1 /*-
2 * Copyright (c) 2012 Stephen Montgomery-Smith <stephen@FreeBSD.ORG>
3 * Copyright (c) 2017 Mahdi Mokhtari <mmokhi@FreeBSD.org>
4 * All rights reserved.
5 *
6 * Redistribution and use in source and binary forms, with or without
7 * modification, are permitted provided that the following conditions
8 * are met:
9 * 1. Redistributions of source code must retain the above copyright
10 * notice, this list of conditions and the following disclaimer.
11 * 2. Redistributions in binary form must reproduce the above copyright
12 * notice, this list of conditions and the following disclaimer in the
13 * documentation and/or other materials provided with the distribution.
14 *
15 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND
16 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
17 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
18 * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
19 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
20 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
21 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
22 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
23 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
24 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
25 * SUCH DAMAGE.
26 */
27
28 /*
29 * The algorithm is very close to that in "Implementing the complex arcsine
30 * and arccosine functions using exception handling" by T. E. Hull, Thomas F.
31 * Fairgrieve, and Ping Tak Peter Tang, published in ACM Transactions on
32 * Mathematical Software, Volume 23 Issue 3, 1997, Pages 299-335,
33 * http://dl.acm.org/citation.cfm?id=275324.
34 *
35 * See catrig.c for complete comments.
36 *
37 * XXX comments were removed automatically, and even short ones on the right
38 * of statements were removed (all of them), contrary to normal style. Only
39 * a few comments on the right of declarations remain.
40 */
41
42 #include <sys/cdefs.h>
43 __FBSDID("$FreeBSD: head/lib/msun/src/catrigl.c 336362 2018-07-17 07:42:14Z bde $");
44
45 #include <complex.h>
46 #include <float.h>
47
48 #include "invtrig.h"
49 #include "math.h"
50 #include "math_private.h"
51
52 #undef isinf
53 #define isinf(x) (fabsl(x) == INFINITY)
54 #undef isnan
55 #define isnan(x) ((x) != (x))
56 #define raise_inexact() do { volatile float junk __unused = 1 + tiny; } while(0)
57 #undef signbit
58 #define signbit(x) (__builtin_signbitl(x))
59
60 #if LDBL_MAX_EXP != 0x4000
61 #error "Unsupported long double format"
62 #endif
63
64 static const long double
65 A_crossover = 10,
66 B_crossover = 0.6417,
67 FOUR_SQRT_MIN = 0x1p-8189L,
68 HALF_MAX = 0x1p16383L,
69 QUARTER_SQRT_MAX = 0x1p8189L,
70 RECIP_EPSILON = 1 / LDBL_EPSILON,
71 SQRT_MIN = 0x1p-8191L;
72
73 #if LDBL_MANT_DIG == 64
74 static const union IEEEl2bits
75 um_e = LD80C(0xadf85458a2bb4a9b, 1, 2.71828182845904523536e+0L),
76 um_ln2 = LD80C(0xb17217f7d1cf79ac, -1, 6.93147180559945309417e-1L);
77 #define m_e um_e.e
78 #define m_ln2 um_ln2.e
79 static const long double
