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,
39 * Double.longBitsToDouble and Double.doubleToRawLongBits, are directly
40 * or indirectly used .
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 private FdLibm() {
56 throw new UnsupportedOperationException("No instances for you.");
57 }
58
59 /**
60 * Return the low-order 32 bits of the double argument as an int.
61 */
62 private static int __LO(double x) {
63 long transducer = Double.doubleToRawLongBits(x);
64 return (int)transducer;
65 }
66
67 /**
68 * Return a double with its low-order bits of the second argument
69 * and the high-order bits of the first argument..
70 */
71 private static double __LO(double x, int low) {
72 long transX = Double.doubleToRawLongBits(x);
73 return Double.longBitsToDouble((transX & 0xFFFF_FFFF_0000_0000L)|low );
74 }
75
76 /**
77 * Return the high-order 32 bits of the double argument as an int.
78 */
79 private static int __HI(double x) {
80 long transducer = Double.doubleToRawLongBits(x);
81 return (int)(transducer >> 32);
82 }
83
84 /**
85 * Return a double with its high-order bits of the second argument
86 * and the low-order bits of the first argument..
87 */
88 private static double __HI(double x, int high) {
89 long transX = Double.doubleToRawLongBits(x);
90 return Double.longBitsToDouble((transX & 0x0000_0000_FFFF_FFFFL)|( ((long)high)) << 32 );
91 }
92
93 /**
94 * hypot(x,y)
95 *
96 * Method :
97 * If (assume round-to-nearest) z=x*x+y*y
98 * has error less than sqrt(2)/2 ulp, than
99 * sqrt(z) has error less than 1 ulp (exercise).
100 *
101 * So, compute sqrt(x*x+y*y) with some care as
102 * follows to get the error below 1 ulp:
103 *
104 * Assume x>y>0;
105 * (if possible, set rounding to round-to-nearest)
106 * 1. if x > 2y use
107 * x1*x1+(y*y+(x2*(x+x1))) for x*x+y*y
108 * where x1 = x with lower 32 bits cleared, x2 = x-x1; else
109 * 2. if x <= 2y use
110 * t1*y1+((x-y)*(x-y)+(t1*y2+t2*y))
111 * where t1 = 2x with lower 32 bits cleared, t2 = 2x-t1,
112 * y1= y with lower 32 bits chopped, y2 = y-y1.
113 *
114 * NOTE: scaling may be necessary if some argument is too
115 * large or too tiny
116 *
117 * Special cases:
118 * hypot(x,y) is INF if x or y is +INF or -INF; else
119 * hypot(x,y) is NAN if x or y is NAN.
120 *
121 * Accuracy:
122 * hypot(x,y) returns sqrt(x^2+y^2) with error less
123 * than 1 ulps (units in the last place)
124 */
125 public static class Hypot {
126 public static double compute(double x, double y) {
127 double a=x,b=y,t1,t2,y1,y2,w;
128 int j,k,ha,hb;
129
130 ha = __HI(x)&0x7fffffff; /* high word of x */
131 hb = __HI(y)&0x7fffffff; /* high word of y */
132 if(hb > ha) {a=y;b=x;j=ha; ha=hb;hb=j;} else {a=x;b=y;}
133 a = __HI(a, ha); /* a <- |a| */
134 b = __HI(b, hb); /* b <- |b| */
