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 <complex.h>
43 #include <float.h>
44
45 #include "invtrig.h"
46 #include "math.h"
47 #include "math_private.h"
48
49 #undef isinf
50 #define isinf(x) (fabsl(x) == INFINITY)
51 #undef isnan
52 #define isnan(x) ((x) != (x))
53 #define raise_inexact() do { volatile float junk __unused = 1 + tiny; } while(0)
54 #undef signbit
55 #define signbit(x) (__builtin_signbitl(x))
56
57 #if LDBL_MAX_EXP != 0x4000
58 #error "Unsupported long double format"
59 #endif
60
61 static const long double
62 A_crossover = 10,
63 B_crossover = 0.6417,
64 FOUR_SQRT_MIN = 0x1p-8189L,
65 HALF_MAX = 0x1p16383L,
66 QUARTER_SQRT_MAX = 0x1p8189L,
67 RECIP_EPSILON = 1 / LDBL_EPSILON,
68 SQRT_MIN = 0x1p-8191L;
69
70 #if LDBL_MANT_DIG == 64
71 static const union IEEEl2bits
72 um_e = LD80C(0xadf85458a2bb4a9b, 1, 2.71828182845904523536e+0L),
73 um_ln2 = LD80C(0xb17217f7d1cf79ac, -1, 6.93147180559945309417e-1L);
74 #define m_e um_e.e
75 #define m_ln2 um_ln2.e
76 static const long double
77 /* The next 2 literals for non-i386. Misrounding them on i386 is harmless. */
78 SQRT_3_EPSILON = 5.70316273435758915310e-10, /* 0x9cc470a0490973e8.0p-94 */
79 SQRT_6_EPSILON = 8.06549008734932771664e-10; /* 0xddb3d742c265539e.0p-94 */
80 #elif LDBL_MANT_DIG == 113
81 static const long double
82 m_e = 2.71828182845904523536028747135266250e0L, /* 0x15bf0a8b1457695355fb8ac404e7a.0p-111 */
83 m_ln2 = 6.93147180559945309417232121458176568e-1L, /* 0x162e42fefa39ef35793c7673007e6.0p-113 */
84 SQRT_3_EPSILON = 2.40370335797945490975336727199878124e-17, /* 0x1bb67ae8584caa73b25742d7078b8.0p-168 */
85 SQRT_6_EPSILON = 3.39934988877629587239082586223300391e-17; /* 0x13988e1409212e7d0321914321a55.0p-167 */
86 #else
87 #error "Unsupported long double format"
88 #endif
89
90 static const volatile float
91 tiny = 0x1p-100;
92
93 static long double complex clog_for_large_values(long double complex z);
94
95 static inline long double
f(long double a,long double b,long double hypot_a_b)96 f(long double a, long double b, long double hypot_a_b)
97 {
98 if (b < 0)
99 return ((hypot_a_b - b) / 2);
100 if (b == 0)
101 return (a / 2);
102 return (a * a / (hypot_a_b + b) / 2);
103 }
104
105 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)106 do_hard_work(long double x, long double y, long double *rx, int *B_is_usable,
107 long double *B, long double *sqrt_A2my2, long double *new_y)
108 {
109 long double R, S, A;
110 long double Am1, Amy;
111
112 R = hypotl(x, y + 1);
113 S = hypotl(x, y - 1);
114
115 A = (R + S) / 2;
116 if (A < 1)
117 A = 1;
118
119 if (A < A_crossover) {
120 if (y == 1 && x < LDBL_EPSILON * LDBL_EPSILON / 128) {
121 *rx = sqrtl(x);
122 } else if (x >= LDBL_EPSILON * fabsl(y - 1)) {
123 Am1 = f(x, 1 + y, R) + f(x, 1 - y, S);
124 *rx = log1pl(Am1 + sqrtl(Am1 * (A + 1)));
125 } else if (y < 1) {
126 *rx = x / sqrtl((1 - y) * (1 + y));
127 } else {
128 *rx = log1pl((y - 1) + sqrtl((y - 1) * (y + 1)));
129 }
130 } else {
131 *rx = logl(A + sqrtl(A * A - 1));
132 }
133
134 *new_y = y;
135
136 if (y < FOUR_SQRT_MIN) {
137 *B_is_usable = 0;
138 *sqrt_A2my2 = A * (2 / LDBL_EPSILON);
139 *new_y = y * (2 / LDBL_EPSILON);
140 return;
141 }
142
143 *B = y / A;
144 *B_is_usable = 1;
145
146 if (*B > B_crossover) {
147 *B_is_usable = 0;
148 if (y == 1 && x < LDBL_EPSILON / 128) {
