1 /*
2 * Copyright (c) 1998, 2015, Oracle and/or its affiliates. All rights reserved.
3 * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.
4 *
5 * This code is free software; you can redistribute it and/or modify it
6 * under the terms of the GNU General Public License version 2 only, as
7 * published by the Free Software Foundation. Oracle designates this
8 * particular file as subject to the "Classpath" exception as provided
9 * by Oracle in the LICENSE file that accompanied this code.
10 *
11 * This code is distributed in the hope that it will be useful, but WITHOUT
12 * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
13 * FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
14 * version 2 for more details (a copy is included in the LICENSE file that
15 * accompanied this code).
16 *
17 * You should have received a copy of the GNU General Public License version
18 * 2 along with this work; if not, write to the Free Software Foundation,
19 * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA.
20 *
21 * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA
22 * or visit www.oracle.com if you need additional information or have any
23 * questions.
24 */
25
26 package java.lang;
27
28 /**
29 * Port of the "Freely Distributable Math Library", version 5.3, from C to Java.
30 *
31 * <p>The C version of fdlibm relied on the idiom of pointer aliasing
32 * a 64-bit double floating-point value as a two-element array of
33 * 32-bit integers and reading and writing the two halves of the
34 * double independently. This coding pattern was problematic to C
35 * optimizers and not directly expressible in Java. Therefore, rather
36 * than a memory level overlay, if portions of a double need to be
37 * operated on as integer values, the standard library methods for
38 * bitwise floating-point to integer conversion are directly or
39 * indirectly used, Double.longBitsToDouble and
40 * Double.doubleToLongBits.
41 *
42 * <p>The C version of fdlibm also took some pains to signal the
43 * correct IEEE 754 exceptional conditions divide by zero, invalid,
44 * overflow and underflow. For example, overflow would be signaled by
45 * {@code huge * huge} where {@code huge} was a large constant that
46 * would overflow when squared. Since IEEE floating-point exceptional
47 * handling is not supported natively in the JVM, such coding patterns
48 * have been omitted from this port. For example, rather than {@code
49 * return huge * huge}, this port will use {@code return INFINITY}.
50 */
51 class FdLibm {
52 // Constants used by multiple algorithms
53 private static final double INFINITY= Double.POSITIVE_INFINITY;
54
55 /**
56 * Return the low-order 32 bits of the double argument as an int.
57 */
58 private static int __LO(double x) {
59 long transducer = Double.doubleToLongBits(x);
60 return (int)transducer;
61 }
62
63 /**
64 * Return a double with its low-order bits of the second argument
65 * and the high-order bits of the first argument..
66 */
67 private static double __LO(double x, int low) {
68 long transX = Double.doubleToLongBits(x);
69 return Double.longBitsToDouble((transX & 0xFFFF_FFFF_0000_0000L)|low );
70 }
71
72 /**
73 * Return the high-order 32 bits of the double argument as an int.
74 */
75 private static int __HI(double x) {
76 long transducer = Double.doubleToLongBits(x);
77 return (int)(transducer >> 32);
78 }
79
80 /**
81 * Return a double with its high-order bits of the second argument
82 * and the low-order bits of the first argument..
83 */
84 private static double __HI(double x, int high) {
85 long transX = Double.doubleToLongBits(x);
86 return Double.longBitsToDouble((transX & 0x0000_0000_FFFF_FFFFL)|( ((long)high)) << 32 );
87 }
88
89 /**
90 * Compute x**y
91 * n
92 * Method: Let x = 2 * (1+f)
93 * 1. Compute and return log2(x) in two pieces:
94 * log2(x) = w1 + w2,
95 * where w1 has 53 - 24 = 29 bit trailing zeros.
96 * 2. Perform y*log2(x) = n+y' by simulating muti-precision
97 * arithmetic, where |y'| <= 0.5.
