glibc/math/math.h

/* Declarations for math functions.
   Copyright (C) 1991-2017 Free Software Foundation, Inc.
   This file is part of the GNU C Library.

   The GNU C Library is free software; you can redistribute it and/or
   modify it under the terms of the GNU Lesser General Public
   License as published by the Free Software Foundation; either
   version 2.1 of the License, or (at your option) any later version.

   The GNU C Library is distributed in the hope that it will be useful,
   but WITHOUT ANY WARRANTY; without even the implied warranty of
   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
   Lesser General Public License for more details.

   You should have received a copy of the GNU Lesser General Public
   License along with the GNU C Library; if not, see
   <http://www.gnu.org/licenses/>.  */

/*
 *	ISO C99 Standard: 7.12 Mathematics	<math.h>
 */

#ifndef	_MATH_H
#define	_MATH_H	1

#define __GLIBC_INTERNAL_STARTING_HEADER_IMPLEMENTATION
#include <bits/libc-header-start.h>

#if defined log && defined __GNUC__
# warning A macro called log was already defined when <math.h> was included.
# warning This will cause compilation problems.
#endif

__BEGIN_DECLS

/* Get definitions of __intmax_t and __uintmax_t.  */
#include <bits/types.h>

/* Get machine-dependent vector math functions declarations.  */
#include <bits/math-vector.h>

/* Gather machine dependent type support.  */
#include <bits/floatn.h>

/* Value returned on overflow.  With IEEE 754 floating point, this is
   +Infinity, otherwise the largest representable positive value.  */
#if __GNUC_PREREQ (3, 3)
# define HUGE_VAL (__builtin_huge_val ())
#else
/* This may provoke compiler warnings, and may not be rounded to
   +Infinity in all IEEE 754 rounding modes, but is the best that can
   be done in ISO C while remaining a constant expression.  10,000 is
   greater than the maximum (decimal) exponent for all supported
   floating-point formats and widths.  */
# define HUGE_VAL 1e10000
#endif
#ifdef __USE_ISOC99
# if __GNUC_PREREQ (3, 3)
#  define HUGE_VALF (__builtin_huge_valf ())
#  define HUGE_VALL (__builtin_huge_vall ())
# else
#  define HUGE_VALF 1e10000f
#  define HUGE_VALL 1e10000L
# endif
#endif
#if __HAVE_FLOAT16 && __GLIBC_USE (IEC_60559_TYPES_EXT)
# define HUGE_VAL_F16 (__builtin_huge_valf16 ())
#endif
#if __HAVE_FLOAT32 && __GLIBC_USE (IEC_60559_TYPES_EXT)
# define HUGE_VAL_F32 (__builtin_huge_valf32 ())
#endif
#if __HAVE_FLOAT64 && __GLIBC_USE (IEC_60559_TYPES_EXT)
# define HUGE_VAL_F64 (__builtin_huge_valf64 ())
#endif
#if __HAVE_FLOAT128 && __GLIBC_USE (IEC_60559_TYPES_EXT)
# define HUGE_VAL_F128 (__builtin_huge_valf128 ())
#endif
#if __HAVE_FLOAT32X && __GLIBC_USE (IEC_60559_TYPES_EXT)
# define HUGE_VAL_F32X (__builtin_huge_valf32x ())
#endif
#if __HAVE_FLOAT64X && __GLIBC_USE (IEC_60559_TYPES_EXT)
# define HUGE_VAL_F64X (__builtin_huge_valf64x ())
#endif
#if __HAVE_FLOAT128X && __GLIBC_USE (IEC_60559_TYPES_EXT)
# define HUGE_VAL_F128X (__builtin_huge_valf128x ())
#endif

#ifdef __USE_ISOC99
/* IEEE positive infinity.  */
# if __GNUC_PREREQ (3, 3)
#  define INFINITY (__builtin_inff ())
# else
#  define INFINITY HUGE_VALF
# endif

/* IEEE Not A Number.  */
# if __GNUC_PREREQ (3, 3)
#  define NAN (__builtin_nanf (""))
# else
/* This will raise an "invalid" exception outside static initializers,
   but is the best that can be done in ISO C while remaining a
   constant expression.  */
#  define NAN (0.0f / 0.0f)
# endif
#endif /* __USE_ISOC99 */

#if __GLIBC_USE (IEC_60559_BFP_EXT)
/* Signaling NaN macros, if supported.  */
# if __GNUC_PREREQ (3, 3)
#  define SNANF (__builtin_nansf (""))
#  define SNAN (__builtin_nans (""))
#  define SNANL (__builtin_nansl (""))
# endif
#endif
#if __HAVE_FLOAT16 && __GLIBC_USE (IEC_60559_TYPES_EXT)
# define SNANF16 (__builtin_nansf16 (""))
#endif
#if __HAVE_FLOAT32 && __GLIBC_USE (IEC_60559_TYPES_EXT)
# define SNANF32 (__builtin_nansf32 (""))
#endif
#if __HAVE_FLOAT64 && __GLIBC_USE (IEC_60559_TYPES_EXT)
# define SNANF64 (__builtin_nansf64 (""))
#endif
#if __HAVE_FLOAT128 && __GLIBC_USE (IEC_60559_TYPES_EXT)
# define SNANF128 (__builtin_nansf128 (""))
#endif
#if __HAVE_FLOAT32X && __GLIBC_USE (IEC_60559_TYPES_EXT)
# define SNANF32X (__builtin_nansf32x (""))
#endif
#if __HAVE_FLOAT64X && __GLIBC_USE (IEC_60559_TYPES_EXT)
# define SNANF64X (__builtin_nansf64x (""))
#endif
#if __HAVE_FLOAT128X && __GLIBC_USE (IEC_60559_TYPES_EXT)
# define SNANF128X (__builtin_nansf128x (""))
#endif

