1 /*
2 * Copyright (c) 1998, 2017, 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
30 * C to Java.
31 *
32 * <p>The C version of fdlibm relied on the idiom of pointer aliasing
33 * a 64-bit double floating-point value as a two-element array of
34 * 32-bit integers and reading and writing the two halves of the
35 * double independently. This coding pattern was problematic to C
36 * optimizers and not directly expressible in Java. Therefore, rather
37 * than a memory level overlay, if portions of a double need to be
38 * operated on as integer values, the standard library methods for
39 * bitwise floating-point to integer conversion,
40 * Double.longBitsToDouble and Double.doubleToRawLongBits, are directly
41 * or indirectly used.
42 *
43 * <p>The C version of fdlibm also took some pains to signal the
44 * correct IEEE 754 exceptional conditions divide by zero, invalid,
45 * overflow and underflow. For example, overflow would be signaled by
46 * {@code huge * huge} where {@code huge} was a large constant that
47 * would overflow when squared. Since IEEE floating-point exceptional
48 * handling is not supported natively in the JVM, such coding patterns
49 * have been omitted from this port. For example, rather than {@code
50 * return huge * huge}, this port will use {@code return INFINITY}.
51 *
52 * <p>Various comparison and arithmetic operations in fdlibm could be
53 * done either based on the integer view of a value or directly on the
54 * floating-point representation. Which idiom is faster may depend on
55 * platform specific factors. However, for code clarity if no other
56 * reason, this port will favor expressing the semantics of those
57 * operations in terms of floating-point operations when convenient to
58 * do so.
59 */
60 class FdLibm {
61 // Constants used by multiple algorithms
62 private static final double INFINITY = Double.POSITIVE_INFINITY;
63
64 private FdLibm() {
65 throw new UnsupportedOperationException("No FdLibm instances for you.");
66 }
67
68 /**
69 * Return the low-order 32 bits of the double argument as an int.
70 */
71 private static int __LO(double x) {
72 long transducer = Double.doubleToRawLongBits(x);
73 return (int)transducer;
74 }
75
76 /**
77 * Return a double with its low-order bits of the second argument
78 * and the high-order bits of the first argument..
79 */
80 private static double __LO(double x, int low) {
81 long transX = Double.doubleToRawLongBits(x);
82 return Double.longBitsToDouble((transX & 0xFFFF_FFFF_0000_0000L) |
83 (low & 0x0000_0000_FFFF_FFFFL));
84 }
85
86 /**
87 * Return the high-order 32 bits of the double argument as an int.
88 */
89 private static int __HI(double x) {
90 long transducer = Double.doubleToRawLongBits(x);
91 return (int)(transducer >> 32);
92 }
93
94 /**
95 * Return a double with its high-order bits of the second argument
96 * and the low-order bits of the first argument..
