1 /* Declarations for math functions.
2 Copyright (C) 1991-2017 Free Software Foundation, Inc.
3 This file is part of the GNU C Library.
5 The GNU C Library is free software; you can redistribute it and/or
6 modify it under the terms of the GNU Lesser General Public
7 License as published by the Free Software Foundation; either
8 version 2.1 of the License, or (at your option) any later version.
10 The GNU C Library is distributed in the hope that it will be useful,
11 but WITHOUT ANY WARRANTY; without even the implied warranty of
12 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
13 Lesser General Public License for more details.
15 You should have received a copy of the GNU Lesser General Public
16 License along with the GNU C Library; if not, see
17 <http://www.gnu.org/licenses/>. */
20 * ISO C99 Standard: 7.12 Mathematics <math.h>
26 #define __GLIBC_INTERNAL_STARTING_HEADER_IMPLEMENTATION
27 #include <bits/libc-header-start.h>
31 /* Get definitions of __intmax_t and __uintmax_t. */
32 #include <bits/types.h>
34 /* Get machine-dependent vector math functions declarations. */
35 #include <bits/math-vector.h>
37 /* Get machine-dependent HUGE_VAL value (returned on overflow).
38 On all IEEE754 machines, this is +Infinity. */
39 #include <bits/huge_val.h>
41 # include <bits/huge_valf.h>
42 # include <bits/huge_vall.h>
44 /* Get machine-dependent INFINITY value. */
45 # include <bits/inf.h>
47 /* Get machine-dependent NAN value (returned for some domain errors). */
48 # include <bits/nan.h>
49 #endif /* __USE_ISOC99 */
51 #if __GLIBC_USE (IEC_60559_BFP_EXT)
52 /* Signaling NaN macros, if supported. */
53 # if __GNUC_PREREQ (3, 3)
54 # define SNANF (__builtin_nansf (""))
55 # define SNAN (__builtin_nans (""))
56 # define SNANL (__builtin_nansl (""))
60 /* Get __GLIBC_FLT_EVAL_METHOD. */
61 #include <bits/flt-eval-method.h>
64 /* Define the following typedefs.
66 float_t floating-point type at least as wide as `float' used
67 to evaluate `float' expressions
68 double_t floating-point type at least as wide as `double' used
69 to evaluate `double' expressions
71 # if __GLIBC_FLT_EVAL_METHOD == 0 || __GLIBC_FLT_EVAL_METHOD == 16
72 typedef float float_t
;
73 typedef double double_t
;
74 # elif __GLIBC_FLT_EVAL_METHOD == 1
75 typedef double float_t
;
76 typedef double double_t
;
77 # elif __GLIBC_FLT_EVAL_METHOD == 2
78 typedef long double float_t
;
79 typedef long double double_t
;
80 # elif __GLIBC_FLT_EVAL_METHOD == 32
81 typedef _Float32 float_t
;
82 typedef double double_t
;
83 # elif __GLIBC_FLT_EVAL_METHOD == 33
84 typedef _Float32x float_t
;
85 typedef _Float32x double_t
;
86 # elif __GLIBC_FLT_EVAL_METHOD == 64
87 typedef _Float64 float_t
;
88 typedef _Float64 double_t
;
89 # elif __GLIBC_FLT_EVAL_METHOD == 65
90 typedef _Float64x float_t
;
91 typedef _Float64x double_t
;
92 # elif __GLIBC_FLT_EVAL_METHOD == 128
93 typedef _Float128 float_t
;
94 typedef _Float128 double_t
;
95 # elif __GLIBC_FLT_EVAL_METHOD == 129
96 typedef _Float128x float_t
;
97 typedef _Float128x double_t
;
99 # error "Unknown __GLIBC_FLT_EVAL_METHOD"
103 /* Define macros for the return values of ilogb and llogb, based on
104 __FP_LOGB0_IS_MIN and __FP_LOGBNAN_IS_MIN.