/* Get __GLIBC_FLT_EVAL_METHOD.  */
#include <bits/flt-eval-method.h>

#ifdef __USE_ISOC99
/* Define the following typedefs.

    float_t	floating-point type at least as wide as `float' used
		to evaluate `float' expressions
    double_t	floating-point type at least as wide as `double' used
		to evaluate `double' expressions
*/
# if __GLIBC_FLT_EVAL_METHOD == 0 || __GLIBC_FLT_EVAL_METHOD == 16
typedef float float_t;
typedef double double_t;
# elif __GLIBC_FLT_EVAL_METHOD == 1
typedef double float_t;
typedef double double_t;
# elif __GLIBC_FLT_EVAL_METHOD == 2
typedef long double float_t;
typedef long double double_t;
# elif __GLIBC_FLT_EVAL_METHOD == 32
typedef _Float32 float_t;
typedef double double_t;
# elif __GLIBC_FLT_EVAL_METHOD == 33
typedef _Float32x float_t;
typedef _Float32x double_t;
# elif __GLIBC_FLT_EVAL_METHOD == 64
typedef _Float64 float_t;
typedef _Float64 double_t;
# elif __GLIBC_FLT_EVAL_METHOD == 65
typedef _Float64x float_t;
typedef _Float64x double_t;
# elif __GLIBC_FLT_EVAL_METHOD == 128
typedef _Float128 float_t;
typedef _Float128 double_t;
# elif __GLIBC_FLT_EVAL_METHOD == 129
typedef _Float128x float_t;
typedef _Float128x double_t;
# else
#  error "Unknown __GLIBC_FLT_EVAL_METHOD"
# endif
#endif

/* Define macros for the return values of ilogb and llogb, based on
   __FP_LOGB0_IS_MIN and __FP_LOGBNAN_IS_MIN.

    FP_ILOGB0	Expands to a value returned by `ilogb (0.0)'.
    FP_ILOGBNAN	Expands to a value returned by `ilogb (NAN)'.
    FP_LLOGB0	Expands to a value returned by `llogb (0.0)'.
    FP_LLOGBNAN	Expands to a value returned by `llogb (NAN)'.

*/

#include <bits/fp-logb.h>
#ifdef __USE_ISOC99
# if __FP_LOGB0_IS_MIN
#  define FP_ILOGB0	(-2147483647 - 1)
# else
#  define FP_ILOGB0	(-2147483647)
# endif
# if __FP_LOGBNAN_IS_MIN
#  define FP_ILOGBNAN	(-2147483647 - 1)
# else
#  define FP_ILOGBNAN	2147483647
# endif
#endif
#if __GLIBC_USE (IEC_60559_BFP_EXT)
# if __WORDSIZE == 32
#  define __FP_LONG_MAX 0x7fffffffL
# else
#  define __FP_LONG_MAX 0x7fffffffffffffffL
# endif
# if __FP_LOGB0_IS_MIN
#  define FP_LLOGB0	(-__FP_LONG_MAX - 1)
# else
#  define FP_LLOGB0	(-__FP_LONG_MAX)
# endif
# if __FP_LOGBNAN_IS_MIN
#  define FP_LLOGBNAN	(-__FP_LONG_MAX - 1)
# else
#  define FP_LLOGBNAN	__FP_LONG_MAX
# endif
#endif

/* Get the architecture specific values describing the floating-point
   evaluation.  The following symbols will get defined:

    FP_FAST_FMA
    FP_FAST_FMAF
    FP_FAST_FMAL
		If defined it indicates that the `fma' function
		generally executes about as fast as a multiply and an add.
		This macro is defined only iff the `fma' function is
		implemented directly with a hardware multiply-add instructions.
*/

#include <bits/fp-fast.h>

#if __GLIBC_USE (IEC_60559_BFP_EXT)
/* Rounding direction macros for fromfp functions.  */
enum
  {
    FP_INT_UPWARD =
# define FP_INT_UPWARD 0
      FP_INT_UPWARD,
    FP_INT_DOWNWARD =
# define FP_INT_DOWNWARD 1
      FP_INT_DOWNWARD,
    FP_INT_TOWARDZERO =
# define FP_INT_TOWARDZERO 2
      FP_INT_TOWARDZERO,
    FP_INT_TONEARESTFROMZERO =
# define FP_INT_TONEARESTFROMZERO 3
      FP_INT_TONEARESTFROMZERO,
    FP_INT_TONEAREST =
# define FP_INT_TONEAREST 4
      FP_INT_TONEAREST,
  };
#endif

/* The file <bits/mathcalls.h> contains the prototypes for all the
   actual math functions.  These macros are used for those prototypes,
   so we can easily declare each function as both `name' and `__name',
   and can declare the float versions `namef' and `__namef'.  */

#define __SIMD_DECL(function) __CONCAT (__DECL_SIMD_, function)

#define __MATHCALL_VEC(function, suffix, args) 	\
  __SIMD_DECL (__MATH_PRECNAME (function, suffix)) \
  __MATHCALL (function, suffix, args)