135 if((ha-hb)>0x3c00000) {return a+b;} /* x/y > 2**60 */
136 k=0;
137 if(ha > 0x5f300000) { /* a>2**500 */
138 if(ha >= 0x7ff00000) { /* Inf or NaN */
139 w = a+b; /* for sNaN */
140 if(((ha&0xfffff)|__LO(a))==0) w = a;
141 if(((hb^0x7ff00000)|__LO(b))==0) w = b;
142 return w;
143 }
144 /* scale a and b by 2**-600 */
145 ha -= 0x25800000; hb -= 0x25800000; k += 600;
146 a = __HI(a, ha);
147 b = __HI(b, hb);
148 }
149 if(hb < 0x20b00000) { /* b < 2**-500 */
150 if(hb <= 0x000fffff) { /* subnormal b or 0 */
151 if((hb|(__LO(b)))==0) return a;
152 t1=0;
153 t1 = __HI(t1, 0x7fd00000); /* t1=2^1022 */
154 b *= t1;
155 a *= t1;
156 k -= 1022;
157 } else { /* scale a and b by 2^600 */
158 ha += 0x25800000; /* a *= 2^600 */
159 hb += 0x25800000; /* b *= 2^600 */
160 k -= 600;
161 a = __HI(a, ha);
162 b = __HI(b, hb);
163 }
164 }
165 /* medium size a and b */
166 w = a-b;
167 if (w>b) {
168 t1 = 0;
169 t1 = __HI(t1, ha);
170 t2 = a-t1;
171 w = Math.sqrt(t1*t1-(b*(-b)-t2*(a+t1)));
172 } else {
173 a = a+a;
174 y1 = 0;
175 y1 = __HI(y1, hb);
176 y2 = b - y1;
177 t1 = 0;
178 t1 = __HI(t1, ha+0x00100000);
179 t2 = a - t1;
180 w = Math.sqrt(t1*y1-(w*(-w)-(t1*y2+t2*b)));
181 }
182 if(k!=0) {
183 t1 = 1.0;
184 int t1_hi = __HI(t1);
185 t1_hi += (k<<20);
186 t1 = __HI(t1, t1_hi);
187 return t1*w;
188 } else return w;
189 }
190 }
191
192 /**
193 * Compute x**y
194 * n
195 * Method: Let x = 2 * (1+f)
196 * 1. Compute and return log2(x) in two pieces:
197 * log2(x) = w1 + w2,
198 * where w1 has 53 - 24 = 29 bit trailing zeros.
199 * 2. Perform y*log2(x) = n+y' by simulating muti-precision
200 * arithmetic, where |y'| <= 0.5.
201 * 3. Return x**y = 2**n*exp(y'*log2)
202 *
203 * Special cases:
204 * 1. (anything) ** 0 is 1
205 * 2. (anything) ** 1 is itself
206 * 3. (anything) ** NAN is NAN
207 * 4. NAN ** (anything except 0) is NAN
208 * 5. +-(|x| > 1) ** +INF is +INF
209 * 6. +-(|x| > 1) ** -INF is +0
210 * 7. +-(|x| < 1) ** +INF is +0
211 * 8. +-(|x| < 1) ** -INF is +INF
212 * 9. +-1 ** +-INF is NAN
213 * 10. +0 ** (+anything except 0, NAN) is +0
214 * 11. -0 ** (+anything except 0, NAN, odd integer) is +0
215 * 12. +0 ** (-anything except 0, NAN) is +INF
216 * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF
217 * 14. -0 ** (odd integer) = -( +0 ** (odd integer) )
218 * 15. +INF ** (+anything except 0,NAN) is +INF
219 * 16. +INF ** (-anything except 0,NAN) is +0
220 * 17. -INF ** (anything) = -0 ** (-anything)
221 * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
222 * 19. (-anything except 0 and inf) ** (non-integer) is NAN
223 *
224 * Accuracy:
225 * pow(x,y) returns x**y nearly rounded. In particular
226 * pow(integer,integer)
227 * always returns the correct integer provided it is
228 * representable.