97 */
98 private static double __HI(double x, int high) {
99 long transX = Double.doubleToRawLongBits(x);
100 return Double.longBitsToDouble((transX & 0x0000_0000_FFFF_FFFFL) |
101 ( ((long)high)) << 32 );
102 }
103
104 /**
105 * cbrt(x)
106 * Return cube root of x
107 */
108 public static class Cbrt {
109 // unsigned
110 private static final int B1 = 715094163; /* B1 = (682-0.03306235651)*2**20 */
111 private static final int B2 = 696219795; /* B2 = (664-0.03306235651)*2**20 */
112
113 private static final double C = 0x1.15f15f15f15f1p-1; // 19/35 ~= 5.42857142857142815906e-01
114 private static final double D = -0x1.691de2532c834p-1; // -864/1225 ~= 7.05306122448979611050e-01
115 private static final double E = 0x1.6a0ea0ea0ea0fp0; // 99/70 ~= 1.41428571428571436819e+00
116 private static final double F = 0x1.9b6db6db6db6ep0; // 45/28 ~= 1.60714285714285720630e+00
117 private static final double G = 0x1.6db6db6db6db7p-2; // 5/14 ~= 3.57142857142857150787e-01
118
119 private Cbrt() {
120 throw new UnsupportedOperationException();
121 }
122
123 public static strictfp double compute(double x) {
124 double t = 0.0;
125 double sign;
126
127 if (x == 0.0 || !Double.isFinite(x))
128 return x; // Handles signed zeros properly
129
130 sign = (x < 0.0) ? -1.0: 1.0;
131
132 x = Math.abs(x); // x <- |x|
133
134 // Rough cbrt to 5 bits
135 if (x < 0x1.0p-1022) { // subnormal number
136 t = 0x1.0p54; // set t= 2**54
137 t *= x;
138 t = __HI(t, __HI(t)/3 + B2);
139 } else {
140 int hx = __HI(x); // high word of x
141 t = __HI(t, hx/3 + B1);
142 }
143
144 // New cbrt to 23 bits, may be implemented in single precision
145 double r, s, w;
146 r = t * t/x;
147 s = C + r*t;
148 t *= G + F/(s + E + D/s);
149
150 // Chopped to 20 bits and make it larger than cbrt(x)
151 t = __LO(t, 0);
152 t = __HI(t, __HI(t) + 0x00000001);
153
154 // One step newton iteration to 53 bits with error less than 0.667 ulps
155 s = t * t; // t*t is exact
156 r = x / s;
157 w = t + t;
158 r = (r - t)/(w + r); // r-s is exact
159 t = t + t*r;
160
161 // Restore the original sign bit
162 return sign * t;
163 }
164 }
165
166 /**
167 * hypot(x,y)
168 *
169 * Method :
170 * If (assume round-to-nearest) z = x*x + y*y
171 * has error less than sqrt(2)/2 ulp, than
172 * sqrt(z) has error less than 1 ulp (exercise).
173 *
174 * So, compute sqrt(x*x + y*y) with some care as
175 * follows to get the error below 1 ulp:
176 *
177 * Assume x > y > 0;
178 * (if possible, set rounding to round-to-nearest)
179 * 1. if x > 2y use
180 * x1*x1 + (y*y + (x2*(x + x1))) for x*x + y*y
181 * where x1 = x with lower 32 bits cleared, x2 = x - x1; else
182 * 2. if x <= 2y use
183 * t1*y1 + ((x-y) * (x-y) + (t1*y2 + t2*y))
184 * where t1 = 2x with lower 32 bits cleared, t2 = 2x - t1,
185 * y1= y with lower 32 bits chopped, y2 = y - y1.
186 *
187 * NOTE: scaling may be necessary if some argument is too
188 * large or too tiny
189 *
190 * Special cases:
191 * hypot(x,y) is INF if x or y is +INF or -INF; else
192 * hypot(x,y) is NAN if x or y is NAN.
193 *
194 * Accuracy:
195 * hypot(x,y) returns sqrt(x^2 + y^2) with error less
196 * than 1 ulp (unit in the last place)
197 */
198 public static class Hypot {
199 public static final double TWO_MINUS_600 = 0x1.0p-600;
200 public static final double TWO_PLUS_600 = 0x1.0p+600;
201
202 private Hypot() {
203 throw new UnsupportedOperationException();
204 }
205
206 public static strictfp double compute(double x, double y) {
207 double a = Math.abs(x);
208 double b = Math.abs(y);
209
210 if (!Double.isFinite(a) || !Double.isFinite(b)) {
211 if (a == INFINITY || b == INFINITY)
212 return INFINITY;
213 else
214 return a + b; // Propagate NaN significand bits
215 }
216
217 if (b > a) {
218 double tmp = a;
219 a = b;
220 b = tmp;
221 }
222 assert a >= b;
223
224 // Doing bitwise conversion after screening for NaN allows
225 // the code to not worry about the possibility of
226 // "negative" NaN values.
