SDL  2.0
Data Structures | Macros | Typedefs | Functions
math_private.h File Reference
#include "SDL_endian.h"
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Data Structures

union  ieee_double_shape_type
union  ieee_float_shape_type

Macros

#define _IEEE_LIBM
#define attribute_hidden
#define libm_hidden_proto(x)
#define libm_hidden_def(x)
#define strong_alias(x, y)
#define atan   SDL_uclibc_atan
#define __ieee754_atan2   SDL_uclibc_atan2
#define copysign   SDL_uclibc_copysign
#define cos   SDL_uclibc_cos
#define __ieee754_exp   SDL_uclibc_exp
#define fabs   SDL_uclibc_fabs
#define floor   SDL_uclibc_floor
#define __ieee754_fmod   SDL_uclibc_fmod
#define __ieee754_log   SDL_uclibc_log
#define __ieee754_log10   SDL_uclibc_log10
#define __ieee754_pow   SDL_uclibc_pow
#define scalbln   SDL_uclibc_scalbln
#define scalbn   SDL_uclibc_scalbn
#define sin   SDL_uclibc_sin
#define __ieee754_sqrt   SDL_uclibc_sqrt
#define tan   SDL_uclibc_tan
#define EXTRACT_WORDS(ix0, ix1, d)
#define GET_HIGH_WORD(i, d)
#define GET_LOW_WORD(i, d)
#define INSERT_WORDS(d, ix0, ix1)
#define SET_HIGH_WORD(d, v)
#define SET_LOW_WORD(d, v)
#define GET_FLOAT_WORD(i, d)
#define SET_FLOAT_WORD(d, i)

Typedefs

typedef unsigned int u_int32_t

Functions

double __ieee754_sqrt (double) attribute_hidden
double __ieee754_acos (double) attribute_hidden
double __ieee754_acosh (double) attribute_hidden
double __ieee754_log (double) attribute_hidden
double __ieee754_atanh (double) attribute_hidden
double __ieee754_asin (double) attribute_hidden
double __ieee754_atan2 (double, double) attribute_hidden
double __ieee754_exp (double) attribute_hidden
double __ieee754_cosh (double) attribute_hidden
double __ieee754_fmod (double, double) attribute_hidden
double __ieee754_pow (double, double) attribute_hidden
double __ieee754_lgamma_r (double, int *) attribute_hidden
double __ieee754_gamma_r (double, int *) attribute_hidden
double __ieee754_lgamma (double) attribute_hidden
double __ieee754_gamma (double) attribute_hidden
double __ieee754_log10 (double) attribute_hidden
double __ieee754_sinh (double) attribute_hidden
double __ieee754_hypot (double, double) attribute_hidden
double __ieee754_j0 (double) attribute_hidden
double __ieee754_j1 (double) attribute_hidden
double __ieee754_y0 (double) attribute_hidden
double __ieee754_y1 (double) attribute_hidden
double __ieee754_jn (int, double) attribute_hidden
double __ieee754_yn (int, double) attribute_hidden
double __ieee754_remainder (double, double) attribute_hidden
int32_t __ieee754_rem_pio2 (double, double *) attribute_hidden
double __ieee754_scalb (double, double) attribute_hidden
double __kernel_sin (double, double, int) attribute_hidden
double __kernel_cos (double, double) attribute_hidden
double __kernel_tan (double, double, int) attribute_hidden
int32_t __kernel_rem_pio2 (double *, double *, int, int, const unsigned int, const int32_t *) attribute_hidden

Macro Definition Documentation

#define __ieee754_atan2   SDL_uclibc_atan2

Definition at line 35 of file math_private.h.

#define __ieee754_exp   SDL_uclibc_exp

Definition at line 38 of file math_private.h.

#define __ieee754_fmod   SDL_uclibc_fmod

Definition at line 41 of file math_private.h.

#define __ieee754_log   SDL_uclibc_log

Definition at line 42 of file math_private.h.

Referenced by __ieee754_log10().

#define __ieee754_log10   SDL_uclibc_log10

Definition at line 43 of file math_private.h.

#define __ieee754_pow   SDL_uclibc_pow

Definition at line 44 of file math_private.h.

#define __ieee754_sqrt   SDL_uclibc_sqrt

Definition at line 48 of file math_private.h.

Referenced by __ieee754_pow().

#define _IEEE_LIBM

Definition at line 24 of file math_private.h.

#define atan   SDL_uclibc_atan

Definition at line 34 of file math_private.h.

#define attribute_hidden

Definition at line 25 of file math_private.h.

#define copysign   SDL_uclibc_copysign

Definition at line 36 of file math_private.h.

#define cos   SDL_uclibc_cos

Definition at line 37 of file math_private.h.

#define EXTRACT_WORDS (   ix0,
  ix1,
  d 
)
Value:
do { \
ieee_double_shape_type ew_u; \
ew_u.value = (d); \
(ix0) = ew_u.parts.msw; \
(ix1) = ew_u.parts.lsw; \
} while (0)

Definition at line 99 of file math_private.h.

Referenced by __ieee754_atan2(), __ieee754_fmod(), __ieee754_log(), __ieee754_log10(), __ieee754_pow(), __ieee754_sqrt(), floor(), and scalbln().

#define fabs   SDL_uclibc_fabs

Definition at line 39 of file math_private.h.

#define floor   SDL_uclibc_floor

Definition at line 40 of file math_private.h.

#define GET_FLOAT_WORD (   i,
  d 
)
Value:
do { \
ieee_float_shape_type gf_u; \
gf_u.value = (d); \
(i) = gf_u.word; \
} while (0)

Definition at line 166 of file math_private.h.

#define GET_HIGH_WORD (   i,
  d 
)
Value:
do { \
ieee_double_shape_type gh_u; \
gh_u.value = (d); \
(i) = gh_u.parts.msw; \
} while (0)

Definition at line 109 of file math_private.h.

