/* @(#)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.
* ====================================================
*/
/* Modified by Naohiko Shimizu/Tokai University, Japan 1997/08/25,
for performance improvement on pipelined processors.
*/
#if defined(LIBM_SCCS) && !defined(lint)
static char rcsid[] = "$NetBSD: e_pow.c,v 1.9 1995/05/12 04:57:32 jtc 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"
#define zero C[0]
#define one C[1]
#define two C[2]
#define two53 C[3]
#define huge C[4]
#define tiny C[5]
#define L1 C[6]
#define L2 C[7]
#define L3 C[8]
#define L4 C[9]
#define L5 C[10]
#define L6 C[11]
#define P1 C[12]
#define P2 C[13]
#define P3 C[14]
#define P4 C[15]
#define P5 C[16]
#define lg2 C[17]
#define lg2_h C[18]
#define lg2_l C[19]
#define ovt C[20]
#define cp C[21]
#define cp_h C[22]
#define cp_l C[23]
#define ivln2 C[24]
#define ivln2_h C[25]
#define ivln2_l C[26]
double scalbn(double,int);
#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)
#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 */
C[] = {
0.0,
1.0,
2.0,
9007199254740992.0 ,
1.0e300,
1.0e-300,
5.99999999999994648725e-01 ,
4.28571428578550184252e-01 ,
3.33333329818377432918e-01 ,
2.72728123808534006489e-01 ,
2.30660745775561754067e-01 ,
2.06975017800338417784e-01 ,
1.66666666666666019037e-01 ,
-2.77777777770155933842e-03 ,
6.61375632143793436117e-05 ,
-1.65339022054652515390e-06 ,
4.13813679705723846039e-08 ,
6.93147180559945286227e-01 ,
6.93147182464599609375e-01 ,
-1.90465429995776804525e-09 ,
8.0085662595372944372e-0017 ,
9.61796693925975554329e-01 ,
9.61796700954437255859e-01 ,
-7.02846165095275826516e-09 ,
1.44269504088896338700e+00 ,
1.44269502162933349609e+00 ,
1.92596299112661746887e-08 };
double pow_test(x,y)
double x, y;
{
double z,ax,z_h,z_l,p_h,p_l;
double y1,t1,t2,r,s,t,u,v,w, t12,t14,r_1,r_2,r_3;
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 C[1];
/* +-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((u_int32_t)(j<<(52-k))==ly) yisint = 2-(j&1);
} else if(ly==0) {
j = iy>>(20-k);
if((int32_t)(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: C[0];
else /* (|x|<1)**-,+inf = inf,0 */
return (hy<0)?-y: C[0];
}
if(iy==0x3ff00000) { /* y is +-1 */
if(hy<0) return C[1]/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 */
}
}
/* 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 = C[1]/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)? C[4]*C[4]:C[5]*C[5];
if(ix>=0x3ff00000) return (hy>0)? C[4]*C[4]:C[5]*C[5];
}
/* over/underflow if x is not close to one */
if(ix<0x3fefffff) return (hy<0)? C[4]*C[4]:C[5]*C[5];
if(ix>0x3ff00000) return (hy>0)? C[4]*C[4]:C[5]*C[5];
/* 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 = C[25]*t; /* ivln2_h has 21 sig. bits */
v = t*C[26]-w*C[24];
t1 = u+v;
SET_LOW_WORD(t1,0);
t2 = v-(t1-u);
} else {
double s2,s_h,s_l,t_h,t_l,s22,s24,s26,r1,r2,r3;
n = 0;
/* take care subnormal number */
if(ix<0x00100000)
{ax *= C[3]; 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 = C[1]/(ax+bp[k]);
s = u*v;
s_h = s;
SET_LOW_WORD(s_h,0);
/* t_h=ax+bp[k] High */
t_h = C[0];
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;
#ifdef DO_NOT_USE_THIS
r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
#else
r1 = C[10]+s2*C[11]; s22=s2*s2;
r2 = C[8]+s2*C[9]; s24=s22*s22;
r3 = C[6]+s2*C[7]; s26=s24*s22;
r = r3*s22 + r2*s24 + r1*s26;
#endif
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 = C[22]*p_h; /* cp_h+cp_l = 2/(3*log2) */
z_l = C[23]*p_h+p_l*C[21]+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 = C[1]; /* s (sign of result -ve**odd) = -1 else = 1 */
if(((((u_int32_t)hx>>31)-1)|(yisint-1))==0)
s = -C[1];/* (-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*C[4]*C[4]; /* overflow */
else {
if(p_l+C[20]>z-p_h) return s*C[4]*C[4]; /* overflow */
}
} else if((j&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */
if(((j-0xc090cc00)|i)!=0) /* z < -1075 */
return s*C[5]*C[5]; /* underflow */
else {
if(p_l<=z-p_h) return s*C[5]*C[5]; /* 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 = C[0];
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*C[18];
v = (p_l-(t-p_h))*C[17]+t*C[19];
z = u+v;
w = v-(z-u);
t = z*z;
#ifdef DO_NOT_USE_THIS
t1 = z - t*(C[12]+t*(C[13]+t*(C[14]+t*(C[15]+t*C[16]))));
#else
r_1 = C[15]+t*C[16]; t12 = t*t;
r_2 = C[13]+t*C[14]; t14 = t12*t12;
r_3 = t*C[12];
t1 = z - r_3 - t12*r_2 - t14*r_1;
#endif
r = (z*t1)/(t1-C[2])-(w+z*w);
z = C[1]-(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;
}