313 lines
9.8 KiB
C
313 lines
9.8 KiB
C
|
|
||
|
|
/* @(#)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.
|
||
|
|
* ====================================================
|
||
|
|
*/
|
||
|
|
|
||
|
|
/* __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 "fdlibm.h"
|
||
|
|
|
||
|
|
#ifndef _DOUBLE_IS_32BITS
|
||
|
|
|
||
|
|
#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;
|
||
|
|
__uint32_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((__uint32_t)(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 __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 */
|
||
|
|
/* 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(((((__uint32_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(((((__uint32_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,(int)n); /* subnormal output */
|
||
|
|
else SET_HIGH_WORD(z,j);
|
||
|
|
return s*z;
|
||
|
|
}
|
||
|
|
|
||
|
|
#endif /* defined(_DOUBLE_IS_32BITS) */
|