106 FP_ILOGB0 Expands to a value returned by `ilogb (0.0)'.
107 FP_ILOGBNAN Expands to a value returned by `ilogb (NAN)'.
108 FP_LLOGB0 Expands to a value returned by `llogb (0.0)'.
109 FP_LLOGBNAN Expands to a value returned by `llogb (NAN)'.
113 #include <bits/fp-logb.h>
115 # if __FP_LOGB0_IS_MIN
116 # define FP_ILOGB0 (-2147483647 - 1)
118 # define FP_ILOGB0 (-2147483647)
120 # if __FP_LOGBNAN_IS_MIN
121 # define FP_ILOGBNAN (-2147483647 - 1)
123 # define FP_ILOGBNAN 2147483647
126 #if __GLIBC_USE (IEC_60559_BFP_EXT)
127 # if __WORDSIZE == 32
128 # define __FP_LONG_MAX 0x7fffffffL
130 # define __FP_LONG_MAX 0x7fffffffffffffffL
132 # if __FP_LOGB0_IS_MIN
133 # define FP_LLOGB0 (-__FP_LONG_MAX - 1)
135 # define FP_LLOGB0 (-__FP_LONG_MAX)
137 # if __FP_LOGBNAN_IS_MIN
138 # define FP_LLOGBNAN (-__FP_LONG_MAX - 1)
140 # define FP_LLOGBNAN __FP_LONG_MAX
144 /* Get the architecture specific values describing the floating-point
145 evaluation. The following symbols will get defined:
150 If defined it indicates that the `fma' function
151 generally executes about as fast as a multiply and an add.
152 This macro is defined only iff the `fma' function is
153 implemented directly with a hardware multiply-add instructions.
156 #include <bits/fp-fast.h>
158 #if __GLIBC_USE (IEC_60559_BFP_EXT)
159 /* Rounding direction macros for fromfp functions. */
163 # define FP_INT_UPWARD 0
166 # define FP_INT_DOWNWARD 1
169 # define FP_INT_TOWARDZERO 2
171 FP_INT_TONEARESTFROMZERO
=
172 # define FP_INT_TONEARESTFROMZERO 3
173 FP_INT_TONEARESTFROMZERO
,
175 # define FP_INT_TONEAREST 4
180 /* The file <bits/mathcalls.h> contains the prototypes for all the
181 actual math functions. These macros are used for those prototypes,
182 so we can easily declare each function as both `name' and `__name',
183 and can declare the float versions `namef' and `__namef'. */
185 #define __SIMD_DECL(function) __CONCAT (__DECL_SIMD_, function)
187 #define __MATHCALL_VEC(function, suffix, args) \
188 __SIMD_DECL (__MATH_PRECNAME (function, suffix)) \
189 __MATHCALL (function, suffix, args)
191 #define __MATHDECL_VEC(type, function,suffix, args) \
192 __SIMD_DECL (__MATH_PRECNAME (function, suffix)) \
193 __MATHDECL(type, function,suffix, args)
195 #define __MATHCALL(function,suffix, args) \
196 __MATHDECL (_Mdouble_,function,suffix, args)
197 #define __MATHDECL(type, function,suffix, args) \
198 __MATHDECL_1(type, function,suffix, args); \
199 __MATHDECL_1(type, __CONCAT(__,function),suffix, args)
200 #define __MATHCALLX(function,suffix, args, attrib) \
201 __MATHDECLX (_Mdouble_,function,suffix, args, attrib)
202 #define __MATHDECLX(type, function,suffix, args, attrib) \
203 __MATHDECL_1(type, function,suffix, args) __attribute__ (attrib); \
204 __MATHDECL_1(type, __CONCAT(__,function),suffix, args) __attribute__ (attrib)
205 #define __MATHDECL_1(type, function,suffix, args) \
206 extern type __MATH_PRECNAME(function,suffix) args __THROW
208 #define _Mdouble_ double
209 #define __MATH_PRECNAME(name,r) __CONCAT(name,r)
210 #define __MATH_DECLARING_DOUBLE 1
211 #include <bits/mathcalls-helper-functions.h>
212 #include <bits/mathcalls.h>
214 #undef __MATH_PRECNAME
215 #undef __MATH_DECLARING_DOUBLE
220 /* Include the file of declarations again, this time using `float'
