1
2 /*
3 * M_APM - mapmasin.c
4 *
5 * Copyright (C) 1999 - 2007 Michael C. Ring
6 *
7 * Permission to use, copy, and distribute this software and its
8 * documentation for any purpose with or without fee is hereby granted,
9 * provided that the above copyright notice appear in all copies and
10 * that both that copyright notice and this permission notice appear
11 * in supporting documentation.
12 *
13 * Permission to modify the software is granted. Permission to distribute
14 * the modified code is granted. Modifications are to be distributed by
15 * using the file 'license.txt' as a template to modify the file header.
16 * 'license.txt' is available in the official MAPM distribution.
17 *
18 * This software is provided "as is" without express or implied warranty.
19 */
20
21 /*
22 * $Id: mapmasin.c,v 1.28 2007/12/03 01:49:10 mike Exp $
23 *
24 * This file contains the 'ARC' family of functions; ARC-SIN, ARC-COS,
25 * ARC-TAN, and ARC-TAN2.
26 *
27 * $Log: mapmasin.c,v $
28 * Revision 1.28 2007/12/03 01:49:10 mike
29 * Update license
30 *
31 * Revision 1.27 2003/07/24 16:34:02 mike
32 * update arctan_large_input
33 *
34 * Revision 1.26 2003/07/21 20:27:48 mike
35 * Modify error messages to be in a consistent format.
36 *
37 * Revision 1.25 2003/07/21 19:19:26 mike
38 * add new arctan with large input value
39 *
40 * Revision 1.24 2003/05/01 21:58:49 mike
41 * remove math.h
42 *
43 * Revision 1.23 2003/04/09 21:43:00 mike
44 * optimize iterative asin & acos with lessons learned
45 * from the new log function
46 *
47 * Revision 1.22 2003/03/31 21:58:11 mike
48 * call generic error handling function
49 *
50 * Revision 1.21 2002/11/03 21:41:54 mike
51 * Updated function parameters to use the modern style
52 *
53 * Revision 1.20 2001/02/07 19:07:07 mike
54 * eliminate MM_skip_limit_PI_check
55 *
56 * Revision 1.19 2001/02/06 21:50:56 mike
57 * don't display accuracy when iteration count maxes out
58 *
59 * Revision 1.18 2000/12/02 20:10:09 mike
60 * add calls to more efficient functions if
61 * the input args are close to zero
62 *
63 * Revision 1.17 2000/09/05 22:18:02 mike
64 * re-arrange code to eliminate goto from atan2
65 *
66 * Revision 1.16 2000/05/28 23:58:41 mike
67 * minor optimization to arc-tan2
68 *
69 * Revision 1.15 2000/05/19 17:13:29 mike
70 * use local copies of PI variables & recompute
71 * on the fly as needed
72 *
73 * Revision 1.14 2000/03/27 21:43:23 mike
74 * dtermine how many iterations should be required at
75 * run time for arc-sin and arc-cos
76 *
77 * Revision 1.13 1999/09/21 21:00:33 mike
78 * make sure the sign of 'sin' from M_cos_to_sin is non-zero
79 * before assigning it from the original angle.
