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1 /*
2 * IBM Accurate Mathematical Library
3 * written by International Business Machines Corp.
4 * Copyright (C) 2001-2015 Free Software Foundation, Inc.
5 *
6 * This program is free software; you can redistribute it and/or modify
7 * it under the terms of the GNU Lesser General Public License as published by
8 * the Free Software Foundation; either version 2.1 of the License, or
9 * (at your option) any later version.
10 *
11 * This program is distributed in the hope that it will be useful,
12 * but WITHOUT ANY WARRANTY; without even the implied warranty of
13 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
14 * GNU Lesser General Public License for more details.
15 *
16 * You should have received a copy of the GNU Lesser General Public License
17 * along with this program; if not, see <http://www.gnu.org/licenses/>.
18 */
19 /****************************************************************************/
20 /* */
21 /* MODULE_NAME:usncs.c */
22 /* */
23 /* FUNCTIONS: usin */
24 /* ucos */
25 /* slow */
26 /* slow1 */
27 /* slow2 */
28 /* sloww */
29 /* sloww1 */
30 /* sloww2 */
31 /* bsloww */
32 /* bsloww1 */
33 /* bsloww2 */
34 /* cslow2 */
35 /* FILES NEEDED: dla.h endian.h mpa.h mydefs.h usncs.h */
36 /* branred.c sincos32.c dosincos.c mpa.c */
37 /* sincos.tbl */
38 /* */
39 /* An ultimate sin and routine. Given an IEEE double machine number x */
40 /* it computes the correctly rounded (to nearest) value of sin(x) or cos(x) */
41 /* Assumption: Machine arithmetic operations are performed in */
42 /* round to nearest mode of IEEE 754 standard. */
43 /* */
44 /****************************************************************************/
45
46
47 #include <errno.h>
48 #include <float.h>
49 #include "endian.h"
50 #include "mydefs.h"
51 #include "usncs.h"
52 #include "MathLib.h"
53 #include <math.h>
54 #include <math_private.h>
55 #include <fenv.h>
56
57 /* Helper macros to compute sin of the input values. */
58 #define POLYNOMIAL2(xx) ((((s5 * (xx) + s4) * (xx) + s3) * (xx) + s2) * (xx))
59
60 #define POLYNOMIAL(xx) (POLYNOMIAL2 (xx) + s1)
61
62 /* The computed polynomial is a variation of the Taylor series expansion for
63 sin(a):
64
65 a - a^3/3! + a^5/5! - a^7/7! + a^9/9! + (1 - a^2) * da / 2
66
67 The constants s1, s2, s3, etc. are pre-computed values of 1/3!, 1/5! and so
68 on. The result is returned to LHS and correction in COR. */
69 #define TAYLOR_SIN(xx, a, da, cor) \
70 ({ \
71 double t = ((POLYNOMIAL (xx) * (a) - 0.5 * (da)) * (xx) + (da)); \
72 double res = (a) + t; \
73 (cor) = ((a) - res) + t; \
74 res; \
75 })
76
77 /* This is again a variation of the Taylor series expansion with the term
78 x^3/3! expanded into the following for better accuracy:
