text: /*
   * ====================================================
   * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   *
   * Developed at SunPro, a Sun Microsystems, Inc. business.
   * Permission to use, copy, modify, and distribute this
   * software is freely granted, provided that this notice
   * is preserved.
   * ====================================================
   */
  
  /*
   * from: @(#)fdlibm.h 5.1 93/09/24
   * $FreeBSD$
   */
  
  #ifndef _MATH_PRIVATE_H_
  #define	_MATH_PRIVATE_H_
  
  #include <sys/types.h>
  
  #if HAVE_MACHINE_ENDIAN_H
  #include <machine/endian.h>
  #elif HAVE_SYS_ENDIAN_H && HOST_ANDROID
  /* Android unified headers don't have machine/endian.h */
  #include <sys/endian.h>
  #endif
  
  /*
   * The original fdlibm code used statements like:
   *	n0 = ((*(int*)&one)>>29)^1;		* index of high word *
   *	ix0 = *(n0+(int*)&x);			* high word of x *
   *	ix1 = *((1-n0)+(int*)&x);		* low word of x *
   * to dig two 32 bit words out of the 64 bit IEEE floating point
   * value.  That is non-ANSI, and, moreover, the gcc instruction
   * scheduler gets it wrong.  We instead use the following macros.
   * Unlike the original code, we determine the endianness at compile
   * time, not at run time; I don't see much benefit to selecting
   * endianness at run time.
   */
  
  /*
   * A union which permits us to convert between a double and two 32 bit
   * ints.
   */
  
  #ifdef __arm__
  #if defined(__VFP_FP__) || defined(__ARM_EABI__)
  #define	IEEE_WORD_ORDER	BYTE_ORDER
  #else
  #define	IEEE_WORD_ORDER	BIG_ENDIAN
  #endif
  #else /* __arm__ */
  #define	IEEE_WORD_ORDER	BYTE_ORDER
  #endif
  
  #if IEEE_WORD_ORDER == BIG_ENDIAN && 0 //!
  
  typedef union
  {
    double value;
    struct
    {
      u_int32_t msw;
      u_int32_t lsw;
    } parts;
    struct
    {
      u_int64_t w;
    } xparts;
  } ieee_double_shape_type;
  
  #endif
  
  #if IEEE_WORD_ORDER == LITTLE_ENDIAN
  
  typedef union
  {
    double value;
    struct
    {
      u_int32_t lsw;
      u_int32_t msw;
    } parts;
    struct
    {
      u_int64_t w;
    } xparts;
  } ieee_double_shape_type;
  
  #endif
  
  /* Get two 32 bit ints from a double.  */
  
  #define EXTRACT_WORDS(ix0,ix1,d)				\
  do {								\
    ieee_double_shape_type ew_u;					\
    ew_u.value = (d);						\
    (ix0) = ew_u.parts.msw;					\
    (ix1) = ew_u.parts.lsw;					\
  } while (0)
  
  /* Get a 64-bit int from a double. */
  #define EXTRACT_WORD64(ix,d)					\
  do {								\
    ieee_double_shape_type ew_u;					\
    ew_u.value = (d);						\
    (ix) = ew_u.xparts.w;						\
  } while (0)
  
  /* Get the more significant 32 bit int from a double.  */
  
  #define GET_HIGH_WORD(i,d)					\
  do {								\
    ieee_double_shape_type gh_u;					\
    gh_u.value = (d);						\
    (i) = gh_u.parts.msw;						\
  } while (0)
  
  /* Get the less significant 32 bit int from a double.  */
  
  #define GET_LOW_WORD(i,d)					\
  do {								\
    ieee_double_shape_type gl_u;					\
    gl_u.value = (d);						\
    (i) = gl_u.parts.lsw;						\
  } while (0)
  
  /* Set a double from two 32 bit ints.  */
  
  #define INSERT_WORDS(d,ix0,ix1)					\
  do {								\
    ieee_double_shape_type iw_u;					\
    iw_u.parts.msw = (ix0);					\
    iw_u.parts.lsw = (ix1);					\
    (d) = iw_u.value;						\
  } while (0)
  