Referenced by __ieee754_atan2(), __ieee754_exp(), __ieee754_log(), __ieee754_log10(), __ieee754_pow(), __ieee754_rem_pio2(), __kernel_cos(), __kernel_sin(), __kernel_tan(), atan(), copysign(), cos(), fabs(), scalbln(), sin(), and tan().

#define GET_LOW_WORD (   i,
  d 
)
Value:
do { \
ieee_double_shape_type gl_u; \
gl_u.value = (d); \
(i) = gl_u.parts.lsw; \
} while (0)

Definition at line 118 of file math_private.h.

Referenced by __ieee754_exp(), __ieee754_rem_pio2(), __kernel_tan(), and atan().

#define INSERT_WORDS (   d,
  ix0,
  ix1 
)
Value:
do { \
ieee_double_shape_type iw_u; \
iw_u.parts.msw = (ix0); \
iw_u.parts.lsw = (ix1); \
(d) = iw_u.value; \
} while (0)

Definition at line 127 of file math_private.h.

Referenced by __ieee754_fmod(), __ieee754_sqrt(), __kernel_cos(), and floor().

#define libm_hidden_def (   x)

Definition at line 27 of file math_private.h.

#define libm_hidden_proto (   x)

Definition at line 26 of file math_private.h.

#define scalbln   SDL_uclibc_scalbln

Definition at line 45 of file math_private.h.

#define scalbn   SDL_uclibc_scalbn

Definition at line 46 of file math_private.h.

Referenced by __ieee754_pow(), __kernel_rem_pio2(), and SDL_scalbn().

#define SET_FLOAT_WORD (   d,
  i 
)
Value:
do { \
ieee_float_shape_type sf_u; \
sf_u.word = (i); \
(d) = sf_u.value; \
} while (0)

Definition at line 175 of file math_private.h.

#define SET_HIGH_WORD (   d,
  v 
)
Value:
do { \
ieee_double_shape_type sh_u; \
sh_u.value = (d); \
sh_u.parts.msw = (v); \
(d) = sh_u.value; \
} while (0)

Definition at line 137 of file math_private.h.

Referenced by __ieee754_atan2(), __ieee754_exp(), __ieee754_log(), __ieee754_log10(), __ieee754_pow(), __ieee754_rem_pio2(), copysign(), fabs(), and scalbln().

#define SET_LOW_WORD (   d,
  v 
)
Value:
do { \
ieee_double_shape_type sl_u; \
sl_u.value = (d); \
sl_u.parts.lsw = (v); \
(d) = sl_u.value; \
} while (0)

Definition at line 147 of file math_private.h.

Referenced by __ieee754_pow(), __ieee754_rem_pio2(), and __kernel_tan().

#define sin   SDL_uclibc_sin

Definition at line 47 of file math_private.h.

#define strong_alias (   x,
  y 
)

Definition at line 28 of file math_private.h.

#define tan   SDL_uclibc_tan

Definition at line 49 of file math_private.h.

Typedef Documentation

typedef unsigned int u_int32_t

Definition at line 31 of file math_private.h.

Function Documentation

double __ieee754_acos ( double  )
double __ieee754_acosh ( double  )
double __ieee754_asin ( double  )
double __ieee754_atan2 ( double  ,
double   
)

Definition at line 50 of file e_atan2.c.

References atan(), EXTRACT_WORDS, fabs(), GET_HIGH_WORD, k, pi, pi_lo, pi_o_2, pi_o_4, SET_HIGH_WORD, tiny, and zero.

{
double z;
int32_t k,m,hx,hy,ix,iy;
u_int32_t lx,ly;
EXTRACT_WORDS(hx,lx,x);
ix = hx&0x7fffffff;
EXTRACT_WORDS(hy,ly,y);
iy = hy&0x7fffffff;
if(((ix|((lx|-(int32_t)lx)>>31))>0x7ff00000)||
((iy|((ly|-(int32_t)ly)>>31))>0x7ff00000)) /* x or y is NaN */
return x+y;
if(((hx-0x3ff00000)|lx)==0) return atan(y); /* x=1.0 */
m = ((hy>>31)&1)|((hx>>30)&2); /* 2*sign(x)+sign(y) */
/* when y = 0 */
if((iy|ly)==0) {
switch(m) {
case 0:
case 1: return y; /* atan(+-0,+anything)=+-0 */
case 2: return pi+tiny;/* atan(+0,-anything) = pi */
case 3: return -pi-tiny;/* atan(-0,-anything) =-pi */
}
}
/* when x = 0 */
if((ix|lx)==0) return (hy<0)? -pi_o_2-tiny: pi_o_2+tiny;
/* when x is INF */
if(ix==0x7ff00000) {
if(iy==0x7ff00000) {
switch(m) {
case 0: return pi_o_4+tiny;/* atan(+INF,+INF) */
case 1: return -pi_o_4-tiny;/* atan(-INF,+INF) */
case 2: return 3.0*pi_o_4+tiny;/*atan(+INF,-INF)*/
case 3: return -3.0*pi_o_4-tiny;/*atan(-INF,-INF)*/
}
} else {
switch(m) {
case 0: return zero ; /* atan(+...,+INF) */
case 1: return -zero ; /* atan(-...,+INF) */
case 2: return pi+tiny ; /* atan(+...,-INF) */
case 3: return -pi-tiny ; /* atan(-...,-INF) */
}
}
}
/* when y is INF */
if(iy==0x7ff00000) return (hy<0)? -pi_o_2-tiny: pi_o_2+tiny;
/* compute y/x */
k = (iy-ix)>>20;
if(k > 60) z=pi_o_2+0.5*pi_lo; /* |y/x| > 2**60 */
else if(hx<0&&k<-60) z=0.0; /* |y|/x < -2**60 */
else z=atan(fabs(y/x)); /* safe to do y/x */
switch (m) {
case 0: return z ; /* atan(+,+) */
case 1: {
u_int32_t zh;
GET_HIGH_WORD(zh,z);
SET_HIGH_WORD(z,zh ^ 0x80000000);
}
return z ; /* atan(-,+) */
case 2: return pi-(z-pi_lo);/* atan(+,-) */
default: /* case 3 */
return (z-pi_lo)-pi;/* atan(-,-) */
}
}
double __ieee754_atanh ( double  )
double __ieee754_cosh ( double  )
double __ieee754_exp ( double  )

Definition at line 96 of file e_exp.c.