79
80 bb * x ^ 3 + 3 * aa * x * x1 * x2 + aa * x1 ^ 3 + aa * x2 ^ 3
81
82 The correction term is dx and bb + aa = -1/3!
83 */
84 #define TAYLOR_SLOW(x0, dx, cor) \
85 ({ \
86 static const double th2_36 = 206158430208.0; /* 1.5*2**37 */ \
87 double xx = (x0) * (x0); \
88 double x1 = ((x0) + th2_36) - th2_36; \
89 double y = aa * x1 * x1 * x1; \
90 double r = (x0) + y; \
91 double x2 = ((x0) - x1) + (dx); \
92 double t = (((POLYNOMIAL2 (xx) + bb) * xx + 3.0 * aa * x1 * x2) \
93 * (x0) + aa * x2 * x2 * x2 + (dx)); \
94 t = (((x0) - r) + y) + t; \
95 double res = r + t; \
96 (cor) = (r - res) + t; \
97 res; \
98 })
99
100 #define SINCOS_TABLE_LOOKUP(u, sn, ssn, cs, ccs) \
101 ({ \
102 int4 k = u.i[LOW_HALF] << 2; \
103 sn = __sincostab.x[k]; \
104 ssn = __sincostab.x[k + 1]; \
105 cs = __sincostab.x[k + 2]; \
106 ccs = __sincostab.x[k + 3]; \
107 })
108
109 #ifndef SECTION
110 # define SECTION
111 #endif
112
113 extern const union
114 {
115 int4 i[880];
116 double x[440];
117 } __sincostab attribute_hidden;
118
119 static const double
120 sn3 = -1.66666666666664880952546298448555E-01,
121 sn5 = 8.33333214285722277379541354343671E-03,
122 cs2 = 4.99999999999999999999950396842453E-01,
123 cs4 = -4.16666666666664434524222570944589E-02,
124 cs6 = 1.38888874007937613028114285595617E-03;
125
126 static const double t22 = 0x1.8p22;
127
128 void __dubsin (double x, double dx, double w[]);
129 void __docos (double x, double dx, double w[]);
130 double __mpsin (double x, double dx, bool reduce_range);
131 double __mpcos (double x, double dx, bool reduce_range);
132 static double slow (double x);
133 static double slow1 (double x);
134 static double slow2 (double x);
135 static double sloww (double x, double dx, double orig, int n);
136 static double sloww1 (double x, double dx, double orig, int m, int n);
137 static double sloww2 (double x, double dx, double orig, int n);
138 static double bsloww (double x, double dx, double orig, int n);
139 static double bsloww1 (double x, double dx, double orig, int n);
140 static double bsloww2 (double x, double dx, double orig, int n);
141 int __branred (double x, double *a, double *aa);
142 static double cslow2 (double x);
143
144 /* Given a number partitioned into U and X such that U is an index into the
145 sin/cos table, this macro computes the cosine of the number by combining
146 the sin and cos of X (as computed by a variation of the Taylor series) with
147 the values looked up from the sin/cos table to get the result in RES and a
148 correction value in COR. */
149 static double
150 do_cos (mynumber u, double x, double *corp)
151 {
152 double xx, s, sn, ssn, c, cs, ccs, res, cor;
153 xx = x * x;
154 s = x + x * xx * (sn3 + xx * sn5);
155 c = xx * (cs2 + xx * (cs4 + xx * cs6));
156 SINCOS_TABLE_LOOKUP (u, sn, ssn, cs, ccs);
157 cor = (ccs - s * ssn - cs * c) - sn * s;
158 res = cs + cor;
159 cor = (cs - res) + cor;
160 *corp = cor;
161 return res;
162 }
163
164 /* A more precise variant of DO_COS where the number is partitioned into U, X
165 and DX. EPS is the adjustment to the correction COR. */
166 static double
167 do_cos_slow (mynumber u, double x, double dx, double eps, double *corp)
168 {
169 double xx, y, x1, x2, e1, e2, res, cor;
170 double s, sn, ssn, c, cs, ccs;
171 xx = x * x;
172 s = x * xx * (sn3 + xx * sn5);
173 c = x * dx + xx * (cs2 + xx * (cs4 + xx * cs6));
174 SINCOS_TABLE_LOOKUP (u, sn, ssn, cs, ccs);
175 x1 = (x + t22) - t22;
176 x2 = (x - x1) + dx;
177 e1 = (sn + t22) - t22;
178 e2 = (sn - e1) + ssn;
179 cor = (ccs - cs * c - e1 * x2 - e2 * x) - sn * s;
180 y = cs - e1 * x1;
181 cor = cor + ((cs - y) - e1 * x1);
182 res = y + cor;
183 cor = (y - res) + cor;
184 if (cor > 0)
185 cor = 1.0005 * cor + eps;
186 else
187 cor = 1.0005 * cor - eps;
188 *corp = cor;
189 return res;
190 }