221 instead of `double' and appending f to each function name. */
224 # define _Mfloat_ float
226 # define _Mdouble_ _Mfloat_
227 # define __MATH_PRECNAME(name,r) name##f##r
228 # define __MATH_DECLARING_DOUBLE 0
229 # include <bits/mathcalls-helper-functions.h>
230 # include <bits/mathcalls.h>
232 # undef __MATH_PRECNAME
233 # undef __MATH_DECLARING_DOUBLE
235 # if !(defined __NO_LONG_DOUBLE_MATH && defined _LIBC) \
236 || defined __LDBL_COMPAT \
237 || defined _LIBC_TEST
238 # ifdef __LDBL_COMPAT
241 extern float __nldbl_nexttowardf (float __x
, long double __y
)
242 __THROW
__attribute__ ((__const__
));
243 # ifdef __REDIRECT_NTH
244 extern float __REDIRECT_NTH (nexttowardf
, (float __x
, long double __y
),
246 __attribute__ ((__const__
));
247 extern double __REDIRECT_NTH (nexttoward
, (double __x
, long double __y
),
248 nextafter
) __attribute__ ((__const__
));
249 extern long double __REDIRECT_NTH (nexttowardl
,
250 (long double __x
, long double __y
),
251 nextafter
) __attribute__ ((__const__
));
256 # define __MATHDECL_2(type, function,suffix, args, alias) \
257 extern type __REDIRECT_NTH(__MATH_PRECNAME(function,suffix), \
259 # define __MATHDECL_1(type, function,suffix, args) \
260 __MATHDECL_2(type, function,suffix, args, __CONCAT(function,suffix))
263 /* Include the file of declarations again, this time using `long double'
264 instead of `double' and appending l to each function name. */
266 # ifndef _Mlong_double_
267 # define _Mlong_double_ long double
269 # define _Mdouble_ _Mlong_double_
270 # define __MATH_PRECNAME(name,r) name##l##r
271 # define __MATH_DECLARING_DOUBLE 0
272 # define __MATH_DECLARE_LDOUBLE 1
273 # include <bits/mathcalls-helper-functions.h>
274 # include <bits/mathcalls.h>
276 # undef __MATH_PRECNAME
277 # undef __MATH_DECLARING_DOUBLE
279 # endif /* !(__NO_LONG_DOUBLE_MATH && _LIBC) || __LDBL_COMPAT */
281 #endif /* Use ISO C99. */
287 #if defined __USE_MISC || defined __USE_XOPEN
288 /* This variable is used by `gamma' and `lgamma'. */
293 /* Depending on the type of TG_ARG, call an appropriately suffixed
294 version of FUNC with arguments (including parentheses) ARGS.
295 Suffixed functions may not exist for long double if it has the same
296 format as double, or for other types with the same format as float,
297 double or long double. The behavior is undefined if the argument
298 does not have a real floating type. The definition may use a
299 conditional expression, so all suffixed versions of FUNC must
300 return the same type (FUNC may include a cast if necessary rather
301 than being a single identifier). */
302 #ifdef __NO_LONG_DOUBLE_MATH
303 # define __MATH_TG(TG_ARG, FUNC, ARGS) \
304 (sizeof (TG_ARG) == sizeof (float) ? FUNC ## f ARGS : FUNC ARGS)
306 # define __MATH_TG(TG_ARG, FUNC, ARGS) \
307 (sizeof (TG_ARG) == sizeof (float) \
309 : sizeof (TG_ARG) == sizeof (double) \
314 /* ISO C99 defines some generic macros which work on any data type. */
317 /* All floating-point numbers can be put in one of these categories. */
324 # define FP_INFINITE 1
330 # define FP_SUBNORMAL 3
337 /* GCC bug 66462 means we cannot use the math builtins with -fsignaling-nan,
338 so disable builtins if this is enabled. When fixed in a newer GCC,
339 the __SUPPORT_SNAN__ check may be skipped for those versions. */