References GET_HIGH_WORD, GET_LOW_WORD, halF, huge, invln2, k, ln2HI, ln2LO, o_threshold, one, P1, P2, P3, P4, P5, SET_HIGH_WORD, twom1000, and u_threshold.

{
double y;
double hi = 0.0;
double lo = 0.0;
double c;
double t;
int32_t k=0;
int32_t xsb;
u_int32_t hx;
GET_HIGH_WORD(hx,x);
xsb = (hx>>31)&1; /* sign bit of x */
hx &= 0x7fffffff; /* high word of |x| */
/* filter out non-finite argument */
if(hx >= 0x40862E42) { /* if |x|>=709.78... */
if(hx>=0x7ff00000) {
u_int32_t lx;
GET_LOW_WORD(lx,x);
if(((hx&0xfffff)|lx)!=0)
return x+x; /* NaN */
else return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */
}
#if 1
if(x > o_threshold) return huge*huge; /* overflow */
#else /* !!! FIXME: check this: "huge * huge" is a compiler warning, maybe they wanted +Inf? */
if(x > o_threshold) return INFINITY; /* overflow */
#endif
if(x < u_threshold) return twom1000*twom1000; /* underflow */
}
/* argument reduction */
if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb;
} else {
k = (int32_t) (invln2*x+halF[xsb]);
t = k;
hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */
lo = t*ln2LO[0];
}
x = hi - lo;
}
else if(hx < 0x3e300000) { /* when |x|<2**-28 */
if(huge+x>one) return one+x;/* trigger inexact */
}
else k = 0;
/* x is now in primary range */
t = x*x;
c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
if(k==0) return one-((x*c)/(c-2.0)-x);
else y = one-((lo-(x*c)/(2.0-c))-hi);
if(k >= -1021) {
u_int32_t hy;
GET_HIGH_WORD(hy,y);
SET_HIGH_WORD(y,hy+(k<<20)); /* add k to y's exponent */
return y;
} else {
u_int32_t hy;
GET_HIGH_WORD(hy,y);
SET_HIGH_WORD(y,hy+((k+1000)<<20)); /* add k to y's exponent */
return y*twom1000;
}
}
double __ieee754_fmod ( double  ,
double   
)

Definition at line 23 of file e_fmod.c.

References EXTRACT_WORDS, i, INSERT_WORDS, one, and Zero.

{
int32_t n,hx,hy,hz,ix,iy,sx,i;
u_int32_t lx,ly,lz;
EXTRACT_WORDS(hx,lx,x);
EXTRACT_WORDS(hy,ly,y);
sx = hx&0x80000000; /* sign of x */
hx ^=sx; /* |x| */
hy &= 0x7fffffff; /* |y| */
/* purge off exception values */
if((hy|ly)==0||(hx>=0x7ff00000)|| /* y=0,or x not finite */
((hy|((ly|-(int32_t)ly)>>31))>0x7ff00000)) /* or y is NaN */
return (x*y)/(x*y);
if(hx<=hy) {
if((hx<hy)||(lx<ly)) return x; /* |x|<|y| return x */
if(lx==ly)
return Zero[(u_int32_t)sx>>31]; /* |x|=|y| return x*0*/
}
/* determine ix = ilogb(x) */
if(hx<0x00100000) { /* subnormal x */
if(hx==0) {
for (ix = -1043, i=lx; i>0; i<<=1) ix -=1;
} else {
for (ix = -1022,i=(hx<<11); i>0; i<<=1) ix -=1;
}
} else ix = (hx>>20)-1023;
/* determine iy = ilogb(y) */
if(hy<0x00100000) { /* subnormal y */
if(hy==0) {
for (iy = -1043, i=ly; i>0; i<<=1) iy -=1;
} else {
for (iy = -1022,i=(hy<<11); i>0; i<<=1) iy -=1;
}
} else iy = (hy>>20)-1023;
/* set up {hx,lx}, {hy,ly} and align y to x */
if(ix >= -1022)
hx = 0x00100000|(0x000fffff&hx);
else { /* subnormal x, shift x to normal */
n = -1022-ix;
if(n<=31) {
hx = (hx<<n)|(lx>>(32-n));
lx <<= n;
} else {
hx = lx<<(n-32);
lx = 0;
}
}
if(iy >= -1022)
hy = 0x00100000|(0x000fffff&hy);
else { /* subnormal y, shift y to normal */
n = -1022-iy;
if(n<=31) {
hy = (hy<<n)|(ly>>(32-n));
ly <<= n;
} else {
hy = ly<<(n-32);
ly = 0;
}
}
/* fix point fmod */
n = ix - iy;
while(n--) {
hz=hx-hy;lz=lx-ly; if(lx<ly) hz -= 1;
if(hz<0){hx = hx+hx+(lx>>31); lx = lx+lx;}
else {
if((hz|lz)==0) /* return sign(x)*0 */
return Zero[(u_int32_t)sx>>31];
hx = hz+hz+(lz>>31); lx = lz+lz;
}
}
hz=hx-hy;lz=lx-ly; if(lx<ly) hz -= 1;
if(hz>=0) {hx=hz;lx=lz;}
/* convert back to floating value and restore the sign */
if((hx|lx)==0) /* return sign(x)*0 */
return Zero[(u_int32_t)sx>>31];
while(hx<0x00100000) { /* normalize x */
hx = hx+hx+(lx>>31); lx = lx+lx;
iy -= 1;
}
if(iy>= -1022) { /* normalize output */
hx = ((hx-0x00100000)|((iy+1023)<<20));
INSERT_WORDS(x,hx|sx,lx);
} else { /* subnormal output */
n = -1022 - iy;
if(n<=20) {
lx = (lx>>n)|((u_int32_t)hx<<(32-n));
hx >>= n;
} else if (n<=31) {
lx = (hx<<(32-n))|(lx>>n); hx = sx;
} else {
lx = hx>>(n-32); hx = sx;
}
INSERT_WORDS(x,hx|sx,lx);
x *= one; /* create necessary signal */
}
return x; /* exact output */
}
double __ieee754_gamma ( double  )
double __ieee754_gamma_r ( double  ,
int *   
)
double __ieee754_hypot ( double  ,
double   
)
double __ieee754_j0 ( double  )
double __ieee754_j1 ( double  )
double __ieee754_jn ( int  ,
double   
)
double __ieee754_lgamma ( double  )
double __ieee754_lgamma_r ( double  ,
int *   
)
double __ieee754_log ( double  )