191
192 /* Given a number partitioned into U and X and DX such that U is an index into
193 the sin/cos table, this macro computes the sine of the number by combining
194 the sin and cos of X (as computed by a variation of the Taylor series) with
195 the values looked up from the sin/cos table to get the result in RES and a
196 correction value in COR. */
197 static double
198 do_sin (mynumber u, double x, double dx, double *corp)
199 {
200 double xx, s, sn, ssn, c, cs, ccs, cor, res;
201 xx = x * x;
202 s = x + (dx + x * xx * (sn3 + xx * sn5));
203 c = x * dx + xx * (cs2 + xx * (cs4 + xx * cs6));
204 SINCOS_TABLE_LOOKUP (u, sn, ssn, cs, ccs);
205 cor = (ssn + s * ccs - sn * c) + cs * s;
206 res = sn + cor;
207 cor = (sn - res) + cor;
208 *corp = cor;
209 return res;
210 }
211
212 /* A more precise variant of res = do_sin where the number is partitioned into U, X
213 and DX. EPS is the adjustment to the correction COR. */
214 static double
215 do_sin_slow (mynumber u, double x, double dx, double eps, double *corp)
216 {
217 double xx, y, x1, x2, c1, c2, res, cor;
218 double s, sn, ssn, c, cs, ccs;
219 xx = x * x;
220 s = x * xx * (sn3 + xx * sn5);
221 c = xx * (cs2 + xx * (cs4 + xx * cs6));
222 SINCOS_TABLE_LOOKUP (u, sn, ssn, cs, ccs);
223 x1 = (x + t22) - t22;
224 x2 = (x - x1) + dx;
225 c1 = (cs + t22) - t22;
226 c2 = (cs - c1) + ccs;
227 cor = (ssn + s * ccs + cs * s + c2 * x + c1 * x2 - sn * x * dx) - sn * c;
228 y = sn + c1 * x1;
229 cor = cor + ((sn - y) + c1 * x1);
230 res = y + cor;
231 cor = (y - res) + cor;
232 if (cor > 0)
233 cor = 1.0005 * cor + eps;
234 else
235 cor = 1.0005 * cor - eps;
236 *corp = cor;
237 return res;
238 }
239
240 /* Reduce range of X and compute sin of a + da. K is the amount by which to
241 rotate the quadrants. This allows us to use the same routine to compute cos
242 by simply rotating the quadrants by 1. */
243 static inline double
244 __always_inline
245 reduce_and_compute (double x, unsigned int k)
246 {
247 double retval = 0, a, da;
248 unsigned int n = __branred (x, &a, &da);
249 k = (n + k) % 4;
250 switch (k)
251 {
252 case 0:
253 if (a * a < 0.01588)
254 retval = bsloww (a, da, x, n);
255 else
256 retval = bsloww1 (a, da, x, n);
257 break;
258 case 2:
259 if (a * a < 0.01588)
260 retval = bsloww (-a, -da, x, n);
261 else
262 retval = bsloww1 (-a, -da, x, n);
263 break;
264
265 case 1:
266 case 3:
267 retval = bsloww2 (a, da, x, n);
268 break;
269 }
270 return retval;
271 }
272
273 static inline int4
274 __always_inline
275 reduce_sincos_1 (double x, double *a, double *da)
276 {
277 mynumber v;
278
279 double t = (x * hpinv + toint);
280 double xn = t - toint;
281 v.x = t;
282 double y = (x - xn * mp1) - xn * mp2;
283 int4 n = v.i[LOW_HALF] & 3;
284 double db = xn * mp3;
285 double b = y - db;
286 db = (y - b) - db;
287
288 *a = b;
289 *da = db;
290
291 return n;
292 }
293
294 /* Compute sin (A + DA). cos can be computed by shifting the quadrant N
295 clockwise. */
296 static double
297 __always_inline
298 do_sincos_1 (double a, double da, double x, int4 n, int4 k)
299 {
300 double xx, retval, res, cor, y;
301 mynumber u;
302 int m;
303 double eps = fabs (x) * 1.2e-30;
304
305 int k1 = (n + k) & 3;
306 switch (k1)
307 { /* quarter of unit circle */
308 case 2:
309 a = -a;
310 da = -da;
311 case 0:
312 xx = a * a;
313 if (xx < 0.01588)
314 {
315 /* Taylor series. */
316 res = TAYLOR_SIN (xx, a, da, cor);
317 cor = (cor > 0) ? 1.02 * cor + eps : 1.02 * cor - eps;
318 retval = (res == res + cor) ? res : sloww (a, da, x, k);
319 }
320 else
321 {
322 if (a > 0)
323 m = 1;
324 else
325 {
326 m = 0;
327 a = -a;
328 da = -da;
329 }
330 u.x = big + a;
331 y = a - (u.x - big);
332 res = do_sin (u, y, da, &cor);
333 cor = (cor > 0) ? 1.035 * cor + eps : 1.035 * cor - eps;
