source: asp3_tinet_ecnl_arm/trunk/musl-1.1.18/src/math/pow.c@ 352

Last change on this file since 352 was 352, checked in by coas-nagasima, 6 years ago

arm向けASP3版ECNLを追加
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1/* origin: FreeBSD /usr/src/lib/msun/src/e_pow.c */
2/*
3 * ====================================================
4 * Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
5 *
6 * Permission to use, copy, modify, and distribute this
7 * software is freely granted, provided that this notice
8 * is preserved.
9 * ====================================================
10 */
11/* pow(x,y) return x**y
12 *
13 * n
14 * Method: Let x = 2 * (1+f)
15 * 1. Compute and return log2(x) in two pieces:
16 * log2(x) = w1 + w2,
17 * where w1 has 53-24 = 29 bit trailing zeros.
18 * 2. Perform y*log2(x) = n+y' by simulating muti-precision
19 * arithmetic, where |y'|<=0.5.
20 * 3. Return x**y = 2**n*exp(y'*log2)
21 *
22 * Special cases:
23 * 1. (anything) ** 0 is 1
24 * 2. 1 ** (anything) is 1
25 * 3. (anything except 1) ** NAN is NAN
26 * 4. NAN ** (anything except 0) is NAN
27 * 5. +-(|x| > 1) ** +INF is +INF
28 * 6. +-(|x| > 1) ** -INF is +0
29 * 7. +-(|x| < 1) ** +INF is +0
30 * 8. +-(|x| < 1) ** -INF is +INF
31 * 9. -1 ** +-INF is 1
32 * 10. +0 ** (+anything except 0, NAN) is +0
33 * 11. -0 ** (+anything except 0, NAN, odd integer) is +0
34 * 12. +0 ** (-anything except 0, NAN) is +INF, raise divbyzero
35 * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF, raise divbyzero
36 * 14. -0 ** (+odd integer) is -0
37 * 15. -0 ** (-odd integer) is -INF, raise divbyzero
38 * 16. +INF ** (+anything except 0,NAN) is +INF
39 * 17. +INF ** (-anything except 0,NAN) is +0
40 * 18. -INF ** (+odd integer) is -INF
41 * 19. -INF ** (anything) = -0 ** (-anything), (anything except odd integer)
42 * 20. (anything) ** 1 is (anything)
43 * 21. (anything) ** -1 is 1/(anything)
44 * 22. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
45 * 23. (-anything except 0 and inf) ** (non-integer) is NAN
46 *
47 * Accuracy:
48 * pow(x,y) returns x**y nearly rounded. In particular
49 * pow(integer,integer)
50 * always returns the correct integer provided it is
51 * representable.
52 *
53 * Constants :
54 * The hexadecimal values are the intended ones for the following
55 * constants. The decimal values may be used, provided that the
56 * compiler will convert from decimal to binary accurately enough
57 * to produce the hexadecimal values shown.
58 */
59
60#include "libm.h"
61
62static const double
63bp[] = {1.0, 1.5,},
64dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
65dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */
66two53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */
67huge = 1.0e300,
68tiny = 1.0e-300,
69/* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
70L1 = 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */
71L2 = 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */
72L3 = 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */
73L4 = 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */
74L5 = 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */
75L6 = 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */
76P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
77P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
78P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
79P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
80P5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */
81lg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
82lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */
83lg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */
84ovt = 8.0085662595372944372e-017, /* -(1024-log2(ovfl+.5ulp)) */
85cp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
86cp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */
87cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/
88ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */
89ivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/
90ivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/
91
92double pow(double x, double y)
93{
94 double z,ax,z_h,z_l,p_h,p_l;
95 double y1,t1,t2,r,s,t,u,v,w;
96 int32_t i,j,k,yisint,n;
97 int32_t hx,hy,ix,iy;
98 uint32_t lx,ly;
99
100 EXTRACT_WORDS(hx, lx, x);
101 EXTRACT_WORDS(hy, ly, y);
102 ix = hx & 0x7fffffff;
103 iy = hy & 0x7fffffff;
104
105 /* x**0 = 1, even if x is NaN */
106 if ((iy|ly) == 0)
107 return 1.0;
108 /* 1**y = 1, even if y is NaN */
109 if (hx == 0x3ff00000 && lx == 0)
110 return 1.0;
111 /* NaN if either arg is NaN */
112 if (ix > 0x7ff00000 || (ix == 0x7ff00000 && lx != 0) ||
113 iy > 0x7ff00000 || (iy == 0x7ff00000 && ly != 0))
114 return x + y;
115
116 /* determine if y is an odd int when x < 0
117 * yisint = 0 ... y is not an integer
118 * yisint = 1 ... y is an odd int
119 * yisint = 2 ... y is an even int
120 */
121 yisint = 0;
122 if (hx < 0) {
123 if (iy >= 0x43400000)
124 yisint = 2; /* even integer y */
125 else if (iy >= 0x3ff00000) {
126 k = (iy>>20) - 0x3ff; /* exponent */
127 if (k > 20) {
128 uint32_t j = ly>>(52-k);
129 if ((j<<(52-k)) == ly)
130 yisint = 2 - (j&1);
131 } else if (ly == 0) {
132 uint32_t j = iy>>(20-k);
133 if ((j<<(20-k)) == iy)
134 yisint = 2 - (j&1);
135 }
136 }
137 }
138
139 /* special value of y */
140 if (ly == 0) {
141 if (iy == 0x7ff00000) { /* y is +-inf */
142 if (((ix-0x3ff00000)|lx) == 0) /* (-1)**+-inf is 1 */
143 return 1.0;
144 else if (ix >= 0x3ff00000) /* (|x|>1)**+-inf = inf,0 */
145 return hy >= 0 ? y : 0.0;
146 else /* (|x|<1)**+-inf = 0,inf */
147 return hy >= 0 ? 0.0 : -y;
148 }
149 if (iy == 0x3ff00000) { /* y is +-1 */
150 if (hy >= 0)
151 return x;
152 y = 1/x;
153#if FLT_EVAL_METHOD!=0
154 {
155 union {double f; uint64_t i;} u = {y};
156 uint64_t i = u.i & -1ULL/2;
157 if (i>>52 == 0 && (i&(i-1)))
158 FORCE_EVAL((float)y);
159 }
160#endif
161 return y;
162 }
163 if (hy == 0x40000000) /* y is 2 */
164 return x*x;
165 if (hy == 0x3fe00000) { /* y is 0.5 */
166 if (hx >= 0) /* x >= +0 */
167 return sqrt(x);
168 }
169 }
170
171 ax = fabs(x);
172 /* special value of x */
173 if (lx == 0) {
174 if (ix == 0x7ff00000 || ix == 0 || ix == 0x3ff00000) { /* x is +-0,+-inf,+-1 */
175 z = ax;
176 if (hy < 0) /* z = (1/|x|) */
177 z = 1.0/z;
178 if (hx < 0) {
179 if (((ix-0x3ff00000)|yisint) == 0) {
180 z = (z-z)/(z-z); /* (-1)**non-int is NaN */
181 } else if (yisint == 1)
182 z = -z; /* (x<0)**odd = -(|x|**odd) */
183 }
184 return z;
185 }
186 }
187
188 s = 1.0; /* sign of result */
189 if (hx < 0) {
190 if (yisint == 0) /* (x<0)**(non-int) is NaN */
191 return (x-x)/(x-x);
192 if (yisint == 1) /* (x<0)**(odd int) */
193 s = -1.0;
194 }
195
196 /* |y| is huge */
197 if (iy > 0x41e00000) { /* if |y| > 2**31 */
198 if (iy > 0x43f00000) { /* if |y| > 2**64, must o/uflow */
199 if (ix <= 0x3fefffff)
200 return hy < 0 ? huge*huge : tiny*tiny;
201 if (ix >= 0x3ff00000)
202 return hy > 0 ? huge*huge : tiny*tiny;
203 }
204 /* over/underflow if x is not close to one */
205 if (ix < 0x3fefffff)
206 return hy < 0 ? s*huge*huge : s*tiny*tiny;