Definition at line 85 of file e_log.c.

References EXTRACT_WORDS, GET_HIGH_WORD, i, j, k, Lg1, Lg2, Lg3, Lg4, Lg5, Lg6, Lg7, ln2_hi, ln2_lo, SET_HIGH_WORD, two54, and zero.

{
double hfsq,f,s,z,R,w,t1,t2,dk;
int32_t k,hx,i,j;
u_int32_t lx;
EXTRACT_WORDS(hx,lx,x);
k=0;
if (hx < 0x00100000) { /* x < 2**-1022 */
if (((hx&0x7fffffff)|lx)==0)
return -two54/zero; /* log(+-0)=-inf */
if (hx<0) return (x-x)/zero; /* log(-#) = NaN */
k -= 54; x *= two54; /* subnormal number, scale up x */
GET_HIGH_WORD(hx,x);
}
if (hx >= 0x7ff00000) return x+x;
k += (hx>>20)-1023;
hx &= 0x000fffff;
i = (hx+0x95f64)&0x100000;
SET_HIGH_WORD(x,hx|(i^0x3ff00000)); /* normalize x or x/2 */
k += (i>>20);
f = x-1.0;
if((0x000fffff&(2+hx))<3) { /* |f| < 2**-20 */
if(f==zero) {if(k==0) return zero; else {dk=(double)k;
return dk*ln2_hi+dk*ln2_lo;}
}
R = f*f*(0.5-0.33333333333333333*f);
if(k==0) return f-R; else {dk=(double)k;
return dk*ln2_hi-((R-dk*ln2_lo)-f);}
}
s = f/(2.0+f);
dk = (double)k;
z = s*s;
i = hx-0x6147a;
w = z*z;
j = 0x6b851-hx;
t1= w*(Lg2+w*(Lg4+w*Lg6));
t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
i |= j;
R = t2+t1;
if(i>0) {
hfsq=0.5*f*f;
if(k==0) return f-(hfsq-s*(hfsq+R)); else
return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f);
} else {
if(k==0) return f-s*(f-R); else
return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f);
}
}
double __ieee754_log10 ( double  )

Definition at line 61 of file e_log10.c.

References __ieee754_log, EXTRACT_WORDS, GET_HIGH_WORD, i, ivln10, k, log10_2hi, log10_2lo, SET_HIGH_WORD, two54, and zero.

{
double y,z;
int32_t i,k,hx;
u_int32_t lx;
EXTRACT_WORDS(hx,lx,x);
k=0;
if (hx < 0x00100000) { /* x < 2**-1022 */
if (((hx&0x7fffffff)|lx)==0)
return -two54/zero; /* log(+-0)=-inf */
if (hx<0) return (x-x)/zero; /* log(-#) = NaN */
k -= 54; x *= two54; /* subnormal number, scale up x */
GET_HIGH_WORD(hx,x);
}
if (hx >= 0x7ff00000) return x+x;
k += (hx>>20)-1023;
i = ((u_int32_t)k&0x80000000)>>31;
hx = (hx&0x000fffff)|((0x3ff-i)<<20);
y = (double)(k+i);
SET_HIGH_WORD(x,hx);
z = y*log10_2lo + ivln10*__ieee754_log(x);
return z+y*log10_2hi;
}
double __ieee754_pow ( double  ,
double   
)

Definition at line 99 of file e_pow.c.

References __ieee754_sqrt, bp, cp, cp_h, cp_l, dp_h, dp_l, EXTRACT_WORDS, fabs(), GET_HIGH_WORD, huge, i, ivln2, ivln2_h, ivln2_l, j, k, L1, L2, L3, L4, L5, L6, lg2, lg2_h, lg2_l, one, ovt, P1, P2, P3, P4, P5, scalbn, SET_HIGH_WORD, SET_LOW_WORD, tiny, two, two53, and zero.