98 * 3. Return x**y = 2**n*exp(y'*log2)
99 *
100 * Special cases:
101 * 1. (anything) ** 0 is 1
102 * 2. (anything) ** 1 is itself
103 * 3. (anything) ** NAN is NAN
104 * 4. NAN ** (anything except 0) is NAN
105 * 5. +-(|x| > 1) ** +INF is +INF
106 * 6. +-(|x| > 1) ** -INF is +0
107 * 7. +-(|x| < 1) ** +INF is +0
108 * 8. +-(|x| < 1) ** -INF is +INF
109 * 9. +-1 ** +-INF is NAN
110 * 10. +0 ** (+anything except 0, NAN) is +0
111 * 11. -0 ** (+anything except 0, NAN, odd integer) is +0
112 * 12. +0 ** (-anything except 0, NAN) is +INF
113 * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF
114 * 14. -0 ** (odd integer) = -( +0 ** (odd integer) )
115 * 15. +INF ** (+anything except 0,NAN) is +INF
116 * 16. +INF ** (-anything except 0,NAN) is +0
117 * 17. -INF ** (anything) = -0 ** (-anything)
118 * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
119 * 19. (-anything except 0 and inf) ** (non-integer) is NAN
120 *
121 * Accuracy:
122 * pow(x,y) returns x**y nearly rounded. In particular
123 * pow(integer,integer)
124 * always returns the correct integer provided it is
125 * representable.
126 */
127 public static class Pow {
128 public static strictfp double compute(final double x, final double y) {
129 double z;
130 double r, s, t, u, v, w;
131 int i, j, k, n;
132
133 // y == zero: x**0 = 1
134 if (y == 0.0)
135 return 1.0;
136
137 // +/-NaN return x + y to propagate NaN significands
138 if (Double.isNaN(x) || Double.isNaN(y))
139 return x + y;
140
141 final double y_abs = Math.abs(y);
142 double x_abs = Math.abs(x);
143 // Special values of y
144 if (y == 2.0) {
145 return x * x;
146 } else if (y == 0.5) {
147 if (x >= -Double.MAX_VALUE) // Handle x == -infinity later
148 return Math.sqrt(x + 0.0); // Add 0.0 to properly handle x == -0.0
149 } else if (y_abs == 1.0) { // y is +/-1
150 return (y == 1.0) ? x : 1.0 / x;
151 } else if (y_abs == INFINITY) { // y is +/-infinity
152 if (x_abs == 1.0)
153 return y - y; // inf**+/-1 is NaN
154 else if (x_abs > 1.0) // (|x| > 1)**+/-inf = inf, 0
155 return (y >= 0) ? y : 0.0;
156 else // (|x| < 1)**-/+inf = inf, 0
157 return (y < 0) ? -y : 0.0;
158 }
159
160 final int hx = __HI(x);
161 int ix = hx & 0x7fffffff;
162
163 /*
164 * When x < 0, determine if y is an odd integer:
165 * y_is_int = 0 ... y is not an integer
166 * y_is_int = 1 ... y is an odd int
167 * y_is_int = 2 ... y is an even int
168 */
169 int y_is_int = 0;
170 if (hx < 0) {
171 if (y_abs >= 0x1.0p53) // |y| >= 2^53 = 9.007199254740992E15
172 y_is_int = 2; // y is an even integer since ulp(2^53) = 2.0
173 else if (y_abs >= 1.0) { // |y| >= 1.0
174 long y_abs_as_long = (long) y_abs;
175 if ( ((double) y_abs_as_long) == y_abs) {
176 y_is_int = 2 - (int)(y_abs_as_long & 0x1L);
177 }
178 }
179 }
180
181 // Special value of x
182 if (x_abs == 0.0 ||
183 x_abs == INFINITY ||
184 x_abs == 1.0) {
185 z = x_abs; // x is +/-0, +/-inf, +/-1
186 if (y < 0.0)
187 z = 1.0/z; // z = (1/|x|)
188 if (hx < 0) {
189 if (((ix - 0x3ff00000) | y_is_int) == 0) {
190 z = (z-z)/(z-z); // (-1)**non-int is NaN
191 } else if (y_is_int == 1)
192 z = -1.0 * z; // (x < 0)**odd = -(|x|**odd)
193 }
194 return z;
195 }
196
197 n = (hx >> 31) + 1;
198
199 // (x < 0)**(non-int) is NaN
200 if ((n | y_is_int) == 0)
201 return (x-x)/(x-x);
202
203 s = 1.0; // s (sign of result -ve**odd) = -1 else = 1