334 retval = ((res == res + cor) ? ((m) ? res : -res)
335 : sloww1 (a, da, x, m, k));
336 }
337 break;
338
339 case 1:
340 case 3:
341 if (a < 0)
342 {
343 a = -a;
344 da = -da;
345 }
346 u.x = big + a;
347 y = a - (u.x - big) + da;
348 res = do_cos (u, y, &cor);
349 cor = (cor > 0) ? 1.025 * cor + eps : 1.025 * cor - eps;
350 retval = ((res == res + cor) ? ((k1 & 2) ? -res : res)
351 : sloww2 (a, da, x, n));
352 break;
353 }
354
355 return retval;
356 }
357
358 static inline int4
359 __always_inline
360 reduce_sincos_2 (double x, double *a, double *da)
361 {
362 mynumber v;
363
364 double t = (x * hpinv + toint);
365 double xn = t - toint;
366 v.x = t;
367 double xn1 = (xn + 8.0e22) - 8.0e22;
368 double xn2 = xn - xn1;
369 double y = ((((x - xn1 * mp1) - xn1 * mp2) - xn2 * mp1) - xn2 * mp2);
370 int4 n = v.i[LOW_HALF] & 3;
371 double db = xn1 * pp3;
372 t = y - db;
373 db = (y - t) - db;
374 db = (db - xn2 * pp3) - xn * pp4;
375 double b = t + db;
376 db = (t - b) + db;
377
378 *a = b;
379 *da = db;
380
381 return n;
382 }
383
384 /* Compute sin (A + DA). cos can be computed by shifting the quadrant N
385 clockwise. */
386 static double
387 __always_inline
388 do_sincos_2 (double a, double da, double x, int4 n, int4 k)
389 {
390 double res, retval, cor, xx;
391 mynumber u;
392
393 double eps = 1.0e-24;
394
395 k = (n + k) & 3;
396
397 switch (k)
398 {
399 case 2:
400 a = -a;
401 da = -da;
402 /* Fall through. */
403 case 0:
404 xx = a * a;
405 if (xx < 0.01588)
406 {
407 /* Taylor series. */
408 res = TAYLOR_SIN (xx, a, da, cor);
409 cor = (cor > 0) ? 1.02 * cor + eps : 1.02 * cor - eps;
410 retval = (res == res + cor) ? res : bsloww (a, da, x, n);
411 }
412 else
413 {
414 double t, db, y;
415 int m;
416 if (a > 0)
417 {
418 m = 1;
419 t = a;
420 db = da;
421 }
422 else
423 {
424 m = 0;
425 t = -a;
426 db = -da;
427 }
428 u.x = big + t;
429 y = t - (u.x - big);
430 res = do_sin (u, y, db, &cor);
431 cor = (cor > 0) ? 1.035 * cor + eps : 1.035 * cor - eps;
432 retval = ((res == res + cor) ? ((m) ? res : -res)
433 : bsloww1 (a, da, x, n));
434 }
435 break;
436
437 case 1:
438 case 3:
439 if (a < 0)
440 {
441 a = -a;
442 da = -da;
443 }
444 u.x = big + a;
445 double y = a - (u.x - big) + da;
446 res = do_cos (u, y, &cor);
447 cor = (cor > 0) ? 1.025 * cor + eps : 1.025 * cor - eps;
448 retval = ((res == res + cor) ? ((n & 2) ? -res : res)
449 : bsloww2 (a, da, x, n));
450 break;
451 }
452
453 return retval;
454 }
455
456 /*******************************************************************/
457 /* An ultimate sin routine. Given an IEEE double machine number x */
458 /* it computes the correctly rounded (to nearest) value of sin(x) */
459 /*******************************************************************/
460 #ifdef IN_SINCOS
461 static double
462 #else
463 double
464 SECTION
465 #endif
466 __sin (double x)
467 {
468 double xx, res, t, cor, y, s, c, sn, ssn, cs, ccs;
469 mynumber u;
470 int4 k, m;
471 double retval = 0;
472
473 #ifndef IN_SINCOS
474 SET_RESTORE_ROUND_53BIT (FE_TONEAREST);
475 #endif
476
477 u.x = x;
478 m = u.i[HIGH_HALF];
479 k = 0x7fffffff & m; /* no sign */
480 if (k < 0x3e500000) /* if x->0 =>sin(x)=x */
481 {
482 math_check_force_underflow (x);
483 retval = x;
484 }
485 /*---------------------------- 2^-26 < |x|< 0.25 ----------------------*/
486 else if (k < 0x3fd00000)
487 {
488 xx = x * x;
489 /* Taylor series. */
490 t = POLYNOMIAL (xx) * (xx * x);
491 res = x + t;
492 cor = (x - res) + t;
493 retval = (res == res + 1.07 * cor) ? res : slow (x);
494 } /* else if (k < 0x3fd00000) */
495 /*---------------------------- 0.25<|x|< 0.855469---------------------- */
496 else if (k < 0x3feb6000)
497 {
498 u.x = (m > 0) ? big + x : big - x;
499 y = (m > 0) ? x - (u.x - big) : x + (u.x - big);
500 xx = y * y;
501 s = y + y * xx * (sn3 + xx * sn5);
