1 /* $NetBSD: catrig.c,v 1.3 2022/04/19 20:32:16 rillig Exp $ */
2 /*-
3 * Copyright (c) 2012 Stephen Montgomery-Smith <stephen@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 #include <sys/cdefs.h>
29 #if 0
30 __FBSDID("$FreeBSD: head/lib/msun/src/catrig.c 275819 2014-12-16 09:21:56Z ed $");
31 #endif
32 __RCSID("$NetBSD: catrig.c,v 1.3 2022/04/19 20:32:16 rillig Exp $");
33
34 #include "namespace.h"
35 #ifdef __weak_alias
36 __weak_alias(casin, _casin)
37 #endif
38 #ifdef __weak_alias
39 __weak_alias(catan, _catan)
40 #endif
41
42 #include <complex.h>
43 #include <float.h>
44
45 #include "math.h"
46 #include "math_private.h"
47
48
49
50 #undef isinf
51 #define isinf(x) (fabs(x) == INFINITY)
52 #undef isnan
53 #define isnan(x) ((x) != (x))
54 #define raise_inexact() do { volatile float junk __unused = /*LINTED*/1 + tiny; } while (0)
55 #undef signbit
56 #define signbit(x) (__builtin_signbit(x))
57
58 /* We need that DBL_EPSILON^2/128 is larger than FOUR_SQRT_MIN. */
59 static const double
60 A_crossover = 10, /* Hull et al suggest 1.5, but 10 works better */
61 B_crossover = 0.6417, /* suggested by Hull et al */
62 m_e = 2.7182818284590452e0, /* 0x15bf0a8b145769.0p-51 */
63 m_ln2 = 6.9314718055994531e-1, /* 0x162e42fefa39ef.0p-53 */
64 pio2_hi = 1.5707963267948966e0, /* 0x1921fb54442d18.0p-52 */
65 RECIP_EPSILON = 1 / DBL_EPSILON,
66 SQRT_3_EPSILON = 2.5809568279517849e-8, /* 0x1bb67ae8584caa.0p-78 */
67 SQRT_6_EPSILON = 3.6500241499888571e-8, /* 0x13988e1409212e.0p-77 */
68 #if DBL_MAX_EXP == 1024 /* IEEE */
69 FOUR_SQRT_MIN = 0x1p-509, /* >= 4 * sqrt(DBL_MIN) */
70 QUARTER_SQRT_MAX = 0x1p509, /* <= sqrt(DBL_MAX) / 4 */
71 SQRT_MIN = 0x1p-511; /* >= sqrt(DBL_MIN) */
72 #elif DBL_MAX_EXP == 127 /* VAX */
73 FOUR_SQRT_MIN = 0x1p-62, /* >= 4 * sqrt(DBL_MIN) */
74 QUARTER_SQRT_MAX = 0x1p62, /* <= sqrt(DBL_MAX) / 4 */
75 SQRT_MIN = 0x1p-64; /* >= sqrt(DBL_MIN) */
76 #else
77 #error "unsupported floating point format"
78 #endif
79
80
81 static const volatile double
82 pio2_lo = 6.1232339957367659e-17; /* 0x11a62633145c07.0p-106 */
83 static const volatile float
84 tiny = 0x1p-100;
85
86 static double complex clog_for_large_values(double complex z);
87
88 /*
89 * Testing indicates that all these functions are accurate up to 4 ULP.
90 * The functions casin(h) and cacos(h) are about 2.5 times slower than asinh.
91 * The functions catan(h) are a little under 2 times slower than atanh.
92 *
93 * The code for casinh, casin, cacos, and cacosh comes first. The code is
94 * rather complicated, and the four functions are highly interdependent.
95 *
96 * The code for catanh and catan comes at the end. It is much simpler than
97 * the other functions, and the code for these can be disconnected from the
98 * rest of the code.
99 */
100
101 /*
102 * ================================
103 * | casinh, casin, cacos, cacosh |
104 * ================================
105 */
106
107 /*
108 * The algorithm is very close to that in "Implementing the complex arcsine
109 * and arccosine functions using exception handling" by T. E. Hull, Thomas F.
110 * Fairgrieve, and Ping Tak Peter Tang, published in ACM Transactions on
111 * Mathematical Software, Volume 23 Issue 3, 1997, Pages 299-335,
112 * http://dl.acm.org/citation.cfm?id=275324.
113 *
114 * Throughout we use the convention z = x + I*y.
