1 /*-
2 * Copyright (c) 2012 Stephen Montgomery-Smith <stephen@FreeBSD.ORG>
3 * All rights reserved.
4 *
5 * Redistribution and use in source and binary forms, with or without
6 * modification, are permitted provided that the following conditions
7 * are met:
8 * 1. Redistributions of source code must retain the above copyright
9 * notice, this list of conditions and the following disclaimer.
10 * 2. Redistributions in binary form must reproduce the above copyright
11 * notice, this list of conditions and the following disclaimer in the
12 * documentation and/or other materials provided with the distribution.
13 *
14 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND
15 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
16 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
17 * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
18 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
19 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
20 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
21 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
22 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
23 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
24 * SUCH DAMAGE.
25 */
26
27 #include <sys/cdefs.h>
28 __FBSDID("$FreeBSD: head/lib/msun/src/catrig.c 275819 2014-12-16 09:21:56Z ed $");
29
30 #include <complex.h>
31 #include <float.h>
32
33 #include "math.h"
34 #include "math_private.h"
35
36 #undef isinf
37 #define isinf(x) (fabs(x) == INFINITY)
38 #undef isnan
39 #define isnan(x) ((x) != (x))
40 #define raise_inexact() do { volatile float junk = 1 + tiny; } while(0)
41 #undef signbit
42 #define signbit(x) (__builtin_signbit(x))
43
44 /* We need that DBL_EPSILON^2/128 is larger than FOUR_SQRT_MIN. */
45 static const double
46 A_crossover = 10, /* Hull et al suggest 1.5, but 10 works better */
47 B_crossover = 0.6417, /* suggested by Hull et al */
48 FOUR_SQRT_MIN = 0x1p-509, /* >= 4 * sqrt(DBL_MIN) */
49 QUARTER_SQRT_MAX = 0x1p509, /* <= sqrt(DBL_MAX) / 4 */
50 m_e = 2.7182818284590452e0, /* 0x15bf0a8b145769.0p-51 */
51 m_ln2 = 6.9314718055994531e-1, /* 0x162e42fefa39ef.0p-53 */
52 pio2_hi = 1.5707963267948966e0, /* 0x1921fb54442d18.0p-52 */
53 RECIP_EPSILON = 1 / DBL_EPSILON,
54 SQRT_3_EPSILON = 2.5809568279517849e-8, /* 0x1bb67ae8584caa.0p-78 */
55 SQRT_6_EPSILON = 3.6500241499888571e-8, /* 0x13988e1409212e.0p-77 */
56 SQRT_MIN = 0x1p-511; /* >= sqrt(DBL_MIN) */
57
58 static const volatile double
59 pio2_lo = 6.1232339957367659e-17; /* 0x11a62633145c07.0p-106 */
60 static const volatile float
61 tiny = 0x1p-100;
62
63 static double complex clog_for_large_values(double complex z);
64
65 /*
66 * Testing indicates that all these functions are accurate up to 4 ULP.
67 * The functions casin(h) and cacos(h) are about 2.5 times slower than asinh.
68 * The functions catan(h) are a little under 2 times slower than atanh.
69 *
70 * The code for casinh, casin, cacos, and cacosh comes first. The code is
71 * rather complicated, and the four functions are highly interdependent.
72 *
73 * The code for catanh and catan comes at the end. It is much simpler than
74 * the other functions, and the code for these can be disconnected from the
75 * rest of the code.
76 */
77
78 /*
79 * ================================
80 * | casinh, casin, cacos, cacosh |
81 * ================================
82 */
83
84 /*
85 * The algorithm is very close to that in "Implementing the complex arcsine
86 * and arccosine functions using exception handling" by T. E. Hull, Thomas F.
87 * Fairgrieve, and Ping Tak Peter Tang, published in ACM Transactions on
88 * Mathematical Software, Volume 23 Issue 3, 1997, Pages 299-335,
89 * http://dl.acm.org/citation.cfm?id=275324.
90 *
91 * Throughout we use the convention z = x + I*y.
92 *
93 * casinh(z) = sign(x)*log(A+sqrt(A*A-1)) + I*asin(B)
94 * where
95 * A = (|z+I| + |z-I|) / 2
96 * B = (|z+I| - |z-I|) / 2 = y/A
97 *
98 * These formulas become numerically unstable:
99 * (a) for Re(casinh(z)) when z is close to the line segment [-I, I] (that
100 * is, Re(casinh(z)) is close to 0);
101 * (b) for Im(casinh(z)) when z is close to either of the intervals
102 * [I, I*infinity) or (-I*infinity, -I] (that is, |Im(casinh(z))| is
103 * close to PI/2).
104 *
105 * These numerical problems are overcome by defining
106 * f(a, b) = (hypot(a, b) - b) / 2 = a*a / (hypot(a, b) + b) / 2
107 * Then if A < A_crossover, we use
108 * log(A + sqrt(A*A-1)) = log1p((A-1) + sqrt((A-1)*(A+1)))
109 * A-1 = f(x, 1+y) + f(x, 1-y)
110 * and if B > B_crossover, we use
111 * asin(B) = atan2(y, sqrt(A*A - y*y)) = atan2(y, sqrt((A+y)*(A-y)))
112 * A-y = f(x, y+1) + f(x, y-1)
113 * where without loss of generality we have assumed that x and y are
114 * non-negative.