149 *sqrt_A2my2 = sqrtl(x) * sqrtl((A + y) / 2);
150 } else if (x >= LDBL_EPSILON * fabsl(y - 1)) {
151 Amy = f(x, y + 1, R) + f(x, y - 1, S);
152 *sqrt_A2my2 = sqrtl(Amy * (A + y));
153 } else if (y > 1) {
154 *sqrt_A2my2 = x * (4 / LDBL_EPSILON / LDBL_EPSILON) * y /
155 sqrtl((y + 1) * (y - 1));
156 *new_y = y * (4 / LDBL_EPSILON / LDBL_EPSILON);
157 } else {
158 *sqrt_A2my2 = sqrtl((1 - y) * (1 + y));
159 }
160 }
161 }
162
163 long double complex
casinhl(long double complex z)164 casinhl(long double complex z)
165 {
166 long double x, y, ax, ay, rx, ry, B, sqrt_A2my2, new_y;
167 int B_is_usable;
168 long double complex w;
169
170 x = creall(z);
171 y = cimagl(z);
172 ax = fabsl(x);
173 ay = fabsl(y);
174
175 if (isnan(x) || isnan(y)) {
176 if (isinf(x))
177 return (CMPLXL(x, y + y));
178 if (isinf(y))
179 return (CMPLXL(y, x + x));
180 if (y == 0)
181 return (CMPLXL(x + x, y));
182 return (CMPLXL(nan_mix(x, y), nan_mix(x, y)));
183 }
184
185 if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) {
186 if (signbit(x) == 0)
187 w = clog_for_large_values(z) + m_ln2;
188 else
189 w = clog_for_large_values(-z) + m_ln2;
190 return (CMPLXL(copysignl(creall(w), x),
191 copysignl(cimagl(w), y)));
192 }
193
194 if (x == 0 && y == 0)
195 return (z);
196
197 raise_inexact();
198
199 if (ax < SQRT_6_EPSILON / 4 && ay < SQRT_6_EPSILON / 4)
200 return (z);
201
202 do_hard_work(ax, ay, &rx, &B_is_usable, &B, &sqrt_A2my2, &new_y);
203 if (B_is_usable)
204 ry = asinl(B);
205 else
206 ry = atan2l(new_y, sqrt_A2my2);
207 return (CMPLXL(copysignl(rx, x), copysignl(ry, y)));
208 }
209
210 long double complex
casinl(long double complex z)211 casinl(long double complex z)
212 {
213 long double complex w;
214
215 w = casinhl(CMPLXL(cimagl(z), creall(z)));
216 return (CMPLXL(cimagl(w), creall(w)));
217 }
218
219 long double complex
cacosl(long double complex z)220 cacosl(long double complex z)
221 {
222 long double x, y, ax, ay, rx, ry, B, sqrt_A2mx2, new_x;
223 int sx, sy;
224 int B_is_usable;
225 long double complex w;
226
227 x = creall(z);
228 y = cimagl(z);
229 sx = signbit(x);
230 sy = signbit(y);
231 ax = fabsl(x);
232 ay = fabsl(y);
233
234 if (isnan(x) || isnan(y)) {
235 if (isinf(x))
236 return (CMPLXL(y + y, -INFINITY));
237 if (isinf(y))
238 return (CMPLXL(x + x, -y));
239 if (x == 0)
240 return (CMPLXL(pio2_hi + pio2_lo, y + y));
241 return (CMPLXL(nan_mix(x, y), nan_mix(x, y)));
242 }
243
244 if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) {
245 w = clog_for_large_values(z);
246 rx = fabsl(cimagl(w));
247 ry = creall(w) + m_ln2;
248 if (sy == 0)
249 ry = -ry;
250 return (CMPLXL(rx, ry));
251 }
252
253 if (x == 1 && y == 0)
254 return (CMPLXL(0, -y));
255
256 raise_inexact();
257
258 if (ax < SQRT_6_EPSILON / 4 && ay < SQRT_6_EPSILON / 4)
259 return (CMPLXL(pio2_hi - (x - pio2_lo), -y));
260
261 do_hard_work(ay, ax, &ry, &B_is_usable, &B, &sqrt_A2mx2, &new_x);
262 if (B_is_usable) {
263 if (sx == 0)
264 rx = acosl(B);
265 else
266 rx = acosl(-B);
267 } else {
268 if (sx == 0)
269 rx = atan2l(sqrt_A2mx2, new_x);
270 else
271 rx = atan2l(sqrt_A2mx2, -new_x);
272 }
273 if (sy == 0)
274 ry = -ry;
275 return (CMPLXL(rx, ry));
276 }
277
278 long double complex
cacoshl(long double complex z)279 cacoshl(long double complex z)
280 {
281 long double complex w;
