1 

        2 /*

        3  * Copyright (c) 1998, 2004, Oracle and/or its affiliates. All rights reserved.

        4  * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.

        5  *

        6  * This code is free software; you can redistribute it and/or modify it

        7  * under the terms of the GNU General Public License version 2 only, as

        8  * published by the Free Software Foundation.  Oracle designates this

        9  * particular file as subject to the "Classpath" exception as provided

       10  * by Oracle in the LICENSE file that accompanied this code.

       11  *

       12  * This code is distributed in the hope that it will be useful, but WITHOUT

       13  * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or

       14  * FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License

       15  * version 2 for more details (a copy is included in the LICENSE file that

       16  * accompanied this code).

       17  *

       18  * You should have received a copy of the GNU General Public License version

       19  * 2 along with this work; if not, write to the Free Software Foundation,

       20  * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA.

       21  *

       22  * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA

       23  * or visit www.oracle.com if you need additional information or have any

       24  * questions.

       25  */

       26 

       27 /* __ieee754_pow(x,y) return x**y

       28  *

       29  *                    n

       30  * Method:  Let x =  2   * (1+f)

       31  *      1. Compute and return log2(x) in two pieces:

       32  *              log2(x) = w1 + w2,

       33  *         where w1 has 53-24 = 29 bit trailing zeros.

       34  *      2. Perform y*log2(x) = n+y' by simulating muti-precision

       35  *         arithmetic, where |y'|<=0.5.

       36  *      3. Return x**y = 2**n*exp(y'*log2)

       37  *

       38  * Special cases:

       39  *      1.  (anything) ** 0  is 1

       40  *      2.  (anything) ** 1  is itself

       41  *      3.  (anything) ** NAN is NAN

       42  *      4.  NAN ** (anything except 0) is NAN

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

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

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

       46  *      8.  +-(|x| < 1) **  -INF is +INF

       47  *      9.  +-1         ** +-INF is NAN

       48  *      10. +0 ** (+anything except 0, NAN)               is +0

       49  *      11. -0 ** (+anything except 0, NAN, odd integer)  is +0

       50  *      12. +0 ** (-anything except 0, NAN)               is +INF

       51  *      13. -0 ** (-anything except 0, NAN, odd integer)  is +INF

       52  *      14. -0 ** (odd integer) = -( +0 ** (odd integer) )

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

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

       55  *      17. -INF ** (anything)  = -0 ** (-anything)

       56  *      18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)

       57  *      19. (-anything except 0 and inf) ** (non-integer) is NAN

       58  *

       59  * Accuracy:

       60  *      pow(x,y) returns x**y nearly rounded. In particular

       61  *                      pow(integer,integer)

       62  *      always returns the correct integer provided it is

       63  *      representable.

       64  *

       65  * Constants :

       66  * The hexadecimal values are the intended ones for the following

       67  * constants. The decimal values may be used, provided that the

       68  * compiler will convert from decimal to binary accurately enough

       69  * to produce the hexadecimal values shown.

       70  */

       71 

       72 #include "fdlibm.h"

       73 

       74 #ifdef __STDC__

       75 static const double

       76 #else

       77 static double

       78 #endif

       79 bp[] = {1.0, 1.5,},

       80 dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */

       81 dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */

       82 zero    =  0.0,

       83 one     =  1.0,

       84 two     =  2.0,

       85 two53   =  9007199254740992.0,  /* 0x43400000, 0x00000000 */

       86 huge    =  1.0e300,

       87 tiny    =  1.0e-300,

       88         /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */

       89 L1  =  5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */

       90 L2  =  4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */

       91 L3  =  3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */

       92 L4  =  2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */

       93 L5  =  2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */

       94 L6  =  2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */

       95 P1   =  1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */

       96 P2   = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */

       97 P3   =  6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */

       98 P4   = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */

       99 P5   =  4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */

      100 lg2  =  6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */

      101 lg2_h  =  6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */

      102 lg2_l  = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */

      103 ovt =  8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */

      104 cp    =  9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */

      105 cp_h  =  9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */

      106 cp_l  = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/

      107 ivln2    =  1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */

      108 ivln2_h  =  1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/

      109 ivln2_l  =  1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/

      110 

      111 #ifdef __STDC__

      112         double __ieee754_pow(double x, double y)

      113 #else

      114         double __ieee754_pow(x,y)

      115         double x, y;

      116 #endif

      117 {

      118         double z,ax,z_h,z_l,p_h,p_l;

      119         double y1,t1,t2,r,s,t,u,v,w;