115 *
116 * Much of the difficulty comes because the intermediate computations may
117 * produce overflows or underflows. This is dealt with in the paper by Hull
118 * et al by using exception handling. We do this by detecting when
119 * computations risk underflow or overflow. The hardest part is handling the
120 * underflows when computing f(a, b).
121 *
122 * Note that the function f(a, b) does not appear explicitly in the paper by
123 * Hull et al, but the idea may be found on pages 308 and 309. Introducing the
124 * function f(a, b) allows us to concentrate many of the clever tricks in this
125 * paper into one function.
126 */
127
128 /*
129 * Function f(a, b, hypot_a_b) = (hypot(a, b) - b) / 2.
130 * Pass hypot(a, b) as the third argument.
131 */
132 static inline double
f(double a,double b,double hypot_a_b)133 f(double a, double b, double hypot_a_b)
134 {
135 if (b < 0)
136 return ((hypot_a_b - b) / 2);
137 if (b == 0)
138 return (a / 2);
139 return (a * a / (hypot_a_b + b) / 2);
140 }
141
142 /*
143 * All the hard work is contained in this function.
144 * x and y are assumed positive or zero, and less than RECIP_EPSILON.
145 * Upon return:
146 * rx = Re(casinh(z)) = -Im(cacos(y + I*x)).
147 * B_is_usable is set to 1 if the value of B is usable.
148 * If B_is_usable is set to 0, sqrt_A2my2 = sqrt(A*A - y*y), and new_y = y.
149 * If returning sqrt_A2my2 has potential to result in an underflow, it is
150 * rescaled, and new_y is similarly rescaled.
151 */
152 static inline void
do_hard_work(double x,double y,double * rx,int * B_is_usable,double * B,double * sqrt_A2my2,double * new_y)153 do_hard_work(double x, double y, double *rx, int *B_is_usable, double *B,
154 double *sqrt_A2my2, double *new_y)
155 {
156 double R, S, A; /* A, B, R, and S are as in Hull et al. */
157 double Am1, Amy; /* A-1, A-y. */
158
159 R = hypot(x, y + 1); /* |z+I| */
160 S = hypot(x, y - 1); /* |z-I| */
161
162 /* A = (|z+I| + |z-I|) / 2 */
163 A = (R + S) / 2;
164 /*
165 * Mathematically A >= 1. There is a small chance that this will not
166 * be so because of rounding errors. So we will make certain it is
167 * so.
168 */
169 if (A < 1)
170 A = 1;
171
172 if (A < A_crossover) {
173 /*
174 * Am1 = fp + fm, where fp = f(x, 1+y), and fm = f(x, 1-y).
175 * rx = log1p(Am1 + sqrt(Am1*(A+1)))
176 */
177 if (y == 1 && x < DBL_EPSILON * DBL_EPSILON / 128) {
178 /*
179 * fp is of order x^2, and fm = x/2.
180 * A = 1 (inexactly).
181 */
182 *rx = sqrt(x);
183 } else if (x >= DBL_EPSILON * fabs(y - 1)) {
184 /*
185 * Underflow will not occur because
186 * x >= DBL_EPSILON^2/128 >= FOUR_SQRT_MIN
187 */
188 Am1 = f(x, 1 + y, R) + f(x, 1 - y, S);
189 *rx = log1p(Am1 + sqrt(Am1 * (A + 1)));
190 } else if (y < 1) {
191 /*
192 * fp = x*x/(1+y)/4, fm = x*x/(1-y)/4, and
193 * A = 1 (inexactly).
194 */
195 *rx = x / sqrt((1 - y) * (1 + y));
196 } else { /* if (y > 1) */
197 /*
198 * A-1 = y-1 (inexactly).
199 */
200 *rx = log1p((y - 1) + sqrt((y - 1) * (y + 1)));
201 }
202 } else {
203 *rx = log(A + sqrt(A * A - 1));
204 }
205
206 *new_y = y;
207
208 if (y < FOUR_SQRT_MIN) {
209 /*
210 * Avoid a possible underflow caused by y/A. For casinh this
211 * would be legitimate, but will be picked up by invoking atan2
212 * later on. For cacos this would not be legitimate.
213 */
214 *B_is_usable = 0;
215 *sqrt_A2my2 = A * (2 / DBL_EPSILON);
216 *new_y = y * (2 / DBL_EPSILON);
217 return;
218 }
219
220 /* B = (|z+I| - |z-I|) / 2 = y/A */
221 *B = y / A;
222 *B_is_usable = 1;
223
224 if (*B > B_crossover) {
225 *B_is_usable = 0;
226 /*
227 * Amy = fp + fm, where fp = f(x, y+1), and fm = f(x, y-1).
228 * sqrt_A2my2 = sqrt(Amy*(A+y))
229 */
230 if (y == 1 && x < DBL_EPSILON / 128) {
231 /*
232 * fp is of order x^2, and fm = x/2.