{
double z,ax,z_h,z_l,p_h,p_l;
double y1,t1,t2,r,s,t,u,v,w;
int32_t i,j,k,yisint,n;
int32_t hx,hy,ix,iy;
u_int32_t lx,ly;
EXTRACT_WORDS(hx,lx,x);
/* x==1: 1**y = 1 (even if y is NaN) */
if (hx==0x3ff00000 && lx==0) {
return x;
}
ix = hx&0x7fffffff;
EXTRACT_WORDS(hy,ly,y);
iy = hy&0x7fffffff;
/* y==zero: x**0 = 1 */
if((iy|ly)==0) return one;
/* +-NaN return x+y */
if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) ||
iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0)))
return x+y;
/* determine if y is an odd int when x < 0
* yisint = 0 ... y is not an integer
* yisint = 1 ... y is an odd int
* yisint = 2 ... y is an even int
*/
yisint = 0;
if(hx<0) {
if(iy>=0x43400000) yisint = 2; /* even integer y */
else if(iy>=0x3ff00000) {
k = (iy>>20)-0x3ff; /* exponent */
if(k>20) {
j = ly>>(52-k);
if((j<<(52-k))==ly) yisint = 2-(j&1);
} else if(ly==0) {
j = iy>>(20-k);
if((j<<(20-k))==iy) yisint = 2-(j&1);
}
}
}
/* special value of y */
if(ly==0) {
if (iy==0x7ff00000) { /* y is +-inf */
if (((ix-0x3ff00000)|lx)==0)
return one; /* +-1**+-inf is 1 (yes, weird rule) */
if (ix >= 0x3ff00000) /* (|x|>1)**+-inf = inf,0 */
return (hy>=0) ? y : zero;
/* (|x|<1)**-,+inf = inf,0 */
return (hy<0) ? -y : zero;
}
if(iy==0x3ff00000) { /* y is +-1 */
if(hy<0) return one/x; else return x;
}
if(hy==0x40000000) return x*x; /* y is 2 */
if(hy==0x3fe00000) { /* y is 0.5 */
if(hx>=0) /* x >= +0 */
return __ieee754_sqrt(x);
}
}
ax = fabs(x);
/* special value of x */
if(lx==0) {
if(ix==0x7ff00000||ix==0||ix==0x3ff00000){
z = ax; /*x is +-0,+-inf,+-1*/
if(hy<0) z = one/z; /* z = (1/|x|) */
if(hx<0) {
if(((ix-0x3ff00000)|yisint)==0) {
z = (z-z)/(z-z); /* (-1)**non-int is NaN */
} else if(yisint==1)
z = -z; /* (x<0)**odd = -(|x|**odd) */
}
return z;
}
}
/* (x<0)**(non-int) is NaN */
if(((((u_int32_t)hx>>31)-1)|yisint)==0) return (x-x)/(x-x);
/* |y| is huge */
if(iy>0x41e00000) { /* if |y| > 2**31 */
if(iy>0x43f00000){ /* if |y| > 2**64, must o/uflow */
if(ix<=0x3fefffff) return (hy<0)? huge*huge:tiny*tiny;
if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny;
}
/* over/underflow if x is not close to one */
if(ix<0x3fefffff) return (hy<0)? huge*huge:tiny*tiny;
if(ix>0x3ff00000) return (hy>0)? huge*huge:tiny*tiny;
/* now |1-x| is tiny <= 2**-20, suffice to compute
log(x) by x-x^2/2+x^3/3-x^4/4 */
t = x-1; /* t has 20 trailing zeros */
w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25));
u = ivln2_h*t; /* ivln2_h has 21 sig. bits */
v = t*ivln2_l-w*ivln2;
t1 = u+v;
SET_LOW_WORD(t1,0);
t2 = v-(t1-u);
} else {
double s2,s_h,s_l,t_h,t_l;
n = 0;
/* take care subnormal number */
if(ix<0x00100000)
{ax *= two53; n -= 53; GET_HIGH_WORD(ix,ax); }
n += ((ix)>>20)-0x3ff;
j = ix&0x000fffff;
/* determine interval */
ix = j|0x3ff00000; /* normalize ix */
if(j<=0x3988E) k=0; /* |x|<sqrt(3/2) */
else if(j<0xBB67A) k=1; /* |x|<sqrt(3) */
else {k=0;n+=1;ix -= 0x00100000;}
SET_HIGH_WORD(ax,ix);
/* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */
v = one/(ax+bp[k]);
s = u*v;
s_h = s;
SET_LOW_WORD(s_h,0);
/* t_h=ax+bp[k] High */
t_h = zero;
SET_HIGH_WORD(t_h,((ix>>1)|0x20000000)+0x00080000+(k<<18));
t_l = ax - (t_h-bp[k]);
s_l = v*((u-s_h*t_h)-s_h*t_l);
/* compute log(ax) */
s2 = s*s;
r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
r += s_l*(s_h+s);
s2 = s_h*s_h;
t_h = 3.0+s2+r;
SET_LOW_WORD(t_h,0);
t_l = r-((t_h-3.0)-s2);
/* u+v = s*(1+...) */
u = s_h*t_h;
v = s_l*t_h+t_l*s;
/* 2/(3log2)*(s+...) */
p_h = u+v;
SET_LOW_WORD(p_h,0);
p_l = v-(p_h-u);
z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */
z_l = cp_l*p_h+p_l*cp+dp_l[k];
/* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
t = (double)n;
t1 = (((z_h+z_l)+dp_h[k])+t);
SET_LOW_WORD(t1,0);
t2 = z_l-(((t1-t)-dp_h[k])-z_h);
}
s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
if(((((u_int32_t)hx>>31)-1)|(yisint-1))==0)
s = -one;/* (-ve)**(odd int) */
/* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
y1 = y;
SET_LOW_WORD(y1,0);
p_l = (y-y1)*t1+y*t2;
p_h = y1*t1;
z = p_l+p_h;
EXTRACT_WORDS(j,i,z);
if (j>=0x40900000) { /* z >= 1024 */
if(((j-0x40900000)|i)!=0) /* if z > 1024 */
return s*huge*huge; /* overflow */
else {
if(p_l+ovt>z-p_h) return s*huge*huge; /* overflow */
}
} else if((j&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */
if(((j-0xc090cc00)|i)!=0) /* z < -1075 */
return s*tiny*tiny; /* underflow */
else {
if(p_l<=z-p_h) return s*tiny*tiny; /* underflow */
}
}
/*
* compute 2**(p_h+p_l)
*/
i = j&0x7fffffff;
k = (i>>20)-0x3ff;
n = 0;
if(i>0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */
n = j+(0x00100000>>(k+1));
k = ((n&0x7fffffff)>>20)-0x3ff; /* new k for n */
t = zero;
SET_HIGH_WORD(t,n&~(0x000fffff>>k));
n = ((n&0x000fffff)|0x00100000)>>(20-k);
if(j<0) n = -n;
p_h -= t;
}
t = p_l+p_h;
SET_LOW_WORD(t,0);
u = t*lg2_h;
v = (p_l-(t-p_h))*lg2+t*lg2_l;
z = u+v;
w = v-(z-u);
t = z*z;
t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
r = (z*t1)/(t1-two)-(w+z*w);
z = one-(r-z);
GET_HIGH_WORD(j,z);
j += (n<<20);
if((j>>20)<=0) z = scalbn(z,n); /* subnormal output */
else SET_HIGH_WORD(z,j);
return s*z;
}
int32_t __ieee754_rem_pio2 ( double  ,
double *   
)