#define __MATHDECL_VEC(type, function,suffix, args) \
  __SIMD_DECL (__MATH_PRECNAME (function, suffix)) \
  __MATHDECL(type, function,suffix, args)

#define __MATHCALL(function,suffix, args)	\
  __MATHDECL (_Mdouble_,function,suffix, args)
#define __MATHDECL(type, function,suffix, args) \
  __MATHDECL_1(type, function,suffix, args); \
  __MATHDECL_1(type, __CONCAT(__,function),suffix, args)
#define __MATHCALLX(function,suffix, args, attrib)	\
  __MATHDECLX (_Mdouble_,function,suffix, args, attrib)
#define __MATHDECLX(type, function,suffix, args, attrib) \
  __MATHDECL_1(type, function,suffix, args) __attribute__ (attrib); \
  __MATHDECL_1(type, __CONCAT(__,function),suffix, args) __attribute__ (attrib)
#define __MATHDECL_1(type, function,suffix, args) \
  extern type __MATH_PRECNAME(function,suffix) args __THROW

#define _Mdouble_		double
#define __MATH_PRECNAME(name,r)	__CONCAT(name,r)
#define __MATH_DECLARING_DOUBLE  1
#define __MATH_DECLARING_FLOATN  0
#include <bits/mathcalls-helper-functions.h>
#include <bits/mathcalls.h>
#undef	_Mdouble_
#undef	__MATH_PRECNAME
#undef __MATH_DECLARING_DOUBLE
#undef __MATH_DECLARING_FLOATN

#ifdef __USE_ISOC99


/* Include the file of declarations again, this time using `float'
   instead of `double' and appending f to each function name.  */

# ifndef _Mfloat_
#  define _Mfloat_		float
# endif
# define _Mdouble_		_Mfloat_
# define __MATH_PRECNAME(name,r) name##f##r
# define __MATH_DECLARING_DOUBLE  0
# define __MATH_DECLARING_FLOATN  0
# include <bits/mathcalls-helper-functions.h>
# include <bits/mathcalls.h>
# undef	_Mdouble_
# undef	__MATH_PRECNAME
# undef __MATH_DECLARING_DOUBLE
# undef __MATH_DECLARING_FLOATN

# if !(defined __NO_LONG_DOUBLE_MATH && defined _LIBC) \
     || defined __LDBL_COMPAT \
     || defined _LIBC_TEST
#  ifdef __LDBL_COMPAT

#   ifdef __USE_ISOC99
extern float __nldbl_nexttowardf (float __x, long double __y)
				  __THROW __attribute__ ((__const__));
#    ifdef __REDIRECT_NTH
extern float __REDIRECT_NTH (nexttowardf, (float __x, long double __y),
			     __nldbl_nexttowardf)
     __attribute__ ((__const__));
extern double __REDIRECT_NTH (nexttoward, (double __x, long double __y),
			      nextafter) __attribute__ ((__const__));
extern long double __REDIRECT_NTH (nexttowardl,
				   (long double __x, long double __y),
				   nextafter) __attribute__ ((__const__));
#    endif
#   endif

#   undef __MATHDECL_1
#   define __MATHDECL_2(type, function,suffix, args, alias) \
  extern type __REDIRECT_NTH(__MATH_PRECNAME(function,suffix), \
			     args, alias)
#   define __MATHDECL_1(type, function,suffix, args) \
  __MATHDECL_2(type, function,suffix, args, __CONCAT(function,suffix))
#  endif

/* Include the file of declarations again, this time using `long double'
   instead of `double' and appending l to each function name.  */

#  ifndef _Mlong_double_
#   define _Mlong_double_	long double
#  endif
#  define _Mdouble_		_Mlong_double_
#  define __MATH_PRECNAME(name,r) name##l##r
#  define __MATH_DECLARING_DOUBLE  0
#  define __MATH_DECLARING_FLOATN  0
#  define __MATH_DECLARE_LDOUBLE   1
#  include <bits/mathcalls-helper-functions.h>
#  include <bits/mathcalls.h>
#  undef _Mdouble_
#  undef __MATH_PRECNAME
#  undef __MATH_DECLARING_DOUBLE
#  undef __MATH_DECLARING_FLOATN

# endif /* !(__NO_LONG_DOUBLE_MATH && _LIBC) || __LDBL_COMPAT */

#endif	/* Use ISO C99.  */

/* Include the file of declarations for _FloatN and _FloatNx
   types.  */

#if __HAVE_DISTINCT_FLOAT16 || (__HAVE_FLOAT16 && !defined _LIBC)
# ifndef _Mfloat16_
#  define _Mfloat16_		_Float16
# endif
# define _Mdouble_		_Mfloat16_
# define __MATH_PRECNAME(name,r) name##f16##r
# define __MATH_DECLARING_DOUBLE  0
# define __MATH_DECLARING_FLOATN  1
# if __HAVE_DISTINCT_FLOAT16
#  include <bits/mathcalls-helper-functions.h>
# endif
# if __GLIBC_USE (IEC_60559_TYPES_EXT)
#  include <bits/mathcalls.h>
# endif
# undef _Mdouble_
# undef __MATH_PRECNAME
# undef __MATH_DECLARING_DOUBLE
# undef __MATH_DECLARING_FLOATN
#endif /* __HAVE_DISTINCT_FLOAT16 || (__HAVE_FLOAT16 && !_LIBC).  */