282 long double rx, ry;
283
284 w = cacosl(z);
285 rx = creall(w);
286 ry = cimagl(w);
287 if (isnan(rx) && isnan(ry))
288 return (CMPLXL(ry, rx));
289 if (isnan(rx))
290 return (CMPLXL(fabsl(ry), rx));
291 if (isnan(ry))
292 return (CMPLXL(ry, ry));
293 return (CMPLXL(fabsl(ry), copysignl(rx, cimagl(z))));
294 }
295
296 static long double complex
clog_for_large_values(long double complex z)297 clog_for_large_values(long double complex z)
298 {
299 long double x, y;
300 long double ax, ay, t;
301
302 x = creall(z);
303 y = cimagl(z);
304 ax = fabsl(x);
305 ay = fabsl(y);
306 if (ax < ay) {
307 t = ax;
308 ax = ay;
309 ay = t;
310 }
311
312 if (ax > HALF_MAX)
313 return (CMPLXL(logl(hypotl(x / m_e, y / m_e)) + 1,
314 atan2l(y, x)));
315
316 if (ax > QUARTER_SQRT_MAX || ay < SQRT_MIN)
317 return (CMPLXL(logl(hypotl(x, y)), atan2l(y, x)));
318
319 return (CMPLXL(logl(ax * ax + ay * ay) / 2, atan2l(y, x)));
320 }
321
322 static inline long double
sum_squares(long double x,long double y)323 sum_squares(long double x, long double y)
324 {
325
326 if (y < SQRT_MIN)
327 return (x * x);
328
329 return (x * x + y * y);
330 }
331
332 static inline long double
real_part_reciprocal(long double x,long double y)333 real_part_reciprocal(long double x, long double y)
334 {
335 long double scale;
336 uint16_t hx, hy;
337 int16_t ix, iy;
338
339 GET_LDBL_EXPSIGN(hx, x);
340 ix = hx & 0x7fff;
341 GET_LDBL_EXPSIGN(hy, y);
342 iy = hy & 0x7fff;
343 #define BIAS (LDBL_MAX_EXP - 1)
344 #define CUTOFF (LDBL_MANT_DIG / 2 + 1)
345 if (ix - iy >= CUTOFF || isinf(x))
346 return (1 / x);
347 if (iy - ix >= CUTOFF)
348 return (x / y / y);
349 if (ix <= BIAS + LDBL_MAX_EXP / 2 - CUTOFF)
350 return (x / (x * x + y * y));
351 scale = 1;
352 SET_LDBL_EXPSIGN(scale, 0x7fff - ix);
353 x *= scale;
354 y *= scale;
355 return (x / (x * x + y * y) * scale);
356 }
357
358 long double complex
catanhl(long double complex z)359 catanhl(long double complex z)
360 {
361 long double x, y, ax, ay, rx, ry;
362
363 x = creall(z);
364 y = cimagl(z);
365 ax = fabsl(x);
366 ay = fabsl(y);
367
368 if (y == 0 && ax <= 1)
369 return (CMPLXL(atanhl(x), y));
370
371 if (x == 0)
372 return (CMPLXL(x, atanl(y)));
373
374 if (isnan(x) || isnan(y)) {
375 if (isinf(x))
376 return (CMPLXL(copysignl(0, x), y + y));
377 if (isinf(y))
378 return (CMPLXL(copysignl(0, x),
379 copysignl(pio2_hi + pio2_lo, y)));
380 return (CMPLXL(nan_mix(x, y), nan_mix(x, y)));
381 }
382
383 if (ax > RECIP_EPSILON || ay > RECIP_EPSILON)
384 return (CMPLXL(real_part_reciprocal(x, y),
385 copysignl(pio2_hi + pio2_lo, y)));
386
387 if (ax < SQRT_3_EPSILON / 2 && ay < SQRT_3_EPSILON / 2) {
388 raise_inexact();
389 return (z);
390 }
391
392 if (ax == 1 && ay < LDBL_EPSILON)
393 rx = (m_ln2 - logl(ay)) / 2;
394 else
395 rx = log1pl(4 * ax / sum_squares(ax - 1, ay)) / 4;
396
397 if (ax == 1)
398 ry = atan2l(2, -ay) / 2;
399 else if (ay < LDBL_EPSILON)
400 ry = atan2l(2 * ay, (1 - ax) * (1 + ax)) / 2;
401 else
402 ry = atan2l(2 * ay, (1 - ax) * (1 + ax) - ay * ay) / 2;
403
404 return (CMPLXL(copysignl(rx, x), copysignl(ry, y)));
405 }
406
407 long double complex
catanl(long double complex z)408 catanl(long double complex z)
409 {
410 long double complex w;
411
412 w = catanhl(CMPLXL(cimagl(z), creall(z)));
413 return (CMPLXL(cimagl(w), creall(w)));
414 }
415