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 * <<Note on avoiding strictfp, worry (or not) about setting the sticky flags>>
32 * <<prefer floating-point operations to integer ones>>
33 * <<can't just return huge * huge since that could live on the stack as an inaccurate finite value>>
34 *
35 */
36 class FdLibm {
37 /**
38 * A field whose purpose is to witness writes of double values
39 * whose computation sets the IEEE 754 exception status,
40 * principally underflow, overflow, and invalid. The actual value
41 * of this field does not matter. Writes can be racy and the field
42 * does *not* need to volatile or otherwise synchronized. The
43 * field is made public (within a package-private class) to thwart
44 * optimizations that a JVM might otherwise be justified in
45 * making.
46 */
47 public static double exceptionWitness;
48
49 /**
50 * Return the low-order 32 bits of the double argument as an int.
51 */
52 private static int __LO(double x) {
53 long transducer = Double.doubleToLongBits(x);
54 return (int)transducer;
55 }
56
57 /**
58 * Return the a double with its low-order bits reset.
59 */
60 private static double __LO(double x, int low) {
61 long transX = Double.doubleToLongBits(x);
62 return Double.longBitsToDouble((transX & 0xFFFF_FFFF_0000_0000L)|low );
63 }
64
65 /**
66 * Return the high-order 32 bits of the double argument as an int.
67 */
68 private static int __HI(double x) {
69 long transducer = Double.doubleToLongBits(x);
70 return (int)(transducer >> 32);
71 }
72 /**
73 * Return the a double with its high-order bits reset.
74 */
75 private static double __HI(double x, int high) {
76 long transX = Double.doubleToLongBits(x);
77 return Double.longBitsToDouble((transX & 0x0000_0000_FFFF_FFFFL)|( ((long)high)) << 32 );
78 }
79
80 /**
81 * Compute x**y
82 * n
83 * Method: Let x = 2 * (1+f)
84 * 1. Compute and return log2(x) in two pieces:
85 * log2(x) = w1 + w2,
86 * where w1 has 53 - 24 = 29 bit trailing zeros.
87 * 2. Perform y*log2(x) = n+y' by simulating muti-precision
88 * arithmetic, where |y'| <= 0.5.
89 * 3. Return x**y = 2**n*exp(y'*log2)
90 *
91 * Special cases:
92 * 1. (anything) ** 0 is 1
93 * 2. (anything) ** 1 is itself
94 * 3. (anything) ** NAN is NAN
95 * 4. NAN ** (anything except 0) is NAN
96 * 5. +-(|x| > 1) ** +INF is +INF
97 * 6. +-(|x| > 1) ** -INF is +0
98 * 7. +-(|x| < 1) ** +INF is +0
99 * 8. +-(|x| < 1) ** -INF is +INF
100 * 9. +-1 ** +-INF is NAN
101 * 10. +0 ** (+anything except 0, NAN) is +0
102 * 11. -0 ** (+anything except 0, NAN, odd integer) is +0
103 * 12. +0 ** (-anything except 0, NAN) is +INF
104 * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF
105 * 14. -0 ** (odd integer) = -( +0 ** (odd integer) )
106 * 15. +INF ** (+anything except 0,NAN) is +INF
107 * 16. +INF ** (-anything except 0,NAN) is +0
108 * 17. -INF ** (anything) = -0 ** (-anything)
109 * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
110 * 19. (-anything except 0 and inf) ** (non-integer) is NAN
111 *
112 * Accuracy:
113 * pow(x,y) returns x**y nearly rounded. In particular
114 * pow(integer,integer)
115 * always returns the correct integer provided it is
116 * representable.
