1
2 /* @(#)e_pow.c 5.1 93/09/24 */
3 /*
4 * ====================================================
5 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
6 *
7 * Developed at SunPro, a Sun Microsystems, Inc. business.
8 * Permission to use, copy, modify, and distribute this
9 * software is freely granted, provided that this notice
10 * is preserved.
11 * ====================================================
12 */
13
14 /* pow(x,y) return x**y
15 *
16 * n
17 * Method: Let x = 2 * (1+f)
18 * 1. Compute and return log2(x) in two pieces:
19 * log2(x) = w1 + w2,
20 * where w1 has 53-24 = 29 bit trailing zeros.
21 * 2. Perform y*log2(x) = n+y' by simulating multi-precision
22 * arithmetic, where |y'|<=0.5.
23 * 3. Return x**y = 2**n*exp(y'*log2)
24 *
25 * Special cases:
26 * 1. (anything) ** 0 is 1
27 * 2. (anything) ** 1 is itself
28 * 3a. (anything) ** NAN is NAN except
29 * 3b. +1 ** NAN is 1
30 * 4. NAN ** (anything except 0) is NAN
31 * 5. +-(|x| > 1) ** +INF is +INF
32 * 6. +-(|x| > 1) ** -INF is +0
33 * 7. +-(|x| < 1) ** +INF is +0
34 * 8. +-(|x| < 1) ** -INF is +INF
35 * 9. +-1 ** +-INF is 1
36 * 10. +0 ** (+anything except 0, NAN) is +0
37 * 11. -0 ** (+anything except 0, NAN, odd integer) is +0
38 * 12. +0 ** (-anything except 0, NAN) is +INF
39 * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF
40 * 14. -0 ** (odd integer) = -( +0 ** (odd integer) )
41 * 15. +INF ** (+anything except 0,NAN) is +INF
42 * 16. +INF ** (-anything except 0,NAN) is +0
43 * 17. -INF ** (anything) = -0 ** (-anything)
44 * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
45 * 19. (-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 "fdlibm.h"
61
62 #if __OBSOLETE_MATH_DOUBLE
63
64 #ifdef _NEED_FLOAT64
65
66 static const __float64
67 bp[] = {_F_64(1.0), _F_64(1.5),},
68 dp_h[] = { _F_64(0.0), _F_64(5.84962487220764160156e-01),}, /* 0x3FE2B803, 0x40000000 */
69 dp_l[] = { _F_64(0.0), _F_64(1.35003920212974897128e-08),}, /* 0x3E4CFDEB, 0x43CFD006 */
70 zero = _F_64(0.0),
71 one = _F_64(1.0),
72 two = _F_64(2.0),
73 two53 = _F_64(9007199254740992.0), /* 0x43400000, 0x00000000 */
74 /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
75 L1 = _F_64(5.99999999999994648725e-01), /* 0x3FE33333, 0x33333303 */
76 L2 = _F_64(4.28571428578550184252e-01), /* 0x3FDB6DB6, 0xDB6FABFF */
77 L3 = _F_64(3.33333329818377432918e-01), /* 0x3FD55555, 0x518F264D */
78 L4 = _F_64(2.72728123808534006489e-01), /* 0x3FD17460, 0xA91D4101 */
79 L5 = _F_64(2.30660745775561754067e-01), /* 0x3FCD864A, 0x93C9DB65 */
80 L6 = _F_64(2.06975017800338417784e-01), /* 0x3FCA7E28, 0x4A454EEF */
81 P1 = _F_64(1.66666666666666019037e-01), /* 0x3FC55555, 0x5555553E */
82 P2 = _F_64(-2.77777777770155933842e-03), /* 0xBF66C16C, 0x16BEBD93 */
83 P3 = _F_64(6.61375632143793436117e-05), /* 0x3F11566A, 0xAF25DE2C */