227
228 // Note: the ha and hb variables are the high-order
229 // 32-bits of a and b stored as integer values. The ha and
230 // hb values are used first for a rough magnitude
231 // comparison of a and b and second for simulating higher
232 // precision by allowing a and b, respectively, to be
233 // decomposed into non-overlapping portions. Both of these
234 // uses could be eliminated. The magnitude comparison
235 // could be eliminated by extracting and comparing the
236 // exponents of a and b or just be performing a
237 // floating-point divide. Splitting a floating-point
238 // number into non-overlapping portions can be
239 // accomplished by judicious use of multiplies and
240 // additions. For details see T. J. Dekker, A Floating
241 // Point Technique for Extending the Available Precision ,
242 // Numerische Mathematik, vol. 18, 1971, pp.224-242 and
243 // subsequent work.
244
245 int ha = __HI(a); // high word of a
246 int hb = __HI(b); // high word of b
247
248 if ((ha - hb) > 0x3c00000) {
249 return a + b; // x / y > 2**60
250 }
251
252 int k = 0;
253 if (a > 0x1.00000_ffff_ffffp500) { // a > ~2**500
254 // scale a and b by 2**-600
255 ha -= 0x25800000;
256 hb -= 0x25800000;
257 a = a * TWO_MINUS_600;
258 b = b * TWO_MINUS_600;
259 k += 600;
260 }
261 double t1, t2;
262 if (b < 0x1.0p-500) { // b < 2**-500
263 if (b < Double.MIN_NORMAL) { // subnormal b or 0 */
264 if (b == 0.0)
265 return a;
266 t1 = 0x1.0p1022; // t1 = 2^1022
267 b *= t1;
268 a *= t1;
269 k -= 1022;
270 } else { // scale a and b by 2^600
271 ha += 0x25800000; // a *= 2^600
272 hb += 0x25800000; // b *= 2^600
273 a = a * TWO_PLUS_600;
274 b = b * TWO_PLUS_600;
275 k -= 600;
276 }
277 }
278 // medium size a and b
279 double w = a - b;
280 if (w > b) {
281 t1 = 0;
282 t1 = __HI(t1, ha);
283 t2 = a - t1;
284 w = Math.sqrt(t1*t1 - (b*(-b) - t2 * (a + t1)));
285 } else {
286 double y1, y2;
287 a = a + a;
288 y1 = 0;
289 y1 = __HI(y1, hb);
290 y2 = b - y1;
291 t1 = 0;
292 t1 = __HI(t1, ha + 0x00100000);
293 t2 = a - t1;
294 w = Math.sqrt(t1*y1 - (w*(-w) - (t1*y2 + t2*b)));
295 }
296 if (k != 0) {
297 return Math.powerOfTwoD(k) * w;
298 } else
299 return w;
300 }
301 }
302
303 /**
304 * Compute x**y
305 * n
306 * Method: Let x = 2 * (1+f)
307 * 1. Compute and return log2(x) in two pieces:
308 * log2(x) = w1 + w2,
309 * where w1 has 53 - 24 = 29 bit trailing zeros.
310 * 2. Perform y*log2(x) = n+y' by simulating multi-precision
311 * arithmetic, where |y'| <= 0.5.