  /* Set a double from a 64-bit int. */
  #define INSERT_WORD64(d,ix)					\
  do {								\
    ieee_double_shape_type iw_u;					\
    iw_u.xparts.w = (ix);						\
    (d) = iw_u.value;						\
  } while (0)
  
  /* Set the more significant 32 bits of a double from an int.  */
  
  #define SET_HIGH_WORD(d,v)					\
  do {								\
    ieee_double_shape_type sh_u;					\
    sh_u.value = (d);						\
    sh_u.parts.msw = (v);						\
    (d) = sh_u.value;						\
  } while (0)
  
  /* Set the less significant 32 bits of a double from an int.  */
  
  #define SET_LOW_WORD(d,v)					\
  do {								\
    ieee_double_shape_type sl_u;					\
    sl_u.value = (d);						\
    sl_u.parts.lsw = (v);						\
    (d) = sl_u.value;						\
  } while (0)
  
  /*
   * A union which permits us to convert between a float and a 32 bit
   * int.
   */
  
  typedef union
  {
    float value;
    /* FIXME: Assumes 32 bit int.  */
    unsigned int word;
  } ieee_float_shape_type;
  
  /* Get a 32 bit int from a float.  */
  
  #define GET_FLOAT_WORD(i,d)					\
  do {								\
    ieee_float_shape_type gf_u;					\
    gf_u.value = (d);						\
    (i) = gf_u.word;						\
  } while (0)
  
  /* Set a float from a 32 bit int.  */
  
  #define SET_FLOAT_WORD(d,i)					\
  do {								\
    ieee_float_shape_type sf_u;					\
    sf_u.word = (i);						\
    (d) = sf_u.value;						\
  } while (0)
  
  /*
   * Get expsign and mantissa as 16 bit and 64 bit ints from an 80 bit long
   * double.
   */
  
  #define	EXTRACT_LDBL80_WORDS(ix0,ix1,d)				\
  do {								\
    union IEEEl2bits ew_u;					\
    ew_u.e = (d);							\
    (ix0) = ew_u.xbits.expsign;					\
    (ix1) = ew_u.xbits.man;					\
  } while (0)
  
  /*
   * Get expsign and mantissa as one 16 bit and two 64 bit ints from a 128 bit
   * long double.
   */
  
  #define	EXTRACT_LDBL128_WORDS(ix0,ix1,ix2,d)			\
  do {								\
    union IEEEl2bits ew_u;					\
    ew_u.e = (d);							\
    (ix0) = ew_u.xbits.expsign;					\
    (ix1) = ew_u.xbits.manh;					\
    (ix2) = ew_u.xbits.manl;					\
  } while (0)
  
  /* Get expsign as a 16 bit int from a long double.  */
  
  #define	GET_LDBL_EXPSIGN(i,d)					\
  do {								\
    union IEEEl2bits ge_u;					\
    ge_u.e = (d);							\
    (i) = ge_u.xbits.expsign;					\
  } while (0)
  
  /*
   * Set an 80 bit long double from a 16 bit int expsign and a 64 bit int
   * mantissa.
   */
  
  #define	INSERT_LDBL80_WORDS(d,ix0,ix1)				\
  do {								\
    union IEEEl2bits iw_u;					\
    iw_u.xbits.expsign = (ix0);					\
    iw_u.xbits.man = (ix1);					\
    (d) = iw_u.e;							\
  } while (0)
  
  /*
   * Set a 128 bit long double from a 16 bit int expsign and two 64 bit ints
   * comprising the mantissa.
   */
  
  #define	INSERT_LDBL128_WORDS(d,ix0,ix1,ix2)			\
  do {								\
    union IEEEl2bits iw_u;					\
    iw_u.xbits.expsign = (ix0);					\
    iw_u.xbits.manh = (ix1);					\
    iw_u.xbits.manl = (ix2);					\
    (d) = iw_u.e;							\
  } while (0)
  
