1 /* origin: FreeBSD /usr/src/lib/msun/src/e_pow.c */
3 * ====================================================
4 * Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
6 * Permission to use, copy, modify, and distribute this
7 * software is freely granted, provided that this notice
9 * ====================================================
11 /* pow(x,y) return x**y
14 * Method: Let x = 2 * (1+f)
15 * 1. Compute and return log2(x) in two pieces:
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)
23 * 1. (anything) ** 0 is 1
24 * 2. (anything) ** 1 is itself
25 * 3. (anything except 1) ** NAN is NAN, 1 ** NAN is 1
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
35 * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF
36 * 14. -0 ** (odd integer) = -( +0 ** (odd integer) )
37 * 15. +INF ** (+anything except 0,NAN) is +INF
38 * 16. +INF ** (-anything except 0,NAN) is +0
39 * 17. -INF ** (anything) = -0 ** (-anything)
40 * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
41 * 19. (-anything except 0 and inf) ** (non-integer) is NAN
44 * pow(x,y) returns x**y nearly rounded. In particular
45 * pow(integer,integer)
46 * always returns the correct integer provided it is
50 * The hexadecimal values are the intended ones for the following
51 * constants. The decimal values may be used, provided that the
52 * compiler will convert from decimal to binary accurately enough
53 * to produce the hexadecimal values shown.
60 dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
61 dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */
62 two53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */
65 /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
66 L1 = 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */
67 L2 = 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */
68 L3 = 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */
69 L4 = 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */
70 L5 = 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */
71 L6 = 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */
72 P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
73 P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
74 P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
75 P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
76 P5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */
77 lg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
78 lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */
79 lg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */
80 ovt = 8.0085662595372944372e-017, /* -(1024-log2(ovfl+.5ulp)) */
81 cp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
82 cp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */
83 cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/
84 ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */
85 ivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/
86 ivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/
88 double pow(double x, double y)
90 double z,ax,z_h,z_l,p_h,p_l;
91 double y1,t1,t2,r,s,t,u,v,w;
92 int32_t i,j,k,yisint,n;
96 EXTRACT_WORDS(hx, lx, x);
97 EXTRACT_WORDS(hy, ly, y);
101 /* y == 0.0: x**0 = 1 */
105 /* x == 1: 1**y = 1, even if y is NaN */
106 if (hx == 0x3ff00000 && lx == 0)
109 /* y != 0.0: result is NaN if either arg is NaN */
110 if (ix > 0x7ff00000 || (ix == 0x7ff00000 && lx != 0) ||
111 iy > 0x7ff00000 || (iy == 0x7ff00000 && ly != 0))
112 return (x+0.0) + (y+0.0);
114 /* determine if y is an odd int when x < 0
115 * yisint = 0 ... y is not an integer
116 * yisint = 1 ... y is an odd int
117 * yisint = 2 ... y is an even int
121 if (iy >= 0x43400000)
122 yisint = 2; /* even integer y */
123 else if (iy >= 0x3ff00000) {
124 k = (iy>>20) - 0x3ff; /* exponent */
127 if ((j<<(52-k)) == ly)
129 } else if (ly == 0) {