84 P4 = _F_64(-1.65339022054652515390e-06), /* 0xBEBBBD41, 0xC5D26BF1 */
85 P5 = _F_64(4.13813679705723846039e-08), /* 0x3E663769, 0x72BEA4D0 */
86 lg2 = _F_64(6.93147180559945286227e-01), /* 0x3FE62E42, 0xFEFA39EF */
87 lg2_h = _F_64(6.93147182464599609375e-01), /* 0x3FE62E43, 0x00000000 */
88 lg2_l = _F_64(-1.90465429995776804525e-09), /* 0xBE205C61, 0x0CA86C39 */
89 ovt = _F_64(8.0085662595372944372e-0017), /* -(1024-log2(ovfl+.5ulp)) */
90 cp = _F_64(9.61796693925975554329e-01), /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
91 cp_h = _F_64(9.61796700954437255859e-01), /* 0x3FEEC709, 0xE0000000 =(float)cp */
92 cp_l = _F_64(-7.02846165095275826516e-09), /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/
93 ivln2 = _F_64(1.44269504088896338700e+00), /* 0x3FF71547, 0x652B82FE =1/ln2 */
94 ivln2_h = _F_64(1.44269502162933349609e+00), /* 0x3FF71547, 0x60000000 =24b 1/ln2*/
95 ivln2_l = _F_64(1.92596299112661746887e-08); /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/
96
97 __float64
pow64(__float64 x,__float64 y)98 pow64(__float64 x, __float64 y)
99 {
100 __float64 z, ax, z_h, z_l, p_h, p_l;
101 __float64 y1, t1, t2, r, s, t, u, v, w;
102 __int32_t i, j, k, yisint, n;
103 __int32_t hx, hy, ix, iy;
104 __uint32_t lx, ly;
105
106 EXTRACT_WORDS(hx, lx, x);
107 EXTRACT_WORDS(hy, ly, y);
108 ix = hx & 0x7fffffff;
109 iy = hy & 0x7fffffff;
110
111 /* y==zero: x**0 = 1 unless x is snan */
112 if ((iy | ly) == 0) {
113 if (issignaling64_inline(x))
114 return x + y;
115 return one;
116 }
117
118 /* x|y==NaN return NaN unless x==1 then return 1 */
119 if (ix > 0x7ff00000 || ((ix == 0x7ff00000) && (lx != 0)) ||
120 iy > 0x7ff00000 || ((iy == 0x7ff00000) && (ly != 0))) {
121 if (((hx - 0x3ff00000) | lx) == 0 && !issignaling64_inline(y))
122 return one;
123 else
124 return x + y;
125 }
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)
135 yisint = 2; /* even integer y */
136 else if (iy >= 0x3ff00000) {
137 k = (iy >> 20) - 0x3ff; /* exponent */
138 if (k > 20) {
139 __uint32_t uj = ly >> (52 - k);
140 if ((uj << (52 - k)) == ly)
141 yisint = 2 - (uj & 1);
142 } else if (ly == 0) {
143 j = iy >> (20 - k);
144 if ((j << (20 - k)) == iy)
145 yisint = 2 - (j & 1);
146 }
147 }
148 }
149
150 /* special value of y */
151 if (ly == 0) {
152 if (iy == 0x7ff00000) { /* y is +-inf */
153 if (((ix - 0x3ff00000) | lx) == 0)
154 return one; /* +-1**+-inf = 1 */
155 else if (ix >= 0x3ff00000) /* (|x|>1)**+-inf = inf,0 */
156 return (hy >= 0) ? y : zero;
157 else /* (|x|<1)**-,+inf = inf,0 */
158 return (hy < 0) ? -y : zero;
159 }
160 if (iy == 0x3ff00000) { /* y is +-1 */
161 if (hy < 0) {
162 if (x == 0)
163 return __math_divzero(hx < 0);
164 return one / x;
165 } else
166 return x;
167 }
168 if (hy == 0x40000000 && ix < 0x5ff00000 && ix > 0x1e500000)
169 return x * x; /* y is 2 */
170 if (hy == 0x3fe00000) { /* y is 0.5 */