  /* Set expsign of a long double from a 16 bit int.  */
  
  #define	SET_LDBL_EXPSIGN(d,v)					\
  do {								\
    union IEEEl2bits se_u;					\
    se_u.e = (d);							\
    se_u.xbits.expsign = (v);					\
    (d) = se_u.e;							\
  } while (0)
  
  #ifdef __i386__
  /* Long double constants are broken on i386. */
  #define	LD80C(m, ex, v) {						\
  	.xbits.man = __CONCAT(m, ULL),					\
  	.xbits.expsign = (0x3fff + (ex)) | ((v) < 0 ? 0x8000 : 0),	\
  }
  #else
  /* The above works on non-i386 too, but we use this to check v. */
  #define	LD80C(m, ex, v)	{ .e = (v), }
  #endif
  
  #ifdef FLT_EVAL_METHOD
  /*
   * Attempt to get strict C99 semantics for assignment with non-C99 compilers.
   */
  #if FLT_EVAL_METHOD == 0 || __GNUC__ == 0
  #define	STRICT_ASSIGN(type, lval, rval)	((lval) = (rval))
  #else
  #define	STRICT_ASSIGN(type, lval, rval) do {	\
  	volatile type __lval;			\
  						\
  	if (sizeof(type) >= sizeof(long double))	\
  		(lval) = (rval);		\
  	else {					\
  		__lval = (rval);		\
  		(lval) = __lval;		\
  	}					\
  } while (0)
  #endif
  #endif /* FLT_EVAL_METHOD */
  
  /* Support switching the mode to FP_PE if necessary. */
  #if defined(__i386__) && !defined(NO_FPSETPREC)
  #define	ENTERI()				\
  	long double __retval;			\
  	fp_prec_t __oprec;			\
  						\
  	if ((__oprec = fpgetprec()) != FP_PE)	\
  		fpsetprec(FP_PE)
  #define	RETURNI(x) do {				\
  	__retval = (x);				\
  	if (__oprec != FP_PE)			\
  		fpsetprec(__oprec);		\
  	RETURNF(__retval);			\
  } while (0)
  #else
  #define	ENTERI(x)
  #define	RETURNI(x)	RETURNF(x)
  #endif
  
  /* Default return statement if hack*_t() is not used. */
  #define      RETURNF(v)      return (v)
  
  /*
   * 2sum gives the same result as 2sumF without requiring |a| >= |b| or
   * a == 0, but is slower.
   */
  #define	_2sum(a, b) do {	\
  	__typeof(a) __s, __w;	\
  				\
  	__w = (a) + (b);	\
  	__s = __w - (a);	\
  	(b) = ((a) - (__w - __s)) + ((b) - __s); \
  	(a) = __w;		\
  } while (0)
  
