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/* @(#)e_pow.c 5.1 93/09/24 */
/*
* ====================================================
* 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.
* ====================================================
*/
#ifndef lint
static char rcsid
[] = "$\Id: e_pow.c,v 1.2 1995/05/30 05:48:34 rgrimes Exp $";
#endif
/* __ieee754_pow(x,y) return x**y
*
* n
* Method: Let x = 2 * (1+f)
* 1. Compute and return log2(x) in two pieces:
* log2(x) = w1 + w2,
* where w1 has 53-24 = 29 bit trailing zeros.
* 2. Perform y*log2(x) = n+y' by simulating muti-precision
* arithmetic, where |y'|<=0.5.
* 3. Return x**y = 2**n*exp(y'*log2)
*
* Special cases:
* 1. (anything) ** 0 is 1
* 2. (anything) ** 1 is itself
* 3. (anything) ** NAN is NAN
* 4. NAN ** (anything except 0) is NAN
* 5. +-(|x| > 1) ** +INF is +INF
* 6. +-(|x| > 1) ** -INF is +0
* 7. +-(|x| < 1) ** +INF is +0
* 8. +-(|x| < 1) ** -INF is +INF
* 9. +-1 ** +-INF is NAN
* 10. +0 ** (+anything except 0, NAN) is +0
* 11. -0 ** (+anything except 0, NAN, odd integer) is +0
* 12. +0 ** (-anything except 0, NAN) is +INF
* 13. -0 ** (-anything except 0, NAN, odd integer) is +INF
* 14. -0 ** (odd integer) = -( +0 ** (odd integer) )
* 15. +INF ** (+anything except 0,NAN) is +INF
* 16. +INF ** (-anything except 0,NAN) is +0
* 17. -INF ** (anything) = -0 ** (-anything)
* 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
* 19. (-anything except 0 and inf) ** (non-integer) is NAN
*
* Accuracy:
* pow(x,y) returns x**y nearly rounded. In particular
* pow(integer,integer)
* always returns the correct integer provided it is
* representable.
*
* Constants :
* The hexadecimal values are the intended ones for the following
* constants. The decimal values may be used, provided that the
* compiler will convert from decimal to binary accurately enough
* to produce the hexadecimal values shown.
*/
#include "math.h"
#include "math_private.h"
#ifdef __STDC__
static const double
#else
static double
#endif
bp
[] = {1.0, 1.5,},
dp_h
[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
dp_l
[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */
zero
= 0.0,
one
= 1.0,
two
= 2.0,
two53
= 9007199254740992.0, /* 0x43400000, 0x00000000 */
huge
= 1.0e300,
tiny
= 1.0e-300,
/* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
L1
= 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */
L2
= 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */
L3
= 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */
L4
= 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */
L5
= 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */
L6
= 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */
P1
= 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
P2
= -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
P3
= 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
P4
= -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
P5
= 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */
lg2
= 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
lg2_h
= 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */
lg2_l
= -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */
ovt
= 8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */
cp
= 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
cp_h
= 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */
cp_l
= -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/
ivln2
= 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */
ivln2_h
= 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/
ivln2_l
= 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/
#ifdef __STDC__
double __ieee754_pow
(double x
, double y
)
#else
double __ieee754_pow
(x
,y
)
double x
, y
;
#endif
{
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
);
EXTRACT_WORDS
(hy
,ly
,y
);
ix
= hx
&0x7fffffff; 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 y
- y
; /* inf**+-1 is NaN */
else if (ix
>= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */
return (hy
>=0)? y
: zero
;
else /* (|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 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 */
/* CYGNUS LOCAL: This used to be
if((((hx>>31)+1)|yisint)==0) return (x-x)/(x-x);
but ANSI C says a right shift of a signed negative quantity is
implementation defined. */
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
;
}