131 if ((j<<(20-k)) == iy)
137 /* special value of y */
139 if (iy == 0x7ff00000) { /* y is +-inf */
140 if (((ix-0x3ff00000)|lx) == 0) /* (-1)**+-inf is 1 */
142 else if (ix >= 0x3ff00000) /* (|x|>1)**+-inf = inf,0 */
143 return hy >= 0 ? y : 0.0;
144 else /* (|x|<1)**+-inf = 0,inf */
145 return hy < 0 ? -y : 0.0;
147 if (iy == 0x3ff00000) { /* y is +-1 */
152 if (hy == 0x40000000) /* y is 2 */
154 if (hy == 0x3fe00000) { /* y is 0.5 */
155 if (hx >= 0) /* x >= +0 */
161 /* special value of x */
163 if (ix == 0x7ff00000 || ix == 0 || ix == 0x3ff00000) { /* x is +-0,+-inf,+-1 */
165 if (hy < 0) /* z = (1/|x|) */
168 if (((ix-0x3ff00000)|yisint) == 0) {
169 z = (z-z)/(z-z); /* (-1)**non-int is NaN */
170 } else if (yisint == 1)
171 z = -z; /* (x<0)**odd = -(|x|**odd) */
177 /* CYGNUS LOCAL + fdlibm-5.3 fix: This used to be
179 but ANSI C says a right shift of a signed negative quantity is
180 implementation defined. */
181 n = ((uint32_t)hx>>31) - 1;
183 /* (x<0)**(non-int) is NaN */
187 s = 1.0; /* s (sign of result -ve**odd) = -1 else = 1 */
188 if ((n|(yisint-1)) == 0)
189 s = -1.0;/* (-ve)**(odd int) */
192 if (iy > 0x41e00000) { /* if |y| > 2**31 */
193 if (iy > 0x43f00000) { /* if |y| > 2**64, must o/uflow */
194 if (ix <= 0x3fefffff)
195 return hy < 0 ? huge*huge : tiny*tiny;
196 if (ix >= 0x3ff00000)
197 return hy > 0 ? huge*huge : tiny*tiny;
199 /* over/underflow if x is not close to one */
201 return hy < 0 ? s*huge*huge : s*tiny*tiny;
203 return hy > 0 ? s*huge*huge : s*tiny*tiny;
204 /* now |1-x| is tiny <= 2**-20, suffice to compute
205 log(x) by x-x^2/2+x^3/3-x^4/4 */
206 t = ax - 1.0; /* t has 20 trailing zeros */
207 w = (t*t)*(0.5 - t*(0.3333333333333333333333-t*0.25));
208 u = ivln2_h*t; /* ivln2_h has 21 sig. bits */
209 v = t*ivln2_l - w*ivln2;
214 double ss,s2,s_h,s_l,t_h,t_l;
216 /* take care subnormal number */
217 if (ix < 0x00100000) {
220 GET_HIGH_WORD(ix,ax);
222 n += ((ix)>>20) - 0x3ff;
224 /* determine interval */
225 ix = j | 0x3ff00000; /* normalize ix */
226 if (j <= 0x3988E) /* |x|<sqrt(3/2) */
228 else if (j < 0xBB67A) /* |x|<sqrt(3) */
235 SET_HIGH_WORD(ax, ix);
237 /* compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
238 u = ax - bp[k]; /* bp[0]=1.0, bp[1]=1.5 */
242 SET_LOW_WORD(s_h, 0);
243 /* t_h=ax+bp[k] High */
245 SET_HIGH_WORD(t_h, ((ix>>1)|0x20000000) + 0x00080000 + (k<<18));
246 t_l = ax - (t_h-bp[k]);
247 s_l = v*((u-s_h*t_h)-s_h*t_l);
248 /* compute log(ax) */
250 r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
254 SET_LOW_WORD(t_h, 0);
255 t_l = r - ((t_h-3.0)-s2);
256 /* u+v = ss*(1+...) */
258 v = s_l*t_h + t_l*ss;
259 /* 2/(3log2)*(ss+...) */
261 SET_LOW_WORD(p_h, 0);
263 z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */
264 z_l = cp_l*p_h+p_l*cp + dp_l[k];
265 /* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */
267 t1 = ((z_h + z_l) + dp_h[k]) + t;
269 t2 = z_l - (((t1 - t) - dp_h[k]) - z_h);
272 /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
275 p_l = (y-y1)*t1 + y*t2;
278 EXTRACT_WORDS(j, i, z);
279 if (j >= 0x40900000) { /* z >= 1024 */
280 if (((j-0x40900000)|i) != 0) /* if z > 1024 */
281 return s*huge*huge; /* overflow */
282 if (p_l + ovt > z - p_h)
283 return s*huge*huge; /* overflow */
284 } else if ((j&0x7fffffff) >= 0x4090cc00) { /* z <= -1075 */ // FIXME: instead of abs(j) use unsigned j
285 if (((j-0xc090cc00)|i) != 0) /* z < -1075 */
286 return s*tiny*tiny; /* underflow */
288 return s*tiny*tiny; /* underflow */
291 * compute 2**(p_h+p_l)
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 */
300 SET_HIGH_WORD(t, n & ~(0x000fffff>>k));
301 n = ((n&0x000fffff)|0x00100000)>>(20-k);
309 v = (p_l-(t-p_h))*lg2 + t*lg2_l;
313 t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
314 r = (z*t1)/(t1-2.0) - (w + z*w);
318 if ((j>>20) <= 0) /* subnormal output */