502 c = xx * (cs2 + xx * (cs4 + xx * cs6));
503 SINCOS_TABLE_LOOKUP (u, sn, ssn, cs, ccs);
504 if (m <= 0)
505 {
506 sn = -sn;
507 ssn = -ssn;
508 }
509 cor = (ssn + s * ccs - sn * c) + cs * s;
510 res = sn + cor;
511 cor = (sn - res) + cor;
512 retval = (res == res + 1.096 * cor) ? res : slow1 (x);
513 } /* else if (k < 0x3feb6000) */
514
515 /*----------------------- 0.855469 <|x|<2.426265 ----------------------*/
516 else if (k < 0x400368fd)
517 {
518
519 y = (m > 0) ? hp0 - x : hp0 + x;
520 if (y >= 0)
521 {
522 u.x = big + y;
523 y = (y - (u.x - big)) + hp1;
524 }
525 else
526 {
527 u.x = big - y;
528 y = (-hp1) - (y + (u.x - big));
529 }
530 res = do_cos (u, y, &cor);
531 retval = (res == res + 1.020 * cor) ? ((m > 0) ? res : -res) : slow2 (x);
532 } /* else if (k < 0x400368fd) */
533
534 #ifndef IN_SINCOS
535 /*-------------------------- 2.426265<|x|< 105414350 ----------------------*/
536 else if (k < 0x419921FB)
537 {
538 double a, da;
539 int4 n = reduce_sincos_1 (x, &a, &da);
540 retval = do_sincos_1 (a, da, x, n, 0);
541 } /* else if (k < 0x419921FB ) */
542
543 /*---------------------105414350 <|x|< 281474976710656 --------------------*/
544 else if (k < 0x42F00000)
545 {
546 double a, da;
547
548 int4 n = reduce_sincos_2 (x, &a, &da);
549 retval = do_sincos_2 (a, da, x, n, 0);
550 } /* else if (k < 0x42F00000 ) */
551
552 /* -----------------281474976710656 <|x| <2^1024----------------------------*/
553 else if (k < 0x7ff00000)
554 retval = reduce_and_compute (x, 0);
555
556 /*--------------------- |x| > 2^1024 ----------------------------------*/
557 else
558 {
559 if (k == 0x7ff00000 && u.i[LOW_HALF] == 0)
560 __set_errno (EDOM);
561 retval = x / x;
562 }
563 #endif
564
565 return retval;
566 }
567
568
569 /*******************************************************************/
570 /* An ultimate cos routine. Given an IEEE double machine number x */
571 /* it computes the correctly rounded (to nearest) value of cos(x) */
572 /*******************************************************************/
573
574 #ifdef IN_SINCOS
575 static double
576 #else
577 double
578 SECTION
579 #endif
580 __cos (double x)
581 {
582 double y, xx, res, cor, a, da;
583 mynumber u;
584 int4 k, m;
585
586 double retval = 0;
587
588 #ifndef IN_SINCOS
589 SET_RESTORE_ROUND_53BIT (FE_TONEAREST);
590 #endif
591
592 u.x = x;
593 m = u.i[HIGH_HALF];
594 k = 0x7fffffff & m;
595
596 /* |x|<2^-27 => cos(x)=1 */
597 if (k < 0x3e400000)
598 retval = 1.0;
599
600 else if (k < 0x3feb6000)
601 { /* 2^-27 < |x| < 0.855469 */
602 y = fabs (x);
603 u.x = big + y;
604 y = y - (u.x - big);
605 res = do_cos (u, y, &cor);
606 retval = (res == res + 1.020 * cor) ? res : cslow2 (x);
607 } /* else if (k < 0x3feb6000) */
608
609 else if (k < 0x400368fd)
610 { /* 0.855469 <|x|<2.426265 */ ;
611 y = hp0 - fabs (x);
612 a = y + hp1;
613 da = (y - a) + hp1;
614 xx = a * a;
615 if (xx < 0.01588)
616 {
617 res = TAYLOR_SIN (xx, a, da, cor);
618 cor = (cor > 0) ? 1.02 * cor + 1.0e-31 : 1.02 * cor - 1.0e-31;
619 retval = (res == res + cor) ? res : sloww (a, da, x, 1);
620 }
621 else
622 {
623 if (a > 0)
624 {
625 m = 1;
626 }
627 else
628 {
629 m = 0;
630 a = -a;
631 da = -da;
632 }
633 u.x = big + a;
634 y = a - (u.x - big);
635 res = do_sin (u, y, da, &cor);
636 cor = (cor > 0) ? 1.035 * cor + 1.0e-31 : 1.035 * cor - 1.0e-31;
637 retval = ((res == res + cor) ? ((m) ? res : -res)
638 : sloww1 (a, da, x, m, 1));
639 }
640
641 } /* else if (k < 0x400368fd) */
642
643
644 #ifndef IN_SINCOS
645 else if (k < 0x419921FB)
646 { /* 2.426265<|x|< 105414350 */
647 double a, da;
648 int4 n = reduce_sincos_1 (x, &a, &da);
649 retval = do_sincos_1 (a, da, x, n, 1);
650 } /* else if (k < 0x419921FB ) */
651
652 else if (k < 0x42F00000)
653 {
654 double a, da;
655