115 *
116 * casinh(z) = sign(x)*log(A+sqrt(A*A-1)) + I*asin(B)
117 * where
118 * A = (|z+I| + |z-I|) / 2
119 * B = (|z+I| - |z-I|) / 2 = y/A
120 *
121 * These formulas become numerically unstable:
122 * (a) for Re(casinh(z)) when z is close to the line segment [-I, I] (that
123 * is, Re(casinh(z)) is close to 0);
124 * (b) for Im(casinh(z)) when z is close to either of the intervals
125 * [I, I*infinity) or (-I*infinity, -I] (that is, |Im(casinh(z))| is
126 * close to PI/2).
127 *
128 * These numerical problems are overcome by defining
129 * f(a, b) = (hypot(a, b) - b) / 2 = a*a / (hypot(a, b) + b) / 2
130 * Then if A < A_crossover, we use
131 * log(A + sqrt(A*A-1)) = log1p((A-1) + sqrt((A-1)*(A+1)))
132 * A-1 = f(x, 1+y) + f(x, 1-y)
133 * and if B > B_crossover, we use
134 * asin(B) = atan2(y, sqrt(A*A - y*y)) = atan2(y, sqrt((A+y)*(A-y)))
135 * A-y = f(x, y+1) + f(x, y-1)
136 * where without loss of generality we have assumed that x and y are
137 * non-negative.
138 *
139 * Much of the difficulty comes because the intermediate computations may
140 * produce overflows or underflows. This is dealt with in the paper by Hull
141 * et al by using exception handling. We do this by detecting when
142 * computations risk underflow or overflow. The hardest part is handling the
143 * underflows when computing f(a, b).
144 *
145 * Note that the function f(a, b) does not appear explicitly in the paper by
146 * Hull et al, but the idea may be found on pages 308 and 309. Introducing the
147 * function f(a, b) allows us to concentrate many of the clever tricks in this
148 * paper into one function.
149 */
150
151 /*
152 * Function f(a, b, hypot_a_b) = (hypot(a, b) - b) / 2.
153 * Pass hypot(a, b) as the third argument.
154 */
155 static inline double
f(double a,double b,double hypot_a_b)156 f(double a, double b, double hypot_a_b)
157 {
158 if (b < 0)
159 return ((hypot_a_b - b) / 2);
160 if (b == 0)
161 return (a / 2);
162 return (a * a / (hypot_a_b + b) / 2);
163 }
164
165 /*
166 * All the hard work is contained in this function.
167 * x and y are assumed positive or zero, and less than RECIP_EPSILON.
168 * Upon return:
169 * rx = Re(casinh(z)) = -Im(cacos(y + I*x)).
170 * B_is_usable is set to 1 if the value of B is usable.
171 * If B_is_usable is set to 0, sqrt_A2my2 = sqrt(A*A - y*y), and new_y = y.
172 * If returning sqrt_A2my2 has potential to result in an underflow, it is
173 * rescaled, and new_y is similarly rescaled.
174 */
175 static inline void
do_hard_work(double x,double y,double * rx,int * B_is_usable,double * B,double * sqrt_A2my2,double * new_y)176 do_hard_work(double x, double y, double *rx, int *B_is_usable, double *B,
177 double *sqrt_A2my2, double *new_y)
178 {
179 double R, S, A; /* A, B, R, and S are as in Hull et al. */
180 double Am1, Amy; /* A-1, A-y. */
181
182 R = hypot(x, y + 1); /* |z+I| */
183 S = hypot(x, y - 1); /* |z-I| */
184
185 /* A = (|z+I| + |z-I|) / 2 */
186 A = (R + S) / 2;
187 /*
188 * Mathematically A >= 1. There is a small chance that this will not
189 * be so because of rounding errors. So we will make certain it is
190 * so.
191 */
192 if (A < 1)
193 A = 1;
194
195 if (A < A_crossover) {
196 /*
197 * Am1 = fp + fm, where fp = f(x, 1+y), and fm = f(x, 1-y).
198 * rx = log1p(Am1 + sqrt(Am1*(A+1)))
199 */
200 if (y == 1 && x < DBL_EPSILON * DBL_EPSILON / 128) {
201 /*
202 * fp is of order x^2, and fm = x/2.
203 * A = 1 (inexactly).