117 */
118 public static class Pow {
119 static final double bp[] = {1.0,
120 1.5};
121 static final double dp_h[] = {0.0,
122 0x1.2b8034p-1}; // 5.84962487220764160156e-01
123 static final double dp_l[] = {0.0,
124 0x1.cfdeb43cfd006p-27};// 1.35003920212974897128e-08
125 static final double zero = 0.0;
126 static final double one = 1.0;
127 static final double two = 2.0;
128 static final double two53 = 0x1.0p53; // 9007199254740992.0
129 static final double huge = 1.0e300;
130 static final double tiny = 1.0e-300;
131 // Poly coefs for (3/2)*(log(x)-2s-2/3*s**3
132 static final double L1 = 0x1.3333333333303p-1; // 5.99999999999994648725e-01
133 static final double L2 = 0x1.b6db6db6fabffp-2; // 4.28571428578550184252e-01
134 static final double L3 = 0x1.55555518f264dp-2; // 3.33333329818377432918e-01
135 static final double L4 = 0x1.17460a91d4101p-2; // 2.72728123808534006489e-01
136 static final double L5 = 0x1.d864a93c9db65p-3; // 2.30660745775561754067e-01
137 static final double L6 = 0x1.a7e284a454eefp-3; // 2.06975017800338417784e-01
138 static final double P1 = 0x1.555555555553ep-3; // 1.66666666666666019037e-01
139 static final double P2 = -0x1.6c16c16bebd93p-9; // -2.77777777770155933842e-03
140 static final double P3 = 0x1.1566aaf25de2cp-14; // 6.61375632143793436117e-05
141 static final double P4 = -0x1.bbd41c5d26bf1p-20; // -1.65339022054652515390e-06
142 static final double P5 = 0x1.6376972bea4d0p-25; // 4.13813679705723846039e-08
143 static final double lg2 = 0x1.62e42fefa39efp-1; // 6.93147180559945286227e-01
144 static final double lg2_h = 0x1.62e43p-1; // 6.93147182464599609375e-01
145 static final double lg2_l = -0x1.05c610ca86c39p-29; // -1.90465429995776804525e-09
146 static final double ovt = 8.0085662595372944372e-0017; // -(1024-log2(ovfl+.5ulp))
147 static final double cp = 0x1.ec709dc3a03fdp-1; // 9.61796693925975554329e-01 = 2/(3ln2)
148 static final double cp_h = 0x1.ec709ep-1; // 9.61796700954437255859e-01 = (float)cp
149 static final double cp_l = -0x1.e2fe0145b01f5p-28; // -7.02846165095275826516e-09 = tail of cp_h
150 static final double ivln2 = 0x1.71547652b82fep0; // 1.44269504088896338700e+00 = 1/ln2
151 static final double ivln2_h = 0x1.715476p0; // 1.44269502162933349609e+00 = 24 bits of 1/ln2
152 static final double ivln2_l = 0x1.4ae0bf85ddf44p-26; // 1.92596299112661746887e-08 = 1/ln2 tail
153
154 public static double compute(final double x, final double y) {
155 double z;
156 double t1, t2, r, s, t, u, v, w;
157 int i, j, k, n;
158
159 // y == zero: x**0 = 1
160 if (y == 0.0)
161 return 1.0;
162
163 // +/-NaN return x + y to propagate NaN significands
164 if (Double.isNaN(x) || Double.isNaN(y))
165 return x + y;
166
167 final double y_abs = Math.abs(y);
168 double x_abs = Math.abs(x);
169 // Special values of y
170 if (y == 2.0) {
171 return x * x;
172 } else if (y == 0.5) {
173 if (x >= -Double.MAX_VALUE) // Handle x == -infinity later
174 return Math.sqrt(x + 0.0); // Add 0.0 to properly handle x == -0.0
175 } else if (y_abs == 1.0) { // y is +/-1
176 return (y == 1.0) ? x : one / x;
177 } else if (y_abs == Double.POSITIVE_INFINITY) { // y is +/-infinity
178 if (x_abs == 1.0)
179 return y - y; // inf**+/-1 is NaN
180 else if (x_abs > 1.0) // (|x| > 1)**+/-inf = inf, 0
181 return (y >= 0) ? y : zero;
182 else // (|x| < 1)**-, +inf = inf, 0
183 return (y < 0) ? -y : zero;
184 }
185
186 final int hx = __HI(x); // Try to replace with copysign usage
187 // final int lx = __LO(x);
188 // final int hy = __HI(y);
189 final int ly = __LO(y);
190 int ix = hx & 0x7fffffff;