229 */
230 public static class Pow {
231 public static strictfp double compute(final double x, final double y) {
232 double z;
233 double r, s, t, u, v, w;
234 int i, j, k, n;
235
236 // y == zero: x**0 = 1
237 if (y == 0.0)
238 return 1.0;
239
240 // +/-NaN return x + y to propagate NaN significands
241 if (Double.isNaN(x) || Double.isNaN(y))
242 return x + y;
243
244 final double y_abs = Math.abs(y);
245 double x_abs = Math.abs(x);
246 // Special values of y
247 if (y == 2.0) {
248 return x * x;
249 } else if (y == 0.5) {
250 if (x >= -Double.MAX_VALUE) // Handle x == -infinity later
251 return Math.sqrt(x + 0.0); // Add 0.0 to properly handle x == -0.0
252 } else if (y_abs == 1.0) { // y is +/-1
253 return (y == 1.0) ? x : 1.0 / x;
254 } else if (y_abs == INFINITY) { // y is +/-infinity
255 if (x_abs == 1.0)
256 return y - y; // inf**+/-1 is NaN
257 else if (x_abs > 1.0) // (|x| > 1)**+/-inf = inf, 0
258 return (y >= 0) ? y : 0.0;
259 else // (|x| < 1)**-/+inf = inf, 0
260 return (y < 0) ? -y : 0.0;
261 }
262
263 final int hx = __HI(x);
264 int ix = hx & 0x7fffffff;
265
266 /*
267 * When x < 0, determine if y is an odd integer:
268 * y_is_int = 0 ... y is not an integer
269 * y_is_int = 1 ... y is an odd int
270 * y_is_int = 2 ... y is an even int
271 */
272 int y_is_int = 0;
273 if (hx < 0) {
274 if (y_abs >= 0x1.0p53) // |y| >= 2^53 = 9.007199254740992E15
275 y_is_int = 2; // y is an even integer since ulp(2^53) = 2.0
276 else if (y_abs >= 1.0) { // |y| >= 1.0
277 long y_abs_as_long = (long) y_abs;
278 if ( ((double) y_abs_as_long) == y_abs) {
279 y_is_int = 2 - (int)(y_abs_as_long & 0x1L);
280 }
281 }
282 }
283
284 // Special value of x
285 if (x_abs == 0.0 ||
286 x_abs == INFINITY ||
287 x_abs == 1.0) {
288 z = x_abs; // x is +/-0, +/-inf, +/-1
289 if (y < 0.0)
290 z = 1.0/z; // z = (1/|x|)
291 if (hx < 0) {
292 if (((ix - 0x3ff00000) | y_is_int) == 0) {
293 z = (z-z)/(z-z); // (-1)**non-int is NaN
294 } else if (y_is_int == 1)
295 z = -1.0 * z; // (x < 0)**odd = -(|x|**odd)
296 }
297 return z;
298 }
299
300 n = (hx >> 31) + 1;
301
302 // (x < 0)**(non-int) is NaN
303 if ((n | y_is_int) == 0)
304 return (x-x)/(x-x);
305
306 s = 1.0; // s (sign of result -ve**odd) = -1 else = 1
307 if ( (n | (y_is_int - 1)) == 0)
308 s = -1.0; // (-ve)**(odd int)
309
310 double p_h, p_l, t1, t2;
311 // |y| is huge
312 if (y_abs > 0x1.0p31) { // if |y| > 2**31
313 final double INV_LN2 = 0x1.7154_7652_b82fep0; // 1.44269504088896338700e+00 = 1/ln2
314 final double INV_LN2_H = 0x1.715476p0; // 1.44269502162933349609e+00 = 24 bits of 1/ln2
315 final double INV_LN2_L = 0x1.4ae0_bf85_ddf44p-26; // 1.92596299112661746887e-08 = 1/ln2 tail
316
317 // Over/underflow if x is not close to one
318 if (x_abs < 0x1.fffffp-1) // |x| < 0.9999995231628418
