/*

* ====================================================

* 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. +-1 ** anything is 1.0

* 2. +-1 ** +-INF is 1.0

* 3. (anything) ** 0 is 1

* 4. (anything) ** 1 is itself

* 5. (anything) ** NAN is NAN

* 6. NAN ** (anything except 0) is NAN

* 7. +-(|x| > 1) ** +INF is +INF

* 8. +-(|x| > 1) ** -INF is +0

* 9. +-(|x| < 1) ** +INF is +0

* 10 +-(|x| < 1) ** -INF is +INF

* 11. +0 ** (+anything except 0, NAN) is +0

* 12. -0 ** (+anything except 0, NAN, odd integer) is +0

* 13. +0 ** (-anything except 0, NAN) is +INF

* 14. -0 ** (-anything except 0, NAN, odd integer) is +INF

* 15. -0 ** (odd integer) = -( +0 ** (odd integer) )

* 16. +INF ** (+anything except 0,NAN) is +INF

* 17. +INF ** (-anything except 0,NAN) is +0

* 18. -INF ** (anything) = -0 ** (-anything)

* 19. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)

* 20. (-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_libm.h"

#include "math_private.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 */

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*/

double attribute_hidden __ieee754_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;

u_int32_t lx,ly;

EXTRACT_WORDS(hx,lx,x);

/* x==1: 1**y = 1 (even if y is NaN) */

if (hx==0x3ff00000 && lx==0) {

return x;

}

ix = hx&0x7fffffff;

EXTRACT_WORDS(hy,ly,y);

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 one; /* +-1**+-inf is 1 (yes, weird rule) */

if (ix >= 0x3ff00000) /* (|x|>1)**+-inf = inf,0 */

return (hy>=0) ? y : zero;

/* (|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 */

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;

}

/*

* wrapper pow(x,y) return x**y

*/

#ifndef _IEEE_LIBM

double pow(double x, double y)

{

double z = __ieee754_pow(x, y);

if (_LIB_VERSION == _IEEE_|| isnan(y))

return z;

if (isnan(x)) {

if (y == 0.0)

return __kernel_standard(x, y, 42); /* pow(NaN,0.0) */

return z;

}

if (x == 0.0) {

if (y == 0.0)

return __kernel_standard(x, y, 20); /* pow(0.0,0.0) */

if (isfinite(y) && y < 0.0)

return __kernel_standard(x,y,23); /* pow(0.0,negative) */

return z;

}

if (!isfinite(z)) {

if (isfinite(x) && isfinite(y)) {

if (isnan(z))

return __kernel_standard(x, y, 24); /* pow neg**non-int */

return __kernel_standard(x, y, 21); /* pow overflow */

}

}

if (z == 0.0 && isfinite(x) && isfinite(y))

return __kernel_standard(x, y, 22); /* pow underflow */

return z;

}

#else

strong_alias(__ieee754_pow, pow)

#endif

libm_hidden_def(pow)