191 final int iy = __HI(y) & 0x7fffffff; // Try to replace with getExponent in yisint
192
193 /*
194 * When x < 0, determine if y is an odd integer:
195 * yisint = 0 ... y is not an integer
196 * yisint = 1 ... y is an odd int
197 * yisint = 2 ... y is an even int
198 */
199 {
200 int yisint = 0;
201 if (hx < 0) {
202 if (y_abs >= 0x1.0p53) // |y| >= 2^53 = 9.007199254740992E15
203 yisint = 2; // even integer y
204 else if (y_abs >= 1.0) { // |y| >= 1.0
205 k = (iy >> 20) - 0x3ff; // exponent
206 if (k > 20) {
207 j = ly >> (52 - k);
208 if ((j << (52 - k) ) == ly)
209 yisint = 2 - (j & 1);
210 } else if (ly == 0) {
211 j = iy >> (20 - k);
212 if ((j << (20 - k)) == iy) {
213 yisint = 2 - (j & 1);
214 }
215 }
216 }
217 }
218
219 // Special value of x
220 if (x_abs == 0.0 ||
221 x_abs == Double.POSITIVE_INFINITY ||
222 x_abs == 1.0) {
223 z = x_abs; // x is +/-0, +/-inf, +/-1
224 if (y < 0.0)
225 z = one/z; // z = (1/|x|)
226 if (hx < 0) {
227 if (((ix - 0x3ff00000) | yisint) == 0) {
228 z = (z-z)/(z-z); // (-1)**non-int is NaN
229 } else if (yisint == 1)
230 z = -1.0 * z; // (x < 0)**odd = -(|x|**odd)
231 }
232 return z;
233 }
234
235 n = (hx >> 31) + 1;
236
237 // (x < 0)**(non-int) is NaN
238 if ((n | yisint) == 0)
239 return (x-x)/(x-x);
240
241 s = one; // s (sign of result -ve**odd) = -1 else = 1
242 if ( (n | (yisint - 1)) == 0)
243 s = -one; // (-ve)**(odd int)
244 }
245 double p_h, p_l;
246 // |y| is huge
247 if (y_abs > 0x1.0p31) { // if |y| > 2**31
248 if (y_abs > 0x1.0p64){ // if |y| > 2**64, must over/underflow
249 if (x_abs <= 0x1.fffffp-1) // |x| <= 0.9999995231628418
250 return (y < 0.0) ? huge*huge : tiny*tiny;
251 if (x_abs >= 1.0) // |x| >= 1.0
252 return (y > 0.0) ? huge*huge : tiny*tiny;
253 }
254 // Over/underflow if x is not close to one
255 if (x_abs < 0x1.fffffp-1) // |x| < 0.9999995231628418
256 return (y < 0.0) ? s*huge*huge : s*tiny*tiny;
257 if (x_abs > 1.0) // |x| > 1.0
258 return (y > 0.0) ? s*huge*huge : s*tiny*tiny;
259 /*
260 * now |1-x| is tiny <= 2**-20, sufficient to compute
261 * log(x) by x - x^2/2 + x^3/3 - x^4/4
262 */
263 t = x_abs - one; // t has 20 trailing zeros
264 w = (t * t) * (0.5 - t * (0.3333333333333333333333 - t * 0.25));
265 u = ivln2_h * t; // ivln2_h has 21 sig. bits
266 v = t * ivln2_l - w * ivln2;
267 t1 = u + v;
268 t1 =__LO(t1, 0);
269 t2 = v - (t1 - u);
270 } else {
271 double z_h, z_l, ss, s2, s_h, s_l, t_h, t_l;
272 n = 0;
273 // Take care of subnormal numbers
274 if (ix < 0x00100000) {
275 x_abs *= two53;
276 n -= 53;
277 ix = __HI(x_abs);
278 }
279 n += ((ix) >> 20) - 0x3ff;
280 j = ix & 0x000fffff;
281 // Determine interval
282 ix = j | 0x3ff00000; // Normalize ix
283 if (j <= 0x3988E)
284 k = 0; // |x| <sqrt(3/2)
285 else if (j < 0xBB67A)
286 k = 1; // |x| <sqrt(3)
287 else {
288 k = 0;
289 n += 1;
290 ix -= 0x00100000;
291 }
292 x_abs = __HI(x_abs, ix);
293
294 // Compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5)
295 u = x_abs - bp[k]; // bp[0]=1.0, bp[1]=1.5
296 v = one / (x_abs + bp[k]);
297 ss = u * v;
298 s_h = ss;
299 s_h = __LO(s_h, 0);
300 // t_h=x_abs+bp[k] High
301 t_h = zero;
302 t_h = __HI(t_h, ((ix >> 1) | 0x20000000) + 0x00080000 + (k << 18) );
303 t_l = x_abs - (t_h - bp[k]);
304 s_l = v * ((u - s_h * t_h) - s_h * t_l);
305 // Compute log(x_abs)
306 s2 = ss * ss;
307 r = s2 * s2* (L1 + s2 * (L2 + s2 * (L3 + s2 * (L4 + s2 * (L5 + s2 * L6)))));
308 r += s_l * (s_h + ss);
309 s2 = s_h * s_h;
310 t_h = 3.0 + s2 + r;
311 t_h = __LO(t_h, 0);
312 t_l = r - ((t_h - 3.0) - s2);
313 // u+v = ss*(1+...)