312 * 3. Return x**y = 2**n*exp(y'*log2)
313 *
314 * Special cases:
315 * 1. (anything) ** 0 is 1
316 * 2. (anything) ** 1 is itself
317 * 3. (anything) ** NAN is NAN
318 * 4. NAN ** (anything except 0) is NAN
319 * 5. +-(|x| > 1) ** +INF is +INF
320 * 6. +-(|x| > 1) ** -INF is +0
321 * 7. +-(|x| < 1) ** +INF is +0
322 * 8. +-(|x| < 1) ** -INF is +INF
323 * 9. +-1 ** +-INF is NAN
324 * 10. +0 ** (+anything except 0, NAN) is +0
325 * 11. -0 ** (+anything except 0, NAN, odd integer) is +0
326 * 12. +0 ** (-anything except 0, NAN) is +INF
327 * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF
328 * 14. -0 ** (odd integer) = -( +0 ** (odd integer) )
329 * 15. +INF ** (+anything except 0,NAN) is +INF
330 * 16. +INF ** (-anything except 0,NAN) is +0
331 * 17. -INF ** (anything) = -0 ** (-anything)
332 * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
333 * 19. (-anything except 0 and inf) ** (non-integer) is NAN
334 *
335 * Accuracy:
336 * pow(x,y) returns x**y nearly rounded. In particular
337 * pow(integer,integer)
338 * always returns the correct integer provided it is
339 * representable.
340 */
341 public static class Pow {
342 private Pow() {
343 throw new UnsupportedOperationException();
344 }
345
346 public static strictfp double compute(final double x, final double y) {
347 double z;
348 double r, s, t, u, v, w;
349 int i, j, k, n;
350
351 // y == zero: x**0 = 1
352 if (y == 0.0)
353 return 1.0;
354
355 // +/-NaN return x + y to propagate NaN significands
356 if (Double.isNaN(x) || Double.isNaN(y))
357 return x + y;
358
359 final double y_abs = Math.abs(y);
360 double x_abs = Math.abs(x);
361 // Special values of y
362 if (y == 2.0) {
363 return x * x;
364 } else if (y == 0.5) {
365 if (x >= -Double.MAX_VALUE) // Handle x == -infinity later
366 return Math.sqrt(x + 0.0); // Add 0.0 to properly handle x == -0.0
367 } else if (y_abs == 1.0) { // y is +/-1
368 return (y == 1.0) ? x : 1.0 / x;
369 } else if (y_abs == INFINITY) { // y is +/-infinity
370 if (x_abs == 1.0)
371 return y - y; // inf**+/-1 is NaN
372 else if (x_abs > 1.0) // (|x| > 1)**+/-inf = inf, 0
373 return (y >= 0) ? y : 0.0;
374 else // (|x| < 1)**-/+inf = inf, 0
375 return (y < 0) ? -y : 0.0;
376 }
377
378 final int hx = __HI(x);
379 int ix = hx & 0x7fffffff;
380
381 /*
382 * When x < 0, determine if y is an odd integer:
383 * y_is_int = 0 ... y is not an integer
384 * y_is_int = 1 ... y is an odd int
385 * y_is_int = 2 ... y is an even int
386 */
387 int y_is_int = 0;
388 if (hx < 0) {
389 if (y_abs >= 0x1.0p53) // |y| >= 2^53 = 9.007199254740992E15
390 y_is_int = 2; // y is an even integer since ulp(2^53) = 2.0
391 else if (y_abs >= 1.0) { // |y| >= 1.0
392 long y_abs_as_long = (long) y_abs;
393 if ( ((double) y_abs_as_long) == y_abs) {
394 y_is_int = 2 - (int)(y_abs_as_long & 0x1L);
395 }
396 }
397 }
398
399 // Special value of x
