1 package java.lang;
2
3 /**
4 * Port of the "Freely Distributable Math Library", version 4.3, from C to Java.
5 */
6 class FdLibm {
7 /**
8 * Return the low-order 32 bits of the double argument as an int.
9 */
10 private static int __LO(double x) {
11 long transducer = Double.doubleToLongBits(x);
12 return (int)transducer;
13 }
14
15 /**
16 * Return the a double with its low-order bits reset.
17 */
18 private static double __LO(double x, int low) {
19 long transX = Double.doubleToLongBits(x);
20 return Double.longBitsToDouble((transX & 0xFFFF_FFFF_0000_0000L)|low );
21 }
22
23 /**
24 * Return the high-order 32 bits of the double argument as an int.
25 */
26 private static int __HI(double x) {
27 long transducer = Double.doubleToLongBits(x);
28 return (int)(transducer >> 32);
29 }
30 /**
31 * Return the a double with its high-order bits reset.
32 */
33 private static double __HI(double x, int high) {
34 long transX = Double.doubleToLongBits(x);
35 return Double.longBitsToDouble((transX & 0x0000_0000_FFFF_FFFFL)|( ((long)high)) << 32 );
36 }
37
38 /**
39 * Return x**y
40 *
41 * n
42 * Method: Let x = 2 * (1+f)
43 * 1. Compute and return log2(x) in two pieces:
44 * log2(x) = w1 + w2,
45 * where w1 has 53-24 = 29 bit trailing zeros.
46 * 2. Perform y*log2(x) = n+y' by simulating muti-precision
47 * arithmetic, where |y'|<=0.5.
48 * 3. Return x**y = 2**n*exp(y'*log2)
49 *
50 * Special cases:
51 * 1. (anything) ** 0 is 1
52 * 2. (anything) ** 1 is itself
53 * 3. (anything) ** NAN is NAN
54 * 4. NAN ** (anything except 0) is NAN
55 * 5. +-(|x| > 1) ** +INF is +INF
56 * 6. +-(|x| > 1) ** -INF is +0
57 * 7. +-(|x| < 1) ** +INF is +0
58 * 8. +-(|x| < 1) ** -INF is +INF
59 * 9. +-1 ** +-INF is NAN
60 * 10. +0 ** (+anything except 0, NAN) is +0
61 * 11. -0 ** (+anything except 0, NAN, odd integer) is +0
62 * 12. +0 ** (-anything except 0, NAN) is +INF
63 * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF
64 * 14. -0 ** (odd integer) = -( +0 ** (odd integer) )
65 * 15. +INF ** (+anything except 0,NAN) is +INF
66 * 16. +INF ** (-anything except 0,NAN) is +0
67 * 17. -INF ** (anything) = -0 ** (-anything)
68 * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
69 * 19. (-anything except 0 and inf) ** (non-integer) is NAN
70 *
71 * Accuracy:
72 * pow(x,y) returns x**y nearly rounded. In particular
73 * pow(integer,integer)
74 * always returns the correct integer provided it is
75 * representable.
76 *
77 * Constants :
78 * The hexadecimal values are the intended ones for the following
79 * constants. The decimal values may be used, provided that the
80 * compiler will convert from decimal to binary accurately enough
81 * to produce the hexadecimal values shown.
82 */
83 public static class Pow {
84 static final double bp[] = {1.0, 1.5,};
85 static final double dp_h[] = { 0.0, 0x1.2b8034p-1,}; // 5.84962487220764160156e-01
86 static final double dp_l[] = { 0.0, 0x1.cfdeb43cfd006p-27,}; // 1.35003920212974897128e-08
87 static final double zero = 0.0;
88 static final double one = 1.0;
89 static final double two = 2.0;
90 static final double two53 = 0x1.0p53; // 9007199254740992.0
91 static final double huge = 1.0e300;
92 static final double tiny = 1.0e-300;
93 /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
94 static final double L1 = 0x1.3333333333303p-1; // 5.99999999999994648725e-01
95 static final double L2 = 0x1.b6db6db6fabffp-2; // 4.28571428578550184252e-01
96 static final double L3 = 0x1.55555518f264dp-2; // 3.33333329818377432918e-01
97 static final double L4 = 0x1.17460a91d4101p-2; // 2.72728123808534006489e-01