Definition at line 69 of file e_rem_pio2.c.

References __kernel_rem_pio2(), fabs(), GET_HIGH_WORD, GET_LOW_WORD, half, i, invpio2, j, npio2_hw, pio2_1, pio2_1t, pio2_2, pio2_2t, pio2_3, pio2_3t, SET_HIGH_WORD, SET_LOW_WORD, two24, two_over_pi, and zero.

Referenced by cos(), sin(), and tan().

{
double z=0.0,w,t,r,fn;
double tx[3];
int32_t e0,i,j,nx,n,ix,hx;
u_int32_t low;
GET_HIGH_WORD(hx,x); /* high word of x */
ix = hx&0x7fffffff;
if(ix<=0x3fe921fb) /* |x| ~<= pi/4 , no need for reduction */
{y[0] = x; y[1] = 0; return 0;}
if(ix<0x4002d97c) { /* |x| < 3pi/4, special case with n=+-1 */
if(hx>0) {
z = x - pio2_1;
if(ix!=0x3ff921fb) { /* 33+53 bit pi is good enough */
y[0] = z - pio2_1t;
y[1] = (z-y[0])-pio2_1t;
} else { /* near pi/2, use 33+33+53 bit pi */
z -= pio2_2;
y[0] = z - pio2_2t;
y[1] = (z-y[0])-pio2_2t;
}
return 1;
} else { /* negative x */
z = x + pio2_1;
if(ix!=0x3ff921fb) { /* 33+53 bit pi is good enough */
y[0] = z + pio2_1t;
y[1] = (z-y[0])+pio2_1t;
} else { /* near pi/2, use 33+33+53 bit pi */
z += pio2_2;
y[0] = z + pio2_2t;
y[1] = (z-y[0])+pio2_2t;
}
return -1;
}
}
if(ix<=0x413921fb) { /* |x| ~<= 2^19*(pi/2), medium size */
t = fabs(x);
n = (int32_t) (t*invpio2+half);
fn = (double)n;
r = t-fn*pio2_1;
w = fn*pio2_1t; /* 1st round good to 85 bit */
if(n<32&&ix!=npio2_hw[n-1]) {
y[0] = r-w; /* quick check no cancellation */
} else {
u_int32_t high;
j = ix>>20;
y[0] = r-w;
GET_HIGH_WORD(high,y[0]);
i = j-((high>>20)&0x7ff);
if(i>16) { /* 2nd iteration needed, good to 118 */
t = r;
w = fn*pio2_2;
r = t-w;
w = fn*pio2_2t-((t-r)-w);
y[0] = r-w;
GET_HIGH_WORD(high,y[0]);
i = j-((high>>20)&0x7ff);
if(i>49) { /* 3rd iteration need, 151 bits acc */
t = r; /* will cover all possible cases */
w = fn*pio2_3;
r = t-w;
w = fn*pio2_3t-((t-r)-w);
y[0] = r-w;
}
}
}
y[1] = (r-y[0])-w;
if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;}
else return n;
}
/*
* all other (large) arguments
*/
if(ix>=0x7ff00000) { /* x is inf or NaN */
y[0]=y[1]=x-x; return 0;
}
/* set z = scalbn(|x|,ilogb(x)-23) */
GET_LOW_WORD(low,x);
SET_LOW_WORD(z,low);
e0 = (ix>>20)-1046; /* e0 = ilogb(z)-23; */
SET_HIGH_WORD(z, ix - ((int32_t)(e0<<20)));
for(i=0;i<2;i++) {
tx[i] = (double)((int32_t)(z));
z = (z-tx[i])*two24;
}
tx[2] = z;
nx = 3;
while((nx > 0) && tx[nx-1]==zero) nx--; /* skip zero term */
n = __kernel_rem_pio2(tx,y,e0,nx,2,two_over_pi);
if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;}
return n;
}
double __ieee754_remainder ( double  ,
double   
)
double __ieee754_scalb ( double  ,
double   
)
double __ieee754_sinh ( double  )
double __ieee754_sqrt ( double  )

Definition at line 87 of file e_sqrt.c.

References EXTRACT_WORDS, i, INSERT_WORDS, one, and tiny.