400 if (x_abs == 0.0 ||
401 x_abs == INFINITY ||
402 x_abs == 1.0) {
403 z = x_abs; // x is +/-0, +/-inf, +/-1
404 if (y < 0.0)
405 z = 1.0/z; // z = (1/|x|)
406 if (hx < 0) {
407 if (((ix - 0x3ff00000) | y_is_int) == 0) {
408 z = (z-z)/(z-z); // (-1)**non-int is NaN
409 } else if (y_is_int == 1)
410 z = -1.0 * z; // (x < 0)**odd = -(|x|**odd)
411 }
412 return z;
413 }
414
415 n = (hx >> 31) + 1;
416
417 // (x < 0)**(non-int) is NaN
418 if ((n | y_is_int) == 0)
419 return (x-x)/(x-x);
420
421 s = 1.0; // s (sign of result -ve**odd) = -1 else = 1
422 if ( (n | (y_is_int - 1)) == 0)
423 s = -1.0; // (-ve)**(odd int)
424
425 double p_h, p_l, t1, t2;
426 // |y| is huge
427 if (y_abs > 0x1.00000_ffff_ffffp31) { // if |y| > ~2**31
428 final double INV_LN2 = 0x1.7154_7652_b82fep0; // 1.44269504088896338700e+00 = 1/ln2
429 final double INV_LN2_H = 0x1.715476p0; // 1.44269502162933349609e+00 = 24 bits of 1/ln2
430 final double INV_LN2_L = 0x1.4ae0_bf85_ddf44p-26; // 1.92596299112661746887e-08 = 1/ln2 tail
431
432 // Over/underflow if x is not close to one
433 if (x_abs < 0x1.fffff_0000_0000p-1) // |x| < ~0.9999995231628418
434 return (y < 0.0) ? s * INFINITY : s * 0.0;
435 if (x_abs > 0x1.00000_ffff_ffffp0) // |x| > ~1.0
436 return (y > 0.0) ? s * INFINITY : s * 0.0;
437 /*
438 * now |1-x| is tiny <= 2**-20, sufficient to compute
439 * log(x) by x - x^2/2 + x^3/3 - x^4/4
440 */
441 t = x_abs - 1.0; // t has 20 trailing zeros
442 w = (t * t) * (0.5 - t * (0.3333333333333333333333 - t * 0.25));
443 u = INV_LN2_H * t; // INV_LN2_H has 21 sig. bits
444 v = t * INV_LN2_L - w * INV_LN2;
445 t1 = u + v;
446 t1 =__LO(t1, 0);
447 t2 = v - (t1 - u);
448 } else {
449 final double CP = 0x1.ec70_9dc3_a03fdp-1; // 9.61796693925975554329e-01 = 2/(3ln2)
450 final double CP_H = 0x1.ec709ep-1; // 9.61796700954437255859e-01 = (float)cp
451 final double CP_L = -0x1.e2fe_0145_b01f5p-28; // -7.02846165095275826516e-09 = tail of CP_H
452
453 double z_h, z_l, ss, s2, s_h, s_l, t_h, t_l;
454 n = 0;
455 // Take care of subnormal numbers
456 if (ix < 0x00100000) {
457 x_abs *= 0x1.0p53; // 2^53 = 9007199254740992.0
458 n -= 53;
459 ix = __HI(x_abs);
460 }
461 n += ((ix) >> 20) - 0x3ff;
462 j = ix & 0x000fffff;
463 // Determine interval
464 ix = j | 0x3ff00000; // Normalize ix
465 if (j <= 0x3988E)
466 k = 0; // |x| <sqrt(3/2)
467 else if (j < 0xBB67A)
468 k = 1; // |x| <sqrt(3)
469 else {
470 k = 0;
471 n += 1;
472 ix -= 0x00100000;
473 }
474 x_abs = __HI(x_abs, ix);
475
476 // Compute ss = s_h + s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5)
477
478 final double[] BP = {1.0,
479 1.5};
480 final double[] DP_H = {0.0,
481 0x1.2b80_34p-1}; // 5.84962487220764160156e-01
482 final double[] DP_L = {0.0,
483 0x1.cfde_b43c_fd006p-27};// 1.35003920212974897128e-08
484
485 // Poly coefs for (3/2)*(log(x)-2s-2/3*s**3
486 final double L1 = 0x1.3333_3333_33303p-1; // 5.99999999999994648725e-01