204 */
205 *rx = sqrt(x);
206 } else if (x >= DBL_EPSILON * fabs(y - 1)) {
207 /*
208 * Underflow will not occur because
209 * x >= DBL_EPSILON^2/128 >= FOUR_SQRT_MIN
210 */
211 Am1 = f(x, 1 + y, R) + f(x, 1 - y, S);
212 *rx = log1p(Am1 + sqrt(Am1 * (A + 1)));
213 } else if (y < 1) {
214 /*
215 * fp = x*x/(1+y)/4, fm = x*x/(1-y)/4, and
216 * A = 1 (inexactly).
217 */
218 *rx = x / sqrt((1 - y) * (1 + y));
219 } else { /* if (y > 1) */
220 /*
221 * A-1 = y-1 (inexactly).
222 */
223 *rx = log1p((y - 1) + sqrt((y - 1) * (y + 1)));
224 }
225 } else {
226 *rx = log(A + sqrt(A * A - 1));
227 }
228
229 *new_y = y;
230
231 if (y < FOUR_SQRT_MIN) {
232 /*
233 * Avoid a possible underflow caused by y/A. For casinh this
234 * would be legitimate, but will be picked up by invoking atan2
235 * later on. For cacos this would not be legitimate.
236 */
237 *B_is_usable = 0;
238 *sqrt_A2my2 = A * (2 / DBL_EPSILON);
239 *new_y = y * (2 / DBL_EPSILON);
240 return;
241 }
242
243 /* B = (|z+I| - |z-I|) / 2 = y/A */
244 *B = y / A;
245 *B_is_usable = 1;
246
247 if (*B > B_crossover) {
248 *B_is_usable = 0;
249 /*
250 * Amy = fp + fm, where fp = f(x, y+1), and fm = f(x, y-1).
251 * sqrt_A2my2 = sqrt(Amy*(A+y))
252 */
253 if (y == 1 && x < DBL_EPSILON / 128) {
254 /*
255 * fp is of order x^2, and fm = x/2.
256 * A = 1 (inexactly).
257 */
258 *sqrt_A2my2 = sqrt(x) * sqrt((A + y) / 2);
259 } else if (x >= DBL_EPSILON * fabs(y - 1)) {
260 /*
261 * Underflow will not occur because
262 * x >= DBL_EPSILON/128 >= FOUR_SQRT_MIN
263 * and
264 * x >= DBL_EPSILON^2 >= FOUR_SQRT_MIN
265 */
266 Amy = f(x, y + 1, R) + f(x, y - 1, S);
267 *sqrt_A2my2 = sqrt(Amy * (A + y));
268 } else if (y > 1) {
269 /*
270 * fp = x*x/(y+1)/4, fm = x*x/(y-1)/4, and
271 * A = y (inexactly).
272 *
273 * y < RECIP_EPSILON. So the following
274 * scaling should avoid any underflow problems.
275 */
276 *sqrt_A2my2 = x * (4 / DBL_EPSILON / DBL_EPSILON) * y /
277 sqrt((y + 1) * (y - 1));
278 *new_y = y * (4 / DBL_EPSILON / DBL_EPSILON);
279 } else { /* if (y < 1) */
280 /*
281 * fm = 1-y >= DBL_EPSILON, fp is of order x^2, and
282 * A = 1 (inexactly).
283 */
284 *sqrt_A2my2 = sqrt((1 - y) * (1 + y));
285 }
286 }
287 }
288
289 /*
290 * casinh(z) = z + O(z^3) as z -> 0
291 *
292 * casinh(z) = sign(x)*clog(sign(x)*z) + O(1/z^2) as z -> infinity
293 * The above formula works for the imaginary part as well, because
294 * Im(casinh(z)) = sign(x)*atan2(sign(x)*y, fabs(x)) + O(y/z^3)
295 * as z -> infinity, uniformly in y
296 */
297 double complex
casinh(double complex z)298 casinh(double complex z)
299 {
300 double x, y, ax, ay, rx, ry, B, sqrt_A2my2, new_y;
301 int B_is_usable;
302 double complex w;
303
304 x = creal(z);
305 y = cimag(z);
306 ax = fabs(x);
307 ay = fabs(y);
308
309 if (isnan(x) || isnan(y)) {
310 /* casinh(+-Inf + I*NaN) = +-Inf + I*NaN */
311 if (isinf(x))
312 return (CMPLX(x, y + y));
313 /* casinh(NaN + I*+-Inf) = opt(+-)Inf + I*NaN */
314 if (isinf(y))
315 return (CMPLX(y, x + x));
316 /* casinh(NaN + I*0) = NaN + I*0 */
317 if (y == 0)
318 return (CMPLX(x + x, y));
319 /*
320 * All other cases involving NaN return NaN + I*NaN.