80 *
81 * Revision 1.12 1999/07/21 03:05:06 mike
82 * added some comments
83 *
84 * Revision 1.11 1999/07/19 02:33:39 mike
85 * reset local precision again
86 *
87 * Revision 1.10 1999/07/19 02:18:05 mike
88 * more fine tuning of local precision
89 *
90 * Revision 1.9 1999/07/19 00:08:34 mike
91 * adjust local precision during iterative loops
92 *
93 * Revision 1.8 1999/07/18 22:35:56 mike
94 * make arc-sin and arc-cos use dynamically changing
95 * precision to speed up iterative routines for large N
96 *
97 * Revision 1.7 1999/07/09 22:52:00 mike
98 * skip limit PI check when not needed
99 *
100 * Revision 1.6 1999/07/09 00:10:39 mike
101 * use better method for arc sin and arc cos
102 *
103 * Revision 1.5 1999/07/08 22:56:20 mike
104 * replace local MAPM constant with a global
105 *
106 * Revision 1.4 1999/06/20 16:55:01 mike
107 * changed local static variables to MAPM stack variables
108 *
109 * Revision 1.3 1999/05/15 02:10:27 mike
110 * add check for number of decimal places
111 *
112 * Revision 1.2 1999/05/10 21:10:21 mike
113 * added some comments
114 *
115 * Revision 1.1 1999/05/10 20:56:31 mike
116 * Initial revision
117 */
118
119 #include "m_apm_lc.h"
120
121 /****************************************************************************/
m_apm_arctan2(M_APM rr,int places,M_APM yy,M_APM xx)122 void m_apm_arctan2(M_APM rr, int places, M_APM yy, M_APM xx)
123 {
124 M_APM tmp5, tmp6, tmp7;
125 int ix, iy;
126
127 iy = yy->m_apm_sign;
128 ix = xx->m_apm_sign;
129
130 if (ix == 0) /* x == 0 */
131 {
132 if (iy == 0) /* y == 0 */
133 {
134 M_apm_log_error_msg(M_APM_RETURN, "\'m_apm_arctan2\', Both Inputs = 0");
135 M_set_to_zero(rr);
136 return;
137 }
138
139 M_check_PI_places(places);
140 m_apm_round(rr, places, MM_lc_HALF_PI);
141 rr->m_apm_sign = iy;
142 return;
143 }
144
145 if (iy == 0)
146 {
147 if (ix == 1)
148 {
149 M_set_to_zero(rr);
150 }
151 else
152 {
153 M_check_PI_places(places);
154 m_apm_round(rr, places, MM_lc_PI);
155 }
156
157 return;
158 }
159
160 /*
161 * the special cases have been handled, now do the real work
162 */
163
164 tmp5 = M_get_stack_var();
165 tmp6 = M_get_stack_var();
166 tmp7 = M_get_stack_var();
167
168 m_apm_divide(tmp6, (places + 6), yy, xx);
169 m_apm_arctan(tmp5, (places + 6), tmp6);
170
171 if (ix == 1) /* 'x' is positive */
172 {
173 m_apm_round(rr, places, tmp5);
174 }
175 else /* 'x' is negative */
176 {
177 M_check_PI_places(places);
178
179 if (iy == 1) /* 'y' is positive */
180 {
181 m_apm_add(tmp7, tmp5, MM_lc_PI);
182 m_apm_round(rr, places, tmp7);
183 }
184 else /* 'y' is negative */
185 {
186 m_apm_subtract(tmp7, tmp5, MM_lc_PI);
187 m_apm_round(rr, places, tmp7);
188 }
189 }
190
191 M_restore_stack(3);
192 }
193 /****************************************************************************/
194 /*
195 Calculate arctan using the identity :
196
197 x
198 arctan (x) == arcsin [ --------------- ]
199 sqrt(1 + x^2)
200
201 */
m_apm_arctan(M_APM rr,int places,M_APM xx)202 void m_apm_arctan(M_APM rr, int places, M_APM xx)
203 {
204 M_APM tmp8, tmp9;
205
206 if (xx->m_apm_sign == 0) /* input == 0 ?? */
207 {
208 M_set_to_zero(rr);
209 return;
210 }
211
212 if (xx->m_apm_exponent <= -4) /* input close to 0 ?? */
213 {
214 M_arctan_near_0(rr, places, xx);
215 return;
216 }
217
218 if (xx->m_apm_exponent >= 4) /* large input */
219 {
220 M_arctan_large_input(rr, places, xx);
221 return;
222 }
223
224 tmp8 = M_get_stack_var();
225 tmp9 = M_get_stack_var();
226
227 m_apm_multiply(tmp9, xx, xx);
