Akos Kiss
2a5d17e736
Most of the legacy version control IDs in fdlibm are already simple comments at the beginning of the source files. However, in some files, there are leftover global variables and pragmas. (The global variable in e_pow.c even increases the .data section in both debug and release builds.) This patch turns all legacy IDs to comments of unified style. JerryScript-DCO-1.0-Signed-off-by: Akos Kiss akiss@inf.u-szeged.hu
307 lines
9.4 KiB
C
307 lines
9.4 KiB
C
|
|
/* @(#)e_pow.c 1.5 04/04/22 */
|
|
/*
|
|
* ====================================================
|
|
* Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
|
|
*
|
|
* 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"
|
|
|
|
#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;
|
|
int i0,i1,i,j,k,yisint,n;
|
|
int hx,hy,ix,iy;
|
|
unsigned lx,ly;
|
|
|
|
i0 = ((*(int*)&one)>>29)^1; i1=1-i0;
|
|
hx = __HI(x); lx = __LO(x);
|
|
hy = __HI(y); ly = __LO(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;
|
|
}
|
|
}
|
|
|
|
n = (hx>>31)+1;
|
|
|
|
/* (x<0)**(non-int) is NaN */
|
|
if((n|yisint)==0) return (x-x)/(x-x);
|
|
|
|
s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
|
|
if((n|(yisint-1))==0) s = -one;/* (-ve)**(odd int) */
|
|
|
|
/* |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)? s*huge*huge:s*tiny*tiny;
|
|
if(ix>0x3ff00000) return (hy>0)? s*huge*huge:s*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 = ax-one; /* 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;
|
|
__LO(t1) = 0;
|
|
t2 = v-(t1-u);
|
|
} else {
|
|
double ss,s2,s_h,s_l,t_h,t_l;
|
|
n = 0;
|
|
/* take care subnormal number */
|
|
if(ix<0x00100000)
|
|
{ax *= two53; n -= 53; ix = __HI(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;}
|
|
__HI(ax) = ix;
|
|
|
|
/* compute ss = 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]);
|
|
ss = u*v;
|
|
s_h = ss;
|
|
__LO(s_h) = 0;
|
|
/* t_h=ax+bp[k] High */
|
|
t_h = zero;
|
|
__HI(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 = ss*ss;
|
|
r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
|
|
r += s_l*(s_h+ss);
|
|
s2 = s_h*s_h;
|
|
t_h = 3.0+s2+r;
|
|
__LO(t_h) = 0;
|
|
t_l = r-((t_h-3.0)-s2);
|
|
/* u+v = ss*(1+...) */
|
|
u = s_h*t_h;
|
|
v = s_l*t_h+t_l*ss;
|
|
/* 2/(3log2)*(ss+...) */
|
|
p_h = u+v;
|
|
__LO(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) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */
|
|
t = (double)n;
|
|
t1 = (((z_h+z_l)+dp_h[k])+t);
|
|
__LO(t1) = 0;
|
|
t2 = z_l-(((t1-t)-dp_h[k])-z_h);
|
|
}
|
|
|
|
/* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
|
|
y1 = y;
|
|
__LO(y1) = 0;
|
|
p_l = (y-y1)*t1+y*t2;
|
|
p_h = y1*t1;
|
|
z = p_l+p_h;
|
|
j = __HI(z);
|
|
i = __LO(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;
|
|
__HI(t) = (n&~(0x000fffff>>k));
|
|
n = ((n&0x000fffff)|0x00100000)>>(20-k);
|
|
if(j<0) n = -n;
|
|
p_h -= t;
|
|
}
|
|
t = p_l+p_h;
|
|
__LO(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);
|
|
j = __HI(z);
|
|
j += (n<<20);
|
|
if((j>>20)<=0) z = scalbn(z,n); /* subnormal output */
|
|
else __HI(z) += (n<<20);
|
|
return s*z;
|
|
}
|