80 /* The next 2 literals for non-i386. Misrounding them on i386 is harmless. */
81 SQRT_3_EPSILON = 5.70316273435758915310e-10, /* 0x9cc470a0490973e8.0p-94 */
82 SQRT_6_EPSILON = 8.06549008734932771664e-10; /* 0xddb3d742c265539e.0p-94 */
83 #elif LDBL_MANT_DIG == 113
84 static const long double
85 m_e = 2.71828182845904523536028747135266250e0L, /* 0x15bf0a8b1457695355fb8ac404e7a.0p-111 */
86 m_ln2 = 6.93147180559945309417232121458176568e-1L, /* 0x162e42fefa39ef35793c7673007e6.0p-113 */
87 SQRT_3_EPSILON = 2.40370335797945490975336727199878124e-17, /* 0x1bb67ae8584caa73b25742d7078b8.0p-168 */
88 SQRT_6_EPSILON = 3.39934988877629587239082586223300391e-17; /* 0x13988e1409212e7d0321914321a55.0p-167 */
89 #else
90 #error "Unsupported long double format"
91 #endif
92
93 static const volatile float
94 tiny = 0x1p-100;
95
96 static long double complex clog_for_large_values(long double complex z);
97
98 static inline long double
f(long double a,long double b,long double hypot_a_b)99 f(long double a, long double b, long double hypot_a_b)
100 {
101 if (b < 0)
102 return ((hypot_a_b - b) / 2);
103 if (b == 0)
104 return (a / 2);
105 return (a * a / (hypot_a_b + b) / 2);
106 }
107
108 static inline void
do_hard_work(long double x,long double y,long double * rx,int * B_is_usable,long double * B,long double * sqrt_A2my2,long double * new_y)109 do_hard_work(long double x, long double y, long double *rx, int *B_is_usable,
110 long double *B, long double *sqrt_A2my2, long double *new_y)
111 {
112 long double R, S, A;
113 long double Am1, Amy;
114
115 R = hypotl(x, y + 1);
116 S = hypotl(x, y - 1);
117
118 A = (R + S) / 2;
119 if (A < 1)
120 A = 1;
121
122 if (A < A_crossover) {
123 if (y == 1 && x < LDBL_EPSILON * LDBL_EPSILON / 128) {
124 *rx = sqrtl(x);
125 } else if (x >= LDBL_EPSILON * fabsl(y - 1)) {
126 Am1 = f(x, 1 + y, R) + f(x, 1 - y, S);
127 *rx = log1pl(Am1 + sqrtl(Am1 * (A + 1)));
128 } else if (y < 1) {
129 *rx = x / sqrtl((1 - y) * (1 + y));
130 } else {
131 *rx = log1pl((y - 1) + sqrtl((y - 1) * (y + 1)));
132 }
133 } else {
134 *rx = logl(A + sqrtl(A * A - 1));
135 }
136
137 *new_y = y;
138
139 if (y < FOUR_SQRT_MIN) {
140 *B_is_usable = 0;
141 *sqrt_A2my2 = A * (2 / LDBL_EPSILON);
142 *new_y = y * (2 / LDBL_EPSILON);
143 return;
144 }
145
146 *B = y / A;
147 *B_is_usable = 1;
148
149 if (*B > B_crossover) {
150 *B_is_usable = 0;
151 if (y == 1 && x < LDBL_EPSILON / 128) {
152 *sqrt_A2my2 = sqrtl(x) * sqrtl((A + y) / 2);
153 } else if (x >= LDBL_EPSILON * fabsl(y - 1)) {
154 Amy = f(x, y + 1, R) + f(x, y - 1, S);
155 *sqrt_A2my2 = sqrtl(Amy * (A + y));
156 } else if (y > 1) {
157 *sqrt_A2my2 = x * (4 / LDBL_EPSILON / LDBL_EPSILON) * y /
158 sqrtl((y + 1) * (y - 1));
159 *new_y = y * (4 / LDBL_EPSILON / LDBL_EPSILON);
160 } else {
161 *sqrt_A2my2 = sqrtl((1 - y) * (1 + y));
162 }
163 }
164 }
165
166 long double complex
casinhl(long double complex z)167 casinhl(long double complex z)
168 {
169 long double x, y, ax, ay, rx, ry, B, sqrt_A2my2, new_y;
170 int B_is_usable;