321 * C99 leaves it optional whether to raise invalid if one of
322 * the arguments is not NaN, so we opt not to raise it.
323 */
324 return (CMPLX(x + 0.0L + (y + 0), x + 0.0L + (y + 0)));
325 }
326
327 if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) {
328 /* clog...() will raise inexact unless x or y is infinite. */
329 if (signbit(x) == 0)
330 w = clog_for_large_values(z) + m_ln2;
331 else
332 w = clog_for_large_values(-z) + m_ln2;
333 return (CMPLX(copysign(creal(w), x), copysign(cimag(w), y)));
334 }
335
336 /* Avoid spuriously raising inexact for z = 0. */
337 if (x == 0 && y == 0)
338 return (z);
339
340 /* All remaining cases are inexact. */
341 raise_inexact();
342
343 if (ax < SQRT_6_EPSILON / 4 && ay < SQRT_6_EPSILON / 4)
344 return (z);
345
346 do_hard_work(ax, ay, &rx, &B_is_usable, &B, &sqrt_A2my2, &new_y);
347 if (B_is_usable)
348 ry = asin(B);
349 else
350 ry = atan2(new_y, sqrt_A2my2);
351 return (CMPLX(copysign(rx, x), copysign(ry, y)));
352 }
353
354 /*
355 * casin(z) = reverse(casinh(reverse(z)))
356 * where reverse(x + I*y) = y + I*x = I*conj(z).
357 */
358 double complex
casin(double complex z)359 casin(double complex z)
360 {
361 double complex w = casinh(CMPLX(cimag(z), creal(z)));
362
363 return (CMPLX(cimag(w), creal(w)));
364 }
365
366 /*
367 * cacos(z) = PI/2 - casin(z)
368 * but do the computation carefully so cacos(z) is accurate when z is
369 * close to 1.
370 *
371 * cacos(z) = PI/2 - z + O(z^3) as z -> 0
372 *
373 * cacos(z) = -sign(y)*I*clog(z) + O(1/z^2) as z -> infinity
374 * The above formula works for the real part as well, because
375 * Re(cacos(z)) = atan2(fabs(y), x) + O(y/z^3)
376 * as z -> infinity, uniformly in y
377 */
378 double complex
cacos(double complex z)379 cacos(double complex z)
380 {
381 double x, y, ax, ay, rx, ry, B, sqrt_A2mx2, new_x;
382 int sx, sy;
383 int B_is_usable;
384 double complex w;
385
386 x = creal(z);
387 y = cimag(z);
388 sx = signbit(x);
389 sy = signbit(y);
390 ax = fabs(x);
391 ay = fabs(y);
392
393 if (isnan(x) || isnan(y)) {
394 /* cacos(+-Inf + I*NaN) = NaN + I*opt(-)Inf */
395 if (isinf(x))
396 return (CMPLX(y + y, -INFINITY));
397 /* cacos(NaN + I*+-Inf) = NaN + I*-+Inf */
398 if (isinf(y))
399 return (CMPLX(x + x, -y));
400 /* cacos(0 + I*NaN) = PI/2 + I*NaN with inexact */
401 if (x == 0)
402 return (CMPLX(pio2_hi + pio2_lo, y + y));
403 /*
404 * All other cases involving NaN return NaN + I*NaN.
405 * C99 leaves it optional whether to raise invalid if one of
406 * the arguments is not NaN, so we opt not to raise it.
407 */
408 return (CMPLX(x + 0.0L + (y + 0), x + 0.0L + (y + 0)));
409 }
410
411 if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) {
412 /* clog...() will raise inexact unless x or y is infinite. */
413 w = clog_for_large_values(z);
414 rx = fabs(cimag(w));
415 ry = creal(w) + m_ln2;
416 if (sy == 0)
417 ry = -ry;
418 return (CMPLX(rx, ry));
419 }
420
421 /* Avoid spuriously raising inexact for z = 1. */
422 if (x == 1 && y == 0)
423 return (CMPLX(0, -y));
424
425 /* All remaining cases are inexact. */
426 raise_inexact();
427
428 if (ax < SQRT_6_EPSILON / 4 && ay < SQRT_6_EPSILON / 4)
429 return (CMPLX(pio2_hi - (x - pio2_lo), -y));
430
431 do_hard_work(ay, ax, &ry, &B_is_usable, &B, &sqrt_A2mx2, &new_x);
432 if (B_is_usable) {
433 if (sx == 0)
434 rx = acos(B);
435 else
436 rx = acos(-B);
437 } else {
438 if (sx == 0)
439 rx = atan2(sqrt_A2mx2, new_x);
440 else
441 rx = atan2(sqrt_A2mx2, -new_x);
442 }
443 if (sy == 0)
444 ry = -ry;
445 return (CMPLX(rx, ry));
446 }
447
448 /*
449 * cacosh(z) = I*cacos(z) or -I*cacos(z)
450 * where the sign is chosen so Re(cacosh(z)) >= 0.
