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
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)|low );
83 }
84
85 /**
86 * Return the high-order 32 bits of the double argument as an int.
87 */
88 private static int __HI(double x) {
89 long transducer = Double.doubleToRawLongBits(x);
90 return (int)(transducer >> 32);
91 }
92
93 /**
94 * Return a double with its high-order bits of the second argument
95 * and the low-order bits of the first argument..
96 */
97 private static double __HI(double x, int high) {
98 long transX = Double.doubleToRawLongBits(x);
99 return Double.longBitsToDouble((transX & 0x0000_0000_FFFF_FFFFL)|( ((long)high)) << 32 );
100 }
101
102 /**
103 * cbrt(x)
104 * Return cube root of x
105 */
106 public static class Cbrt {
107 // unsigned
108 private static final int B1 = 715094163; /* B1 = (682-0.03306235651)*2**20 */
109 private static final int B2 = 696219795; /* B2 = (664-0.03306235651)*2**20 */
110
111 private static final double C = 5.42857142857142815906e-01; /* 19/35 = 0x3FE15F15, 0xF15F15F1 */
112 private static final double D = -7.05306122448979611050e-01; /* -864/1225 = 0xBFE691DE, 0x2532C834 */
113 private static final double E = 1.41428571428571436819e+00; /* 99/70 = 0x3FF6A0EA, 0x0EA0EA0F */
114 private static final double F = 1.60714285714285720630e+00; /* 45/28 = 0x3FF9B6DB, 0x6DB6DB6E */
115 private static final double G = 3.57142857142857150787e-01; /* 5/14 = 0x3FD6DB6D, 0xB6DB6DB7 */
116
117 public static strictfp double compute(double x) {
118 int hx;
119 double r, s, t=0.0, w;
120 double sign;
121
122 if (!Double.isFinite(x) || x == 0.0)
123 return x; // Handles signed zeros properly
124
125 sign = (x < 0.0) ? -1.0: 1.0;
126
127 x = Math.abs(x); // x <- |x|
128 hx = __HI(x); // high word of x
129
130 // Rough cbrt to 5 bits
131 if (hx < 0x00100000) { // subnormal number
132 t = __HI(t, 0x43500000); // set t= 2**54
133 t *= x;
134 t = __HI(t, __HI(t)/3 + B2);
135 } else {
136 t = __HI(t, hx/3 + B1);
137 }
138
139 // New cbrt to 23 bits, may be implemented in single precision
140 r = t * t/x;
141 s = C + r*t;
142 t *= G + F/(s + E + D/s);
143
144 // Chopped to 20 bits and make it larger than cbrt(x)
145 t = __LO(t, 0);
146 t = __HI(t, __HI(t) + 0x00000001);
147
148 // One step newton iteration to 53 bits with error less than 0.667 ulps
149 s = t * t; // t*t is exact
150 r = x / s;
151 w = t + t;
152 r= (r - t)/(w + r); // r-s is exact
153 t= t + t*r;
154
155 // Restore the original sign bit
156 return sign * t;
157 }
158 }
159
160 /**
161 * hypot(x,y)
162 *
163 * Method :
164 * If (assume round-to-nearest) z = x*x + y*y
165 * has error less than sqrt(2)/2 ulp, than
166 * sqrt(z) has error less than 1 ulp (exercise).
167 *
168 * So, compute sqrt(x*x + y*y) with some care as
169 * follows to get the error below 1 ulp:
170 *
171 * Assume x > y > 0;
172 * (if possible, set rounding to round-to-nearest)
173 * 1. if x > 2y use
174 * x1*x1 + (y*y + (x2*(x + x1))) for x*x + y*y
175 * where x1 = x with lower 32 bits cleared, x2 = x - x1; else
176 * 2. if x <= 2y use
177 * t1*y1 + ((x-y) * (x-y) + (t1*y2 + t2*y))
178 * where t1 = 2x with lower 32 bits cleared, t2 = 2x - t1,
179 * y1= y with lower 32 bits chopped, y2 = y - y1.
180 *
181 * NOTE: scaling may be necessary if some argument is too
182 * large or too tiny
183 *
184 * Special cases:
185 * hypot(x,y) is INF if x or y is +INF or -INF; else
186 * hypot(x,y) is NAN if x or y is NAN.