228 m_apm_add(tmp8, tmp9, MM_One);
229 m_apm_sqrt(tmp9, (places + 6), tmp8);
230 m_apm_divide(tmp8, (places + 6), xx, tmp9);
231 m_apm_arcsin(rr, places, tmp8);
232 M_restore_stack(2);
233 }
234 /****************************************************************************/
235 /*
236
237 for large input values use :
238
239 arctan(x) = (PI / 2) - arctan(1 / |x|)
240
241 and sign of result = sign of original input
242
243 */
M_arctan_large_input(M_APM rr,int places,M_APM xx)244 void M_arctan_large_input(M_APM rr, int places, M_APM xx)
245 {
246 M_APM tmp1, tmp2;
247
248 tmp1 = M_get_stack_var();
249 tmp2 = M_get_stack_var();
250
251 M_check_PI_places(places);
252
253 m_apm_divide(tmp1, (places + 6), MM_One, xx); /* 1 / xx */
254 tmp1->m_apm_sign = 1; /* make positive */
255 m_apm_arctan(tmp2, (places + 6), tmp1);
256 m_apm_subtract(tmp1, MM_lc_HALF_PI, tmp2);
257 m_apm_round(rr, places, tmp1);
258
259 rr->m_apm_sign = xx->m_apm_sign; /* fix final sign */
260
261 M_restore_stack(2);
262 }
263 /****************************************************************************/
m_apm_arcsin(M_APM r,int places,M_APM x)264 void m_apm_arcsin(M_APM r, int places, M_APM x)
265 {
266 M_APM tmp0, tmp1, tmp2, tmp3, current_x;
267 int ii, maxiter, maxp, tolerance, local_precision;
268
269 current_x = M_get_stack_var();
270 tmp0 = M_get_stack_var();
271 tmp1 = M_get_stack_var();
272 tmp2 = M_get_stack_var();
273 tmp3 = M_get_stack_var();
274
275 m_apm_absolute_value(tmp0, x);
276
277 ii = m_apm_compare(tmp0, MM_One);
278
279 if (ii == 1) /* |x| > 1 */
280 {
281 M_apm_log_error_msg(M_APM_RETURN, "\'m_apm_arcsin\', |Argument| > 1");
282 M_set_to_zero(r);
283 M_restore_stack(5);
284 return;
285 }
286
287 if (ii == 0) /* |x| == 1, arcsin = +/- PI / 2 */
288 {
289 M_check_PI_places(places);
290 m_apm_round(r, places, MM_lc_HALF_PI);
291 r->m_apm_sign = x->m_apm_sign;
292
293 M_restore_stack(5);
294 return;
295 }
296
297 if (m_apm_compare(tmp0, MM_0_85) == 1) /* check if > 0.85 */
298 {
299 M_cos_to_sin(tmp2, (places + 4), x);
300 m_apm_arccos(r, places, tmp2);
301 r->m_apm_sign = x->m_apm_sign;
302
303 M_restore_stack(5);
304 return;
305 }
306
307 if (x->m_apm_sign == 0) /* input == 0 ?? */
308 {
309 M_set_to_zero(r);
310 M_restore_stack(5);
311 return;
312 }
313
314 if (x->m_apm_exponent <= -4) /* input close to 0 ?? */
315 {
316 M_arcsin_near_0(r, places, x);
317 M_restore_stack(5);
318 return;
319 }
320
321 tolerance = -(places + 4);
322 maxp = places + 8 - x->m_apm_exponent;
323 local_precision = 20 - x->m_apm_exponent;
324
325 /*
326 * compute the maximum number of iterations
327 * that should be needed to calculate to
328 * the desired accuracy. [ constant below ~= 1 / log(2) ]
329 */
330
331 maxiter = (int)(log((double)(places + 2)) * 1.442695) + 3;
332
333 if (maxiter < 5)
334 maxiter = 5;
335
336 M_get_asin_guess(current_x, x);
337
338 /* Use the following iteration to solve for arc-sin :
339
340 sin(X) - N
341 X = X - ------------
342 n+1 cos(X)
343 */
344
345 ii = 0;
346
347 while (TRUE)
348 {
349 M_4x_cos(tmp1, local_precision, current_x);
350
351 M_cos_to_sin(tmp2, local_precision, tmp1);
352 if (tmp2->m_apm_sign != 0)
353 tmp2->m_apm_sign = current_x->m_apm_sign;
354
355 m_apm_subtract(tmp3, tmp2, x);
356 m_apm_divide(tmp0, local_precision, tmp3, tmp1);
357
358 m_apm_subtract(tmp2, current_x, tmp0);
359 m_apm_copy(current_x, tmp2);
360
361 if (ii != 0)
362 {
363 if (((2 * tmp0->m_apm_exponent) < tolerance) || (tmp0->m_apm_sign == 0))