171 if (hx >= 0) /* x >= +0 */
172 return sqrt(x);
173 }
174 }
175
176 ax = fabs64(x);
177 /* special value of x */
178 if (lx == 0) {
179 if (ix == 0x7ff00000 || ix == 0x3ff00000) {
180 z = ax; /*x is +-inf,+-1*/
181 if (hy < 0)
182 z = one / z; /* z = (1/|x|) */
183 if (hx < 0) {
184 if (((ix - 0x3ff00000) | yisint) == 0) {
185 return __math_invalid(x); /* (-1)**non-int is NaN */
186 } else if (yisint == 1)
187 z = -z; /* (x<0)**odd = -(|x|**odd) */
188 }
189 return z;
190 }
191
192 if (ix == 0) {
193 if (hy < 0)
194 return __math_divzero(hx < 0 && yisint == 1);
195 if (yisint != 1)
196 x = ax;
197 return x;
198 }
199 }
200
201 /* (x<0)**(non-int) is NaN */
202 /* REDHAT LOCAL: This used to be
203 if((((hx>>31)+1)|yisint)==0) return __math_invalid(x);
204 but ANSI C says a right shift of a signed negative quantity is
205 implementation defined. */
206 if (((((__uint32_t)hx >> 31) - 1) | yisint) == 0)
207 return __math_invalid(x);
208
209 /* |y| is huge */
210 if (iy > 0x42000000) { /* if |y| > ~2**33 */
211 if (iy > 0x43f00000) { /* if |y| > ~2**64, must o/uflow */
212 if (ix <= 0x3fefffff)
213 return (hy < 0) ? __math_oflow(0) : __math_uflow(0);
214 else
215 return (hy > 0) ? __math_oflow(0) : __math_uflow(0);
216 }
217 /* over/underflow if x is not close to one */
218 if (ix < 0x3fefffff) {
219 int sign = yisint & ((__uint32_t)hx>>31);
220 return (hy < 0) ? __math_oflow(sign) : __math_uflow(sign);
221 }
222 if (ix > 0x3ff00000) {
223 int sign = yisint & ((__uint32_t)hx>>31);
224 return (hy > 0) ? __math_oflow(sign) : __math_uflow(sign);
225 }
226 /* now |1-x| is tiny <= 2**-20, suffice to compute
227 log(x) by x-x^2/2+x^3/3-x^4/4 */
228 t = ax - 1; /* t has 20 trailing zeros */
229 w = (t * t) * (_F_64(0.5) - t * (_F_64(0.3333333333333333333333) - t * _F_64(0.25)));
230 u = ivln2_h * t; /* ivln2_h has 21 sig. bits */
231 v = t * ivln2_l - w * ivln2;
232 t1 = u + v;
233 SET_LOW_WORD(t1, 0);
234 t2 = v - (t1 - u);
235 } else {
236 __float64 s2, s_h, s_l, t_h, t_l;
237 n = 0;
238 /* take care subnormal number */
239 if (ix < 0x00100000) {
240 ax *= two53;
241 n -= 53;
242 GET_HIGH_WORD(ix, ax);
243 }
244 n += ((ix) >> 20) - 0x3ff;
245 j = ix & 0x000fffff;
246 /* determine interval */
247 ix = j | 0x3ff00000; /* normalize ix */
248 if (j <= 0x3988E)
249 k = 0; /* |x|<sqrt(3/2) */
250 else if (j < 0xBB67A)
251 k = 1; /* |x|<sqrt(3) */
252 else {
253 k = 0;
254 n += 1;
255 ix -= 0x00100000;
256 }
257 SET_HIGH_WORD(ax, ix);
258
259 /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
260 u = ax - bp[k]; /* bp[0]=1.0, bp[1]=1.5 */
261 v = one / (ax + bp[k]);
262 s = u * v;
263 s_h = s;
264 SET_LOW_WORD(s_h, 0);
265 /* t_h=ax+bp[k] High */
266 t_h = zero;
267 SET_HIGH_WORD(t_h, ((ix >> 1) | 0x20000000) + 0x00080000 + (k << 18));
268 t_l = ax - (t_h - bp[k]);
269 s_l = v * ((u - s_h * t_h) - s_h * t_l);