341 /* Return number of classification appropriate for X. */
342 # if __GNUC_PREREQ (4,4) && !defined __SUPPORT_SNAN__ \
343 && !defined __OPTIMIZE_SIZE__
344 # define fpclassify(x) __builtin_fpclassify (FP_NAN, FP_INFINITE, \
345 FP_NORMAL, FP_SUBNORMAL, FP_ZERO, x)
347 # define fpclassify(x) __MATH_TG ((x), __fpclassify, (x))
350 /* Return nonzero value if sign of X is negative. */
351 # if __GNUC_PREREQ (4,0)
352 # define signbit(x) __MATH_TG ((x), __builtin_signbit, (x))
354 # define signbit(x) __MATH_TG ((x), __signbit, (x))
357 /* Return nonzero value if X is not +-Inf or NaN. */
358 # if __GNUC_PREREQ (4,4) && !defined __SUPPORT_SNAN__
359 # define isfinite(x) __builtin_isfinite (x)
361 # define isfinite(x) __MATH_TG ((x), __finite, (x))
364 /* Return nonzero value if X is neither zero, subnormal, Inf, nor NaN. */
365 # if __GNUC_PREREQ (4,4) && !defined __SUPPORT_SNAN__
366 # define isnormal(x) __builtin_isnormal (x)
368 # define isnormal(x) (fpclassify (x) == FP_NORMAL)
371 /* Return nonzero value if X is a NaN. We could use `fpclassify' but
372 we already have this functions `__isnan' and it is faster. */
373 # if __GNUC_PREREQ (4,4) && !defined __SUPPORT_SNAN__
374 # define isnan(x) __builtin_isnan (x)
376 # define isnan(x) __MATH_TG ((x), __isnan, (x))
379 /* Return nonzero value if X is positive or negative infinity. */
380 # if __GNUC_PREREQ (4,4) && !defined __SUPPORT_SNAN__
381 # define isinf(x) __builtin_isinf_sign (x)
383 # define isinf(x) __MATH_TG ((x), __isinf, (x))
386 /* Bitmasks for the math_errhandling macro. */
387 # define MATH_ERRNO 1 /* errno set by math functions. */
388 # define MATH_ERREXCEPT 2 /* Exceptions raised by math functions. */
390 /* By default all functions support both errno and exception handling.
391 In gcc's fast math mode and if inline functions are defined this
392 might not be true. */
393 # ifndef __FAST_MATH__
394 # define math_errhandling (MATH_ERRNO | MATH_ERREXCEPT)
397 #endif /* Use ISO C99. */
399 #if __GLIBC_USE (IEC_60559_BFP_EXT)
400 # include <bits/iscanonical.h>
402 /* Return nonzero value if X is a signaling NaN. */
403 # define issignaling(x) __MATH_TG ((x), __issignaling, (x))
405 /* Return nonzero value if X is subnormal. */
406 # define issubnormal(x) (fpclassify (x) == FP_SUBNORMAL)
408 /* Return nonzero value if X is zero. */
410 # ifdef __SUPPORT_SNAN__
411 # define iszero(x) (fpclassify (x) == FP_ZERO)
413 # define iszero(x) (((__typeof (x)) (x)) == 0)
415 # else /* __cplusplus */
417 template <class __T
> inline bool
420 # ifdef __SUPPORT_SNAN__
421 return fpclassify (__val
) == FP_ZERO
;
427 # endif /* __cplusplus */
428 #endif /* Use IEC_60559_BFP_EXT. */
431 /* Support for various different standard error handling behaviors. */
434 _IEEE_
= -1, /* According to IEEE 754/IEEE 854. */
435 _SVID_
, /* According to System V, release 4. */
436 _XOPEN_
, /* Nowadays also Unix98. */
438 _ISOC_
/* Actually this is ISO C99. */
441 /* This variable can be changed at run-time to any of the values above to
442 affect floating point error handling behavior (it may also be necessary
443 to change the hardware FPU exception settings). */
444 extern _LIB_VERSION_TYPE _LIB_VERSION
;
449 /* In SVID error handling, `matherr' is called with this description
450 of the exceptional condition.
