/* origin: FreeBSD /usr/src/lib/msun/src/e_pow.c */
/*
* ====================================================
* 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.
* ====================================================
*/
/* 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. 1 ** (anything) is 1
* 3. (anything except 1) ** 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 1
* 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, raise divbyzero
* 13. -0 ** (-anything except 0, NAN, odd integer) is +INF, raise divbyzero
* 14. -0 ** (+odd integer) is -0
* 15. -0 ** (-odd integer) is -INF, raise divbyzero
* 16. +INF ** (+anything except 0,NAN) is +INF
* 17. +INF ** (-anything except 0,NAN) is +0
* 18. -INF ** (+odd integer) is -INF
* 19. -INF ** (anything) = -0 ** (-anything), (anything except odd integer)
* 20. (anything) ** 1 is (anything)
* 21. (anything) ** -1 is 1/(anything)
* 22. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
* 23. (-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 "libm.h"
static const double
bp[] = {1.0, 1.5,},
dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */
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-017, /* -(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*/
double pow(double x, double y)
{
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;
/* x**0 = 1, even if x is NaN */
if ((iy|ly) == 0)
return 1.0;
/* 1**y = 1, even if y is NaN */
if (hx == 0x3ff00000 && lx == 0)
return 1.0;
/* NaN if either arg is NaN */
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) {
uint32_t j = ly>>(52-k);
if ((j<<(52-k)) == ly)
yisint = 2 - (j&1);
} else if (ly == 0) {
uint32_t 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) /* (-1)**+-inf is 1 */
return 1.0;
else if (ix >= 0x3ff00000) /* (|x|>1)**+-inf = inf,0 */
return hy >= 0 ? y : 0.0;
else /* (|x|<1)**+-inf = 0,inf */
return hy >= 0 ? 0.0 : -y;
}
if (iy == 0x3ff00000) { /* y is +-1 */
if (hy >= 0)
return x;
y = 1/x;
#if FLT_EVAL_METHOD!=0
{
union {double f; uint64_t i;} u = {y};
uint64_t i = u.i & -1ULL/2;
if (i>>52 == 0 && (i&(i-1)))
FORCE_EVAL((float)y);
}
#endif
return y;
}
if (hy == 0x40000000) /* y is 2 */
return x*x;
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) { /* x is +-0,+-inf,+-1 */
z = ax;
if (hy < 0) /* z = (1/|x|) */
z = 1.0/z;
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;
}
}
s = 1.0; /* sign of result */
if (hx < 0) {
if (yisint == 0) /* (x<0)**(non-int) is NaN */
return (x-x)/(x-x);
if (yisint == 1) /* (x<0)**(odd int) */
s = -1.0;
}
/* |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 - 1.0; /* 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 ss,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) /* |x|<sqrt(3/2) */
k = 0;
else if (j < 0xBB67A) /* |x|<sqrt(3) */
k = 1;
else {
k = 0;
n += 1;
ix -= 0x00100000;
}
SET_HIGH_WORD(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 = 1.0/(ax+bp[k]);
ss = u*v;
s_h = ss;
SET_LOW_WORD(s_h, 0);
/* t_h=ax+bp[k] High */
t_h = 0.0;
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 = 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;
SET_LOW_WORD(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;
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) = (ss+..)*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);
}
/* 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 */
if (p_l + ovt > z - p_h)
return s*huge*huge; /* overflow */
} else if ((j&0x7fffffff) >= 0x4090cc00) { /* z <= -1075 */ // FIXME: instead of abs(j) use unsigned j
if (((j-0xc090cc00)|i) != 0) /* z < -1075 */
return s*tiny*tiny; /* underflow */
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 = 0.0;
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*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)/(t1-2.0) - (w + z*w);
z = 1.0 - (r-z);
GET_HIGH_WORD(j, z);
j += n<<20;
if ((j>>20) <= 0) /* subnormal output */
z = scalbn(z,n);
else
SET_HIGH_WORD(z, j);
return s*z;
}