204 if ( (n | (y_is_int - 1)) == 0)
205 s = -1.0; // (-ve)**(odd int)
206
207 double p_h, p_l, t1, t2;
208 // |y| is huge
209 if (y_abs > 0x1.0p31) { // if |y| > 2**31
210 final double INV_LN2 = 0x1.7154_7652_b82fep0; // 1.44269504088896338700e+00 = 1/ln2
211 final double INV_LN2_H = 0x1.715476p0; // 1.44269502162933349609e+00 = 24 bits of 1/ln2
212 final double INV_LN2_L = 0x1.4ae0_bf85_ddf44p-26; // 1.92596299112661746887e-08 = 1/ln2 tail
213
214 // Over/underflow if x is not close to one
215 if (x_abs < 0x1.fffffp-1) // |x| < 0.9999995231628418
216 return (y < 0.0) ? s * INFINITY : s * 0.0;
217 if (x_abs > 1.0) // |x| > 1.0
218 return (y > 0.0) ? s * INFINITY : s * 0.0;
219 /*
220 * now |1-x| is tiny <= 2**-20, sufficient to compute
221 * log(x) by x - x^2/2 + x^3/3 - x^4/4
222 */
223 t = x_abs - 1.0; // t has 20 trailing zeros
224 w = (t * t) * (0.5 - t * (0.3333333333333333333333 - t * 0.25));
225 u = INV_LN2_H * t; // INV_LN2_H has 21 sig. bits
226 v = t * INV_LN2_L - w * INV_LN2;
227 t1 = u + v;
228 t1 =__LO(t1, 0);
229 t2 = v - (t1 - u);
230 } else {
231 final double CP = 0x1.ec70_9dc3_a03fdp-1; // 9.61796693925975554329e-01 = 2/(3ln2)
232 final double CP_H = 0x1.ec709ep-1; // 9.61796700954437255859e-01 = (float)cp
233 final double CP_L = -0x1.e2fe_0145_b01f5p-28; // -7.02846165095275826516e-09 = tail of CP_H
234
235 double z_h, z_l, ss, s2, s_h, s_l, t_h, t_l;
236 n = 0;
237 // Take care of subnormal numbers
238 if (ix < 0x00100000) {
239 x_abs *= 0x1.0p53; // 2^53 = 9007199254740992.0
240 n -= 53;
241 ix = __HI(x_abs);
242 }
243 n += ((ix) >> 20) - 0x3ff;
244 j = ix & 0x000fffff;
245 // Determine interval
246 ix = j | 0x3ff00000; // Normalize ix
247 if (j <= 0x3988E)
248 k = 0; // |x| <sqrt(3/2)
249 else if (j < 0xBB67A)
250 k = 1; // |x| <sqrt(3)
251 else {
252 k = 0;
253 n += 1;
254 ix -= 0x00100000;
255 }
256 x_abs = __HI(x_abs, ix);
257
258 // Compute ss = s_h + s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5)
259
260 final double BP[] = {1.0,
261 1.5};
262 final double DP_H[] = {0.0,
263 0x1.2b80_34p-1}; // 5.84962487220764160156e-01
264 final double DP_L[] = {0.0,
265 0x1.cfde_b43c_fd006p-27};// 1.35003920212974897128e-08
266
267 // Poly coefs for (3/2)*(log(x)-2s-2/3*s**3
268 final double L1 = 0x1.3333_3333_33303p-1; // 5.99999999999994648725e-01
269 final double L2 = 0x1.b6db_6db6_fabffp-2; // 4.28571428578550184252e-01
270 final double L3 = 0x1.5555_5518_f264dp-2; // 3.33333329818377432918e-01
271 final double L4 = 0x1.1746_0a91_d4101p-2; // 2.72728123808534006489e-01
272 final double L5 = 0x1.d864_a93c_9db65p-3; // 2.30660745775561754067e-01
273 final double L6 = 0x1.a7e2_84a4_54eefp-3; // 2.06975017800338417784e-01
274 u = x_abs - BP[k]; // BP[0]=1.0, BP[1]=1.5
275 v = 1.0 / (x_abs + BP[k]);
276 ss = u * v;
277 s_h = ss;
278 s_h = __LO(s_h, 0);
279 // t_h=x_abs + BP[k] High
280 t_h = 0.0;
281 t_h = __HI(t_h, ((ix >> 1) | 0x20000000) + 0x00080000 + (k << 18) );
282 t_l = x_abs - (t_h - BP[k]);
283 s_l = v * ((u - s_h * t_h) - s_h * t_l);
284 // Compute log(x_abs)
285 s2 = ss * ss;
286 r = s2 * s2* (L1 + s2 * (L2 + s2 * (L3 + s2 * (L4 + s2 * (L5 + s2 * L6)))));
287 r += s_l * (s_h + ss);
288 s2 = s_h * s_h;
289 t_h = 3.0 + s2 + r;
290 t_h = __LO(t_h, 0);
291 t_l = r - ((t_h - 3.0) - s2);
292 // u+v = ss*(1+...)
