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 */
65 two53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */
68 /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
69 L1 = 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */
70 L2 = 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */
71 L3 = 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */
72 L4 = 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */
73 L5 = 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */
74 L6 = 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */
75 P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
76 P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
77 P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
78 P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
79 P5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */
80 lg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
81 lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */
82 lg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */
83 ovt = 8.0085662595372944372e-017, /* -(1024-log2(ovfl+.5ulp)) */
84 cp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
85 cp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */
86 cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/
87 ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */
88 ivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/
89 ivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/
91 double pow(double x, double y)
93 double z,ax,z_h,z_l,p_h,p_l;
94 double y1,t1,t2,r,s,t,u,v,w;
95 int32_t i,j,k,yisint,n;
99 EXTRACT_WORDS(hx, lx, x);
100 EXTRACT_WORDS(hy, ly, y);
101 ix = hx & 0x7fffffff;
102 iy = hy & 0x7fffffff;
104 /* y == zero: x**0 = 1 */
108 /* x == 1: 1**y = 1, even if y is NaN */
109 if (hx == 0x3ff00000 && lx == 0)
112 /* y != zero: result is NaN if either arg is NaN */
113 if (ix > 0x7ff00000 || (ix == 0x7ff00000 && lx != 0) ||
114 iy > 0x7ff00000 || (iy == 0x7ff00000 && ly != 0))
115 return (x+0.0) + (y+0.0);
117 /* determine if y is an odd int when x < 0
118 * yisint = 0 ... y is not an integer
119 * yisint = 1 ... y is an odd int
120 * yisint = 2 ... y is an even int
124 if (iy >= 0x43400000)
125 yisint = 2; /* even integer y */
126 else if (iy >= 0x3ff00000) {
127 k = (iy>>20) - 0x3ff; /* exponent */
130 if ((j<<(52-k)) == ly)
132 } else if (ly == 0) {
134 if ((j<<(20-k)) == iy)
140 /* special value of y */
142 if (iy == 0x7ff00000) { /* y is +-inf */
143 if (((ix-0x3ff00000)|lx) == 0) /* (-1)**+-inf is 1 */
145 else if (ix >= 0x3ff00000) /* (|x|>1)**+-inf = inf,0 */
146 return hy >= 0 ? y : zero;
147 else /* (|x|<1)**+-inf = 0,inf */
148 return hy < 0 ? -y : zero;
150 if (iy == 0x3ff00000) { /* y is +-1 */
155 if (hy == 0x40000000) /* y is 2 */
157 if (hy == 0x3fe00000) { /* y is 0.5 */
158 if (hx >= 0) /* x >= +0 */
164 /* special value of x */
166 if (ix == 0x7ff00000 || ix == 0 || ix == 0x3ff00000) { /* x is +-0,+-inf,+-1 */
168 if (hy < 0) /* z = (1/|x|) */
171 if (((ix-0x3ff00000)|yisint) == 0) {
172 z = (z-z)/(z-z); /* (-1)**non-int is NaN */
173 } else if (yisint == 1)
174 z = -z; /* (x<0)**odd = -(|x|**odd) */
180 /* CYGNUS LOCAL + fdlibm-5.3 fix: This used to be
182 but ANSI C says a right shift of a signed negative quantity is
183 implementation defined. */
184 n = ((uint32_t)hx>>31) - 1;
186 /* (x<0)**(non-int) is NaN */
190 s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
191 if ((n|(yisint-1)) == 0)
192 s = -one;/* (-ve)**(odd int) */
195 if (iy > 0x41e00000) { /* if |y| > 2**31 */
196 if (iy > 0x43f00000) { /* if |y| > 2**64, must o/uflow */
197 if (ix <= 0x3fefffff)
198 return hy < 0 ? huge*huge : tiny*tiny;
199 if (ix >= 0x3ff00000)
200 return hy > 0 ? huge*huge : tiny*tiny;
202 /* over/underflow if x is not close to one */
204 return hy < 0 ? s*huge*huge : s*tiny*tiny;
206 return hy > 0 ? s*huge*huge : s*tiny*tiny;
207 /* now |1-x| is tiny <= 2**-20, suffice to compute
208 log(x) by x-x^2/2+x^3/3-x^4/4 */
209 t = ax - one; /* t has 20 trailing zeros */
210 w = (t*t)*(0.5 - t*(0.3333333333333333333333-t*0.25));
211 u = ivln2_h*t; /* ivln2_h has 21 sig. bits */
212 v = t*ivln2_l - w*ivln2;
217 double ss,s2,s_h,s_l,t_h,t_l;
219 /* take care subnormal number */
220 if (ix < 0x00100000) {
223 GET_HIGH_WORD(ix,ax);
225 n += ((ix)>>20) - 0x3ff;
227 /* determine interval */
228 ix = j | 0x3ff00000; /* normalize ix */
229 if (j <= 0x3988E) /* |x|<sqrt(3/2) */
231 else if (j < 0xBB67A) /* |x|<sqrt(3) */
238 SET_HIGH_WORD(ax, ix);
240 /* compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
241 u = ax - bp[k]; /* bp[0]=1.0, bp[1]=1.5 */
245 SET_LOW_WORD(s_h, 0);
246 /* t_h=ax+bp[k] High */
248 SET_HIGH_WORD(t_h, ((ix>>1)|0x20000000) + 0x00080000 + (k<<18));
249 t_l = ax - (t_h-bp[k]);
250 s_l = v*((u-s_h*t_h)-s_h*t_l);
251 /* compute log(ax) */
253 r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
257 SET_LOW_WORD(t_h, 0);
258 t_l = r - ((t_h-3.0)-s2);
259 /* u+v = ss*(1+...) */
261 v = s_l*t_h + t_l*ss;
262 /* 2/(3log2)*(ss+...) */
264 SET_LOW_WORD(p_h, 0);
266 z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */
267 z_l = cp_l*p_h+p_l*cp + dp_l[k];
268 /* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */
270 t1 = ((z_h + z_l) + dp_h[k]) + t;
272 t2 = z_l - (((t1 - t) - dp_h[k]) - z_h);
275 /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
278 p_l = (y-y1)*t1 + y*t2;
281 EXTRACT_WORDS(j, i, z);
282 if (j >= 0x40900000) { /* z >= 1024 */
283 if (((j-0x40900000)|i) != 0) /* if z > 1024 */
284 return s*huge*huge; /* overflow */
285 if (p_l + ovt > z - p_h)
286 return s*huge*huge; /* overflow */
287 } else if ((j&0x7fffffff) >= 0x4090cc00) { /* z <= -1075 */ // FIXME: instead of abs(j) use unsigned j
288 if (((j-0xc090cc00)|i) != 0) /* z < -1075 */
289 return s*tiny*tiny; /* underflow */
291 return s*tiny*tiny; /* underflow */
294 * compute 2**(p_h+p_l)
299 if (i > 0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */
300 n = j + (0x00100000>>(k+1));
301 k = ((n&0x7fffffff)>>20) - 0x3ff; /* new k for n */
303 SET_HIGH_WORD(t, n & ~(0x000fffff>>k));
304 n = ((n&0x000fffff)|0x00100000)>>(20-k);
312 v = (p_l-(t-p_h))*lg2 + t*lg2_l;
316 t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
317 r = (z*t1)/(t1-two) - (w + z*w);
321 if ((j>>20) <= 0) /* subnormal output */