  /*
   * 2sumF algorithm.
   *
   * "Normalize" the terms in the infinite-precision expression a + b for
   * the sum of 2 floating point values so that b is as small as possible
   * relative to 'a'.  (The resulting 'a' is the value of the expression in
   * the same precision as 'a' and the resulting b is the rounding error.)
   * |a| must be >= |b| or 0, b's type must be no larger than 'a's type, and
   * exponent overflow or underflow must not occur.  This uses a Theorem of
   * Dekker (1971).  See Knuth (1981) 4.2.2 Theorem C.  The name "TwoSum"
   * is apparently due to Skewchuk (1997).
   *
   * For this to always work, assignment of a + b to 'a' must not retain any
   * extra precision in a + b.  This is required by C standards but broken
   * in many compilers.  The brokenness cannot be worked around using
   * STRICT_ASSIGN() like we do elsewhere, since the efficiency of this
   * algorithm would be destroyed by non-null strict assignments.  (The
   * compilers are correct to be broken -- the efficiency of all floating
   * point code calculations would be destroyed similarly if they forced the
   * conversions.)
   *
   * Fortunately, a case that works well can usually be arranged by building
   * any extra precision into the type of 'a' -- 'a' should have type float_t,
   * double_t or long double.  b's type should be no larger than 'a's type.
   * Callers should use these types with scopes as large as possible, to
   * reduce their own extra-precision and efficiciency problems.  In
   * particular, they shouldn't convert back and forth just to call here.
   */
  #ifdef DEBUG
  #define	_2sumF(a, b) do {				\
  	__typeof(a) __w;				\
  	volatile __typeof(a) __ia, __ib, __r, __vw;	\
  							\
  	__ia = (a);					\
  	__ib = (b);					\
  	assert(__ia == 0 || fabsl(__ia) >= fabsl(__ib));	\
  							\
  	__w = (a) + (b);				\
  	(b) = ((a) - __w) + (b);			\
  	(a) = __w;					\
  							\
  	/* The next 2 assertions are weak if (a) is already long double. */ \
  	assert((long double)__ia + __ib == (long double)(a) + (b));	\
  	__vw = __ia + __ib;				\
  	__r = __ia - __vw;				\
  	__r += __ib;					\
  	assert(__vw == (a) && __r == (b));		\
  } while (0)
  #else /* !DEBUG */
  #define	_2sumF(a, b) do {	\
  	__typeof(a) __w;	\
  				\
  	__w = (a) + (b);	\
  	(b) = ((a) - __w) + (b); \
  	(a) = __w;		\
  } while (0)
  #endif /* DEBUG */
  
  /*
   * Set x += c, where x is represented in extra precision as a + b.
   * x must be sufficiently normalized and sufficiently larger than c,
   * and the result is then sufficiently normalized.
   *
   * The details of ordering are that |a| must be >= |c| (so that (a, c)
   * can be normalized without extra work to swap 'a' with c).  The details of
   * the normalization are that b must be small relative to the normalized 'a'.
   * Normalization of (a, c) makes the normalized c tiny relative to the
   * normalized a, so b remains small relative to 'a' in the result.  However,
   * b need not ever be tiny relative to 'a'.  For example, b might be about
   * 2**20 times smaller than 'a' to give about 20 extra bits of precision.
   * That is usually enough, and adding c (which by normalization is about
   * 2**53 times smaller than a) cannot change b significantly.  However,
   * cancellation of 'a' with c in normalization of (a, c) may reduce 'a'
   * significantly relative to b.  The caller must ensure that significant
   * cancellation doesn't occur, either by having c of the same sign as 'a',
   * or by having |c| a few percent smaller than |a|.  Pre-normalization of
   * (a, b) may help.
   *
   * This is is a variant of an algorithm of Kahan (see Knuth (1981) 4.2.2
   * exercise 19).  We gain considerable efficiency by requiring the terms to
   * be sufficiently normalized and sufficiently increasing.
   */
  #define	_3sumF(a, b, c) do {	\
  	__typeof(a) __tmp;	\
  				\
  	__tmp = (c);		\
  	_2sumF(__tmp, (a));	\
  	(b) += (a);		\
  	(a) = __tmp;		\
  } while (0)
  
  /*
   * Common routine to process the arguments to nan(), nanf(), and nanl().
   */
  void _scan_nan(uint32_t *__words, int __num_words, const char *__s);
  
  #ifdef _COMPLEX_H
  
  /*
   * C99 specifies that complex numbers have the same representation as
   * an array of two elements, where the first element is the real part
   * and the second element is the imaginary part.
   */
  typedef union {
  	float complex f;
  	float a[2];
  } float_complex;
  typedef union {
  	double complex f;
  	double a[2];
  } double_complex;
  typedef union {
  	long double complex f;
  	long double a[2];
  } long_double_complex;
  #define	REALPART(z)	((z).a[0])
  #define	IMAGPART(z)	((z).a[1])
  
