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1 | /* @(#)e_pow.c 5.1 93/09/24 */ |
2 | /* | |
3 | * ==================================================== | |
4 | * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. | |
5 | * | |
6 | * Developed at SunPro, a Sun Microsystems, Inc. business. | |
7 | * Permission to use, copy, modify, and distribute this | |
8 | * software is freely granted, provided that this notice | |
9 | * is preserved. | |
10 | * ==================================================== | |
11 | */ | |
12 | ||
13 | #if defined(LIBM_SCCS) && !defined(lint) | |
14 | static char rcsid[] = "$NetBSD: e_pow.c,v 1.9 1995/05/12 04:57:32 jtc Exp $"; | |
15 | #endif | |
16 | ||
17 | /* __ieee754_pow(x,y) return x**y | |
18 | * | |
19 | * n | |
20 | * Method: Let x = 2 * (1+f) | |
21 | * 1. Compute and return log2(x) in two pieces: | |
22 | * log2(x) = w1 + w2, | |
23 | * where w1 has 53-24 = 29 bit trailing zeros. | |
24 | * 2. Perform y*log2(x) = n+y' by simulating muti-precision | |
25 | * arithmetic, where |y'|<=0.5. | |
26 | * 3. Return x**y = 2**n*exp(y'*log2) | |
27 | * | |
28 | * Special cases: | |
29 | * 1. (anything) ** 0 is 1 | |
30 | * 2. (anything) ** 1 is itself | |
31 | * 3. (anything) ** NAN is NAN | |
32 | * 4. NAN ** (anything except 0) is NAN | |
33 | * 5. +-(|x| > 1) ** +INF is +INF | |
34 | * 6. +-(|x| > 1) ** -INF is +0 | |
35 | * 7. +-(|x| < 1) ** +INF is +0 | |
36 | * 8. +-(|x| < 1) ** -INF is +INF | |
37 | * 9. +-1 ** +-INF is NAN | |
38 | * 10. +0 ** (+anything except 0, NAN) is +0 | |
39 | * 11. -0 ** (+anything except 0, NAN, odd integer) is +0 | |
40 | * 12. +0 ** (-anything except 0, NAN) is +INF | |
41 | * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF | |
42 | * 14. -0 ** (odd integer) = -( +0 ** (odd integer) ) | |
43 | * 15. +INF ** (+anything except 0,NAN) is +INF | |
44 | * 16. +INF ** (-anything except 0,NAN) is +0 | |
45 | * 17. -INF ** (anything) = -0 ** (-anything) | |
46 | * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer) | |
47 | * 19. (-anything except 0 and inf) ** (non-integer) is NAN | |
48 | * | |
49 | * Accuracy: | |
50 | * pow(x,y) returns x**y nearly rounded. In particular | |
51 | * pow(integer,integer) | |
52 | * always returns the correct integer provided it is | |
53 | * representable. | |
54 | * | |
55 | * Constants : | |
56 | * The hexadecimal values are the intended ones for the following | |
57 | * constants. The decimal values may be used, provided that the | |
58 | * compiler will convert from decimal to binary accurately enough | |
59 | * to produce the hexadecimal values shown. | |
60 | */ | |
61 | ||
62 | #include "math.h" | |
63 | #include "math_private.h" | |
64 | ||
65 | #ifdef __STDC__ | |
66 | static const double | |
67 | #else | |
68 | static double | |
69 | #endif | |
70 | bp[] = {1.0, 1.5,}, | |
71 | dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */ | |
72 | dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */ | |
73 | zero = 0.0, | |
74 | one = 1.0, | |
75 | two = 2.0, | |
76 | two53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */ | |
77 | huge = 1.0e300, | |
78 | tiny = 1.0e-300, | |
79 | /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */ | |
80 | L1 = 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */ | |
81 | L2 = 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */ | |
82 | L3 = 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */ | |
83 | L4 = 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */ | |
84 | L5 = 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */ | |
85 | L6 = 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */ | |
86 | P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */ | |
87 | P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */ | |
88 | P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */ | |
89 | P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */ | |
90 | P5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */ | |
91 | lg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */ | |
92 | lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */ | |
93 | lg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */ | |
94 | ovt = 8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */ | |
95 | cp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */ | |
96 | cp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */ | |
97 | cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/ | |
98 | ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */ | |
99 | ivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/ | |
100 | ivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/ | |
101 | ||
102 | #ifdef __STDC__ | |
103 | double __ieee754_pow(double x, double y) | |
104 | #else | |
105 | double __ieee754_pow(x,y) | |
106 | double x, y; | |
107 | #endif | |
108 | { | |
109 | double z,ax,z_h,z_l,p_h,p_l; | |
110 | double y1,t1,t2,r,s,t,u,v,w; | |
111 | int32_t i,j,k,yisint,n; | |
112 | int32_t hx,hy,ix,iy; | |
113 | u_int32_t lx,ly; | |
114 | ||
115 | EXTRACT_WORDS(hx,lx,x); | |
116 | EXTRACT_WORDS(hy,ly,y); | |
117 | ix = hx&0x7fffffff; iy = hy&0x7fffffff; | |
118 | ||
119 | /* y==zero: x**0 = 1 */ | |
120 | if((iy|ly)==0) return one; | |
121 | ||
122 | /* +-NaN return x+y */ | |
123 | if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) || | |
124 | iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0))) | |
125 | return x+y; | |
126 | ||
127 | /* determine if y is an odd int when x < 0 | |
128 | * yisint = 0 ... y is not an integer | |
129 | * yisint = 1 ... y is an odd int | |
130 | * yisint = 2 ... y is an even int | |
131 | */ | |
132 | yisint = 0; | |
133 | if(hx<0) { | |
134 | if(iy>=0x43400000) yisint = 2; /* even integer y */ | |
135 | else if(iy>=0x3ff00000) { | |
136 | k = (iy>>20)-0x3ff; /* exponent */ | |
137 | if(k>20) { | |
138 | j = ly>>(52-k); | |
139 | if((j<<(52-k))==ly) yisint = 2-(j&1); | |
140 | } else if(ly==0) { | |
141 | j = iy>>(20-k); | |
142 | if((j<<(20-k))==iy) yisint = 2-(j&1); | |
143 | } | |
144 | } | |
145 | } | |
146 | ||
147 | /* special value of y */ | |
148 | if(ly==0) { | |
149 | if (iy==0x7ff00000) { /* y is +-inf */ | |
150 | if(((ix-0x3ff00000)|lx)==0) | |
151 | return y - y; /* inf**+-1 is NaN */ | |
152 | else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */ | |
153 | return (hy>=0)? y: zero; | |
154 | else /* (|x|<1)**-,+inf = inf,0 */ | |
155 | return (hy<0)?-y: zero; | |
156 | } | |
157 | if(iy==0x3ff00000) { /* y is +-1 */ | |
158 | if(hy<0) return one/x; else return x; | |
159 | } | |
160 | if(hy==0x40000000) return x*x; /* y is 2 */ | |
161 | if(hy==0x3fe00000) { /* y is 0.5 */ | |
162 | if(hx>=0) /* x >= +0 */ | |
163 | return __ieee754_sqrt(x); | |
164 | } | |
165 | } | |
166 | ||
167 | ax = fabs(x); | |
168 | /* special value of x */ | |
169 | if(lx==0) { | |
170 | if(ix==0x7ff00000||ix==0||ix==0x3ff00000){ | |
171 | z = ax; /*x is +-0,+-inf,+-1*/ | |
172 | if(hy<0) z = one/z; /* z = (1/|x|) */ | |
173 | if(hx<0) { | |
174 | if(((ix-0x3ff00000)|yisint)==0) { | |
175 | z = (z-z)/(z-z); /* (-1)**non-int is NaN */ | |
176 | } else if(yisint==1) | |
177 | z = -z; /* (x<0)**odd = -(|x|**odd) */ | |
178 | } | |
179 | return z; | |
180 | } | |
181 | } | |
182 | ||
183 | /* (x<0)**(non-int) is NaN */ | |
184 | if(((((u_int32_t)hx>>31)-1)|yisint)==0) return (x-x)/(x-x); | |
185 | ||
186 | /* |y| is huge */ | |
187 | if(iy>0x41e00000) { /* if |y| > 2**31 */ | |
188 | if(iy>0x43f00000){ /* if |y| > 2**64, must o/uflow */ | |
189 | if(ix<=0x3fefffff) return (hy<0)? huge*huge:tiny*tiny; | |
190 | if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny; | |
191 | } | |
192 | /* over/underflow if x is not close to one */ | |
193 | if(ix<0x3fefffff) return (hy<0)? huge*huge:tiny*tiny; | |
194 | if(ix>0x3ff00000) return (hy>0)? huge*huge:tiny*tiny; | |
195 | /* now |1-x| is tiny <= 2**-20, suffice to compute | |
196 | log(x) by x-x^2/2+x^3/3-x^4/4 */ | |