364 break;
365 }
366
367 if (++ii == maxiter)
368 {
369 M_apm_log_error_msg(M_APM_RETURN,
370 "\'m_apm_arcsin\', max iteration count reached");
371 break;
372 }
373
374 local_precision *= 2;
375
376 if (local_precision > maxp)
377 local_precision = maxp;
378 }
379
380 m_apm_round(r, places, current_x);
381 M_restore_stack(5);
382 }
383 /****************************************************************************/
m_apm_arccos(M_APM r,int places,M_APM x)384 void m_apm_arccos(M_APM r, int places, M_APM x)
385 {
386 M_APM tmp0, tmp1, tmp2, tmp3, current_x;
387 int ii, maxiter, maxp, tolerance, local_precision;
388
389 current_x = M_get_stack_var();
390 tmp0 = M_get_stack_var();
391 tmp1 = M_get_stack_var();
392 tmp2 = M_get_stack_var();
393 tmp3 = M_get_stack_var();
394
395 m_apm_absolute_value(tmp0, x);
396
397 ii = m_apm_compare(tmp0, MM_One);
398
399 if (ii == 1) /* |x| > 1 */
400 {
401 M_apm_log_error_msg(M_APM_RETURN, "\'m_apm_arccos\', |Argument| > 1");
402 M_set_to_zero(r);
403 M_restore_stack(5);
404 return;
405 }
406
407 if (ii == 0) /* |x| == 1, arccos = 0, PI */
408 {
409 if (x->m_apm_sign == 1)
410 {
411 M_set_to_zero(r);
412 }
413 else
414 {
415 M_check_PI_places(places);
416 m_apm_round(r, places, MM_lc_PI);
417 }
418
419 M_restore_stack(5);
420 return;
421 }
422
423 if (m_apm_compare(tmp0, MM_0_85) == 1) /* check if > 0.85 */
424 {
425 M_cos_to_sin(tmp2, (places + 4), x);
426
427 if (x->m_apm_sign == 1)
428 {
429 m_apm_arcsin(r, places, tmp2);
430 }
431 else
432 {
433 M_check_PI_places(places);
434 m_apm_arcsin(tmp3, (places + 4), tmp2);
435 m_apm_subtract(tmp1, MM_lc_PI, tmp3);
436 m_apm_round(r, places, tmp1);
437 }
438
439 M_restore_stack(5);
440 return;
441 }
442
443 if (x->m_apm_sign == 0) /* input == 0 ?? */
444 {
445 M_check_PI_places(places);
446 m_apm_round(r, places, MM_lc_HALF_PI);
447 M_restore_stack(5);
448 return;
449 }
450
451 if (x->m_apm_exponent <= -4) /* input close to 0 ?? */
452 {
453 M_arccos_near_0(r, places, x);
454 M_restore_stack(5);
455 return;
456 }
457
458 tolerance = -(places + 4);
459 maxp = places + 8;
460 local_precision = 18;
461
462 /*
463 * compute the maximum number of iterations
464 * that should be needed to calculate to
465 * the desired accuracy. [ constant below ~= 1 / log(2) ]
466 */
467
468 maxiter = (int)(log((double)(places + 2)) * 1.442695) + 3;
469
470 if (maxiter < 5)
471 maxiter = 5;
472
473 M_get_acos_guess(current_x, x);
474
475 /* Use the following iteration to solve for arc-cos :
476
477 cos(X) - N
478 X = X + ------------
479 n+1 sin(X)
480 */
481
482 ii = 0;
483
484 while (TRUE)
485 {
486 M_4x_cos(tmp1, local_precision, current_x);
487
488 M_cos_to_sin(tmp2, local_precision, tmp1);
489 if (tmp2->m_apm_sign != 0)
490 tmp2->m_apm_sign = current_x->m_apm_sign;
491
492 m_apm_subtract(tmp3, tmp1, x);
493 m_apm_divide(tmp0, local_precision, tmp3, tmp2);
494
495 m_apm_add(tmp2, current_x, tmp0);
496 m_apm_copy(current_x, tmp2);
497
498 if (ii != 0)
499 {
500 if (((2 * tmp0->m_apm_exponent) < tolerance) || (tmp0->m_apm_sign == 0))
501 break;
502 }
503
504 if (++ii == maxiter)
505 {
506 M_apm_log_error_msg(M_APM_RETURN,
507 "\'m_apm_arccos\', max iteration count reached");
508 break;
509 }
510
511 local_precision *= 2;
512
513 if (local_precision > maxp)
514 local_precision = maxp;
515 }
516
517 m_apm_round(r, places, current_x);
518 M_restore_stack(5);
519 }
520 /****************************************************************************/
521