  /*
   * Inline functions that can be used to construct complex values.
   *
   * The C99 standard intends x+I*y to be used for this, but x+I*y is
   * currently unusable in general since gcc introduces many overflow,
   * underflow, sign and efficiency bugs by rewriting I*y as
   * (0.0+I)*(y+0.0*I) and laboriously computing the full complex product.
   * In particular, I*Inf is corrupted to NaN+I*Inf, and I*-0 is corrupted
   * to -0.0+I*0.0.
   */
  static __inline float complex
  cpackf(float x, float y)
  {
  	float_complex z;
  
  	REALPART(z) = x;
  	IMAGPART(z) = y;
  	return (z.f);
  }
  
  static __inline double complex
  cpack(double x, double y)
  {
  	double_complex z;
  
  	REALPART(z) = x;
  	IMAGPART(z) = y;
  	return (z.f);
  }
  
  static __inline long double complex
  cpackl(long double x, long double y)
  {
  	long_double_complex z;
  
  	REALPART(z) = x;
  	IMAGPART(z) = y;
  	return (z.f);
  }
  #endif /* _COMPLEX_H */
   
  #ifdef __GNUCLIKE_ASM
  
  /* Asm versions of some functions. */
  
  #ifdef __amd64__
  static __inline int
  irint(double x)
  {
  	int n;
  
  	asm("cvtsd2si %1,%0" : "=r" (n) : "x" (x));
  	return (n);
  }
  #define	HAVE_EFFICIENT_IRINT
  #endif
  
  #ifdef __i386__
  static __inline int
  irint(double x)
  {
  	int n;
  
  	asm("fistl %0" : "=m" (n) : "t" (x));
  	return (n);
  }
  #define	HAVE_EFFICIENT_IRINT
  #endif
  
  #if defined(__amd64__) || defined(__i386__)
  static __inline int
  irintl(long double x)
  {
  	int n;
  
  	asm("fistl %0" : "=m" (n) : "t" (x));
  	return (n);
  }
  #define	HAVE_EFFICIENT_IRINTL
  #endif
  
  #endif /* __GNUCLIKE_ASM */
  
  #ifdef DEBUG
  #if defined(__amd64__) || defined(__i386__)
  #define	breakpoint()	asm("int $3")
  #else
  #include <signal.h>
  
  #define	breakpoint()	raise(SIGTRAP)
  #endif
  #endif
  
  /* Write a pari script to test things externally. */
  #ifdef DOPRINT
  #include <stdio.h>
  
  #ifndef DOPRINT_SWIZZLE
  #define	DOPRINT_SWIZZLE		0
  #endif
  
  #ifdef DOPRINT_LD80
  
  #define	DOPRINT_START(xp) do {						\
  	uint64_t __lx;							\
  	uint16_t __hx;							\
  									\
  	/* Hack to give more-problematic args. */			\
  	EXTRACT_LDBL80_WORDS(__hx, __lx, *xp);				\
  	__lx ^= DOPRINT_SWIZZLE;					\
  	INSERT_LDBL80_WORDS(*xp, __hx, __lx);				\
  	printf("x = %.21Lg; ", (long double)*xp);			\
  } while (0)
  #define	DOPRINT_END1(v)							\
  	printf("y = %.21Lg; z = 0; show(x, y, z);\n", (long double)(v))
  #define	DOPRINT_END2(hi, lo)						\
  	printf("y = %.21Lg; z = %.21Lg; show(x, y, z);\n",		\
  	    (long double)(hi), (long double)(lo))
  
  #elif defined(DOPRINT_D64)
  
  #define	DOPRINT_START(xp) do {						\
  	uint32_t __hx, __lx;						\
  									\
  	EXTRACT_WORDS(__hx, __lx, *xp);					\
  	__lx ^= DOPRINT_SWIZZLE;					\
  	INSERT_WORDS(*xp, __hx, __lx);					\
  	printf("x = %.21Lg; ", (long double)*xp);			\
  } while (0)
  #define	DOPRINT_END1(v)							\
  	printf("y = %.21Lg; z = 0; show(x, y, z);\n", (long double)(v))
  #define	DOPRINT_END2(hi, lo)						\
  	printf("y = %.21Lg; z = %.21Lg; show(x, y, z);\n",		\
  	    (long double)(hi), (long double)(lo))
  
