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40 /* @(#)e_pow.c 1.3 95/01/18 */
42 * ====================================================
43 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
45 * Developed at SunSoft, a Sun Microsystems, Inc. business.
46 * Permission to use, copy, modify, and distribute this
47 * software is freely granted, provided that this notice
49 * ====================================================
52 /* __ieee754_pow(x,y) return x**y
55 * Method: Let x = 2 * (1+f)
56 * 1. Compute and return log2(x) in two pieces:
58 * where w1 has 53-24 = 29 bit trailing zeros.
59 * 2. Perform y*log2(x) = n+y' by simulating muti-precision
60 * arithmetic, where |y'|<=0.5.
61 * 3. Return x**y = 2**n*exp(y'*log2)
64 * 1. (anything) ** 0 is 1
65 * 2. (anything) ** 1 is itself
66 * 3. (anything) ** NAN is NAN
67 * 4. NAN ** (anything except 0) is NAN
68 * 5. +-(|x| > 1) ** +INF is +INF
69 * 6. +-(|x| > 1) ** -INF is +0
70 * 7. +-(|x| < 1) ** +INF is +0
71 * 8. +-(|x| < 1) ** -INF is +INF
72 * 9. +-1 ** +-INF is NAN
73 * 10. +0 ** (+anything except 0, NAN) is +0
74 * 11. -0 ** (+anything except 0, NAN, odd integer) is +0
75 * 12. +0 ** (-anything except 0, NAN) is +INF
76 * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF
77 * 14. -0 ** (odd integer) = -( +0 ** (odd integer) )
78 * 15. +INF ** (+anything except 0,NAN) is +INF
79 * 16. +INF ** (-anything except 0,NAN) is +0
80 * 17. -INF ** (anything) = -0 ** (-anything)
81 * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
82 * 19. (-anything except 0 and inf) ** (non-integer) is NAN
85 * pow(x,y) returns x**y nearly rounded. In particular
86 * pow(integer,integer)
87 * always returns the correct integer provided it is
91 * The hexadecimal values are the intended ones for the following
92 * constants. The decimal values may be used, provided that the
93 * compiler will convert from decimal to binary accurately enough
94 * to produce the hexadecimal values shown.
100 /* Microsoft Compiler */
101 #pragma warning( disable : 4723 ) /* disables potential divide by 0 warning */
110 dp_h
[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
111 dp_l
[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */
115 two53
= 9007199254740992.0, /* 0x43400000, 0x00000000 */
116 really_big
= 1.0e300
,
118 /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
119 L1
= 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */
120 L2
= 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */
121 L3
= 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */
122 L4
= 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */
123 L5
= 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */
124 L6
= 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */
125 P1
= 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
126 P2
= -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
127 P3
= 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
128 P4
= -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
129 P5
= 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */
130 lg2
= 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
131 lg2_h
= 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */
132 lg2_l
= -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */
133 ovt
= 8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */
134 cp
= 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
135 cp_h
= 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */
136 cp_l
= -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/
137 ivln2
= 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */
138 ivln2_h
= 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/
139 ivln2_l
= 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/
142 double __ieee754_pow(double x
, double y
)
144 double __ieee754_pow(x
,y
)
148 fd_twoints ux
, uy
, uz
;
149 double y1
,t1
,p_h
,t
,z
,ax
;
157 hx
= __HI(ux
); lx
= __LO(ux
);
158 hy
= __HI(uy
); ly
= __LO(uy
);
159 ix
= hx
&0x7fffffff; iy
= hy
&0x7fffffff;
161 /* y==zero: x**0 = 1 */
162 if((iy
|ly
)==0) return one
;
164 /* +-NaN return x+y */
165 if(ix
> 0x7ff00000 || ((ix
==0x7ff00000)&&(lx
!=0)) ||
166 iy
> 0x7ff00000 || ((iy
==0x7ff00000)&&(ly
!=0)))
169 /* determine if y is an odd int when x < 0
170 * yisint = 0 ... y is not an integer
171 * yisint = 1 ... y is an odd int