487 final double L2 = 0x1.b6db_6db6_fabffp-2; // 4.28571428578550184252e-01
488 final double L3 = 0x1.5555_5518_f264dp-2; // 3.33333329818377432918e-01
489 final double L4 = 0x1.1746_0a91_d4101p-2; // 2.72728123808534006489e-01
490 final double L5 = 0x1.d864_a93c_9db65p-3; // 2.30660745775561754067e-01
491 final double L6 = 0x1.a7e2_84a4_54eefp-3; // 2.06975017800338417784e-01
492 u = x_abs - BP[k]; // BP[0]=1.0, BP[1]=1.5
493 v = 1.0 / (x_abs + BP[k]);
494 ss = u * v;
495 s_h = ss;
496 s_h = __LO(s_h, 0);
497 // t_h=x_abs + BP[k] High
498 t_h = 0.0;
499 t_h = __HI(t_h, ((ix >> 1) | 0x20000000) + 0x00080000 + (k << 18) );
500 t_l = x_abs - (t_h - BP[k]);
501 s_l = v * ((u - s_h * t_h) - s_h * t_l);
502 // Compute log(x_abs)
503 s2 = ss * ss;
504 r = s2 * s2* (L1 + s2 * (L2 + s2 * (L3 + s2 * (L4 + s2 * (L5 + s2 * L6)))));
505 r += s_l * (s_h + ss);
506 s2 = s_h * s_h;
507 t_h = 3.0 + s2 + r;
508 t_h = __LO(t_h, 0);
509 t_l = r - ((t_h - 3.0) - s2);
510 // u+v = ss*(1+...)
511 u = s_h * t_h;
512 v = s_l * t_h + t_l * ss;
513 // 2/(3log2)*(ss + ...)
514 p_h = u + v;
515 p_h = __LO(p_h, 0);
516 p_l = v - (p_h - u);
517 z_h = CP_H * p_h; // CP_H + CP_L = 2/(3*log2)
518 z_l = CP_L * p_h + p_l * CP + DP_L[k];
519 // log2(x_abs) = (ss + ..)*2/(3*log2) = n + DP_H + z_h + z_l
520 t = (double)n;
521 t1 = (((z_h + z_l) + DP_H[k]) + t);
522 t1 = __LO(t1, 0);
523 t2 = z_l - (((t1 - t) - DP_H[k]) - z_h);
524 }
525
526 // Split up y into (y1 + y2) and compute (y1 + y2) * (t1 + t2)
527 double y1 = y;
528 y1 = __LO(y1, 0);
529 p_l = (y - y1) * t1 + y * t2;
530 p_h = y1 * t1;
531 z = p_l + p_h;
532 j = __HI(z);
533 i = __LO(z);
534 if (j >= 0x40900000) { // z >= 1024
535 if (((j - 0x40900000) | i)!=0) // if z > 1024
536 return s * INFINITY; // Overflow
537 else {
538 final double OVT = 8.0085662595372944372e-0017; // -(1024-log2(ovfl+.5ulp))
539 if (p_l + OVT > z - p_h)
540 return s * INFINITY; // Overflow
541 }
542 } else if ((j & 0x7fffffff) >= 0x4090cc00 ) { // z <= -1075
543 if (((j - 0xc090cc00) | i)!=0) // z < -1075
544 return s * 0.0; // Underflow
545 else {
546 if (p_l <= z - p_h)
547 return s * 0.0; // Underflow
548 }
549 }
550 /*
551 * Compute 2**(p_h+p_l)
552 */
553 // Poly coefs for (3/2)*(log(x)-2s-2/3*s**3
554 final double P1 = 0x1.5555_5555_5553ep-3; // 1.66666666666666019037e-01
555 final double P2 = -0x1.6c16_c16b_ebd93p-9; // -2.77777777770155933842e-03
556 final double P3 = 0x1.1566_aaf2_5de2cp-14; // 6.61375632143793436117e-05
557 final double P4 = -0x1.bbd4_1c5d_26bf1p-20; // -1.65339022054652515390e-06
558 final double P5 = 0x1.6376_972b_ea4d0p-25; // 4.13813679705723846039e-08
559 final double LG2 = 0x1.62e4_2fef_a39efp-1; // 6.93147180559945286227e-01
560 final double LG2_H = 0x1.62e43p-1; // 6.93147182464599609375e-01
561 final double LG2_L = -0x1.05c6_10ca_86c39p-29; // -1.90465429995776804525e-09