452 We have a problem when using C++ since `exception' is a reserved
468 extern int matherr (struct __exception
*__exc
) throw ();
470 extern int matherr (struct exception
*__exc
);
473 # define X_TLOSS 1.41484755040568800000e+16
475 /* Types of exceptions in the `type' field. */
483 /* SVID mode specifies returning this large value instead of infinity. */
484 # define HUGE 3.40282347e+38F
489 /* X/Open wants another strange constant. */
490 # define MAXFLOAT 3.40282347e+38F
496 /* Some useful constants. */
497 #if defined __USE_MISC || defined __USE_XOPEN
498 # define M_E 2.7182818284590452354 /* e */
499 # define M_LOG2E 1.4426950408889634074 /* log_2 e */
500 # define M_LOG10E 0.43429448190325182765 /* log_10 e */
501 # define M_LN2 0.69314718055994530942 /* log_e 2 */
502 # define M_LN10 2.30258509299404568402 /* log_e 10 */
503 # define M_PI 3.14159265358979323846 /* pi */
504 # define M_PI_2 1.57079632679489661923 /* pi/2 */
505 # define M_PI_4 0.78539816339744830962 /* pi/4 */
506 # define M_1_PI 0.31830988618379067154 /* 1/pi */
507 # define M_2_PI 0.63661977236758134308 /* 2/pi */
508 # define M_2_SQRTPI 1.12837916709551257390 /* 2/sqrt(pi) */
509 # define M_SQRT2 1.41421356237309504880 /* sqrt(2) */
510 # define M_SQRT1_2 0.70710678118654752440 /* 1/sqrt(2) */
513 /* The above constants are not adequate for computation using `long double's.
514 Therefore we provide as an extension constants with similar names as a
515 GNU extension. Provide enough digits for the 128-bit IEEE quad. */
517 # define M_El 2.718281828459045235360287471352662498L /* e */
518 # define M_LOG2El 1.442695040888963407359924681001892137L /* log_2 e */
519 # define M_LOG10El 0.434294481903251827651128918916605082L /* log_10 e */
520 # define M_LN2l 0.693147180559945309417232121458176568L /* log_e 2 */
521 # define M_LN10l 2.302585092994045684017991454684364208L /* log_e 10 */
522 # define M_PIl 3.141592653589793238462643383279502884L /* pi */
523 # define M_PI_2l 1.570796326794896619231321691639751442L /* pi/2 */
524 # define M_PI_4l 0.785398163397448309615660845819875721L /* pi/4 */
525 # define M_1_PIl 0.318309886183790671537767526745028724L /* 1/pi */
526 # define M_2_PIl 0.636619772367581343075535053490057448L /* 2/pi */
527 # define M_2_SQRTPIl 1.128379167095512573896158903121545172L /* 2/sqrt(pi) */
528 # define M_SQRT2l 1.414213562373095048801688724209698079L /* sqrt(2) */
529 # define M_SQRT1_2l 0.707106781186547524400844362104849039L /* 1/sqrt(2) */
533 /* When compiling in strict ISO C compatible mode we must not use the
534 inline functions since they, among other things, do not set the
535 `errno' variable correctly. */
536 #if defined __STRICT_ANSI__ && !defined __NO_MATH_INLINES
537 # define __NO_MATH_INLINES 1
540 #if defined __USE_ISOC99 && __GNUC_PREREQ(2,97)
541 /* ISO C99 defines some macros to compare number while taking care for
542 unordered numbers. Many FPUs provide special instructions to support
543 these operations. Generic support in GCC for these as builtins went
544 in before 3.0.0, but not all cpus added their patterns. We define
545 versions that use the builtins here, and <bits/mathinline.h> will
546 undef/redefine as appropriate for the specific GCC version in use. */
547 # define isgreater(x, y) __builtin_isgreater(x, y)
548 # define isgreaterequal(x, y) __builtin_isgreaterequal(x, y)
549 # define isless(x, y) __builtin_isless(x, y)
550 # define islessequal(x, y) __builtin_islessequal(x, y)