#if __HAVE_DISTINCT_FLOAT32 || (__HAVE_FLOAT32 && !defined _LIBC)
# ifndef _Mfloat32_
#  define _Mfloat32_		_Float32
# endif
# define _Mdouble_		_Mfloat32_
# define __MATH_PRECNAME(name,r) name##f32##r
# define __MATH_DECLARING_DOUBLE  0
# define __MATH_DECLARING_FLOATN  1
# if __HAVE_DISTINCT_FLOAT32
#  include <bits/mathcalls-helper-functions.h>
# endif
# if __GLIBC_USE (IEC_60559_TYPES_EXT)
#  include <bits/mathcalls.h>
# endif
# undef _Mdouble_
# undef __MATH_PRECNAME
# undef __MATH_DECLARING_DOUBLE
# undef __MATH_DECLARING_FLOATN
#endif /* __HAVE_DISTINCT_FLOAT32 || (__HAVE_FLOAT32 && !_LIBC).  */

#if __HAVE_DISTINCT_FLOAT64 || (__HAVE_FLOAT64 && !defined _LIBC)
# ifndef _Mfloat64_
#  define _Mfloat64_		_Float64
# endif
# define _Mdouble_		_Mfloat64_
# define __MATH_PRECNAME(name,r) name##f64##r
# define __MATH_DECLARING_DOUBLE  0
# define __MATH_DECLARING_FLOATN  1
# if __HAVE_DISTINCT_FLOAT64
#  include <bits/mathcalls-helper-functions.h>
# endif
# if __GLIBC_USE (IEC_60559_TYPES_EXT)
#  include <bits/mathcalls.h>
# endif
# undef _Mdouble_
# undef __MATH_PRECNAME
# undef __MATH_DECLARING_DOUBLE
# undef __MATH_DECLARING_FLOATN
#endif /* __HAVE_DISTINCT_FLOAT64 || (__HAVE_FLOAT64 && !_LIBC).  */

#if __HAVE_DISTINCT_FLOAT128 || (__HAVE_FLOAT128 && !defined _LIBC)
# ifndef _Mfloat128_
#  define _Mfloat128_		_Float128
# endif
# define _Mdouble_		_Mfloat128_
# define __MATH_PRECNAME(name,r) name##f128##r
# define __MATH_DECLARING_DOUBLE  0
# define __MATH_DECLARING_FLOATN  1
# if __HAVE_DISTINCT_FLOAT128
#  include <bits/mathcalls-helper-functions.h>
# endif
# if __GLIBC_USE (IEC_60559_TYPES_EXT)
#  include <bits/mathcalls.h>
# endif
# undef _Mdouble_
# undef __MATH_PRECNAME
# undef __MATH_DECLARING_DOUBLE
# undef __MATH_DECLARING_FLOATN
#endif /* __HAVE_DISTINCT_FLOAT128 || (__HAVE_FLOAT128 && !_LIBC).  */

#if __HAVE_DISTINCT_FLOAT32X || (__HAVE_FLOAT32X && !defined _LIBC)
# ifndef _Mfloat32x_
#  define _Mfloat32x_		_Float32x
# endif
# define _Mdouble_		_Mfloat32x_
# define __MATH_PRECNAME(name,r) name##f32x##r
# define __MATH_DECLARING_DOUBLE  0
# define __MATH_DECLARING_FLOATN  1
# if __HAVE_DISTINCT_FLOAT32X
#  include <bits/mathcalls-helper-functions.h>
# endif
# if __GLIBC_USE (IEC_60559_TYPES_EXT)
#  include <bits/mathcalls.h>
# endif
# undef _Mdouble_
# undef __MATH_PRECNAME
# undef __MATH_DECLARING_DOUBLE
# undef __MATH_DECLARING_FLOATN
#endif /* __HAVE_DISTINCT_FLOAT32X || (__HAVE_FLOAT32X && !_LIBC).  */

#if __HAVE_DISTINCT_FLOAT64X || (__HAVE_FLOAT64X && !defined _LIBC)
# ifndef _Mfloat64x_
#  define _Mfloat64x_		_Float64x
# endif
# define _Mdouble_		_Mfloat64x_
# define __MATH_PRECNAME(name,r) name##f64x##r
# define __MATH_DECLARING_DOUBLE  0
# define __MATH_DECLARING_FLOATN  1
# if __HAVE_DISTINCT_FLOAT64X
#  include <bits/mathcalls-helper-functions.h>
# endif
# if __GLIBC_USE (IEC_60559_TYPES_EXT)
#  include <bits/mathcalls.h>
# endif
# undef _Mdouble_
# undef __MATH_PRECNAME
# undef __MATH_DECLARING_DOUBLE
# undef __MATH_DECLARING_FLOATN
#endif /* __HAVE_DISTINCT_FLOAT64X || (__HAVE_FLOAT64X && !_LIBC).  */