293 u = s_h * t_h;
294 v = s_l * t_h + t_l * ss;
295 // 2/(3log2)*(ss + ...)
296 p_h = u + v;
297 p_h = __LO(p_h, 0);
298 p_l = v - (p_h - u);
299 z_h = CP_H * p_h; // CP_H + CP_L = 2/(3*log2)
300 z_l = CP_L * p_h + p_l * CP + DP_L[k];
301 // log2(x_abs) = (ss + ..)*2/(3*log2) = n + DP_H + z_h + z_l
302 t = (double)n;
303 t1 = (((z_h + z_l) + DP_H[k]) + t);
304 t1 = __LO(t1, 0);
305 t2 = z_l - (((t1 - t) - DP_H[k]) - z_h);
306 }
307
308 // Split up y into (y1 + y2) and compute (y1 + y2) * (t1 + t2)
309 double y1 = y;
310 y1 = __LO(y1, 0);
311 p_l = (y - y1) * t1 + y * t2;
312 p_h = y1 * t1;
313 z = p_l + p_h;
314 j = __HI(z);
315 i = __LO(z);
316 if (j >= 0x40900000) { // z >= 1024
317 if (((j - 0x40900000) | i)!=0) // if z > 1024
318 return s * INFINITY; // Overflow
319 else {
320 final double OVT = 8.0085662595372944372e-0017; // -(1024-log2(ovfl+.5ulp))
321 if (p_l + OVT > z - p_h)
322 return s * INFINITY; // Overflow
323 }
324 } else if ((j & 0x7fffffff) >= 0x4090cc00 ) { // z <= -1075
325 if (((j - 0xc090cc00) | i)!=0) // z < -1075
326 return s * 0.0; // Underflow
327 else {
328 if (p_l <= z - p_h)
329 return s * 0.0; // Underflow
330 }
331 }
332 /*
333 * Compute 2**(p_h+p_l)
334 */
335 // Poly coefs for (3/2)*(log(x)-2s-2/3*s**3
336 final double P1 = 0x1.5555_5555_5553ep-3; // 1.66666666666666019037e-01
337 final double P2 = -0x1.6c16_c16b_ebd93p-9; // -2.77777777770155933842e-03
338 final double P3 = 0x1.1566_aaf2_5de2cp-14; // 6.61375632143793436117e-05
339 final double P4 = -0x1.bbd4_1c5d_26bf1p-20; // -1.65339022054652515390e-06
340 final double P5 = 0x1.6376_972b_ea4d0p-25; // 4.13813679705723846039e-08
341 final double LG2 = 0x1.62e4_2fef_a39efp-1; // 6.93147180559945286227e-01
342 final double LG2_H = 0x1.62e43p-1; // 6.93147182464599609375e-01
343 final double LG2_L = -0x1.05c6_10ca_86c39p-29; // -1.90465429995776804525e-09
344 i = j & 0x7fffffff;
345 k = (i >> 20) - 0x3ff;
346 n = 0;
347 if (i > 0x3fe00000) { // if |z| > 0.5, set n = [z + 0.5]
348 n = j + (0x00100000 >> (k + 1));
349 k = ((n & 0x7fffffff) >> 20) - 0x3ff; // new k for n
350 t = 0.0;
351 t = __HI(t, (n & ~(0x000fffff >> k)) );
352 n = ((n & 0x000fffff) | 0x00100000) >> (20 - k);
353 if (j < 0)
354 n = -n;
355 p_h -= t;
356 }
357 t = p_l + p_h;
358 t = __LO(t, 0);
359 u = t * LG2_H;
360 v = (p_l - (t - p_h)) * LG2 + t * LG2_L;
361 z = u + v;
362 w = v - (z - u);
363 t = z * z;
364 t1 = z - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5))));
365 r = (z * t1)/(t1 - 2.0) - (w + z * w);
366 z = 1.0 - (r - z);
367 j = __HI(z);
368 j += (n << 20);
369 if ((j >> 20) <= 0)
370 z = Math.scalb(z, n); // subnormal output
371 else {
372 int z_hi = __HI(z);
373 z_hi += (n << 20);
374 z = __HI(z, z_hi);
375 }
376 return s * z;
377 }
378 }
379 }