187 *
188 * Accuracy:
189 * hypot(x,y) returns sqrt(x^2 + y^2) with error less
190 * than 1 ulp (unit in the last place)
191 */
192 public static class Hypot {
193 public static final double TWO_MINUS_600 = 0x1.0p-600;
194 public static final double TWO_PLUS_600 = 0x1.0p+600;
195
196 public static strictfp double compute(double x, double y) {
197 double a = Math.abs(x);
198 double b = Math.abs(y);
199
200 if (!Double.isFinite(a) || !Double.isFinite(b)) {
201 if (a == INFINITY || b == INFINITY)
202 return INFINITY;
203 else
204 return a + b; // Propagate NaN significand bits
205 }
206
207 if (b > a) {
208 double tmp = a;
209 a = b;
210 b = tmp;
211 }
212 assert a >= b;
213
214 // Doing bitwise conversion after screening for NaN allows
215 // the code to not worry about the possibility of
216 // "negative" NaN values.
217
218 // Note: the ha and hb variables are the high-order
219 // 32-bits of a and b stored as integer values. The ha and
220 // hb values are used first for a rough magnitude
221 // comparison of a and b and second for simulating higher
222 // precision by allowing a and b, respectively, to be
223 // decomposed into non-overlapping portions. Both of these
224 // uses could be eliminated. The magnitude comparison
225 // could be eliminated by extracting and comparing the
226 // exponents of a and b or just be performing a
227 // floating-point divide. Splitting a floating-point
228 // number into non-overlapping portions can be
229 // accomplished by judicious use of multiplies and
230 // additions. For details see T. J. Dekker, A Floating
231 // Point Technique for Extending the Available Precision ,
232 // Numerische Mathematik, vol. 18, 1971, pp.224-242 and
233 // subsequent work.
234
235 int ha = __HI(a); // high word of a
236 int hb = __HI(b); // high word of b
237
238 if ((ha - hb) > 0x3c00000) {
239 return a + b; // x / y > 2**60
240 }
241
242 int k = 0;
243 if (a > 0x1.00000_ffff_ffffp500) { // a > ~2**500
244 // scale a and b by 2**-600
245 ha -= 0x25800000;
246 hb -= 0x25800000;
247 a = a * TWO_MINUS_600;
248 b = b * TWO_MINUS_600;
249 k += 600;
250 }
251 double t1, t2;
252 if (b < 0x1.0p-500) { // b < 2**-500
253 if (b < Double.MIN_NORMAL) { // subnormal b or 0 */
254 if (b == 0.0)
255 return a;
256 t1 = 0x1.0p1022; // t1 = 2^1022
257 b *= t1;
258 a *= t1;
259 k -= 1022;
260 } else { // scale a and b by 2^600
261 ha += 0x25800000; // a *= 2^600
262 hb += 0x25800000; // b *= 2^600
263 a = a * TWO_PLUS_600;
264 b = b * TWO_PLUS_600;
265 k -= 600;
266 }
267 }
268 // medium size a and b
269 double w = a - b;
270 if (w > b) {
271 t1 = 0;
272 t1 = __HI(t1, ha);
273 t2 = a - t1;
274 w = Math.sqrt(t1*t1 - (b*(-b) - t2 * (a + t1)));
275 } else {
276 double y1, y2;
277 a = a + a;
278 y1 = 0;
279 y1 = __HI(y1, hb);
280 y2 = b - y1;
281 t1 = 0;
282 t1 = __HI(t1, ha + 0x00100000);
283 t2 = a - t1;
284 w = Math.sqrt(t1*y1 - (w*(-w) - (t1*y2 + t2*b)));
285 }
286 if (k != 0) {
287 return Math.powerOfTwoD(k) * w;
288 } else
289 return w;
290 }
291 }
292
293 /**
294 * Compute x**y
295 * n
296 * Method: Let x = 2 * (1+f)
297 * 1. Compute and return log2(x) in two pieces:
298 * log2(x) = w1 + w2,
299 * where w1 has 53 - 24 = 29 bit trailing zeros.
300 * 2. Perform y*log2(x) = n+y' by simulating multi-precision
301 * arithmetic, where |y'| <= 0.5.