270 /* compute log(ax) */
271 s2 = s * s;
272 r = s2 * s2 *
273 (L1 + s2 * (L2 + s2 * (L3 + s2 * (L4 + s2 * (L5 + s2 * L6)))));
274 r += s_l * (s_h + s);
275 s2 = s_h * s_h;
276 t_h = _F_64(3.0) + s2 + r;
277 SET_LOW_WORD(t_h, 0);
278 t_l = r - ((t_h - _F_64(3.0)) - s2);
279 /* u+v = s*(1+...) */
280 u = s_h * t_h;
281 v = s_l * t_h + t_l * s;
282 /* 2/(3log2)*(s+...) */
283 p_h = u + v;
284 SET_LOW_WORD(p_h, 0);
285 p_l = v - (p_h - u);
286 z_h = cp_h * p_h; /* cp_h+cp_l = 2/(3*log2) */
287 z_l = cp_l * p_h + p_l * cp + dp_l[k];
288 /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
289 t = (__float64)n;
290 t1 = (((z_h + z_l) + dp_h[k]) + t);
291 SET_LOW_WORD(t1, 0);
292 t2 = z_l - (((t1 - t) - dp_h[k]) - z_h);
293 }
294
295 s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
296 if (((((__uint32_t)hx >> 31) - 1) | (yisint - 1)) == 0)
297 s = -one; /* (-ve)**(odd int) */
298
299 /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
300 y1 = y;
301 SET_LOW_WORD(y1, 0);
302 p_l = (y - y1) * t1 + y * t2;
303 p_h = y1 * t1;
304 z = p_l + p_h;
305 EXTRACT_WORDS(j, i, z);
306 if (j >= 0x40900000) { /* z >= 1024 */
307 if (((j - 0x40900000) | i) != 0) /* if z > 1024 */
308 return __math_oflow(s < 0); /* overflow */
309 else {
310 if (p_l + ovt > z - p_h)
311 return __math_oflow(s < 0); /* overflow */
312 }
313 } else if ((j & 0x7fffffff) >= 0x4090cc00) { /* z <= -1075 */
314 if (((j - 0xc090cc00) | i) != 0) /* z < -1075 */
315 return __math_uflow(s < 0); /* underflow */
316 else {
317 if (p_l <= z - p_h)
318 return __math_uflow(s < 0); /* underflow */
319 }
320 }
321 /*
322 * compute 2**(p_h+p_l)
323 */
324 i = j & 0x7fffffff;
325 k = (i >> 20) - 0x3ff;
326 n = 0;
327 if (i > 0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */
328 n = j + (0x00100000 >> (k + 1));
329 k = ((n & 0x7fffffff) >> 20) - 0x3ff; /* new k for n */
330 t = zero;
331 SET_HIGH_WORD(t, n & ~(0x000fffff >> k));
332 n = ((n & 0x000fffff) | 0x00100000) >> (20 - k);
333 if (j < 0)
334 n = -n;
335 p_h -= t;
336 }
337 t = p_l + p_h;
338 SET_LOW_WORD(t, 0);
339 u = t * lg2_h;
340 v = (p_l - (t - p_h)) * lg2 + t * lg2_l;
341 z = u + v;
342 w = v - (z - u);
343 t = z * z;
344 t1 = z - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5))));
345 r = (z * t1) / (t1 - two) - (w + z * w);
346 z = one - (r - z);
347 GET_HIGH_WORD(j, z);
348 j += (n << 20);
349 if ((j >> 20) <= 0)
350 z = scalbn(z, (int)n); /* subnormal output */
351 else
352 SET_HIGH_WORD(z, j);
353 return s * z;
354 }
355
356 #if defined(_HAVE_ALIAS_ATTRIBUTE)
357 #ifndef __clang__
358 #pragma GCC diagnostic ignored "-Wmissing-attributes"
359 #endif
360 __strong_reference(pow64, _pow64);
361 #endif
362
363 _MATH_ALIAS_d_dd(pow)
364
365 #endif /* _NEED_FLOAT64 */
366 #else
367 #include "../common/pow.c"
368 #endif /* __OBSOLETE_MATH_DOUBLE */
369
