/*
* ====================================================
* 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>
#include <machine/endian.h>
/*
* 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
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_ */