{
double z;
int32_t sign = (int)0x80000000;
int32_t ix0,s0,q,m,t,i;
u_int32_t r,t1,s1,ix1,q1;
EXTRACT_WORDS(ix0,ix1,x);
/* take care of Inf and NaN */
if((ix0&0x7ff00000)==0x7ff00000) {
return x*x+x; /* sqrt(NaN)=NaN, sqrt(+inf)=+inf
sqrt(-inf)=sNaN */
}
/* take care of zero */
if(ix0<=0) {
if(((ix0&(~sign))|ix1)==0) return x;/* sqrt(+-0) = +-0 */
else if(ix0<0)
return (x-x)/(x-x); /* sqrt(-ve) = sNaN */
}
/* normalize x */
m = (ix0>>20);
if(m==0) { /* subnormal x */
while(ix0==0) {
m -= 21;
ix0 |= (ix1>>11); ix1 <<= 21;
}
for(i=0;(ix0&0x00100000)==0;i++) ix0<<=1;
m -= i-1;
ix0 |= (ix1>>(32-i));
ix1 <<= i;
}
m -= 1023; /* unbias exponent */
ix0 = (ix0&0x000fffff)|0x00100000;
if(m&1){ /* odd m, double x to make it even */
ix0 += ix0 + ((ix1&sign)>>31);
ix1 += ix1;
}
m >>= 1; /* m = [m/2] */
/* generate sqrt(x) bit by bit */
ix0 += ix0 + ((ix1&sign)>>31);
ix1 += ix1;
q = q1 = s0 = s1 = 0; /* [q,q1] = sqrt(x) */
r = 0x00200000; /* r = moving bit from right to left */
while(r!=0) {
t = s0+r;
if(t<=ix0) {
s0 = t+r;
ix0 -= t;
q += r;
}
ix0 += ix0 + ((ix1&sign)>>31);
ix1 += ix1;
r>>=1;
}
r = sign;
while(r!=0) {
t1 = s1+r;
t = s0;
if((t<ix0)||((t==ix0)&&(t1<=ix1))) {
s1 = t1+r;
if(((t1&sign)==sign)&&(s1&sign)==0) s0 += 1;
ix0 -= t;
if (ix1 < t1) ix0 -= 1;
ix1 -= t1;
q1 += r;
}
ix0 += ix0 + ((ix1&sign)>>31);
ix1 += ix1;
r>>=1;
}
/* use floating add to find out rounding direction */
if((ix0|ix1)!=0) {
z = one-tiny; /* trigger inexact flag */
if (z>=one) {
z = one+tiny;
if (q1==(u_int32_t)0xffffffff) { q1=0; q += 1;}
else if (z>one) {
if (q1==(u_int32_t)0xfffffffe) q+=1;
q1+=2;
} else
q1 += (q1&1);
}
}
ix0 = (q>>1)+0x3fe00000;
ix1 = q1>>1;
if ((q&1)==1) ix1 |= sign;
ix0 += (m <<20);
INSERT_WORDS(z,ix0,ix1);
return z;
}
double __ieee754_y0 ( double  )
double __ieee754_y1 ( double  )
double __ieee754_yn ( int  ,
double   
)
double __kernel_cos ( double  ,
double   
)

Definition at line 59 of file k_cos.c.

References C1, C2, C3, C4, C5, C6, GET_HIGH_WORD, INSERT_WORDS, and one.

Referenced by cos(), and sin().

{
double a,hz,z,r,qx;
int32_t ix;
GET_HIGH_WORD(ix,x);
ix &= 0x7fffffff; /* ix = |x|'s high word*/
if(ix<0x3e400000) { /* if x < 2**27 */
if(((int)x)==0) return one; /* generate inexact */
}
z = x*x;
r = z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*C6)))));
if(ix < 0x3FD33333) /* if |x| < 0.3 */
return one - (0.5*z - (z*r - x*y));
else {
if(ix > 0x3fe90000) { /* x > 0.78125 */
qx = 0.28125;
} else {
INSERT_WORDS(qx,ix-0x00200000,0); /* x/4 */
}
hz = 0.5*z-qx;
a = one-qx;
return a - (hz - (z*r-x*y));
}
}
int32_t __kernel_rem_pio2 ( double *  ,
double *  ,
int  ,
int  ,
const unsigned  int,
const int32_t *   
)

Definition at line 152 of file k_rem_pio2.c.

References floor(), i, init_jk, j, k, one, PIo2, scalbn, SDL_arraysize, SDL_assert, SDL_memset, two24, twon24, and zero.

Referenced by __ieee754_rem_pio2().