562 i = j & 0x7fffffff;
563 k = (i >> 20) - 0x3ff;
564 n = 0;
565 if (i > 0x3fe00000) { // if |z| > 0.5, set n = [z + 0.5]
566 n = j + (0x00100000 >> (k + 1));
567 k = ((n & 0x7fffffff) >> 20) - 0x3ff; // new k for n
568 t = 0.0;
569 t = __HI(t, (n & ~(0x000fffff >> k)) );
570 n = ((n & 0x000fffff) | 0x00100000) >> (20 - k);
571 if (j < 0)
572 n = -n;
573 p_h -= t;
574 }
575 t = p_l + p_h;
576 t = __LO(t, 0);
577 u = t * LG2_H;
578 v = (p_l - (t - p_h)) * LG2 + t * LG2_L;
579 z = u + v;
580 w = v - (z - u);
581 t = z * z;
582 t1 = z - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5))));
583 r = (z * t1)/(t1 - 2.0) - (w + z * w);
584 z = 1.0 - (r - z);
585 j = __HI(z);
586 j += (n << 20);
587 if ((j >> 20) <= 0)
588 z = Math.scalb(z, n); // subnormal output
589 else {
590 int z_hi = __HI(z);
591 z_hi += (n << 20);
592 z = __HI(z, z_hi);
593 }
594 return s * z;
595 }
596 }
597
598 /**
599 * Returns the exponential of x.
600 *
601 * Method
602 * 1. Argument reduction:
603 * Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
604 * Given x, find r and integer k such that
605 *
606 * x = k*ln2 + r, |r| <= 0.5*ln2.
607 *
608 * Here r will be represented as r = hi-lo for better
609 * accuracy.
610 *
611 * 2. Approximation of exp(r) by a special rational function on
612 * the interval [0,0.34658]:
613 * Write
614 * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
615 * We use a special Reme algorithm on [0,0.34658] to generate
616 * a polynomial of degree 5 to approximate R. The maximum error
617 * of this polynomial approximation is bounded by 2**-59. In
618 * other words,
619 * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
620 * (where z=r*r, and the values of P1 to P5 are listed below)
621 * and
622 * | 5 | -59
623 * | 2.0+P1*z+...+P5*z - R(z) | <= 2
624 * | |
625 * The computation of exp(r) thus becomes
626 * 2*r
627 * exp(r) = 1 + -------
628 * R - r
629 * r*R1(r)
630 * = 1 + r + ----------- (for better accuracy)
631 * 2 - R1(r)
632 * where
633 * 2 4 10
634 * R1(r) = r - (P1*r + P2*r + ... + P5*r ).
635 *
636 * 3. Scale back to obtain exp(x):
637 * From step 1, we have
638 * exp(x) = 2^k * exp(r)
639 *
640 * Special cases:
641 * exp(INF) is INF, exp(NaN) is NaN;
642 * exp(-INF) is 0, and
643 * for finite argument, only exp(0)=1 is exact.
644 *
645 * Accuracy:
646 * according to an error analysis, the error is always less than
647 * 1 ulp (unit in the last place).
648 *
649 * Misc. info.
650 * For IEEE double
651 * if x > 7.09782712893383973096e+02 then exp(x) overflow
652 * if x < -7.45133219101941108420e+02 then exp(x) underflow
653 *
654 * Constants:
655 * The hexadecimal values are the intended ones for the following
656 * constants. The decimal values may be used, provided that the
657 * compiler will convert from decimal to binary accurately enough
658 * to produce the hexadecimal values shown.