  #elif defined(DOPRINT_F32)
  
  #define	DOPRINT_START(xp) do {						\
  	uint32_t __hx;							\
  									\
  	GET_FLOAT_WORD(__hx, *xp);					\
  	__hx ^= DOPRINT_SWIZZLE;					\
  	SET_FLOAT_WORD(*xp, __hx);					\
  	printf("x = %.21Lg; ", (long double)*xp);			\
  } while (0)
  #define	DOPRINT_END1(v)							\
  	printf("y = %.21Lg; z = 0; show(x, y, z);\n", (long double)(v))
  #define	DOPRINT_END2(hi, lo)						\
  	printf("y = %.21Lg; z = %.21Lg; show(x, y, z);\n",		\
  	    (long double)(hi), (long double)(lo))
  
  #else /* !DOPRINT_LD80 && !DOPRINT_D64 (LD128 only) */
  
  #ifndef DOPRINT_SWIZZLE_HIGH
  #define	DOPRINT_SWIZZLE_HIGH	0
  #endif
  
  #define	DOPRINT_START(xp) do {						\
  	uint64_t __lx, __llx;						\
  	uint16_t __hx;							\
  									\
  	EXTRACT_LDBL128_WORDS(__hx, __lx, __llx, *xp);			\
  	__llx ^= DOPRINT_SWIZZLE;					\
  	__lx ^= DOPRINT_SWIZZLE_HIGH;					\
  	INSERT_LDBL128_WORDS(*xp, __hx, __lx, __llx);			\
  	printf("x = %.36Lg; ", (long double)*xp);					\
  } while (0)
  #define	DOPRINT_END1(v)							\
  	printf("y = %.36Lg; z = 0; show(x, y, z);\n", (long double)(v))
  #define	DOPRINT_END2(hi, lo)						\
  	printf("y = %.36Lg; z = %.36Lg; show(x, y, z);\n",		\
  	    (long double)(hi), (long double)(lo))
  
  #endif /* DOPRINT_LD80 */
  
  #else /* !DOPRINT */
  #define	DOPRINT_START(xp)
  #define	DOPRINT_END1(v)
  #define	DOPRINT_END2(hi, lo)
  #endif /* DOPRINT */
  
  #define	RETURNP(x) do {			\
  	DOPRINT_END1(x);		\
  	RETURNF(x);			\
  } while (0)
  #define	RETURNPI(x) do {		\
  	DOPRINT_END1(x);		\
  	RETURNI(x);			\
  } while (0)
  #define	RETURN2P(x, y) do {		\
  	DOPRINT_END2((x), (y));		\
  	RETURNF((x) + (y));		\
  } while (0)
  #define	RETURN2PI(x, y) do {		\
  	DOPRINT_END2((x), (y));		\
  	RETURNI((x) + (y));		\
  } while (0)
  #ifdef STRUCT_RETURN
  #define	RETURNSP(rp) do {		\
  	if (!(rp)->lo_set)		\
  		RETURNP((rp)->hi);	\
  	RETURN2P((rp)->hi, (rp)->lo);	\
  } while (0)
  #define	RETURNSPI(rp) do {		\
  	if (!(rp)->lo_set)		\
  		RETURNPI((rp)->hi);	\
  	RETURN2PI((rp)->hi, (rp)->lo);	\
  } while (0)
  #endif
  #define	SUM2P(x, y) ({			\
  	const __typeof (x) __x = (x);	\
  	const __typeof (y) __y = (y);	\
  					\
  	DOPRINT_END2(__x, __y);		\
  	__x + __y;			\
  })
  