{
int32_t jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih;
double z,fw,f[20],fq[20],q[20];
if (nx < 1) {
return 0;
}
/* initialize jk*/
SDL_assert(prec < SDL_arraysize(init_jk));
jk = init_jk[prec];
SDL_assert(jk > 0);
jp = jk;
/* determine jx,jv,q0, note that 3>q0 */
jx = nx-1;
jv = (e0-3)/24; if(jv<0) jv=0;
q0 = e0-24*(jv+1);
/* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
j = jv-jx; m = jx+jk;
for(i=0;i<=m;i++,j++) f[i] = (j<0)? zero : (double) ipio2[j];
if ((m+1) < SDL_arraysize(f)) {
SDL_memset(&f[m+1], 0, sizeof (f) - ((m+1) * sizeof (f[0])));
}
/* compute q[0],q[1],...q[jk] */
for (i=0;i<=jk;i++) {
for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j];
q[i] = fw;
}
jz = jk;
recompute:
/* distill q[] into iq[] reversingly */
for(i=0,j=jz,z=q[jz];j>0;i++,j--) {
fw = (double)((int32_t)(twon24* z));
iq[i] = (int32_t)(z-two24*fw);
z = q[j-1]+fw;
}
if (jz < SDL_arraysize(iq)) {
SDL_memset(&iq[jz], 0, sizeof (q) - (jz * sizeof (iq[0])));
}
/* compute n */
z = scalbn(z,q0); /* actual value of z */
z -= 8.0*floor(z*0.125); /* trim off integer >= 8 */
n = (int32_t) z;
z -= (double)n;
ih = 0;
if(q0>0) { /* need iq[jz-1] to determine n */
i = (iq[jz-1]>>(24-q0)); n += i;
iq[jz-1] -= i<<(24-q0);
ih = iq[jz-1]>>(23-q0);
}
else if(q0==0) ih = iq[jz-1]>>23;
else if(z>=0.5) ih=2;
if(ih>0) { /* q > 0.5 */
n += 1; carry = 0;
for(i=0;i<jz ;i++) { /* compute 1-q */
j = iq[i];
if(carry==0) {
if(j!=0) {
carry = 1; iq[i] = 0x1000000- j;
}
} else iq[i] = 0xffffff - j;
}
if(q0>0) { /* rare case: chance is 1 in 12 */
switch(q0) {
case 1:
iq[jz-1] &= 0x7fffff; break;
case 2:
iq[jz-1] &= 0x3fffff; break;
}
}
if(ih==2) {
z = one - z;
if(carry!=0) z -= scalbn(one,q0);
}
}
/* check if recomputation is needed */
if(z==zero) {
j = 0;
for (i=jz-1;i>=jk;i--) j |= iq[i];
if(j==0) { /* need recomputation */
for(k=1;iq[jk-k]==0;k++); /* k = no. of terms needed */
for(i=jz+1;i<=jz+k;i++) { /* add q[jz+1] to q[jz+k] */
f[jx+i] = (double) ipio2[jv+i];
for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j];
q[i] = fw;
}
jz += k;
goto recompute;
}
}
/* chop off zero terms */
if(z==0.0) {
jz -= 1; q0 -= 24;
SDL_assert(jz >= 0);
while(iq[jz]==0) { jz--; SDL_assert(jz >= 0); q0-=24;}
} else { /* break z into 24-bit if necessary */
z = scalbn(z,-q0);
if(z>=two24) {
fw = (double)((int32_t)(twon24*z));
iq[jz] = (int32_t)(z-two24*fw);
jz += 1; q0 += 24;
iq[jz] = (int32_t) fw;
} else iq[jz] = (int32_t) z ;
}
/* convert integer "bit" chunk to floating-point value */
fw = scalbn(one,q0);
for(i=jz;i>=0;i--) {
q[i] = fw*(double)iq[i]; fw*=twon24;
}
/* compute PIo2[0,...,jp]*q[jz,...,0] */
for(i=jz;i>=0;i--) {
for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k];
fq[jz-i] = fw;
}
if ((jz+1) < SDL_arraysize(f)) {
SDL_memset(&fq[jz+1], 0, sizeof (fq) - ((jz+1) * sizeof (fq[0])));
}
/* compress fq[] into y[] */
switch(prec) {
case 0:
fw = 0.0;
for (i=jz;i>=0;i--) fw += fq[i];
y[0] = (ih==0)? fw: -fw;
break;
case 1:
case 2:
fw = 0.0;
for (i=jz;i>=0;i--) fw += fq[i];
y[0] = (ih==0)? fw: -fw;
fw = fq[0]-fw;
for (i=1;i<=jz;i++) fw += fq[i];
y[1] = (ih==0)? fw: -fw;
break;
case 3: /* painful */
for (i=jz;i>0;i--) {
fw = fq[i-1]+fq[i];
fq[i] += fq[i-1]-fw;
fq[i-1] = fw;
}
for (i=jz;i>1;i--) {
fw = fq[i-1]+fq[i];
fq[i] += fq[i-1]-fw;
fq[i-1] = fw;
}
for (fw=0.0,i=jz;i>=2;i--) fw += fq[i];
if(ih==0) {
y[0] = fq[0]; y[1] = fq[1]; y[2] = fw;
} else {
y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw;
}
}
return n&7;
}
double __kernel_sin ( double  ,
double  ,
int   
)

Definition at line 52 of file k_sin.c.

References GET_HIGH_WORD, half, S1, S2, S3, S4, S5, and S6.

Referenced by cos(), and sin().

{
double z,r,v;
int32_t ix;
GET_HIGH_WORD(ix,x);
ix &= 0x7fffffff; /* high word of x */
if(ix<0x3e400000) /* |x| < 2**-27 */
{if((int)x==0) return x;} /* generate inexact */
z = x*x;
v = z*x;
r = S2+z*(S3+z*(S4+z*(S5+z*S6)));
if(iy==0) return x+v*(S1+z*r);
else return x-((z*(half*y-v*r)-y)-v*S1);
}
double __kernel_tan ( double  ,
double  ,
int   
)

Definition at line 69 of file k_tan.c.

References fabs(), GET_HIGH_WORD, GET_LOW_WORD, one, pio4, pio4lo, SET_LOW_WORD, and T.

Referenced by tan().

{
double z,r,v,w,s;
int32_t ix,hx;
GET_HIGH_WORD(hx,x);
ix = hx&0x7fffffff; /* high word of |x| */
if(ix<0x3e300000) /* x < 2**-28 */
{if((int)x==0) { /* generate inexact */
u_int32_t low;
GET_LOW_WORD(low,x);
if(((ix|low)|(iy+1))==0) return one/fabs(x);
else return (iy==1)? x: -one/x;
}
}
if(ix>=0x3FE59428) { /* |x|>=0.6744 */
if(hx<0) {x = -x; y = -y;}
z = pio4-x;
w = pio4lo-y;
x = z+w; y = 0.0;
}
z = x*x;
w = z*z;
/* Break x^5*(T[1]+x^2*T[2]+...) into
* x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
* x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
*/
r = T[1]+w*(T[3]+w*(T[5]+w*(T[7]+w*(T[9]+w*T[11]))));
v = z*(T[2]+w*(T[4]+w*(T[6]+w*(T[8]+w*(T[10]+w*T[12])))));
s = z*x;
r = y + z*(s*(r+v)+y);
r += T[0]*s;
w = x+r;
if(ix>=0x3FE59428) {
v = (double)iy;
return (double)(1-((hx>>30)&2))*(v-2.0*(x-(w*w/(w+v)-r)));
}
if(iy==1) return w;
else { /* if allow error up to 2 ulp,
simply return -1.0/(x+r) here */
/* compute -1.0/(x+r) accurately */
double a,t;
z = w;
SET_LOW_WORD(z,0);
v = r-(z - x); /* z+v = r+x */
t = a = -1.0/w; /* a = -1.0/w */
SET_LOW_WORD(t,0);
s = 1.0+t*z;
return t+a*(s+t*v);
}
}