314 u = s_h * t_h;
315 v = s_l * t_h + t_l * ss;
316 // 2/(3log2)*(ss+...)
317 p_h = u + v;
318 p_h = __LO(p_h, 0);
319 p_l = v - (p_h - u);
320 z_h = cp_h * p_h; // cp_h+cp_l = 2/(3*log2)
321 z_l = cp_l * p_h + p_l * cp + dp_l[k];
322 // log2(x_abs) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l
323 t = (double)n;
324 t1 = (((z_h + z_l) + dp_h[k]) + t);
325 t1 = __LO(t1, 0);
326 t2 = z_l - (((t1 - t) - dp_h[k]) - z_h);
327 }
328
329 // Split up y into (y1 + y2) and compute (y1 + y2) * (t1 + t2)
330 double y1 = y;
331 y1 = __LO(y1, 0);
332 p_l = (y - y1) * t1 + y * t2;
333 p_h = y1 * t1;
334 z = p_l + p_h;
335 j = __HI(z);
336 i = __LO(z);
337 if (j >= 0x40900000) { // z >= 1024
338 if (((j - 0x40900000) | i)!=0) // if z > 1024
339 return s*huge*huge; // Overflow
340 else {
341 if (p_l + ovt > z - p_h)
342 return s*huge*huge; // Overflow
343 }
344 } else if ((j & 0x7fffffff) >= 0x4090cc00 ) { // z <= -1075
345 if (((j - 0xc090cc00) | i)!=0) // z < -1075
346 return s*tiny*tiny; // Underflow
347 else {
348 if (p_l <= z - p_h)
349 return s*tiny*tiny; // Underflow
350 }
351 }
352 /*
353 * Compute 2**(p_h+p_l)
354 */
355 i = j & 0x7fffffff;
356 k = (i >> 20) - 0x3ff;
357 n = 0;
358 if (i > 0x3fe00000) { // if |z| > 0.5, set n = [z+0.5]
359 n = j + (0x00100000 >> (k + 1));
360 k = ((n & 0x7fffffff) >> 20) - 0x3ff; // new k for n
361 t = zero;
362 t = __HI(t, (n & ~(0x000fffff >> k)) );
363 n = ((n & 0x000fffff) | 0x00100000) >> (20 - k);
364 if (j < 0)
365 n = -n;
366 p_h -= t;
367 }
368 t = p_l + p_h;
369 t = __LO(t, 0);
370 u = t * lg2_h;
371 v = (p_l - (t - p_h)) * lg2 + t * lg2_l;
372 z = u + v;
373 w = v - (z - u);
374 t = z * z;
375 t1 = z - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5))));
376 r = (z * t1)/(t1 - two) - (w + z * w);
377 z = one - (r - z);
378 j = __HI(z);
379 j += (n << 20);
380 if ((j >> 20) <= 0)
381 z = Math.scalb(z, n); // subnormal output
382 else {
383 int z_hi = __HI(z);
384 z_hi += (n << 20);
385 z = __HI(z, z_hi);
386 }
387 return s * z;
388 }
389 }
390 }