451 */
452 double complex
cacosh(double complex z)453 cacosh(double complex z)
454 {
455 double complex w;
456 double rx, ry;
457
458 w = cacos(z);
459 rx = creal(w);
460 ry = cimag(w);
461 /* cacosh(NaN + I*NaN) = NaN + I*NaN */
462 if (isnan(rx) && isnan(ry))
463 return (CMPLX(ry, rx));
464 /* cacosh(NaN + I*+-Inf) = +Inf + I*NaN */
465 /* cacosh(+-Inf + I*NaN) = +Inf + I*NaN */
466 if (isnan(rx))
467 return (CMPLX(fabs(ry), rx));
468 /* cacosh(0 + I*NaN) = NaN + I*NaN */
469 if (isnan(ry))
470 return (CMPLX(ry, ry));
471 return (CMPLX(fabs(ry), copysign(rx, cimag(z))));
472 }
473
474 /*
475 * Optimized version of clog() for |z| finite and larger than ~RECIP_EPSILON.
476 */
477 static double complex
clog_for_large_values(double complex z)478 clog_for_large_values(double complex z)
479 {
480 double x, y;
481 double ax, ay, t;
482
483 x = creal(z);
484 y = cimag(z);
485 ax = fabs(x);
486 ay = fabs(y);
487 if (ax < ay) {
488 t = ax;
489 ax = ay;
490 ay = t;
491 }
492
493 /*
494 * Avoid overflow in hypot() when x and y are both very large.
495 * Divide x and y by E, and then add 1 to the logarithm. This depends
496 * on E being larger than sqrt(2).
497 * Dividing by E causes an insignificant loss of accuracy; however
498 * this method is still poor since it is uneccessarily slow.
499 */
500 if (ax > DBL_MAX / 2)
501 return (CMPLX(log(hypot(x / m_e, y / m_e)) + 1, atan2(y, x)));
502
503 /*
504 * Avoid overflow when x or y is large. Avoid underflow when x or
505 * y is small.
506 */
507 if (ax > QUARTER_SQRT_MAX || ay < SQRT_MIN)
508 return (CMPLX(log(hypot(x, y)), atan2(y, x)));
509
510 return (CMPLX(log(ax * ax + ay * ay) / 2, atan2(y, x)));
511 }
512
513 /*
514 * =================
515 * | catanh, catan |
516 * =================
517 */
518
519 /*
520 * sum_squares(x,y) = x*x + y*y (or just x*x if y*y would underflow).
521 * Assumes x*x and y*y will not overflow.
522 * Assumes x and y are finite.
523 * Assumes y is non-negative.
524 * Assumes fabs(x) >= DBL_EPSILON.
525 */
526 static inline double
sum_squares(double x,double y)527 sum_squares(double x, double y)
528 {
529
530 /* Avoid underflow when y is small. */
531 if (y < SQRT_MIN)
532 return (x * x);
533
534 return (x * x + y * y);
535 }
536
537 /*
538 * real_part_reciprocal(x, y) = Re(1/(x+I*y)) = x/(x*x + y*y).
539 * Assumes x and y are not NaN, and one of x and y is larger than
540 * RECIP_EPSILON. We avoid unwarranted underflow. It is important to not use
541 * the code creal(1/z), because the imaginary part may produce an unwanted
542 * underflow.
543 * This is only called in a context where inexact is always raised before
544 * the call, so no effort is made to avoid or force inexact.
545 */
546 static inline double
real_part_reciprocal(double x,double y)547 real_part_reciprocal(double x, double y)
548 {
549 double scale;
550 uint32_t hx, hy;
551 int32_t ix, iy;
552
553 /*
554 * This code is inspired by the C99 document n1124.pdf, Section G.5.1,
555 * example 2.