656 int4 n = reduce_sincos_2 (x, &a, &da);
657 retval = do_sincos_2 (a, da, x, n, 1);
658 } /* else if (k < 0x42F00000 ) */
659
660 /* 281474976710656 <|x| <2^1024 */
661 else if (k < 0x7ff00000)
662 retval = reduce_and_compute (x, 1);
663
664 else
665 {
666 if (k == 0x7ff00000 && u.i[LOW_HALF] == 0)
667 __set_errno (EDOM);
668 retval = x / x; /* |x| > 2^1024 */
669 }
670 #endif
671
672 return retval;
673 }
674
675 /************************************************************************/
676 /* Routine compute sin(x) for 2^-26 < |x|< 0.25 by Taylor with more */
677 /* precision and if still doesn't accurate enough by mpsin or dubsin */
678 /************************************************************************/
679
680 static double
681 SECTION
682 slow (double x)
683 {
684 double res, cor, w[2];
685 res = TAYLOR_SLOW (x, 0, cor);
686 if (res == res + 1.0007 * cor)
687 return res;
688
689 __dubsin (fabs (x), 0, w);
690 if (w[0] == w[0] + 1.000000001 * w[1])
691 return (x > 0) ? w[0] : -w[0];
692
693 return (x > 0) ? __mpsin (x, 0, false) : -__mpsin (-x, 0, false);
694 }
695
696 /*******************************************************************************/
697 /* Routine compute sin(x) for 0.25<|x|< 0.855469 by __sincostab.tbl and Taylor */
698 /* and if result still doesn't accurate enough by mpsin or dubsin */
699 /*******************************************************************************/
700
701 static double
702 SECTION
703 slow1 (double x)
704 {
705 mynumber u;
706 double w[2], y, cor, res;
707 y = fabs (x);
708 u.x = big + y;
709 y = y - (u.x - big);
710 res = do_sin_slow (u, y, 0, 0, &cor);
711 if (res == res + cor)
712 return (x > 0) ? res : -res;
713
714 __dubsin (fabs (x), 0, w);
715 if (w[0] == w[0] + 1.000000005 * w[1])
716 return (x > 0) ? w[0] : -w[0];
717
718 return (x > 0) ? __mpsin (x, 0, false) : -__mpsin (-x, 0, false);
719 }
720
721 /**************************************************************************/
722 /* Routine compute sin(x) for 0.855469 <|x|<2.426265 by __sincostab.tbl */
723 /* and if result still doesn't accurate enough by mpsin or dubsin */
724 /**************************************************************************/
725 static double
726 SECTION
727 slow2 (double x)
728 {
729 mynumber u;
730 double w[2], y, y1, y2, cor, res, del;
731
732 y = fabs (x);
733 y = hp0 - y;
734 if (y >= 0)
735 {
736 u.x = big + y;
737 y = y - (u.x - big);
738 del = hp1;
739 }
740 else
741 {
742 u.x = big - y;
743 y = -(y + (u.x - big));
744 del = -hp1;
745 }
746 res = do_cos_slow (u, y, del, 0, &cor);
747 if (res == res + cor)
748 return (x > 0) ? res : -res;
749
750 y = fabs (x) - hp0;
751 y1 = y - hp1;
752 y2 = (y - y1) - hp1;
753 __docos (y1, y2, w);
754 if (w[0] == w[0] + 1.000000005 * w[1])
755 return (x > 0) ? w[0] : -w[0];
756
757 return (x > 0) ? __mpsin (x, 0, false) : -__mpsin (-x, 0, false);
758 }
759
760 /***************************************************************************/
761 /* Routine compute sin(x+dx) (Double-Length number) where x is small enough*/
762 /* to use Taylor series around zero and (x+dx) */
763 /* in first or third quarter of unit circle.Routine receive also */
764 /* (right argument) the original value of x for computing error of */
765 /* result.And if result not accurate enough routine calls mpsin1 or dubsin */
766 /***************************************************************************/
767
768 static double
769 SECTION
770 sloww (double x, double dx, double orig, int k)
771 {
772 double y, t, res, cor, w[2], a, da, xn;
773 mynumber v;
774 int4 n;
775 res = TAYLOR_SLOW (x, dx, cor);
776
777 if (cor > 0)
778 cor = 1.0005 * cor + fabs (orig) * 3.1e-30;
779 else
780 cor = 1.0005 * cor - fabs (orig) * 3.1e-30;
781
782 if (res == res + cor)
783 return res;
784