#if __HAVE_DISTINCT_FLOAT128X || (__HAVE_FLOAT128X && !defined _LIBC)
# ifndef _Mfloat128x_
#  define _Mfloat128x_		_Float128x
# endif
# define _Mdouble_		_Mfloat128x_
# define __MATH_PRECNAME(name,r) name##f128x##r
# define __MATH_DECLARING_DOUBLE  0
# define __MATH_DECLARING_FLOATN  1
# if __HAVE_DISTINCT_FLOAT128X
#  include <bits/mathcalls-helper-functions.h>
# endif
# if __GLIBC_USE (IEC_60559_TYPES_EXT)
#  include <bits/mathcalls.h>
# endif
# undef _Mdouble_
# undef __MATH_PRECNAME
# undef __MATH_DECLARING_DOUBLE
# undef __MATH_DECLARING_FLOATN
#endif /* __HAVE_DISTINCT_FLOAT128X || (__HAVE_FLOAT128X && !_LIBC).  */

#undef	__MATHDECL_1
#undef	__MATHDECL
#undef	__MATHCALL


#if defined __USE_MISC || defined __USE_XOPEN
/* This variable is used by `gamma' and `lgamma'.  */
extern int signgam;
#endif


/* Depending on the type of TG_ARG, call an appropriately suffixed
   version of FUNC with arguments (including parentheses) ARGS.
   Suffixed functions may not exist for long double if it has the same
   format as double, or for other types with the same format as float,
   double or long double.  The behavior is undefined if the argument
   does not have a real floating type.  The definition may use a
   conditional expression, so all suffixed versions of FUNC must
   return the same type (FUNC may include a cast if necessary rather
   than being a single identifier).  */
#ifdef __NO_LONG_DOUBLE_MATH
# define __MATH_TG(TG_ARG, FUNC, ARGS)					\
  (sizeof (TG_ARG) == sizeof (float) ? FUNC ## f ARGS : FUNC ARGS)
#elif __HAVE_DISTINCT_FLOAT128
# if __HAVE_GENERIC_SELECTION
#  define __MATH_TG(TG_ARG, FUNC, ARGS)		\
     _Generic ((TG_ARG),			\
	       float: FUNC ## f ARGS,		\
	       default: FUNC ARGS,		\
	       long double: FUNC ## l ARGS,	\
	       _Float128: FUNC ## f128 ARGS)
# else
#  define __MATH_TG(TG_ARG, FUNC, ARGS)					\
     __builtin_choose_expr						\
     (__builtin_types_compatible_p (__typeof (TG_ARG), float),		\
      FUNC ## f ARGS,							\
      __builtin_choose_expr						\
      (__builtin_types_compatible_p (__typeof (TG_ARG), double),	\
       FUNC ARGS,							\
       __builtin_choose_expr						\
       (__builtin_types_compatible_p (__typeof (TG_ARG), long double),	\
	FUNC ## l ARGS,							\
	FUNC ## f128 ARGS)))
# endif
#else
# define __MATH_TG(TG_ARG, FUNC, ARGS)		\
  (sizeof (TG_ARG) == sizeof (float)		\
   ? FUNC ## f ARGS				\
   : sizeof (TG_ARG) == sizeof (double)		\
   ? FUNC ARGS					\
   : FUNC ## l ARGS)
#endif

/* ISO C99 defines some generic macros which work on any data type.  */
#ifdef __USE_ISOC99

/* All floating-point numbers can be put in one of these categories.  */
enum
  {
    FP_NAN =
# define FP_NAN 0
      FP_NAN,
    FP_INFINITE =
# define FP_INFINITE 1
      FP_INFINITE,
    FP_ZERO =
# define FP_ZERO 2
      FP_ZERO,
    FP_SUBNORMAL =
# define FP_SUBNORMAL 3
      FP_SUBNORMAL,
    FP_NORMAL =
# define FP_NORMAL 4
      FP_NORMAL
  };

/* GCC bug 66462 means we cannot use the math builtins with -fsignaling-nan,
   so disable builtins if this is enabled.  When fixed in a newer GCC,
   the __SUPPORT_SNAN__ check may be skipped for those versions.  */

/* Return number of classification appropriate for X.  */
# if __GNUC_PREREQ (4,4) && !defined __SUPPORT_SNAN__			      \
     && (!defined __OPTIMIZE_SIZE__ || defined __cplusplus)
     /* The check for __cplusplus allows the use of the builtin, even
	when optimization for size is on.  This is provided for
	libstdc++, only to let its configure test work when it is built
	with -Os.  No further use of this definition of fpclassify is
	expected in C++ mode, since libstdc++ provides its own version
	of fpclassify in cmath (which undefines fpclassify).  */
#  define fpclassify(x) __builtin_fpclassify (FP_NAN, FP_INFINITE,	      \
     FP_NORMAL, FP_SUBNORMAL, FP_ZERO, x)
# else
#  define fpclassify(x) __MATH_TG ((x), __fpclassify, (x))
# endif

/* Return nonzero value if sign of X is negative.  */
# if __GNUC_PREREQ (6,0)
#  define signbit(x) __builtin_signbit (x)
# elif defined __cplusplus
  /* In C++ mode, __MATH_TG cannot be used, because it relies on
     __builtin_types_compatible_p, which is a C-only builtin.
     The check for __cplusplus allows the use of the builtin instead of
     __MATH_TG. This is provided for libstdc++, only to let its configure
     test work. No further use of this definition of signbit is expected
     in C++ mode, since libstdc++ provides its own version of signbit
     in cmath (which undefines signbit). */
#  define signbit(x) __builtin_signbitl (x)
# elif __GNUC_PREREQ (4,0)
#  define signbit(x) __MATH_TG ((x), __builtin_signbit, (x))
# else
#  define signbit(x) __MATH_TG ((x), __signbit, (x))
# endif