98 static final double L5 = 0x1.d864a93c9db65p-3; // 2.30660745775561754067e-01
99 static final double L6 = 0x1.a7e284a454eefp-3; // 2.06975017800338417784e-01
100 static final double P1 = 0x1.555555555553ep-3; // 1.66666666666666019037e-01
101 static final double P2 = -0x1.6c16c16bebd93p-9; // -2.77777777770155933842e-03
102 static final double P3 = 0x1.1566aaf25de2cp-14; // 6.61375632143793436117e-05
103 static final double P4 = -0x1.bbd41c5d26bf1p-20; // -1.65339022054652515390e-06
104 static final double P5 = 0x1.6376972bea4d0p-25; // 4.13813679705723846039e-08
105 static final double lg2 = 0x1.62e42fefa39efp-1; // 6.93147180559945286227e-01
106 static final double lg2_h = 0x1.62e43p-1; // 6.93147182464599609375e-01
107 static final double lg2_l = -0x1.05c610ca86c39p-29; // -1.90465429995776804525e-09
108 static final double ovt = 8.0085662595372944372e-0017; // -(1024-log2(ovfl+.5ulp))
109 static final double cp = 0x1.ec709dc3a03fdp-1; // 9.61796693925975554329e-01 = 2/(3ln2)
110 static final double cp_h = 0x1.ec709ep-1; // 9.61796700954437255859e-01 = (float)cp
111 static final double cp_l = -0x1.e2fe0145b01f5p-28; // -7.02846165095275826516e-09 = tail of cp_h
112 static final double ivln2 = 0x1.71547652b82fep0; // 1.44269504088896338700e+00 = 1/ln2
113 static final double ivln2_h = 0x1.715476p0; // 1.44269502162933349609e+00 = 24 bits of 1/ln2
114 static final double ivln2_l = 0x1.4ae0bf85ddf44p-26; // 1.92596299112661746887e-08 = 1/ln2 tail
115
116 public static double pow(double x, double y) {
117 double z, ax, z_h, z_l, p_h, p_l;
118 double y1, t1, t2, r, s, t, u, v, w;
119 int i, j, k, yisint, n;
120 int hx, hy, ix, iy;
121 /*unsigned*/ int lx, ly;
122
123 hx = __HI(x);
124 lx = __LO(x);
125 hy = __HI(y);
126 ly = __LO(y);
127 ix = hx & 0x7fffffff;
128 iy = hy & 0x7fffffff;
129
130 /* y==zero: x**0 = 1 */
131 if (y == 0.0)
132 return 1.0;
133
134 /* +-NaN return x+y */
135 if (Double.isNaN(x) || Double.isNaN(y))
136 return x + y;
137
138 /* determine if y is an odd int when x < 0
139 * yisint = 0 ... y is not an integer
140 * yisint = 1 ... y is an odd int
141 * yisint = 2 ... y is an even int
142 */
143 yisint = 0;
144 if (hx < 0) {
145 if (iy >= 0x43400000)
146 yisint = 2; /* even integer y */
147 else if (iy >= 0x3ff00000) {
148 k = (iy >> 20) - 0x3ff; /* exponent */
149 if (k > 20) {
150 j = ly >> (52-k);
151 if ((j << (52-k) )==ly)
152 yisint = 2 - (j&1);
153 } else if (ly == 0) {
154 j = iy >> (20-k);
155 if ((j << (20-k))==iy) {
156 yisint = 2-(j & 1);
157 }
158 }
159 }
160 }
161
162 /* special value of y */
163 if ( ly == 0 ) {
164 if (iy == 0x7ff00000) { /* y is +-inf */
165 if (((ix - 0x3ff00000) | lx) == 0)
166 return y - y; /* inf**+-1 is NaN */
167 else if (ix >= 0x3ff00000) /* (|x| > 1)**+-inf = inf,0 */
168 return (hy >= 0) ? y: zero;
169 else /* (|x| < 1)**-,+inf = inf,0 */
170 return (hy < 0) ? -y: zero;
171 }
172 if (iy == 0x3ff00000) { /* y is +-1 */
173 if (hy < 0)
174 return one/x;
175 else
176 return x;
177 }
178 if (hy == 0x40000000)
179 return x*x; /* y is 2 */
180 if (hy == 0x3fe00000) { /* y is 0.5 */
181 if (hx >= 0) /* x >= +0 */
182 return Math.sqrt(x);
183 }
184 }
185
186 ax = Math.abs(x);
187 /* special value of x */
188 if (lx == 0) {
189 if (ix == 0x7ff00000 || ix==0 || ix == 0x3ff00000){
190 z = ax; /*x is +-0,+-inf,+-1*/
191 if (hy < 0)
192 z = one/z; /* z = (1/|x|) */
193 if (hx < 0) {
194 if (((ix - 0x3ff00000) | yisint)==0) {
195 z = (z-z)/(z-z); /* (-1)**non-int is NaN */
196 } else if (yisint == 1)
197 z = -1.0*z; /* (x<0)**odd = -(|x|**odd) */
198 }
199 return z;
200 }