      120         int i0,i1,i,j,k,yisint,n;

      121         int hx,hy,ix,iy;

      122         unsigned lx,ly;

      123 

      124         i0 = ((*(int*)&one)>>29)^1; i1=1-i0;

      125         hx = __HI(x); lx = __LO(x);

      126         hy = __HI(y); ly = __LO(y);

      127         ix = hx&0x7fffffff;  iy = hy&0x7fffffff;

      128 

      129     /* y==zero: x**0 = 1 */

      130         if((iy|ly)==0) return one;

      131 

      132     /* +-NaN return x+y */

      133         if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) ||

      134            iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0)))

      135                 return x+y;

      136 

      137     /* determine if y is an odd int when x < 0

      138      * yisint = 0       ... y is not an integer

      139      * yisint = 1       ... y is an odd int

      140      * yisint = 2       ... y is an even int

      141      */

      142         yisint  = 0;

      143         if(hx<0) {

      144             if(iy>=0x43400000) yisint = 2; /* even integer y */

      145             else if(iy>=0x3ff00000) {

      146                 k = (iy>>20)-0x3ff;        /* exponent */

      147                 if(k>20) {

      148                     j = ly>>(52-k);

      149                     if((j<<(52-k))==ly) yisint = 2-(j&1);

      150                 } else if(ly==0) {

      151                     j = iy>>(20-k);

      152                     if((j<<(20-k))==iy) yisint = 2-(j&1);

      153                 }

      154             }

      155         }

      156 

      157     /* special value of y */

      158         if(ly==0) {

      159             if (iy==0x7ff00000) {       /* y is +-inf */

      160                 if(((ix-0x3ff00000)|lx)==0)

      161                     return  y - y;      /* inf**+-1 is NaN */

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

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

      164                 else                    /* (|x|<1)**-,+inf = inf,0 */

      165                     return (hy<0)?-y: zero;

      166             }

      167             if(iy==0x3ff00000) {        /* y is  +-1 */

      168                 if(hy<0) return one/x; else return x;

      169             }

      170             if(hy==0x40000000) return x*x; /* y is  2 */

      171             if(hy==0x3fe00000) {        /* y is  0.5 */

      172                 if(hx>=0)       /* x >= +0 */

      173                 return sqrt(x);

      174             }

      175         }

      176 

      177         ax   = fabs(x);

      178     /* special value of x */

      179         if(lx==0) {

      180             if(ix==0x7ff00000||ix==0||ix==0x3ff00000){

      181                 z = ax;                 /*x is +-0,+-inf,+-1*/

      182                 if(hy<0) z = one/z;     /* z = (1/|x|) */

      183                 if(hx<0) {

      184                     if(((ix-0x3ff00000)|yisint)==0) {

      185                         z = (z-z)/(z-z); /* (-1)**non-int is NaN */

      186                     } else if(yisint==1)

      187                         z = -1.0*z;             /* (x<0)**odd = -(|x|**odd) */

      188                 }

      189                 return z;

      190             }

      191         }

      192 

      193         n = (hx>>31)+1;

      194 

      195     /* (x<0)**(non-int) is NaN */

      196         if((n|yisint)==0) return (x-x)/(x-x);

      197 

      198         s = one; /* s (sign of result -ve**odd) = -1 else = 1 */

      199         if((n|(yisint-1))==0) s = -one;/* (-ve)**(odd int) */

      200 

      201     /* |y| is huge */

      202         if(iy>0x41e00000) { /* if |y| > 2**31 */

      203             if(iy>0x43f00000){  /* if |y| > 2**64, must o/uflow */

      204                 if(ix<=0x3fefffff) return (hy<0)? huge*huge:tiny*tiny;

      205                 if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny;

      206             }

      207         /* over/underflow if x is not close to one */

      208             if(ix<0x3fefffff) return (hy<0)? s*huge*huge:s*tiny*tiny;

      209             if(ix>0x3ff00000) return (hy>0)? s*huge*huge:s*tiny*tiny;

      210         /* now |1-x| is tiny <= 2**-20, suffice to compute

      211            log(x) by x-x^2/2+x^3/3-x^4/4 */

      212             t = ax-one;         /* t has 20 trailing zeros */

      213             w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25));

      214             u = ivln2_h*t;      /* ivln2_h has 21 sig. bits */

      215             v = t*ivln2_l-w*ivln2;

      216             t1 = u+v;

      217             __LO(t1) = 0;

      218             t2 = v-(t1-u);

      219         } else {

      220             double ss,s2,s_h,s_l,t_h,t_l;

      221             n = 0;