/* Return nonzero value if X is not +-Inf or NaN.  */
# if __GNUC_PREREQ (4,4) && !defined __SUPPORT_SNAN__
#  define isfinite(x) __builtin_isfinite (x)
# else
#  define isfinite(x) __MATH_TG ((x), __finite, (x))
# endif

/* Return nonzero value if X is neither zero, subnormal, Inf, nor NaN.  */
# if __GNUC_PREREQ (4,4) && !defined __SUPPORT_SNAN__
#  define isnormal(x) __builtin_isnormal (x)
# else
#  define isnormal(x) (fpclassify (x) == FP_NORMAL)
# endif

/* Return nonzero value if X is a NaN.  We could use `fpclassify' but
   we already have this functions `__isnan' and it is faster.  */
# if __GNUC_PREREQ (4,4) && !defined __SUPPORT_SNAN__
#  define isnan(x) __builtin_isnan (x)
# else
#  define isnan(x) __MATH_TG ((x), __isnan, (x))
# endif

/* Return nonzero value if X is positive or negative infinity.  */
# if __HAVE_DISTINCT_FLOAT128 && !__GNUC_PREREQ (7,0) \
     && !defined __SUPPORT_SNAN__ && !defined __cplusplus
   /* Since __builtin_isinf_sign is broken for float128 before GCC 7.0,
      use the helper function, __isinff128, with older compilers.  This is
      only provided for C mode, because in C++ mode, GCC has no support
      for __builtin_types_compatible_p (and when in C++ mode, this macro is
      not used anyway, because libstdc++ headers undefine it).  */
#  define isinf(x) \
    (__builtin_types_compatible_p (__typeof (x), _Float128) \
     ? __isinff128 (x) : __builtin_isinf_sign (x))
# elif __GNUC_PREREQ (4,4) && !defined __SUPPORT_SNAN__
#  define isinf(x) __builtin_isinf_sign (x)
# else
#  define isinf(x) __MATH_TG ((x), __isinf, (x))
# endif

/* Bitmasks for the math_errhandling macro.  */
# define MATH_ERRNO	1	/* errno set by math functions.  */
# define MATH_ERREXCEPT	2	/* Exceptions raised by math functions.  */

/* By default all functions support both errno and exception handling.
   In gcc's fast math mode and if inline functions are defined this
   might not be true.  */
# ifndef __FAST_MATH__
#  define math_errhandling	(MATH_ERRNO | MATH_ERREXCEPT)
# endif

#endif /* Use ISO C99.  */

#if __GLIBC_USE (IEC_60559_BFP_EXT)
# include <bits/iscanonical.h>

/* Return nonzero value if X is a signaling NaN.  */
# ifndef __cplusplus
#  define issignaling(x) __MATH_TG ((x), __issignaling, (x))
# else
   /* In C++ mode, __MATH_TG cannot be used, because it relies on
      __builtin_types_compatible_p, which is a C-only builtin.  On the
      other hand, overloading provides the means to distinguish between
      the floating-point types.  The overloading resolution will match
      the correct parameter (regardless of type qualifiers (i.e.: const
      and volatile)).  */
extern "C++" {
inline int issignaling (float __val) { return __issignalingf (__val); }
inline int issignaling (double __val) { return __issignaling (__val); }
inline int
issignaling (long double __val)
{
#  ifdef __NO_LONG_DOUBLE_MATH
  return __issignaling (__val);
#  else
  return __issignalingl (__val);
#  endif
}
#  if __HAVE_DISTINCT_FLOAT128
inline int issignaling (_Float128 __val) { return __issignalingf128 (__val); }
#  endif
} /* extern C++ */
# endif

/* Return nonzero value if X is subnormal.  */
# define issubnormal(x) (fpclassify (x) == FP_SUBNORMAL)

/* Return nonzero value if X is zero.  */
# ifndef __cplusplus
#  ifdef __SUPPORT_SNAN__
#   define iszero(x) (fpclassify (x) == FP_ZERO)
#  else
#   define iszero(x) (((__typeof (x)) (x)) == 0)
#  endif
# else	/* __cplusplus */
extern "C++" {
#  ifdef __SUPPORT_SNAN__
inline int
iszero (float __val)
{
  return __fpclassifyf (__val) == FP_ZERO;
}
inline int
iszero (double __val)
{
  return __fpclassify (__val) == FP_ZERO;
}
inline int
iszero (long double __val)
{
#   ifdef __NO_LONG_DOUBLE_MATH
  return __fpclassify (__val) == FP_ZERO;
#   else
  return __fpclassifyl (__val) == FP_ZERO;
#   endif
}
#   if __HAVE_DISTINCT_FLOAT128
inline int
iszero (_Float128 __val)
{
  return __fpclassifyf128 (__val) == FP_ZERO;
}
#   endif
#  else
template <class __T> inline bool
iszero (__T __val)
{
  return __val == 0;
}
#  endif
} /* extern C++ */
# endif	/* __cplusplus */
#endif /* Use IEC_60559_BFP_EXT.  */

#ifdef __USE_XOPEN
/* X/Open wants another strange constant.  */
# define MAXFLOAT	3.40282347e+38F
#endif