201 }
202
203 n = (hx >> 31)+1;
204
205 /* (x<0)**(non-int) is NaN */
206 if ((n | yisint) == 0)
207 return (x-x)/(x-x);
208
209 s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
210 if ( (n | (yisint-1)) == 0)
211 s = -one;/* (-ve)**(odd int) */
212
213 /* |y| is huge */
214 if(iy > 0x41e00000) { /* if |y| > 2**31 */
215 if(iy > 0x43f00000){ /* if |y| > 2**64, must o/uflow */
216 if (ix <= 0x3fefffff)
217 return (hy < 0) ? huge*huge : tiny*tiny;
218 if (ix >= 0x3ff00000)
219 return (hy > 0) ? huge*huge : tiny*tiny;
220 }
221 /* over/underflow if x is not close to one */
222 if (ix < 0x3fefffff)
223 return (hy < 0) ? s*huge*huge : s*tiny*tiny;
224 if (ix > 0x3ff00000)
225 return (hy > 0) ? s*huge*huge : s*tiny*tiny;
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-one; /* t has 20 trailing zeros */
229 w = (t*t) * (0.5-t*(0.3333333333333333333333-t*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 t1 =__LO(t1, 0);
234 t2 = v-(t1-u);
235 } else {
236 double ss, 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 ix = __HI(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 ax = __HI(ax, ix);
258
259 /* compute ss = 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 ss = u * v;
263 s_h = ss;
264 s_h = __LO(s_h, 0);
265 /* t_h=ax+bp[k] High */
266 t_h = zero;
267 t_h = __HI(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 = ss * ss;
272 r = s2 * s2* (L1 + s2 * (L2 + s2 * (L3 + s2 * (L4 + s2 * (L5 + s2 * L6)))));
273 r += s_l * (s_h + ss);
274 s2 = s_h * s_h;
275 t_h = 3.0 + s2 + r;
276 t_h = __LO(t_h, 0);
277 t_l = r-((t_h - 3.0)-s2);
278 /* u+v = ss*(1+...) */
279 u = s_h * t_h;
280 v = s_l * t_h + t_l * ss;
281 /* 2/(3log2)*(ss+...) */
282 p_h = u + v;
283 p_h = __LO(p_h, 0);
284 p_l = v-(p_h-u);
285 z_h = cp_h * p_h; /* cp_h+cp_l = 2/(3*log2) */
286 z_l = cp_l * p_h + p_l * cp + dp_l[k];
287 /* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */
288 t = (double)n;
289 t1 = (((z_h + z_l) + dp_h[k]) + t);
290 t1 = __LO(t1, 0);
291 t2 = z_l - (((t1-t)-dp_h[k])-z_h);
292 }
293
294 /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
295 y1 = y;
296 y1 = __LO(y1, 0);
297 p_l = (y-y1) * t1 + y *t2;
298 p_h = y1 * t1;
299 z = p_l + p_h;
300 j = __HI(z);
301 i = __LO(z);
302 if (j >= 0x40900000) { /* z >= 1024 */
303 if (((j - 0x40900000) | i)!=0) /* if z > 1024 */
304 return s*huge*huge; /* overflow */
305 else {
306 if (p_l+ovt>z-p_h)
307 return s*huge*huge; /* overflow */
308 }
309 } else if ((j & 0x7fffffff) >= 0x4090cc00 ) { /* z <= -1075 */
310 if (((j-0xc090cc00)|i)!=0) /* z < -1075 */
311 return s*tiny*tiny; /* underflow */
312 else {
313 if(p_l<=z-p_h)
314 return s*tiny*tiny; /* underflow */
315 }
316 }
317 /*
318 * compute 2**(p_h+p_l)
319 */
320 i = j & 0x7fffffff;
321 k = (i >> 20)-0x3ff;
322 n = 0;
323 if (i > 0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */
324 n = j + (0x00100000 >> (k+1));
325 k = ((n & 0x7fffffff) >> 20)-0x3ff; /* new k for n */
326 t = zero;
327 t = __HI(t, (n & ~(0x000fffff >> k)) );
328 n = ((n & 0x000fffff)|0x00100000) >> (20-k);
329 if (j < 0)
330 n = -n;
331 p_h -= t;
332 }
333 t = p_l+p_h;
334 t = __LO(t, 0);
335 u = t * lg2_h;
336 v = (p_l-(t-p_h))* lg2 + t * lg2_l;
337 z = u + v;
338 w = v-(z-u);
339 t = z * z;
340 t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
341 r = (z*t1)/(t1-two)-(w+z*w);
342 z = one - (r-z);
343 j = __HI(z);
344 j += (n << 20);
345 if ((j >> 20) <= 0)
346 z = Math.scalb(z, n); /* subnormal output */
347 else {
348 int z_hi = __HI(z);
349 z_hi += (n << 20);
350 z = __HI(z, z_hi);
351 }
352 return s * z;
353 }
354 }
355 }