302 * 3. Return x**y = 2**n*exp(y'*log2)
303 *
304 * Special cases:
305 * 1. (anything) ** 0 is 1
306 * 2. (anything) ** 1 is itself
307 * 3. (anything) ** NAN is NAN
308 * 4. NAN ** (anything except 0) is NAN
309 * 5. +-(|x| > 1) ** +INF is +INF
310 * 6. +-(|x| > 1) ** -INF is +0
311 * 7. +-(|x| < 1) ** +INF is +0
312 * 8. +-(|x| < 1) ** -INF is +INF
313 * 9. +-1 ** +-INF is NAN
314 * 10. +0 ** (+anything except 0, NAN) is +0
315 * 11. -0 ** (+anything except 0, NAN, odd integer) is +0
316 * 12. +0 ** (-anything except 0, NAN) is +INF
317 * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF
318 * 14. -0 ** (odd integer) = -( +0 ** (odd integer) )
319 * 15. +INF ** (+anything except 0,NAN) is +INF
320 * 16. +INF ** (-anything except 0,NAN) is +0
321 * 17. -INF ** (anything) = -0 ** (-anything)
322 * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
323 * 19. (-anything except 0 and inf) ** (non-integer) is NAN
324 *
325 * Accuracy:
326 * pow(x,y) returns x**y nearly rounded. In particular
327 * pow(integer,integer)
328 * always returns the correct integer provided it is
329 * representable.
330 */
331 public static class Pow {
332 public static strictfp double compute(final double x, final double y) {
333 double z;
334 double r, s, t, u, v, w;
335 int i, j, k, n;
336
337 // y == zero: x**0 = 1
338 if (y == 0.0)
339 return 1.0;
340
341 // +/-NaN return x + y to propagate NaN significands
342 if (Double.isNaN(x) || Double.isNaN(y))
343 return x + y;
344
345 final double y_abs = Math.abs(y);
346 double x_abs = Math.abs(x);
347 // Special values of y
348 if (y == 2.0) {
349 return x * x;
350 } else if (y == 0.5) {
351 if (x >= -Double.MAX_VALUE) // Handle x == -infinity later
352 return Math.sqrt(x + 0.0); // Add 0.0 to properly handle x == -0.0
353 } else if (y_abs == 1.0) { // y is +/-1
354 return (y == 1.0) ? x : 1.0 / x;
355 } else if (y_abs == INFINITY) { // y is +/-infinity
356 if (x_abs == 1.0)
357 return y - y; // inf**+/-1 is NaN
358 else if (x_abs > 1.0) // (|x| > 1)**+/-inf = inf, 0
359 return (y >= 0) ? y : 0.0;
360 else // (|x| < 1)**-/+inf = inf, 0
361 return (y < 0) ? -y : 0.0;
362 }
363
364 final int hx = __HI(x);
365 int ix = hx & 0x7fffffff;
366
367 /*
368 * When x < 0, determine if y is an odd integer:
369 * y_is_int = 0 ... y is not an integer
370 * y_is_int = 1 ... y is an odd int
371 * y_is_int = 2 ... y is an even int
372 */
373 int y_is_int = 0;
374 if (hx < 0) {
375 if (y_abs >= 0x1.0p53) // |y| >= 2^53 = 9.007199254740992E15
376 y_is_int = 2; // y is an even integer since ulp(2^53) = 2.0
377 else if (y_abs >= 1.0) { // |y| >= 1.0
378 long y_abs_as_long = (long) y_abs;
379 if ( ((double) y_abs_as_long) == y_abs) {
380 y_is_int = 2 - (int)(y_abs_as_long & 0x1L);
381 }
382 }
383 }
384
385 // Special value of x
386 if (x_abs == 0.0 ||
387 x_abs == INFINITY ||
388 x_abs == 1.0) {
389 z = x_abs; // x is +/-0, +/-inf, +/-1
390 if (y < 0.0)
391 z = 1.0/z; // z = (1/|x|)
392 if (hx < 0) {
393 if (((ix - 0x3ff00000) | y_is_int) == 0) {
394 z = (z-z)/(z-z); // (-1)**non-int is NaN
395 } else if (y_is_int == 1)
396 z = -1.0 * z; // (x < 0)**odd = -(|x|**odd)
397 }
398 return z;
399 }
400
401 n = (hx >> 31) + 1;
402
403 // (x < 0)**(non-int) is NaN
404 if ((n | y_is_int) == 0)
405 return (x-x)/(x-x);
406
407 s = 1.0; // s (sign of result -ve**odd) = -1 else = 1