556 */
557 GET_HIGH_WORD(hx, x);
558 ix = hx & 0x7ff00000;
559 GET_HIGH_WORD(hy, y);
560 iy = hy & 0x7ff00000;
561 #define BIAS (DBL_MAX_EXP - 1)
562 /* XXX more guard digits are useful iff there is extra precision. */
563 #define CUTOFF (DBL_MANT_DIG / 2 + 1) /* just half or 1 guard digit */
564 if (ix - iy >= CUTOFF << 20 || isinf(x))
565 return (1 / x); /* +-Inf -> +-0 is special */
566 if (iy - ix >= CUTOFF << 20)
567 return (x / y / y); /* should avoid double div, but hard */
568 if (ix <= (BIAS + DBL_MAX_EXP / 2 - CUTOFF) << 20)
569 return (x / (x * x + y * y));
570 scale = 1;
571 SET_HIGH_WORD(scale, 0x7ff00000 - ix); /* 2**(1-ilogb(x)) */
572 x *= scale;
573 y *= scale;
574 return (x / (x * x + y * y) * scale);
575 }
576
577 /*
578 * catanh(z) = log((1+z)/(1-z)) / 2
579 * = log1p(4*x / |z-1|^2) / 4
580 * + I * atan2(2*y, (1-x)*(1+x)-y*y) / 2
581 *
582 * catanh(z) = z + O(z^3) as z -> 0
583 *
584 * catanh(z) = 1/z + sign(y)*I*PI/2 + O(1/z^3) as z -> infinity
585 * The above formula works for the real part as well, because
586 * Re(catanh(z)) = x/|z|^2 + O(x/z^4)
587 * as z -> infinity, uniformly in x
588 */
589 double complex
catanh(double complex z)590 catanh(double complex z)
591 {
592 double x, y, ax, ay, rx, ry;
593
594 x = creal(z);
595 y = cimag(z);
596 ax = fabs(x);
597 ay = fabs(y);
598
599 /* This helps handle many cases. */
600 if (y == 0 && ax <= 1)
601 return (CMPLX(atanh(x), y));
602
603 /* To ensure the same accuracy as atan(), and to filter out z = 0. */
604 if (x == 0)
605 return (CMPLX(x, atan(y)));
606
607 if (isnan(x) || isnan(y)) {
608 /* catanh(+-Inf + I*NaN) = +-0 + I*NaN */
609 if (isinf(x))
610 return (CMPLX(copysign(0, x), y + y));
611 /* catanh(NaN + I*+-Inf) = sign(NaN)0 + I*+-PI/2 */
612 if (isinf(y))
613 return (CMPLX(copysign(0, x),
614 copysign(pio2_hi + pio2_lo, y)));
615 /*
616 * All other cases involving NaN return NaN + I*NaN.
617 * C99 leaves it optional whether to raise invalid if one of
618 * the arguments is not NaN, so we opt not to raise it.
619 */
620 return (CMPLX(x + 0.0L + (y + 0), x + 0.0L + (y + 0)));
621 }
622
623 if (ax > RECIP_EPSILON || ay > RECIP_EPSILON)
624 return (CMPLX(real_part_reciprocal(x, y),
625 copysign(pio2_hi + pio2_lo, y)));
626
627 if (ax < SQRT_3_EPSILON / 2 && ay < SQRT_3_EPSILON / 2) {
628 /*
629 * z = 0 was filtered out above. All other cases must raise
630 * inexact, but this is the only only that needs to do it
631 * explicitly.
632 */
633 raise_inexact();
634 return (z);
635 }
636
637 if (ax == 1 && ay < DBL_EPSILON)
638 rx = (m_ln2 - log(ay)) / 2;
639 else
640 rx = log1p(4 * ax / sum_squares(ax - 1, ay)) / 4;
641
642 if (ax == 1)
643 ry = atan2(2, -ay) / 2;
644 else if (ay < DBL_EPSILON)
645 ry = atan2(2 * ay, (1 - ax) * (1 + ax)) / 2;
646 else
647 ry = atan2(2 * ay, (1 - ax) * (1 + ax) - ay * ay) / 2;
648
649 return (CMPLX(copysign(rx, x), copysign(ry, y)));
650 }
651
652 /*
653 * catan(z) = reverse(catanh(reverse(z)))
654 * where reverse(x + I*y) = y + I*x = I*conj(z).
655 */
656 double complex
catan(double complex z)657 catan(double complex z)
658 {
659 double complex w = catanh(CMPLX(cimag(z), creal(z)));
660
661 return (CMPLX(cimag(w), creal(w)));
662 }
663