207 if (ix > 0x3ff00000)
208 return hy > 0 ? s*huge*huge : s*tiny*tiny;
209 /* now |1-x| is tiny <= 2**-20, suffice to compute
210 log(x) by x-x^2/2+x^3/3-x^4/4 */
211 t = ax - 1.0; /* t has 20 trailing zeros */
212 w = (t*t)*(0.5 - t*(0.3333333333333333333333-t*0.25));
213 u = ivln2_h*t; /* ivln2_h has 21 sig. bits */
214 v = t*ivln2_l - w*ivln2;
215 t1 = u + v;
216 SET_LOW_WORD(t1, 0);
217 t2 = v - (t1-u);
218 } else {
219 double ss,s2,s_h,s_l,t_h,t_l;
220 n = 0;
221 /* take care subnormal number */
222 if (ix < 0x00100000) {
223 ax *= two53;
224 n -= 53;
225 GET_HIGH_WORD(ix,ax);
226 }
227 n += ((ix)>>20) - 0x3ff;
228 j = ix & 0x000fffff;
229 /* determine interval */
230 ix = j | 0x3ff00000; /* normalize ix */
231 if (j <= 0x3988E) /* |x|<sqrt(3/2) */
232 k = 0;
233 else if (j < 0xBB67A) /* |x|<sqrt(3) */
234 k = 1;
235 else {
236 k = 0;
237 n += 1;
238 ix -= 0x00100000;
239 }
240 SET_HIGH_WORD(ax, ix);
241
242 /* compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
243 u = ax - bp[k]; /* bp[0]=1.0, bp[1]=1.5 */
244 v = 1.0/(ax+bp[k]);
245 ss = u*v;
246 s_h = ss;
247 SET_LOW_WORD(s_h, 0);
248 /* t_h=ax+bp[k] High */
249 t_h = 0.0;
250 SET_HIGH_WORD(t_h, ((ix>>1)|0x20000000) + 0x00080000 + (k<<18));
251 t_l = ax - (t_h-bp[k]);
252 s_l = v*((u-s_h*t_h)-s_h*t_l);
253 /* compute log(ax) */
254 s2 = ss*ss;
255 r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
256 r += s_l*(s_h+ss);
257 s2 = s_h*s_h;
258 t_h = 3.0 + s2 + r;
259 SET_LOW_WORD(t_h, 0);
260 t_l = r - ((t_h-3.0)-s2);
261 /* u+v = ss*(1+...) */
262 u = s_h*t_h;
263 v = s_l*t_h + t_l*ss;
264 /* 2/(3log2)*(ss+...) */
265 p_h = u + v;
266 SET_LOW_WORD(p_h, 0);
267 p_l = v - (p_h-u);
268 z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */
269 z_l = cp_l*p_h+p_l*cp + dp_l[k];
270 /* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */
271 t = (double)n;
272 t1 = ((z_h + z_l) + dp_h[k]) + t;
273 SET_LOW_WORD(t1, 0);
274 t2 = z_l - (((t1 - t) - dp_h[k]) - z_h);
275 }
276
277 /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
278 y1 = y;
279 SET_LOW_WORD(y1, 0);
280 p_l = (y-y1)*t1 + y*t2;
281 p_h = y1*t1;
282 z = p_l + p_h;
283 EXTRACT_WORDS(j, i, z);
284 if (j >= 0x40900000) { /* z >= 1024 */
285 if (((j-0x40900000)|i) != 0) /* if z > 1024 */
286 return s*huge*huge; /* overflow */
287 if (p_l + ovt > z - p_h)
288 return s*huge*huge; /* overflow */
289 } else if ((j&0x7fffffff) >= 0x4090cc00) { /* z <= -1075 */ // FIXME: instead of abs(j) use unsigned j
290 if (((j-0xc090cc00)|i) != 0) /* z < -1075 */
291 return s*tiny*tiny; /* underflow */
292 if (p_l <= z - p_h)
293 return s*tiny*tiny; /* underflow */
294 }
295 /*
296 * compute 2**(p_h+p_l)
297 */
298 i = j & 0x7fffffff;
299 k = (i>>20) - 0x3ff;
300 n = 0;
301 if (i > 0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */
302 n = j + (0x00100000>>(k+1));
303 k = ((n&0x7fffffff)>>20) - 0x3ff; /* new k for n */
304 t = 0.0;
305 SET_HIGH_WORD(t, n & ~(0x000fffff>>k));
306 n = ((n&0x000fffff)|0x00100000)>>(20-k);
307 if (j < 0)
308 n = -n;
309 p_h -= t;
310 }
311 t = p_l + p_h;
312 SET_LOW_WORD(t, 0);
313 u = t*lg2_h;
314 v = (p_l-(t-p_h))*lg2 + t*lg2_l;
315 z = u + v;
316 w = v - (z-u);
317 t = z*z;
318 t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
319 r = (z*t1)/(t1-2.0) - (w + z*w);
320 z = 1.0 - (r-z);
321 GET_HIGH_WORD(j, z);
322 j += n<<20;
323 if ((j>>20) <= 0) /* subnormal output */
324 z = scalbn(z,n);
325 else
326 SET_HIGH_WORD(z, j);
327 return s*z;
328}
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