551 # define islessgreater(x, y) __builtin_islessgreater(x, y)
552 # define isunordered(u, v) __builtin_isunordered(u, v)
555 /* Get machine-dependent inline versions (if there are any). */
556 #ifdef __USE_EXTERN_INLINES
557 # include <bits/mathinline.h>
560 /* Define special entry points to use when the compiler got told to
561 only expect finite results. */
562 #if defined __FINITE_MATH_ONLY__ && __FINITE_MATH_ONLY__ > 0
564 /* Include bits/math-finite.h for double. */
565 # define _Mdouble_ double
566 # define __MATH_DECLARING_DOUBLE 1
567 # define __MATH_DECLARING_LDOUBLE 0
569 # include <bits/math-finite.h>
571 # undef __MATH_DECLARING_DOUBLE
572 # undef __MATH_DECLARING_LDOUBLE
575 /* When __USE_ISOC99 is defined, include math-finite for float and
576 long double, as well. */
579 /* Include bits/math-finite.h for float. */
580 # define _Mdouble_ float
581 # define __MATH_DECLARING_DOUBLE 0
582 # define __MATH_DECLARING_LDOUBLE 0
584 # include <bits/math-finite.h>
586 # undef __MATH_DECLARING_DOUBLE
587 # undef __MATH_DECLARING_LDOUBLE
590 /* Include bits/math-finite.h for long double. */
591 # ifdef __MATH_DECLARE_LDOUBLE
592 # define _Mdouble_ long double
593 # define __MATH_DECLARING_DOUBLE 0
594 # define __MATH_DECLARING_LDOUBLE 1
596 # include <bits/math-finite.h>
598 # undef __MATH_DECLARING_DOUBLE
599 # undef __MATH_DECLARING_LDOUBLE
603 # endif /* __USE_ISOC99. */
604 #endif /* __FINITE_MATH_ONLY__ > 0. */
607 /* If we've still got undefined comparison macros, provide defaults. */
609 /* Return nonzero value if X is greater than Y. */
611 # define isgreater(x, y) \
613 ({ __typeof__(x) __x = (x); __typeof__(y) __y = (y); \
614 !isunordered (__x, __y) && __x > __y; }))
617 /* Return nonzero value if X is greater than or equal to Y. */
618 # ifndef isgreaterequal
619 # define isgreaterequal(x, y) \
621 ({ __typeof__(x) __x = (x); __typeof__(y) __y = (y); \
622 !isunordered (__x, __y) && __x >= __y; }))
625 /* Return nonzero value if X is less than Y. */
627 # define isless(x, y) \
629 ({ __typeof__(x) __x = (x); __typeof__(y) __y = (y); \
630 !isunordered (__x, __y) && __x < __y; }))
633 /* Return nonzero value if X is less than or equal to Y. */
635 # define islessequal(x, y) \
637 ({ __typeof__(x) __x = (x); __typeof__(y) __y = (y); \
638 !isunordered (__x, __y) && __x <= __y; }))
641 /* Return nonzero value if either X is less than Y or Y is less than X. */
642 # ifndef islessgreater
643 # define islessgreater(x, y) \
645 ({ __typeof__(x) __x = (x); __typeof__(y) __y = (y); \
646 !isunordered (__x, __y) && (__x < __y || __y < __x); }))
649 /* Return nonzero value if arguments are unordered. */
651 # define isunordered(u, v) \
653 ({ __typeof__(u) __u = (u); __typeof__(v) __v = (v); \
654 fpclassify (__u) == FP_NAN || fpclassify (__v) == FP_NAN; }))
659 #if __GLIBC_USE (IEC_60559_BFP_EXT)
660 /* An expression whose type has the widest of the evaluation formats
661 of X and Y (which are of floating-point types). */
662 # if __FLT_EVAL_METHOD__ == 2 || __FLT_EVAL_METHOD__ > 64
663 # define __MATH_EVAL_FMT2(x, y) ((x) + (y) + 0.0L)
664 # elif __FLT_EVAL_METHOD__ == 1 || __FLT_EVAL_METHOD__ > 32
665 # define __MATH_EVAL_FMT2(x, y) ((x) + (y) + 0.0)
667 # define __MATH_EVAL_FMT2(x, y) ((x) + (y))
670 /* Return X == Y but raising "invalid" and setting errno if X or Y is
672 # define iseqsig(x, y) \
673 __MATH_TG (__MATH_EVAL_FMT2 (x, y), __iseqsig, ((x), (y)))