319 return (y < 0.0) ? s * INFINITY : s * 0.0;
320 if (x_abs > 1.0) // |x| > 1.0
321 return (y > 0.0) ? s * INFINITY : s * 0.0;
322 /*
323 * now |1-x| is tiny <= 2**-20, sufficient to compute
324 * log(x) by x - x^2/2 + x^3/3 - x^4/4
325 */
326 t = x_abs - 1.0; // t has 20 trailing zeros
327 w = (t * t) * (0.5 - t * (0.3333333333333333333333 - t * 0.25));
328 u = INV_LN2_H * t; // INV_LN2_H has 21 sig. bits
329 v = t * INV_LN2_L - w * INV_LN2;
330 t1 = u + v;
331 t1 =__LO(t1, 0);
332 t2 = v - (t1 - u);
333 } else {
334 final double CP = 0x1.ec70_9dc3_a03fdp-1; // 9.61796693925975554329e-01 = 2/(3ln2)
335 final double CP_H = 0x1.ec709ep-1; // 9.61796700954437255859e-01 = (float)cp
336 final double CP_L = -0x1.e2fe_0145_b01f5p-28; // -7.02846165095275826516e-09 = tail of CP_H
337
338 double z_h, z_l, ss, s2, s_h, s_l, t_h, t_l;
339 n = 0;
340 // Take care of subnormal numbers
341 if (ix < 0x00100000) {
342 x_abs *= 0x1.0p53; // 2^53 = 9007199254740992.0
343 n -= 53;
344 ix = __HI(x_abs);
345 }
346 n += ((ix) >> 20) - 0x3ff;
347 j = ix & 0x000fffff;
348 // Determine interval
349 ix = j | 0x3ff00000; // Normalize ix
350 if (j <= 0x3988E)
351 k = 0; // |x| <sqrt(3/2)
352 else if (j < 0xBB67A)
353 k = 1; // |x| <sqrt(3)
354 else {
355 k = 0;
356 n += 1;
357 ix -= 0x00100000;
358 }
359 x_abs = __HI(x_abs, ix);
360
361 // Compute ss = s_h + s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5)
362
363 final double BP[] = {1.0,
364 1.5};
365 final double DP_H[] = {0.0,
366 0x1.2b80_34p-1}; // 5.84962487220764160156e-01
367 final double DP_L[] = {0.0,
368 0x1.cfde_b43c_fd006p-27};// 1.35003920212974897128e-08
369
370 // Poly coefs for (3/2)*(log(x)-2s-2/3*s**3
371 final double L1 = 0x1.3333_3333_33303p-1; // 5.99999999999994648725e-01
372 final double L2 = 0x1.b6db_6db6_fabffp-2; // 4.28571428578550184252e-01
373 final double L3 = 0x1.5555_5518_f264dp-2; // 3.33333329818377432918e-01
374 final double L4 = 0x1.1746_0a91_d4101p-2; // 2.72728123808534006489e-01
375 final double L5 = 0x1.d864_a93c_9db65p-3; // 2.30660745775561754067e-01
376 final double L6 = 0x1.a7e2_84a4_54eefp-3; // 2.06975017800338417784e-01
377 u = x_abs - BP[k]; // BP[0]=1.0, BP[1]=1.5
378 v = 1.0 / (x_abs + BP[k]);
379 ss = u * v;
380 s_h = ss;
381 s_h = __LO(s_h, 0);
382 // t_h=x_abs + BP[k] High
383 t_h = 0.0;
384 t_h = __HI(t_h, ((ix >> 1) | 0x20000000) + 0x00080000 + (k << 18) );
385 t_l = x_abs - (t_h - BP[k]);
386 s_l = v * ((u - s_h * t_h) - s_h * t_l);
387 // Compute log(x_abs)
388 s2 = ss * ss;
389 r = s2 * s2* (L1 + s2 * (L2 + s2 * (L3 + s2 * (L4 + s2 * (L5 + s2 * L6)))));
390 r += s_l * (s_h + ss);
391 s2 = s_h * s_h;
392 t_h = 3.0 + s2 + r;
393 t_h = __LO(t_h, 0);
394 t_l = r - ((t_h - 3.0) - s2);
395 // u+v = ss*(1+...)
396 u = s_h * t_h;
397 v = s_l * t_h + t_l * ss;
398 // 2/(3log2)*(ss + ...)