785 (x > 0) ? __dubsin (x, dx, w) : __dubsin (-x, -dx, w);
786 if (w[1] > 0)
787 cor = 1.000000001 * w[1] + fabs (orig) * 1.1e-30;
788 else
789 cor = 1.000000001 * w[1] - fabs (orig) * 1.1e-30;
790
791 if (w[0] == w[0] + cor)
792 return (x > 0) ? w[0] : -w[0];
793
794 t = (orig * hpinv + toint);
795 xn = t - toint;
796 v.x = t;
797 y = (orig - xn * mp1) - xn * mp2;
798 n = (v.i[LOW_HALF] + k) & 3;
799 da = xn * pp3;
800 t = y - da;
801 da = (y - t) - da;
802 y = xn * pp4;
803 a = t - y;
804 da = ((t - a) - y) + da;
805
806 if (n == 2 || n == 1)
807 {
808 a = -a;
809 da = -da;
810 }
811 (a > 0) ? __dubsin (a, da, w) : __dubsin (-a, -da, w);
812 if (w[1] > 0)
813 cor = 1.000000001 * w[1] + fabs (orig) * 1.1e-40;
814 else
815 cor = 1.000000001 * w[1] - fabs (orig) * 1.1e-40;
816
817 if (w[0] == w[0] + cor)
818 return (a > 0) ? w[0] : -w[0];
819
820 return (n & 1) ? __mpcos (orig, 0, true) : __mpsin (orig, 0, true);
821 }
822
823 /***************************************************************************/
824 /* Routine compute sin(x+dx) (Double-Length number) where x in first or */
825 /* third quarter of unit circle.Routine receive also (right argument) the */
826 /* original value of x for computing error of result.And if result not */
827 /* accurate enough routine calls mpsin1 or dubsin */
828 /***************************************************************************/
829
830 static double
831 SECTION
832 sloww1 (double x, double dx, double orig, int m, int k)
833 {
834 mynumber u;
835 double w[2], y, cor, res;
836
837 u.x = big + x;
838 y = x - (u.x - big);
839 res = do_sin_slow (u, y, dx, 3.1e-30 * fabs (orig), &cor);
840
841 if (res == res + cor)
842 return (m > 0) ? res : -res;
843
844 __dubsin (x, dx, w);
845
846 if (w[1] > 0)
847 cor = 1.000000005 * w[1] + 1.1e-30 * fabs (orig);
848 else
849 cor = 1.000000005 * w[1] - 1.1e-30 * fabs (orig);
850
851 if (w[0] == w[0] + cor)
852 return (m > 0) ? w[0] : -w[0];
853
854 return (k == 1) ? __mpcos (orig, 0, true) : __mpsin (orig, 0, true);
855 }
856
857 /***************************************************************************/
858 /* Routine compute sin(x+dx) (Double-Length number) where x in second or */
859 /* fourth quarter of unit circle.Routine receive also the original value */
860 /* and quarter(n= 1or 3)of x for computing error of result.And if result not*/
861 /* accurate enough routine calls mpsin1 or dubsin */
862 /***************************************************************************/
863
864 static double
865 SECTION
866 sloww2 (double x, double dx, double orig, int n)
867 {
868 mynumber u;
869 double w[2], y, cor, res;
870
871 u.x = big + x;
872 y = x - (u.x - big);
873 res = do_cos_slow (u, y, dx, 3.1e-30 * fabs (orig), &cor);
874
875 if (res == res + cor)
876 return (n & 2) ? -res : res;
877
878 __docos (x, dx, w);
879
880 if (w[1] > 0)
881 cor = 1.000000005 * w[1] + 1.1e-30 * fabs (orig);
882 else
883 cor = 1.000000005 * w[1] - 1.1e-30 * fabs (orig);
884
885 if (w[0] == w[0] + cor)
886 return (n & 2) ? -w[0] : w[0];
887
888 return (n & 1) ? __mpsin (orig, 0, true) : __mpcos (orig, 0, true);
889 }
890
891 /***************************************************************************/
892 /* Routine compute sin(x+dx) or cos(x+dx) (Double-Length number) where x */
893 /* is small enough to use Taylor series around zero and (x+dx) */
894 /* in first or third quarter of unit circle.Routine receive also */
895 /* (right argument) the original value of x for computing error of */
896 /* result.And if result not accurate enough routine calls other routines */
897 /***************************************************************************/
898
899 static double
900 SECTION
901 bsloww (double x, double dx, double orig, int n)
902 {
903 double res, cor, w[2];
904
905 res = TAYLOR_SLOW (x, dx, cor);