172 * yisint = 2 ... y is an even int
176 if(iy
>=0x43400000) yisint
= 2; /* even integer y */
177 else if(iy
>=0x3ff00000) {
178 k
= (iy
>>20)-0x3ff; /* exponent */
181 if((j
<<(52-k
))==(int)ly
) yisint
= 2-(j
&1);
184 if((j
<<(20-k
))==iy
) yisint
= 2-(j
&1);
189 /* special value of y */
191 if (iy
==0x7ff00000) { /* y is +-inf */
192 if(((ix
-0x3ff00000)|lx
)==0)
194 /* VC++ optimizer reduces y - y to 0 */
197 return y
- y
; /* inf**+-1 is NaN */
199 else if (ix
>= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */
200 return (hy
>=0)? y
: zero
;
201 else /* (|x|<1)**-,+inf = inf,0 */
202 return (hy
<0)?-y
: zero
;
204 if(iy
==0x3ff00000) { /* y is +-1 */
205 if(hy
<0) return one
/x
; else return x
;
207 if(hy
==0x40000000) return x
*x
; /* y is 2 */
208 if(hy
==0x3fe00000) { /* y is 0.5 */
209 if(hx
>=0) /* x >= +0 */
215 /* special value of x */
217 if(ix
==0x7ff00000||ix
==0||ix
==0x3ff00000){
218 z
= ax
; /*x is +-0,+-inf,+-1*/
219 if(hy
<0) z
= one
/z
; /* z = (1/|x|) */
221 if(((ix
-0x3ff00000)|yisint
)==0) {
222 z
= (z
-z
)/(z
-z
); /* (-1)**non-int is NaN */
223 } else if(yisint
==1) {
226 __HI(uz
) ^= 1<<31; /* some HPUXes cannot negate 0.. */
229 z
= -z
; /* (x<0)**odd = -(|x|**odd) */
237 /* (x<0)**(non-int) is NaN */
238 if((((hx
>>31)+1)|yisint
)==0) return (x
-x
)/(x
-x
);
240 /* |y| is really_big */
241 if(iy
>0x41e00000) { /* if |y| > 2**31 */
242 if(iy
>0x43f00000){ /* if |y| > 2**64, must o/uflow */
243 if(ix
<=0x3fefffff) return (hy
<0)? really_big
*really_big
:tiny
*tiny
;
244 if(ix
>=0x3ff00000) return (hy
>0)? really_big
*really_big
:tiny
*tiny
;
246 /* over/underflow if x is not close to one */
247 if(ix
<0x3fefffff) return (hy
<0)? really_big
*really_big
:tiny
*tiny
;
248 if(ix
>0x3ff00000) return (hy
>0)? really_big
*really_big
:tiny
*tiny
;
249 /* now |1-x| is tiny <= 2**-20, suffice to compute
250 log(x) by x-x^2/2+x^3/3-x^4/4 */
251 t
= x
-1; /* t has 20 trailing zeros */
252 w
= (t
*t
)*(0.5-t
*(0.3333333333333333333333-t
*0.25));
253 u
= ivln2_h
*t
; /* ivln2_h has 21 sig. bits */
254 v
= t
*ivln2_l
-w
*ivln2
;
264 /* take care subnormal number */
266 {ax
*= two53
; n
-= 53; uz
.d
= ax
; ix
= __HI(uz
); }
267 n
+= ((ix
)>>20)-0x3ff;
269 /* determine interval */
270 ix
= j
|0x3ff00000; /* normalize ix */
271 if(j
<=0x3988E) k
=0; /* |x|<sqrt(3/2) */
272 else if(j
<0xBB67A) k
=1; /* |x|<sqrt(3) */
273 else {k
=0;n
+=1;ix
-= 0x00100000;}
278 /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
279 u
= ax
-bp
[k
]; /* bp[0]=1.0, bp[1]=1.5 */
286 /* t_h=ax+bp[k] High */
289 __HI(uz
)=((ix
>>1)|0x20000000)+0x00080000+(k
<<18);
291 t_l
= ax
- (t_h
-bp
[k
]);
292 s_l
= v
*((u
-s_h
*t_h
)-s_h
*t_l
);
293 /* compute log(ax) */
295 r
= s2
*s2
*(L1
+s2
*(L2
+s2
*(L3
+s2
*(L4
+s2
*(L5
+s2
*L6
)))));
302 t_l
= r
-((t_h
-3.0)-s2
);
303 /* u+v = s*(1+...) */
306 /* 2/(3log2)*(s+...) */
312 z_h
= cp_h
*p_h
; /* cp_h+cp_l = 2/(3*log2) */
313 z_l
= cp_l
*p_h
+p_l
*cp
+dp_l
[k
];
314 /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
316 t1
= (((z_h
+z_l
)+dp_h
[k
])+t
);
320 t2
= z_l
-(((t1
-t
)-dp_h
[k
])-z_h
);
323 s
= one
; /* s (sign of result -ve**odd) = -1 else = 1 */
324 if((((hx
>>31)+1)|(yisint
-1))==0) s
= -one
;/* (-ve)**(odd int) */
326 /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
331 p_l
= (y
-y1
)*t1
+y
*t2
;
338 if (j
>=0x40900000) { /* z >= 1024 */
339 if(((j
-0x40900000)|i
)!=0) /* if z > 1024 */
340 return s
*really_big
*really_big
; /* overflow */
342 if(p_l
+ovt
>z
-p_h
) return s
*really_big
*really_big
; /* overflow */
344 } else if((j
&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */
345 if(((j
-0xc090cc00)|i
)!=0) /* z < -1075 */
346 return s
*tiny
*tiny
; /* underflow */
348 if(p_l
<=z
-p_h
) return s
*tiny
*tiny
; /* underflow */
352 * compute 2**(p_h+p_l)
357 if(i
>0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */
358 n
= j
+(0x00100000>>(k
+1));
359 k
= ((n
&0x7fffffff)>>20)-0x3ff; /* new k for n */
362 __HI(uz
) = (n
&~(0x000fffff>>k
));
364 n
= ((n
&0x000fffff)|0x00100000)>>(20-k
);
373 v
= (p_l
-(t
-p_h
))*lg2
+t
*lg2_l
;
377 t1
= z
- t
*(P1
+t
*(P2
+t
*(P3
+t
*(P4
+t
*P5
))));
378 r
= (z
*t1
)/(t1
-two
)-(w
+z
*w
);
383 if((j
>>20)<=0) z
= fd_scalbn(z
,n
); /* subnormal output */
384 else { uz
.d
= z
; __HI(uz
) += (n
<<20); z
= uz
.d
; }