      222         /* take care subnormal number */

      223             if(ix<0x00100000)

      224                 {ax *= two53; n -= 53; ix = __HI(ax); }

      225             n  += ((ix)>>20)-0x3ff;

      226             j  = ix&0x000fffff;

      227         /* determine interval */

      228             ix = j|0x3ff00000;          /* normalize ix */

      229             if(j<=0x3988E) k=0;         /* |x|<sqrt(3/2) */

      230             else if(j<0xBB67A) k=1;     /* |x|<sqrt(3)   */

      231             else {k=0;n+=1;ix -= 0x00100000;}

      232             __HI(ax) = ix;

      233 

      234         /* compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */

      235             u = ax-bp[k];               /* bp[0]=1.0, bp[1]=1.5 */

      236             v = one/(ax+bp[k]);

      237             ss = u*v;

      238             s_h = ss;

      239             __LO(s_h) = 0;

      240         /* t_h=ax+bp[k] High */

      241             t_h = zero;

      242             __HI(t_h)=((ix>>1)|0x20000000)+0x00080000+(k<<18);

      243             t_l = ax - (t_h-bp[k]);

      244             s_l = v*((u-s_h*t_h)-s_h*t_l);

      245         /* compute log(ax) */

      246             s2 = ss*ss;

      247             r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));

      248             r += s_l*(s_h+ss);

      249             s2  = s_h*s_h;

      250             t_h = 3.0+s2+r;

      251             __LO(t_h) = 0;

      252             t_l = r-((t_h-3.0)-s2);

      253         /* u+v = ss*(1+...) */

      254             u = s_h*t_h;

      255             v = s_l*t_h+t_l*ss;

      256         /* 2/(3log2)*(ss+...) */

      257             p_h = u+v;

      258             __LO(p_h) = 0;

      259             p_l = v-(p_h-u);

      260             z_h = cp_h*p_h;             /* cp_h+cp_l = 2/(3*log2) */

      261             z_l = cp_l*p_h+p_l*cp+dp_l[k];

      262         /* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */

      263             t = (double)n;

      264             t1 = (((z_h+z_l)+dp_h[k])+t);

      265             __LO(t1) = 0;

      266             t2 = z_l-(((t1-t)-dp_h[k])-z_h);

      267         }

      268 

      269     /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */

      270         y1  = y;

      271         __LO(y1) = 0;

      272         p_l = (y-y1)*t1+y*t2;

      273         p_h = y1*t1;

      274         z = p_l+p_h;

      275         j = __HI(z);

      276         i = __LO(z);

      277         if (j>=0x40900000) {                            /* z >= 1024 */

      278             if(((j-0x40900000)|i)!=0)                   /* if z > 1024 */

      279                 return s*huge*huge;                     /* overflow */

      280             else {

      281                 if(p_l+ovt>z-p_h) return s*huge*huge;   /* overflow */

      282             }

      283         } else if((j&0x7fffffff)>=0x4090cc00 ) {        /* z <= -1075 */

      284             if(((j-0xc090cc00)|i)!=0)           /* z < -1075 */

      285                 return s*tiny*tiny;             /* underflow */

      286             else {

      287                 if(p_l<=z-p_h) return s*tiny*tiny;      /* underflow */

      288             }

      289         }

      290     /*

      291      * compute 2**(p_h+p_l)

      292      */

      293         i = j&0x7fffffff;

      294         k = (i>>20)-0x3ff;

      295         n = 0;

      296         if(i>0x3fe00000) {              /* if |z| > 0.5, set n = [z+0.5] */

      297             n = j+(0x00100000>>(k+1));

      298             k = ((n&0x7fffffff)>>20)-0x3ff;     /* new k for n */

      299             t = zero;

      300             __HI(t) = (n&~(0x000fffff>>k));

      301             n = ((n&0x000fffff)|0x00100000)>>(20-k);

      302             if(j<0) n = -n;

      303             p_h -= t;

      304         }

      305         t = p_l+p_h;

      306         __LO(t) = 0;

      307         u = t*lg2_h;

      308         v = (p_l-(t-p_h))*lg2+t*lg2_l;

      309         z = u+v;

      310         w = v-(z-u);

      311         t  = z*z;

      312         t1  = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));

      313         r  = (z*t1)/(t1-two)-(w+z*w);

      314         z  = one-(r-z);

      315         j  = __HI(z);

      316         j += (n<<20);

      317         if((j>>20)<=0) z = scalbn(z,n); /* subnormal output */

      318         else __HI(z) += (n<<20);

      319         return s*z;

      320 }