399 p_h = u + v;
400 p_h = __LO(p_h, 0);
401 p_l = v - (p_h - u);
402 z_h = CP_H * p_h; // CP_H + CP_L = 2/(3*log2)
403 z_l = CP_L * p_h + p_l * CP + DP_L[k];
404 // log2(x_abs) = (ss + ..)*2/(3*log2) = n + DP_H + z_h + z_l
405 t = (double)n;
406 t1 = (((z_h + z_l) + DP_H[k]) + t);
407 t1 = __LO(t1, 0);
408 t2 = z_l - (((t1 - t) - DP_H[k]) - z_h);
409 }
410
411 // Split up y into (y1 + y2) and compute (y1 + y2) * (t1 + t2)
412 double y1 = y;
413 y1 = __LO(y1, 0);
414 p_l = (y - y1) * t1 + y * t2;
415 p_h = y1 * t1;
416 z = p_l + p_h;
417 j = __HI(z);
418 i = __LO(z);
419 if (j >= 0x40900000) { // z >= 1024
420 if (((j - 0x40900000) | i)!=0) // if z > 1024
421 return s * INFINITY; // Overflow
422 else {
423 final double OVT = 8.0085662595372944372e-0017; // -(1024-log2(ovfl+.5ulp))
424 if (p_l + OVT > z - p_h)
425 return s * INFINITY; // Overflow
426 }
427 } else if ((j & 0x7fffffff) >= 0x4090cc00 ) { // z <= -1075
428 if (((j - 0xc090cc00) | i)!=0) // z < -1075
429 return s * 0.0; // Underflow
430 else {
431 if (p_l <= z - p_h)
432 return s * 0.0; // Underflow
433 }
434 }
435 /*
436 * Compute 2**(p_h+p_l)
437 */
438 // Poly coefs for (3/2)*(log(x)-2s-2/3*s**3
439 final double P1 = 0x1.5555_5555_5553ep-3; // 1.66666666666666019037e-01
440 final double P2 = -0x1.6c16_c16b_ebd93p-9; // -2.77777777770155933842e-03
441 final double P3 = 0x1.1566_aaf2_5de2cp-14; // 6.61375632143793436117e-05
442 final double P4 = -0x1.bbd4_1c5d_26bf1p-20; // -1.65339022054652515390e-06
443 final double P5 = 0x1.6376_972b_ea4d0p-25; // 4.13813679705723846039e-08
444 final double LG2 = 0x1.62e4_2fef_a39efp-1; // 6.93147180559945286227e-01
445 final double LG2_H = 0x1.62e43p-1; // 6.93147182464599609375e-01
446 final double LG2_L = -0x1.05c6_10ca_86c39p-29; // -1.90465429995776804525e-09
447 i = j & 0x7fffffff;
448 k = (i >> 20) - 0x3ff;
449 n = 0;
450 if (i > 0x3fe00000) { // if |z| > 0.5, set n = [z + 0.5]
451 n = j + (0x00100000 >> (k + 1));
452 k = ((n & 0x7fffffff) >> 20) - 0x3ff; // new k for n
453 t = 0.0;
454 t = __HI(t, (n & ~(0x000fffff >> k)) );
455 n = ((n & 0x000fffff) | 0x00100000) >> (20 - k);
456 if (j < 0)
457 n = -n;
458 p_h -= t;
459 }
460 t = p_l + p_h;
461 t = __LO(t, 0);
462 u = t * LG2_H;
463 v = (p_l - (t - p_h)) * LG2 + t * LG2_L;
464 z = u + v;
465 w = v - (z - u);
466 t = z * z;
467 t1 = z - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5))));
468 r = (z * t1)/(t1 - 2.0) - (w + z * w);
469 z = 1.0 - (r - z);
470 j = __HI(z);
471 j += (n << 20);
472 if ((j >> 20) <= 0)
473 z = Math.scalb(z, n); // subnormal output
474 else {
475 int z_hi = __HI(z);
476 z_hi += (n << 20);
477 z = __HI(z, z_hi);
478 }
479 return s * z;
480 }
481 }
482 }