/* Some useful constants.  */
#if defined __USE_MISC || defined __USE_XOPEN
# define M_E		2.7182818284590452354	/* e */
# define M_LOG2E	1.4426950408889634074	/* log_2 e */
# define M_LOG10E	0.43429448190325182765	/* log_10 e */
# define M_LN2		0.69314718055994530942	/* log_e 2 */
# define M_LN10		2.30258509299404568402	/* log_e 10 */
# define M_PI		3.14159265358979323846	/* pi */
# define M_PI_2		1.57079632679489661923	/* pi/2 */
# define M_PI_4		0.78539816339744830962	/* pi/4 */
# define M_1_PI		0.31830988618379067154	/* 1/pi */
# define M_2_PI		0.63661977236758134308	/* 2/pi */
# define M_2_SQRTPI	1.12837916709551257390	/* 2/sqrt(pi) */
# define M_SQRT2	1.41421356237309504880	/* sqrt(2) */
# define M_SQRT1_2	0.70710678118654752440	/* 1/sqrt(2) */
#endif

/* The above constants are not adequate for computation using `long double's.
   Therefore we provide as an extension constants with similar names as a
   GNU extension.  Provide enough digits for the 128-bit IEEE quad.  */
#ifdef __USE_GNU
# define M_El		2.718281828459045235360287471352662498L /* e */
# define M_LOG2El	1.442695040888963407359924681001892137L /* log_2 e */
# define M_LOG10El	0.434294481903251827651128918916605082L /* log_10 e */
# define M_LN2l		0.693147180559945309417232121458176568L /* log_e 2 */
# define M_LN10l	2.302585092994045684017991454684364208L /* log_e 10 */
# define M_PIl		3.141592653589793238462643383279502884L /* pi */
# define M_PI_2l	1.570796326794896619231321691639751442L /* pi/2 */
# define M_PI_4l	0.785398163397448309615660845819875721L /* pi/4 */
# define M_1_PIl	0.318309886183790671537767526745028724L /* 1/pi */
# define M_2_PIl	0.636619772367581343075535053490057448L /* 2/pi */
# define M_2_SQRTPIl	1.128379167095512573896158903121545172L /* 2/sqrt(pi) */
# define M_SQRT2l	1.414213562373095048801688724209698079L /* sqrt(2) */
# define M_SQRT1_2l	0.707106781186547524400844362104849039L /* 1/sqrt(2) */
#endif

#if __HAVE_FLOAT128 && defined __USE_GNU
# define M_Ef128	__f128 (2.718281828459045235360287471352662498) /* e */
# define M_LOG2Ef128	__f128 (1.442695040888963407359924681001892137) /* log_2 e */
# define M_LOG10Ef128	__f128 (0.434294481903251827651128918916605082) /* log_10 e */
# define M_LN2f128	__f128 (0.693147180559945309417232121458176568) /* log_e 2 */
# define M_LN10f128	__f128 (2.302585092994045684017991454684364208) /* log_e 10 */
# define M_PIf128	__f128 (3.141592653589793238462643383279502884) /* pi */
# define M_PI_2f128	__f128 (1.570796326794896619231321691639751442) /* pi/2 */
# define M_PI_4f128	__f128 (0.785398163397448309615660845819875721) /* pi/4 */
# define M_1_PIf128	__f128 (0.318309886183790671537767526745028724) /* 1/pi */
# define M_2_PIf128	__f128 (0.636619772367581343075535053490057448) /* 2/pi */
# define M_2_SQRTPIf128	__f128 (1.128379167095512573896158903121545172) /* 2/sqrt(pi) */
# define M_SQRT2f128	__f128 (1.414213562373095048801688724209698079) /* sqrt(2) */
# define M_SQRT1_2f128	__f128 (0.707106781186547524400844362104849039) /* 1/sqrt(2) */
#endif

/* When compiling in strict ISO C compatible mode we must not use the
   inline functions since they, among other things, do not set the
   `errno' variable correctly.  */
#if defined __STRICT_ANSI__ && !defined __NO_MATH_INLINES
# define __NO_MATH_INLINES	1
#endif

#ifdef __USE_ISOC99
# if __GNUC_PREREQ (3, 1)
/* ISO C99 defines some macros to compare number while taking care for
   unordered numbers.  Many FPUs provide special instructions to support
   these operations.  Generic support in GCC for these as builtins went
   in 2.97, but not all cpus added their patterns until 3.1.  Therefore
   we enable the builtins from 3.1 onwards and use a generic implementation
   othwerwise.  */
#  define isgreater(x, y)	__builtin_isgreater(x, y)
#  define isgreaterequal(x, y)	__builtin_isgreaterequal(x, y)
#  define isless(x, y)		__builtin_isless(x, y)
#  define islessequal(x, y)	__builtin_islessequal(x, y)
#  define islessgreater(x, y)	__builtin_islessgreater(x, y)
#  define isunordered(x, y)	__builtin_isunordered(x, y)
# else
#  define isgreater(x, y) \
  (__extension__ ({ __typeof__ (x) __x = (x); __typeof__ (y) __y = (y); \
		    !isunordered (__x, __y) && __x > __y; }))
#  define isgreaterequal(x, y) \
  (__extension__ ({ __typeof__ (x) __x = (x); __typeof__ (y) __y = (y); \
		    !isunordered (__x, __y) && __x >= __y; }))
#  define isless(x, y) \
  (__extension__ ({ __typeof__ (x) __x = (x); __typeof__ (y) __y = (y); \
		    !isunordered (__x, __y) && __x < __y; }))
#  define islessequal(x, y) \
  (__extension__ ({ __typeof__ (x) __x = (x); __typeof__ (y) __y = (y); \
		    !isunordered (__x, __y) && __x <= __y; }))
#  define islessgreater(x, y) \
  (__extension__ ({ __typeof__ (x) __x = (x); __typeof__ (y) __y = (y); \
		    !isunordered (__x, __y) && __x != __y; }))
/* isunordered must always check both operands first for signaling NaNs.  */
#  define isunordered(x, y) \
  (__extension__ ({ __typeof__ (x) __u = (x); __typeof__ (y) __v = (y); \
		    __u != __v && (__u != __u || __v != __v); }))
# endif
#endif

