f atan2f
 852 #define __ieee754_expf  expf
 853 #define __ieee754_coshf coshf
 854 #define __ieee754_fmodf fmodf
 855 #define __ieee754_powf  powf
 856 #define __ieee754_lgammaf lgammaf
 857 #define __ieee754_gammaf gammaf
 858 #define __ieee754_lgammaf_r lgammaf_r
 859 #define __ieee754_gammaf_r gammaf_r
 860 #define __ieee754_log10f log10f
 861 #define __ieee754_log2f log2f
 862 #define __ieee754_sinhf sinhf
 863 #define __ieee754_hypotf hypotf
 864 #define __ieee754_j0f   j0f
 865 #define __ieee754_j1f   j1f
 866 #define __ieee754_y0f   y0f
 867 #define __ieee754_y1f   y1f
 868 #define __ieee754_jnf   jnf
 869 #define __ieee754_ynf   ynf
 870 #define __ieee754_remainderf remainderf
 871 #define __ieee754_scalbf scalbf
 872
 873 #define acos fdlibm_acos
 874 #define acosf fdlibm_acosf
 875 #define asin fdlibm_asin
 876 #define asinf fdlibm_asinf
 877 #define atan fdlibm_atan
 878 #define atanf fdlibm_atanf
 879 #define atan2 fdlibm_atan2
 880 #define cos fdlibm_cos
 881 #define cosf fdlibm_cosf
 882 #define sin fdlibm_sin
 883 #define sinf fdlibm_sinf
 884 #define tan fdlibm_tan
 885 #define tanf fdlibm_tanf
 886 #define cosh fdlibm_cosh
 887 #define sinh fdlibm_sinh
 888 #define tanh fdlibm_tanh
 889 #define exp fdlibm_exp
 890 #define expf fdlibm_expf
 891 #define exp2 fdlibm_exp2
 892 #define exp2f fdlibm_exp2f
 893 #define log fdlibm_log
 894 #define logf fdlibm_logf
 895 #define log10 fdlibm_log10
 896 #define log10f fdlibm_log10f
 897 #define pow fdlibm_pow
 898 #define powf fdlibm_powf
 899 #define ceil fdlibm_ceil
 900 #define ceilf fdlibm_ceilf
 901 #define fabs fdlibm_fabs
 902 #define fabsf fdlibm_fabsf
 903 #define floor fdlibm_floor
 904 #define acosh fdlibm_acosh
 905 #define asinh fdlibm_asinh
 906 #define atanh fdlibm_atanh
 907 #define cbrt fdlibm_cbrt
 908 #define expm1 fdlibm_expm1
 909 #define hypot fdlibm_hypot
 910 #define hypotf fdlibm_hypotf
 911 #define log1p fdlibm_log1p
 912 #define log2 fdlibm_log2
 913 #define scalb fdlibm_scalb
 914 #define copysign fdlibm_copysign
 915 #define scalbn fdlibm_scalbn
 916 #define scalbnf fdlibm_scalbnf
 917 #define trunc fdlibm_trunc
 918 #define truncf fdlibm_truncf
 919 #define floorf fdlibm_floorf
 920 #define nearbyint fdlibm_nearbyint
 921 #define nearbyintf fdlibm_nearbyintf
 922 #define rint fdlibm_rint
 923 #define rintf fdlibm_rintf
 924 #define sqrtf fdlibm_sqrtf
 925
 926 /* fdlibm kernel function */
 927 int     __kernel_rem_pio2(double*,double*,int,int,int);
 928
 929 /* double precision kernel functions */
 930 #ifndef INLINE_REM_PIO2
 931 int     __ieee754_rem_pio2(double,double*);
 932 #endif
 933 double  __kernel_sin(double,double,int);
 934 double  __kernel_cos(double,double);
 935 double  __kernel_tan(double,double,int);
 936 double  __ldexp_exp(double,int);
 937 #ifdef _COMPLEX_H
 938 double complex __ldexp_cexp(double complex,int);
 939 #endif
 940
 941 /* float precision kernel functions */
 942 #ifndef INLINE_REM_PIO2F
 943 int     __ieee754_rem_pio2f(float,double*);
 944 #endif
 945 #ifndef INLINE_KERNEL_SINDF
 946 float   __kernel_sindf(double);
 947 #endif
 948 #ifndef INLINE_KERNEL_COSDF
 949 float   __kernel_cosdf(double);
 950 #endif
 951 #ifndef INLINE_KERNEL_TANDF
 952 float   __kernel_tandf(double,int);
 953 #endif
 954 float   __ldexp_expf(float,int);
 955 #ifdef _COMPLEX_H
 956 float complex __ldexp_cexpf(float complex,int);
 957 #endif
 958
 959 /* long double precision kernel functions */
 960 long double __kernel_sinl(long double, long double, int);
 961 long double __kernel_cosl(long double, long double);
 962 long double __kernel_tanl(long double, long double, int);
 963
 964 #endif /* !_MATH_PRIVATE_H_ */