408 if ( (n | (y_is_int - 1)) == 0)
409 s = -1.0; // (-ve)**(odd int)
410
411 double p_h, p_l, t1, t2;
412 // |y| is huge
413 if (y_abs > 0x1.00000_ffff_ffffp31) { // if |y| > ~2**31
414 final double INV_LN2 = 0x1.7154_7652_b82fep0; // 1.44269504088896338700e+00 = 1/ln2
415 final double INV_LN2_H = 0x1.715476p0; // 1.44269502162933349609e+00 = 24 bits of 1/ln2
416 final double INV_LN2_L = 0x1.4ae0_bf85_ddf44p-26; // 1.92596299112661746887e-08 = 1/ln2 tail
417
418 // Over/underflow if x is not close to one
419 if (x_abs < 0x1.fffff_0000_0000p-1) // |x| < ~0.9999995231628418
420 return (y < 0.0) ? s * INFINITY : s * 0.0;
421 if (x_abs > 0x1.00000_ffff_ffffp0) // |x| > ~1.0
422 return (y > 0.0) ? s * INFINITY : s * 0.0;
423 /*
424 * now |1-x| is tiny <= 2**-20, sufficient to compute
425 * log(x) by x - x^2/2 + x^3/3 - x^4/4
426 */
427 t = x_abs - 1.0; // t has 20 trailing zeros
428 w = (t * t) * (0.5 - t * (0.3333333333333333333333 - t * 0.25));
429 u = INV_LN2_H * t; // INV_LN2_H has 21 sig. bits
430 v = t * INV_LN2_L - w * INV_LN2;
431 t1 = u + v;
432 t1 =__LO(t1, 0);
433 t2 = v - (t1 - u);
434 } else {
435 final double CP = 0x1.ec70_9dc3_a03fdp-1; // 9.61796693925975554329e-01 = 2/(3ln2)
436 final double CP_H = 0x1.ec709ep-1; // 9.61796700954437255859e-01 = (float)cp
437 final double CP_L = -0x1.e2fe_0145_b01f5p-28; // -7.02846165095275826516e-09 = tail of CP_H
438
439 double z_h, z_l, ss, s2, s_h, s_l, t_h, t_l;
440 n = 0;
441 // Take care of subnormal numbers
442 if (ix < 0x00100000) {
443 x_abs *= 0x1.0p53; // 2^53 = 9007199254740992.0
444 n -= 53;
445 ix = __HI(x_abs);
446 }
447 n += ((ix) >> 20) - 0x3ff;
448 j = ix & 0x000fffff;
449 // Determine interval
450 ix = j | 0x3ff00000; // Normalize ix
451 if (j <= 0x3988E)
452 k = 0; // |x| <sqrt(3/2)
453 else if (j < 0xBB67A)
454 k = 1; // |x| <sqrt(3)
455 else {
456 k = 0;
457 n += 1;
458 ix -= 0x00100000;
459 }
460 x_abs = __HI(x_abs, ix);
461
462 // Compute ss = s_h + s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5)
463
464 final double BP[] = {1.0,
465 1.5};
466 final double DP_H[] = {0.0,
467 0x1.2b80_34p-1}; // 5.84962487220764160156e-01
468 final double DP_L[] = {0.0,
469 0x1.cfde_b43c_fd006p-27};// 1.35003920212974897128e-08
470
471 // Poly coefs for (3/2)*(log(x)-2s-2/3*s**3
472 final double L1 = 0x1.3333_3333_33303p-1; // 5.99999999999994648725e-01
473 final double L2 = 0x1.b6db_6db6_fabffp-2; // 4.28571428578550184252e-01
474 final double L3 = 0x1.5555_5518_f264dp-2; // 3.33333329818377432918e-01
475 final double L4 = 0x1.1746_0a91_d4101p-2; // 2.72728123808534006489e-01
476 final double L5 = 0x1.d864_a93c_9db65p-3; // 2.30660745775561754067e-01
477 final double L6 = 0x1.a7e2_84a4_54eefp-3; // 2.06975017800338417784e-01
478 u = x_abs - BP[k]; // BP[0]=1.0, BP[1]=1.5
479 v = 1.0 / (x_abs + BP[k]);
480 ss = u * v;
481 s_h = ss;
482 s_h = __LO(s_h, 0);
483 // t_h=x_abs + BP[k] High
484 t_h = 0.0;
485 t_h = __HI(t_h, ((ix >> 1) | 0x20000000) + 0x00080000 + (k << 18) );
486 t_l = x_abs - (t_h - BP[k]);
487 s_l = v * ((u - s_h * t_h) - s_h * t_l);
488 // Compute log(x_abs)
489 s2 = ss * ss;
490 r = s2 * s2* (L1 + s2 * (L2 + s2 * (L3 + s2 * (L4 + s2 * (L5 + s2 * L6)))));
491 r += s_l * (s_h + ss);
492 s2 = s_h * s_h;
493 t_h = 3.0 + s2 + r;
494 t_h = __LO(t_h, 0);
495 t_l = r - ((t_h - 3.0) - s2);
496 // u+v = ss*(1+...)