171 long double complex w;
172
173 x = creall(z);
174 y = cimagl(z);
175 ax = fabsl(x);
176 ay = fabsl(y);
177
178 if (isnan(x) || isnan(y)) {
179 if (isinf(x))
180 return (CMPLXL(x, y + y));
181 if (isinf(y))
182 return (CMPLXL(y, x + x));
183 if (y == 0)
184 return (CMPLXL(x + x, y));
185 return (CMPLXL(nan_mix(x, y), nan_mix(x, y)));
186 }
187
188 if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) {
189 if (signbit(x) == 0)
190 w = clog_for_large_values(z) + m_ln2;
191 else
192 w = clog_for_large_values(-z) + m_ln2;
193 return (CMPLXL(copysignl(creall(w), x),
194 copysignl(cimagl(w), y)));
195 }
196
197 if (x == 0 && y == 0)
198 return (z);
199
200 raise_inexact();
201
202 if (ax < SQRT_6_EPSILON / 4 && ay < SQRT_6_EPSILON / 4)
203 return (z);
204
205 do_hard_work(ax, ay, &rx, &B_is_usable, &B, &sqrt_A2my2, &new_y);
206 if (B_is_usable)
207 ry = asinl(B);
208 else
209 ry = atan2l(new_y, sqrt_A2my2);
210 return (CMPLXL(copysignl(rx, x), copysignl(ry, y)));
211 }
212
213 long double complex
casinl(long double complex z)214 casinl(long double complex z)
215 {
216 long double complex w;
217
218 w = casinhl(CMPLXL(cimagl(z), creall(z)));
219 return (CMPLXL(cimagl(w), creall(w)));
220 }
221
222 long double complex
cacosl(long double complex z)223 cacosl(long double complex z)
224 {
225 long double x, y, ax, ay, rx, ry, B, sqrt_A2mx2, new_x;
226 int sx, sy;
227 int B_is_usable;
228 long double complex w;
229
230 x = creall(z);
231 y = cimagl(z);
232 sx = signbit(x);
233 sy = signbit(y);
234 ax = fabsl(x);
235 ay = fabsl(y);
236
237 if (isnan(x) || isnan(y)) {
238 if (isinf(x))
239 return (CMPLXL(y + y, -INFINITY));
240 if (isinf(y))
241 return (CMPLXL(x + x, -y));
242 if (x == 0)
243 return (CMPLXL(pio2_hi + pio2_lo, y + y));
244 return (CMPLXL(nan_mix(x, y), nan_mix(x, y)));
245 }
246
247 if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) {
248 w = clog_for_large_values(z);
249 rx = fabsl(cimagl(w));
250 ry = creall(w) + m_ln2;
251 if (sy == 0)
252 ry = -ry;
253 return (CMPLXL(rx, ry));
254 }
255
256 if (x == 1 && y == 0)
257 return (CMPLXL(0, -y));
258
259 raise_inexact();
260
261 if (ax < SQRT_6_EPSILON / 4 && ay < SQRT_6_EPSILON / 4)
262 return (CMPLXL(pio2_hi - (x - pio2_lo), -y));
263
264 do_hard_work(ay, ax, &ry, &B_is_usable, &B, &sqrt_A2mx2, &new_x);
265 if (B_is_usable) {
266 if (sx == 0)
267 rx = acosl(B);
268 else
269 rx = acosl(-B);
270 } else {
271 if (sx == 0)
272 rx = atan2l(sqrt_A2mx2, new_x);
273 else
274 rx = atan2l(sqrt_A2mx2, -new_x);
275 }
276 if (sy == 0)
277 ry = -ry;
278 return (CMPLXL(rx, ry));
279 }
280
281 long double complex
cacoshl(long double complex z)282 cacoshl(long double complex z)
283 {
284 long double complex w;
285 long double rx, ry;
286
287 w = cacosl(z);
288 rx = creall(w);
289 ry = cimagl(w);
290 if (isnan(rx) && isnan(ry))
291 return (CMPLXL(ry, rx));
292 if (isnan(rx))
293 return (CMPLXL(fabsl(ry), rx));
294 if (isnan(ry))
295 return (CMPLXL(ry, ry));
296 return (CMPLXL(fabsl(ry), copysignl(rx, cimagl(z))));