659 */
660 static class Exp {
661 private static final double one = 1.0;
662 private static final double[] half = {0.5, -0.5,};
663 private static final double huge = 1.0e+300;
664 private static final double twom1000= 0x1.0p-1000; // 9.33263618503218878990e-302 = 2^-1000
665 private static final double o_threshold= 0x1.62e42fefa39efp9; // 7.09782712893383973096e+02
666 private static final double u_threshold= -0x1.74910d52d3051p9; // -7.45133219101941108420e+02;
667 private static final double[] ln2HI ={ 0x1.62e42feep-1, // 6.93147180369123816490e-01
668 -0x1.62e42feep-1}; // -6.93147180369123816490e-01
669 private static final double[] ln2LO ={ 0x1.a39ef35793c76p-33, // 1.90821492927058770002e-10
670 -0x1.a39ef35793c76p-33}; // -1.90821492927058770002e-10
671 private static final double invln2 = 0x1.71547652b82fep0; // 1.44269504088896338700e+00
672
673 private static final double P1 = 0x1.555555555553ep-3; // 1.66666666666666019037e-01
674 private static final double P2 = -0x1.6c16c16bebd93p-9; // -2.77777777770155933842e-03
675 private static final double P3 = 0x1.1566aaf25de2cp-14; // 6.61375632143793436117e-05
676 private static final double P4 = -0x1.bbd41c5d26bf1p-20; // -1.65339022054652515390e-06
677 private static final double P5 = 0x1.6376972bea4d0p-25; // 4.13813679705723846039e-08
678
679 private Exp() {
680 throw new UnsupportedOperationException();
681 }
682
683 // should be able to forgo strictfp due to controlled over/underflow
684 public static strictfp double compute(double x) {
685 double y;
686 double hi = 0.0;
687 double lo = 0.0;
688 double c;
689 double t;
690 int k = 0;
691 int xsb;
692 /*unsigned*/ int hx;
693
694 hx = __HI(x); /* high word of x */
695 xsb = (hx >> 31) & 1; /* sign bit of x */
696 hx &= 0x7fffffff; /* high word of |x| */
697
698 /* filter out non-finite argument */
699 if (hx >= 0x40862E42) { /* if |x| >= 709.78... */
700 if (hx >= 0x7ff00000) {
701 if (((hx & 0xfffff) | __LO(x)) != 0)
702 return x + x; /* NaN */
703 else
704 return (xsb == 0) ? x : 0.0; /* exp(+-inf) = {inf, 0} */
705 }
706 if (x > o_threshold)
707 return huge * huge; /* overflow */
708 if (x < u_threshold) // unsigned compare needed here?
709 return twom1000 * twom1000; /* underflow */
710 }
711
712 /* argument reduction */
713 if (hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
714 if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
715 hi = x - ln2HI[xsb];
716 lo=ln2LO[xsb];
717 k = 1 - xsb - xsb;
718 } else {
719 k = (int)(invln2 * x + half[xsb]);
720 t = k;
721 hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */
722 lo = t*ln2LO[0];
723 }
724 x = hi - lo;
725 } else if (hx < 0x3e300000) { /* when |x|<2**-28 */
726 if (huge + x > one)
727 return one + x; /* trigger inexact */
728 } else {
729 k = 0;
730 }
731
732 /* x is now in primary range */
733 t = x * x;
734 c = x - t*(P1 + t*(P2 + t*(P3 + t*(P4 + t*P5))));
735 if (k == 0)
736 return one - ((x*c)/(c - 2.0) - x);
737 else
738 y = one - ((lo - (x*c)/(2.0 - c)) - hi);
739
740 if(k >= -1021) {
741 y = __HI(y, __HI(y) + (k << 20)); /* add k to y's exponent */
742 return y;
743 } else {
744 y = __HI(y, __HI(y) + ((k + 1000) << 20)); /* add k to y's exponent */
745 return y * twom1000;
746 }
747 }
748 }
749 }