/* Get machine-dependent inline versions (if there are any).  */
#ifdef __USE_EXTERN_INLINES
# include <bits/mathinline.h>
#endif

/* Define special entry points to use when the compiler got told to
   only expect finite results.  */
#if defined __FINITE_MATH_ONLY__ && __FINITE_MATH_ONLY__ > 0

/* Include bits/math-finite.h for double.  */
# define _Mdouble_ double
# define __MATH_DECLARING_DOUBLE 1
# define __MATH_DECLARING_FLOATN 0
# define __REDIRFROM_X(function, reentrant) \
  function ## reentrant
# define __REDIRTO_X(function, reentrant) \
   __ ## function ## reentrant ## _finite
# include <bits/math-finite.h>
# undef _Mdouble_
# undef __MATH_DECLARING_DOUBLE
# undef __MATH_DECLARING_FLOATN
# undef __REDIRFROM_X
# undef __REDIRTO_X

/* When __USE_ISOC99 is defined, include math-finite for float and
   long double, as well.  */
# ifdef __USE_ISOC99

/* Include bits/math-finite.h for float.  */
#  define _Mdouble_ float
#  define __MATH_DECLARING_DOUBLE 0
#  define __MATH_DECLARING_FLOATN 0
#  define __REDIRFROM_X(function, reentrant) \
  function ## f ## reentrant
#  define __REDIRTO_X(function, reentrant) \
   __ ## function ## f ## reentrant ## _finite
#  include <bits/math-finite.h>
#  undef _Mdouble_
#  undef __MATH_DECLARING_DOUBLE
#  undef __MATH_DECLARING_FLOATN
#  undef __REDIRFROM_X
#  undef __REDIRTO_X

/* Include bits/math-finite.h for long double.  */
#  ifdef __MATH_DECLARE_LDOUBLE
#   define _Mdouble_ long double
#   define __MATH_DECLARING_DOUBLE 0
#   define __MATH_DECLARING_FLOATN 0
#   define __REDIRFROM_X(function, reentrant) \
  function ## l ## reentrant
#   ifdef __NO_LONG_DOUBLE_MATH
#    define __REDIRTO_X(function, reentrant) \
   __ ## function ## reentrant ## _finite
#   else
#    define __REDIRTO_X(function, reentrant) \
   __ ## function ## l ## reentrant ## _finite
#   endif
#   include <bits/math-finite.h>
#   undef _Mdouble_
#   undef __MATH_DECLARING_DOUBLE
#   undef __MATH_DECLARING_FLOATN
#   undef __REDIRFROM_X
#   undef __REDIRTO_X
#  endif

# endif /* __USE_ISOC99.  */

/* Include bits/math-finite.h for float128.  */
# if (__HAVE_DISTINCT_FLOAT128 || (__HAVE_FLOAT128 && !defined _LIBC)) \
      && __GLIBC_USE (IEC_60559_TYPES_EXT)
#  define _Mdouble_ _Float128
#  define __MATH_DECLARING_DOUBLE 0
#  define __MATH_DECLARING_FLOATN 1
#  define __REDIRFROM_X(function, reentrant) \
  function ## f128 ## reentrant
#  if __HAVE_DISTINCT_FLOAT128
#   define __REDIRTO_X(function, reentrant) \
   __ ## function ## f128 ## reentrant ## _finite
#  else
#   define __REDIRTO_X(function, reentrant) \
   __ ## function ## l ## reentrant ## _finite
#  endif
#  include <bits/math-finite.h>
#  undef _Mdouble_
#  undef __MATH_DECLARING_DOUBLE
#  undef __MATH_DECLARING_FLOATN
#  undef __REDIRFROM_X
#  undef __REDIRTO_X
# endif
#endif /* __FINITE_MATH_ONLY__ > 0.  */

#if __GLIBC_USE (IEC_60559_BFP_EXT)
/* An expression whose type has the widest of the evaluation formats
   of X and Y (which are of floating-point types).  */
# if __FLT_EVAL_METHOD__ == 2 || __FLT_EVAL_METHOD__ > 64
#  define __MATH_EVAL_FMT2(x, y) ((x) + (y) + 0.0L)
# elif __FLT_EVAL_METHOD__ == 1 || __FLT_EVAL_METHOD__ > 32
#  define __MATH_EVAL_FMT2(x, y) ((x) + (y) + 0.0)
# else
#  define __MATH_EVAL_FMT2(x, y) ((x) + (y))
# endif

/* Return X == Y but raising "invalid" and setting errno if X or Y is
   a NaN.  */
# define iseqsig(x, y) \
  __MATH_TG (__MATH_EVAL_FMT2 (x, y), __iseqsig, ((x), (y)))
#endif

__END_DECLS


#endif /* math.h  */