  /*
   * ieee style elementary functions
   *
   * We rename functions here to improve other sources' diffability
   * against fdlibm.
   */
  #define	__ieee754_sqrt	sqrt
  #define	__ieee754_acos	acos
  #define	__ieee754_acosh	acosh
  #define	__ieee754_log	log
  #define	__ieee754_log2	log2
  #define	__ieee754_atanh	atanh
  #define	__ieee754_asin	asin
  #define	__ieee754_atan2	atan2
  #define	__ieee754_exp	exp
  #define	__ieee754_cosh	cosh
  #define	__ieee754_fmod	fmod
  #define	__ieee754_pow	pow
  #define	__ieee754_lgamma lgamma
  #define	__ieee754_gamma	gamma
  #define	__ieee754_lgamma_r lgamma_r
  #define	__ieee754_gamma_r gamma_r
  #define	__ieee754_log10	log10
  #define	__ieee754_sinh	sinh
  #define	__ieee754_hypot	hypot
  #define	__ieee754_j0	j0
  #define	__ieee754_j1	j1
  #define	__ieee754_y0	y0
  #define	__ieee754_y1	y1
  #define	__ieee754_jn	jn
  #define	__ieee754_yn	yn
  #define	__ieee754_remainder remainder
  #define	__ieee754_scalb	scalb
  #define	__ieee754_sqrtf	sqrtf
  #define	__ieee754_acosf	acosf
  #define	__ieee754_acoshf acoshf
  #define	__ieee754_logf	logf
  #define	__ieee754_atanhf atanhf
  #define	__ieee754_asinf	asinf
  #define	__ieee754_atan2f atan2f
  #define	__ieee754_expf	expf
  #define	__ieee754_coshf	coshf
  #define	__ieee754_fmodf	fmodf
  #define	__ieee754_powf	powf
  #define	__ieee754_lgammaf lgammaf
  #define	__ieee754_gammaf gammaf
  #define	__ieee754_lgammaf_r lgammaf_r
  #define	__ieee754_gammaf_r gammaf_r
  #define	__ieee754_log10f log10f
  #define	__ieee754_log2f log2f
  #define	__ieee754_sinhf	sinhf
  #define	__ieee754_hypotf hypotf
  #define	__ieee754_j0f	j0f
  #define	__ieee754_j1f	j1f
  #define	__ieee754_y0f	y0f
  #define	__ieee754_y1f	y1f
  #define	__ieee754_jnf	jnf
  #define	__ieee754_ynf	ynf
  #define	__ieee754_remainderf remainderf
  #define	__ieee754_scalbf scalbf
  
  /* fdlibm kernel function */
  int	__kernel_rem_pio2(double*,double*,int,int,int);
  
  /* double precision kernel functions */
  #ifndef INLINE_REM_PIO2
  int	__ieee754_rem_pio2(double,double*);
  #endif
  double	__kernel_sin(double,double,int);
  double	__kernel_cos(double,double);
  double	__kernel_tan(double,double,int);
  double	__ldexp_exp(double,int);
  #ifdef _COMPLEX_H
  double complex __ldexp_cexp(double complex,int);
  #endif
  
  /* float precision kernel functions */
  #ifndef INLINE_REM_PIO2F
  int	__ieee754_rem_pio2f(float,double*);
  #endif
  #ifndef INLINE_KERNEL_SINDF
  float	__kernel_sindf(double);
  #endif
  #ifndef INLINE_KERNEL_COSDF
  float	__kernel_cosdf(double);
  #endif
  #ifndef INLINE_KERNEL_TANDF
  float	__kernel_tandf(double,int);
  #endif
  float	__ldexp_expf(float,int);
  #ifdef _COMPLEX_H
  float complex __ldexp_cexpf(float complex,int);
  #endif
  
  /* long double precision kernel functions */
  long double __kernel_sinl(long double, long double, int);
  long double __kernel_cosl(long double, long double);
  long double __kernel_tanl(long double, long double, int);
  
  #endif /* !_MATH_PRIVATE_H_ */