497 u = s_h * t_h;
498 v = s_l * t_h + t_l * ss;
499 // 2/(3log2)*(ss + ...)
500 p_h = u + v;
501 p_h = __LO(p_h, 0);
502 p_l = v - (p_h - u);
503 z_h = CP_H * p_h; // CP_H + CP_L = 2/(3*log2)
504 z_l = CP_L * p_h + p_l * CP + DP_L[k];
505 // log2(x_abs) = (ss + ..)*2/(3*log2) = n + DP_H + z_h + z_l
506 t = (double)n;
507 t1 = (((z_h + z_l) + DP_H[k]) + t);
508 t1 = __LO(t1, 0);
509 t2 = z_l - (((t1 - t) - DP_H[k]) - z_h);
510 }
511
512 // Split up y into (y1 + y2) and compute (y1 + y2) * (t1 + t2)
513 double y1 = y;
514 y1 = __LO(y1, 0);
515 p_l = (y - y1) * t1 + y * t2;
516 p_h = y1 * t1;
517 z = p_l + p_h;
518 j = __HI(z);
519 i = __LO(z);
520 if (j >= 0x40900000) { // z >= 1024
521 if (((j - 0x40900000) | i)!=0) // if z > 1024
522 return s * INFINITY; // Overflow
523 else {
524 final double OVT = 8.0085662595372944372e-0017; // -(1024-log2(ovfl+.5ulp))
525 if (p_l + OVT > z - p_h)
526 return s * INFINITY; // Overflow
527 }
528 } else if ((j & 0x7fffffff) >= 0x4090cc00 ) { // z <= -1075
529 if (((j - 0xc090cc00) | i)!=0) // z < -1075
530 return s * 0.0; // Underflow
531 else {
532 if (p_l <= z - p_h)
533 return s * 0.0; // Underflow
534 }
535 }
536 /*
537 * Compute 2**(p_h+p_l)
538 */
539 // Poly coefs for (3/2)*(log(x)-2s-2/3*s**3
540 final double P1 = 0x1.5555_5555_5553ep-3; // 1.66666666666666019037e-01
541 final double P2 = -0x1.6c16_c16b_ebd93p-9; // -2.77777777770155933842e-03
542 final double P3 = 0x1.1566_aaf2_5de2cp-14; // 6.61375632143793436117e-05
543 final double P4 = -0x1.bbd4_1c5d_26bf1p-20; // -1.65339022054652515390e-06
544 final double P5 = 0x1.6376_972b_ea4d0p-25; // 4.13813679705723846039e-08
545 final double LG2 = 0x1.62e4_2fef_a39efp-1; // 6.93147180559945286227e-01
546 final double LG2_H = 0x1.62e43p-1; // 6.93147182464599609375e-01
547 final double LG2_L = -0x1.05c6_10ca_86c39p-29; // -1.90465429995776804525e-09
548 i = j & 0x7fffffff;
549 k = (i >> 20) - 0x3ff;
550 n = 0;
551 if (i > 0x3fe00000) { // if |z| > 0.5, set n = [z + 0.5]
552 n = j + (0x00100000 >> (k + 1));
553 k = ((n & 0x7fffffff) >> 20) - 0x3ff; // new k for n
554 t = 0.0;
555 t = __HI(t, (n & ~(0x000fffff >> k)) );
556 n = ((n & 0x000fffff) | 0x00100000) >> (20 - k);
557 if (j < 0)
558 n = -n;
559 p_h -= t;
560 }
561 t = p_l + p_h;
562 t = __LO(t, 0);
563 u = t * LG2_H;
564 v = (p_l - (t - p_h)) * LG2 + t * LG2_L;
565 z = u + v;
566 w = v - (z - u);
567 t = z * z;
568 t1 = z - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5))));
569 r = (z * t1)/(t1 - 2.0) - (w + z * w);
570 z = 1.0 - (r - z);
571 j = __HI(z);
572 j += (n << 20);
573 if ((j >> 20) <= 0)
574 z = Math.scalb(z, n); // subnormal output
575 else {
576 int z_hi = __HI(z);
577 z_hi += (n << 20);
578 z = __HI(z, z_hi);
579 }
580 return s * z;
581 }
582 }
583 }