197 | t = x-1; /* t has 20 trailing zeros */ | |
198 | w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25)); | |
199 | u = ivln2_h*t; /* ivln2_h has 21 sig. bits */ | |
200 | v = t*ivln2_l-w*ivln2; | |
201 | t1 = u+v; | |
202 | SET_LOW_WORD(t1,0); | |
203 | t2 = v-(t1-u); | |
204 | } else { | |
205 | double s2,s_h,s_l,t_h,t_l; | |
206 | n = 0; | |
207 | /* take care subnormal number */ | |
208 | if(ix<0x00100000) | |
209 | {ax *= two53; n -= 53; GET_HIGH_WORD(ix,ax); } | |
210 | n += ((ix)>>20)-0x3ff; | |
211 | j = ix&0x000fffff; | |
212 | /* determine interval */ | |
213 | ix = j|0x3ff00000; /* normalize ix */ | |
214 | if(j<=0x3988E) k=0; /* |x|<sqrt(3/2) */ | |
215 | else if(j<0xBB67A) k=1; /* |x|<sqrt(3) */ | |
216 | else {k=0;n+=1;ix -= 0x00100000;} | |
217 | SET_HIGH_WORD(ax,ix); | |
218 | ||
219 | /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */ | |
220 | u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */ | |
221 | v = one/(ax+bp[k]); | |
222 | s = u*v; | |
223 | s_h = s; | |
224 | SET_LOW_WORD(s_h,0); | |
225 | /* t_h=ax+bp[k] High */ | |
226 | t_h = zero; | |
227 | SET_HIGH_WORD(t_h,((ix>>1)|0x20000000)+0x00080000+(k<<18)); | |
228 | t_l = ax - (t_h-bp[k]); | |
229 | s_l = v*((u-s_h*t_h)-s_h*t_l); | |
230 | /* compute log(ax) */ | |
231 | s2 = s*s; | |
232 | r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6))))); | |
233 | r += s_l*(s_h+s); | |
234 | s2 = s_h*s_h; | |
235 | t_h = 3.0+s2+r; | |
236 | SET_LOW_WORD(t_h,0); | |
237 | t_l = r-((t_h-3.0)-s2); | |
238 | /* u+v = s*(1+...) */ | |
239 | u = s_h*t_h; | |
240 | v = s_l*t_h+t_l*s; | |
241 | /* 2/(3log2)*(s+...) */ | |
242 | p_h = u+v; | |
243 | SET_LOW_WORD(p_h,0); | |
244 | p_l = v-(p_h-u); | |
245 | z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */ | |
246 | z_l = cp_l*p_h+p_l*cp+dp_l[k]; | |
247 | /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */ | |
248 | t = (double)n; | |
249 | t1 = (((z_h+z_l)+dp_h[k])+t); | |
250 | SET_LOW_WORD(t1,0); | |
251 | t2 = z_l-(((t1-t)-dp_h[k])-z_h); | |
252 | } | |
253 | ||
254 | s = one; /* s (sign of result -ve**odd) = -1 else = 1 */ | |
255 | if(((((u_int32_t)hx>>31)-1)|(yisint-1))==0) | |
256 | s = -one;/* (-ve)**(odd int) */ | |
257 | ||
258 | /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */ | |
259 | y1 = y; | |
260 | SET_LOW_WORD(y1,0); | |
261 | p_l = (y-y1)*t1+y*t2; | |
262 | p_h = y1*t1; | |
263 | z = p_l+p_h; | |
264 | EXTRACT_WORDS(j,i,z); | |
265 | if (j>=0x40900000) { /* z >= 1024 */ | |
266 | if(((j-0x40900000)|i)!=0) /* if z > 1024 */ | |
267 | return s*huge*huge; /* overflow */ | |
268 | else { | |
269 | if(p_l+ovt>z-p_h) return s*huge*huge; /* overflow */ | |
270 | } | |
271 | } else if((j&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */ | |
272 | if(((j-0xc090cc00)|i)!=0) /* z < -1075 */ | |
273 | return s*tiny*tiny; /* underflow */ | |
274 | else { | |
275 | if(p_l<=z-p_h) return s*tiny*tiny; /* underflow */ | |
276 | } | |
277 | } | |
278 | /* | |
279 | * compute 2**(p_h+p_l) | |
280 | */ | |
281 | i = j&0x7fffffff; | |
282 | k = (i>>20)-0x3ff; | |
283 | n = 0; | |
284 | if(i>0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */ | |
285 | n = j+(0x00100000>>(k+1)); | |
286 | k = ((n&0x7fffffff)>>20)-0x3ff; /* new k for n */ | |
287 | t = zero; | |
288 | SET_HIGH_WORD(t,n&~(0x000fffff>>k)); | |
289 | n = ((n&0x000fffff)|0x00100000)>>(20-k); | |
290 | if(j<0) n = -n; | |
291 | p_h -= t; | |
292 | } | |
293 | t = p_l+p_h; | |
294 | SET_LOW_WORD(t,0); | |
295 | u = t*lg2_h; | |
296 | v = (p_l-(t-p_h))*lg2+t*lg2_l; | |
297 | z = u+v; | |
298 | w = v-(z-u); | |
299 | t = z*z; | |
300 | t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); | |
301 | r = (z*t1)/(t1-two)-(w+z*w); | |
302 | z = one-(r-z); | |
303 | GET_HIGH_WORD(j,z); | |
304 | j += (n<<20); | |
305 | if((j>>20)<=0) z = __scalbn(z,n); /* subnormal output */ | |
306 | else SET_HIGH_WORD(z,j); | |
307 | return s*z; | |
308 | } |