```   1
2 /*
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
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  *
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 }
```