906 cor = (cor > 0) ? 1.0005 * cor + 1.1e-24 : 1.0005 * cor - 1.1e-24;
907 if (res == res + cor)
908 return res;
909
910 (x > 0) ? __dubsin (x, dx, w) : __dubsin (-x, -dx, w);
911 if (w[1] > 0)
912 cor = 1.000000001 * w[1] + 1.1e-24;
913 else
914 cor = 1.000000001 * w[1] - 1.1e-24;
915
916 if (w[0] == w[0] + cor)
917 return (x > 0) ? w[0] : -w[0];
918
919 return (n & 1) ? __mpcos (orig, 0, true) : __mpsin (orig, 0, true);
920 }
921
922 /***************************************************************************/
923 /* Routine compute sin(x+dx) or cos(x+dx) (Double-Length number) where x */
924 /* in first or third quarter of unit circle.Routine receive also */
925 /* (right argument) the original value of x for computing error of result.*/
926 /* And if result not accurate enough routine calls other routines */
927 /***************************************************************************/
928
929 static double
930 SECTION
931 bsloww1 (double x, double dx, double orig, int n)
932 {
933 mynumber u;
934 double w[2], y, cor, res;
935
936 y = fabs (x);
937 u.x = big + y;
938 y = y - (u.x - big);
939 dx = (x > 0) ? dx : -dx;
940 res = do_sin_slow (u, y, dx, 1.1e-24, &cor);
941 if (res == res + cor)
942 return (x > 0) ? res : -res;
943
944 __dubsin (fabs (x), dx, w);
945
946 if (w[1] > 0)
947 cor = 1.000000005 * w[1] + 1.1e-24;
948 else
949 cor = 1.000000005 * w[1] - 1.1e-24;
950
951 if (w[0] == w[0] + cor)
952 return (x > 0) ? w[0] : -w[0];
953
954 return (n & 1) ? __mpcos (orig, 0, true) : __mpsin (orig, 0, true);
955 }
956
957 /***************************************************************************/
958 /* Routine compute sin(x+dx) or cos(x+dx) (Double-Length number) where x */
959 /* in second or fourth quarter of unit circle.Routine receive also the */
960 /* original value and quarter(n= 1or 3)of x for computing error of result. */
961 /* And if result not accurate enough routine calls other routines */
962 /***************************************************************************/
963
964 static double
965 SECTION
966 bsloww2 (double x, double dx, double orig, int n)
967 {
968 mynumber u;
969 double w[2], y, cor, res;
970
971 y = fabs (x);
972 u.x = big + y;
973 y = y - (u.x - big);
974 dx = (x > 0) ? dx : -dx;
975 res = do_cos_slow (u, y, dx, 1.1e-24, &cor);
976 if (res == res + cor)
977 return (n & 2) ? -res : res;
978
979 __docos (fabs (x), dx, w);
980
981 if (w[1] > 0)
982 cor = 1.000000005 * w[1] + 1.1e-24;
983 else
984 cor = 1.000000005 * w[1] - 1.1e-24;
985
986 if (w[0] == w[0] + cor)
987 return (n & 2) ? -w[0] : w[0];
988
989 return (n & 1) ? __mpsin (orig, 0, true) : __mpcos (orig, 0, true);
990 }
991
992 /************************************************************************/
993 /* Routine compute cos(x) for 2^-27 < |x|< 0.25 by Taylor with more */
994 /* precision and if still doesn't accurate enough by mpcos or docos */
995 /************************************************************************/
996
997 static double
998 SECTION
999 cslow2 (double x)
1000 {
1001 mynumber u;
1002 double w[2], y, cor, res;
1003
1004 y = fabs (x);
1005 u.x = big + y;
1006 y = y - (u.x - big);
1007 res = do_cos_slow (u, y, 0, 0, &cor);
1008 if (res == res + cor)
1009 return res;
1010
1011 y = fabs (x);
1012 __docos (y, 0, w);
1013 if (w[0] == w[0] + 1.000000005 * w[1])
1014 return w[0];
1015
1016 return __mpcos (x, 0, false);
1017 }
1018
1019 #ifndef __cos
1020 weak_alias (__cos, cos)
1021 # ifdef NO_LONG_DOUBLE
1022 strong_alias (__cos, __cosl)
1023 weak_alias (__cos, cosl)
1024 # endif
1025 #endif
1026 #ifndef __sin
1027 weak_alias (__sin, sin)
1028 # ifdef NO_LONG_DOUBLE
1029 strong_alias (__sin, __sinl)
1030 weak_alias (__sin, sinl)
1031 # endif
1032 #endif
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