297 }
298
299 static long double complex
clog_for_large_values(long double complex z)300 clog_for_large_values(long double complex z)
301 {
302 long double x, y;
303 long double ax, ay, t;
304
305 x = creall(z);
306 y = cimagl(z);
307 ax = fabsl(x);
308 ay = fabsl(y);
309 if (ax < ay) {
310 t = ax;
311 ax = ay;
312 ay = t;
313 }
314
315 if (ax > HALF_MAX)
316 return (CMPLXL(logl(hypotl(x / m_e, y / m_e)) + 1,
317 atan2l(y, x)));
318
319 if (ax > QUARTER_SQRT_MAX || ay < SQRT_MIN)
320 return (CMPLXL(logl(hypotl(x, y)), atan2l(y, x)));
321
322 return (CMPLXL(logl(ax * ax + ay * ay) / 2, atan2l(y, x)));
323 }
324
325 static inline long double
sum_squares(long double x,long double y)326 sum_squares(long double x, long double y)
327 {
328
329 if (y < SQRT_MIN)
330 return (x * x);
331
332 return (x * x + y * y);
333 }
334
335 static inline long double
real_part_reciprocal(long double x,long double y)336 real_part_reciprocal(long double x, long double y)
337 {
338 long double scale;
339 uint16_t hx, hy;
340 int16_t ix, iy;
341
342 GET_LDBL_EXPSIGN(hx, x);
343 ix = hx & 0x7fff;
344 GET_LDBL_EXPSIGN(hy, y);
345 iy = hy & 0x7fff;
346 #define BIAS (LDBL_MAX_EXP - 1)
347 #define CUTOFF (LDBL_MANT_DIG / 2 + 1)
348 if (ix - iy >= CUTOFF || isinf(x))
349 return (1 / x);
350 if (iy - ix >= CUTOFF)
351 return (x / y / y);
352 if (ix <= BIAS + LDBL_MAX_EXP / 2 - CUTOFF)
353 return (x / (x * x + y * y));
354 scale = 1;
355 SET_LDBL_EXPSIGN(scale, 0x7fff - ix);
356 x *= scale;
357 y *= scale;
358 return (x / (x * x + y * y) * scale);
359 }
360
361 long double complex
catanhl(long double complex z)362 catanhl(long double complex z)
363 {
364 long double x, y, ax, ay, rx, ry;
365
366 x = creall(z);
367 y = cimagl(z);
368 ax = fabsl(x);
369 ay = fabsl(y);
370
371 if (y == 0 && ax <= 1)
372 return (CMPLXL(atanhl(x), y));
373
374 if (x == 0)
375 return (CMPLXL(x, atanl(y)));
376
377 if (isnan(x) || isnan(y)) {
378 if (isinf(x))
379 return (CMPLXL(copysignl(0, x), y + y));
380 if (isinf(y))
381 return (CMPLXL(copysignl(0, x),
382 copysignl(pio2_hi + pio2_lo, y)));
383 return (CMPLXL(nan_mix(x, y), nan_mix(x, y)));
384 }
385
386 if (ax > RECIP_EPSILON || ay > RECIP_EPSILON)
387 return (CMPLXL(real_part_reciprocal(x, y),
388 copysignl(pio2_hi + pio2_lo, y)));
389
390 if (ax < SQRT_3_EPSILON / 2 && ay < SQRT_3_EPSILON / 2) {
391 raise_inexact();
392 return (z);
393 }
394
395 if (ax == 1 && ay < LDBL_EPSILON)
396 rx = (m_ln2 - logl(ay)) / 2;
397 else
398 rx = log1pl(4 * ax / sum_squares(ax - 1, ay)) / 4;
399
400 if (ax == 1)
401 ry = atan2l(2, -ay) / 2;
402 else if (ay < LDBL_EPSILON)
403 ry = atan2l(2 * ay, (1 - ax) * (1 + ax)) / 2;
404 else
405 ry = atan2l(2 * ay, (1 - ax) * (1 + ax) - ay * ay) / 2;
406
407 return (CMPLXL(copysignl(rx, x), copysignl(ry, y)));
408 }
409
410 long double complex
catanl(long double complex z)411 catanl(long double complex z)
412 {
413 long double complex w;
414
415 w = catanhl(CMPLXL(cimagl(z), creall(z)));
416 return (CMPLXL(cimagl(w), creall(w)));
417 }
418