233 * A = 1 (inexactly).
234 */
235 *sqrt_A2my2 = sqrt(x) * sqrt((A + y) / 2);
236 } else if (x >= DBL_EPSILON * fabs(y - 1)) {
237 /*
238 * Underflow will not occur because
239 * x >= DBL_EPSILON/128 >= FOUR_SQRT_MIN
240 * and
241 * x >= DBL_EPSILON^2 >= FOUR_SQRT_MIN
242 */
243 Amy = f(x, y + 1, R) + f(x, y - 1, S);
244 *sqrt_A2my2 = sqrt(Amy * (A + y));
245 } else if (y > 1) {
246 /*
247 * fp = x*x/(y+1)/4, fm = x*x/(y-1)/4, and
248 * A = y (inexactly).
249 *
250 * y < RECIP_EPSILON. So the following
251 * scaling should avoid any underflow problems.
252 */
253 *sqrt_A2my2 = x * (4 / DBL_EPSILON / DBL_EPSILON) * y /
254 sqrt((y + 1) * (y - 1));
255 *new_y = y * (4 / DBL_EPSILON / DBL_EPSILON);
256 } else { /* if (y < 1) */
257 /*
258 * fm = 1-y >= DBL_EPSILON, fp is of order x^2, and
259 * A = 1 (inexactly).
260 */
261 *sqrt_A2my2 = sqrt((1 - y) * (1 + y));
262 }
263 }
264 }
265
266 /*
267 * casinh(z) = z + O(z^3) as z -> 0
268 *
269 * casinh(z) = sign(x)*clog(sign(x)*z) + O(1/z^2) as z -> infinity
270 * The above formula works for the imaginary part as well, because
271 * Im(casinh(z)) = sign(x)*atan2(sign(x)*y, fabs(x)) + O(y/z^3)
272 * as z -> infinity, uniformly in y
273 */
274 double complex
casinh(double complex z)275 casinh(double complex z)
276 {
277 double x, y, ax, ay, rx, ry, B, sqrt_A2my2, new_y;
278 int B_is_usable;
279 double complex w;
280
281 x = creal(z);
282 y = cimag(z);
283 ax = fabs(x);
284 ay = fabs(y);
285
286 if (isnan(x) || isnan(y)) {
287 /* casinh(+-Inf + I*NaN) = +-Inf + I*NaN */
288 if (isinf(x))
289 return (CMPLX(x, y + y));
290 /* casinh(NaN + I*+-Inf) = opt(+-)Inf + I*NaN */
291 if (isinf(y))
292 return (CMPLX(y, x + x));
293 /* casinh(NaN + I*0) = NaN + I*0 */
294 if (y == 0)
295 return (CMPLX(x + x, y));
296 /*
297 * All other cases involving NaN return NaN + I*NaN.
298 * C99 leaves it optional whether to raise invalid if one of
299 * the arguments is not NaN, so we opt not to raise it.
300 */
301 return (CMPLX(x + 0.0L + (y + 0), x + 0.0L + (y + 0)));
302 }
303
304 if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) {
305 /* clog...() will raise inexact unless x or y is infinite. */
306 if (signbit(x) == 0)
307 w = clog_for_large_values(z) + m_ln2;
308 else
309 w = clog_for_large_values(-z) + m_ln2;
310 return (CMPLX(copysign(creal(w), x), copysign(cimag(w), y)));
311 }
312
313 /* Avoid spuriously raising inexact for z = 0. */
314 if (x == 0 && y == 0)
315 return (z);
316
317 /* All remaining cases are inexact. */
318 raise_inexact();
319
320 if (ax < SQRT_6_EPSILON / 4 && ay < SQRT_6_EPSILON / 4)
321 return (z);
322
323 do_hard_work(ax, ay, &rx, &B_is_usable, &B, &sqrt_A2my2, &new_y);
324 if (B_is_usable)
325 ry = asin(B);
326 else
327 ry = atan2(new_y, sqrt_A2my2);
328 return (CMPLX(copysign(rx, x), copysign(ry, y)));
329 }
330
331 /*
332 * casin(z) = reverse(casinh(reverse(z)))
333 * where reverse(x + I*y) = y + I*x = I*conj(z).
334 */
335 double complex
casin(double complex z)336 casin(double complex z)
337 {
338 double complex w = casinh(CMPLX(cimag(z), creal(z)));
339
340 return (CMPLX(cimag(w), creal(w)));
341 }
342
343 /*
344 * cacos(z) = PI/2 - casin(z)
345 * but do the computation carefully so cacos(z) is accurate when z is
346 * close to 1.
347 *
348 * cacos(z) = PI/2 - z + O(z^3) as z -> 0
349 *
350 * cacos(z) = -sign(y)*I*clog(z) + O(1/z^2) as z -> infinity
351 * The above formula works for the real part as well, because
352 * Re(cacos(z)) = atan2(fabs(y), x) + O(y/z^3)
353 * as z -> infinity, uniformly in y
354 */
355 double complex
cacos(double complex z)356 cacos(double complex z)
357 {
358 double x, y, ax, ay, rx, ry, B, sqrt_A2mx2, new_x;
359 int sx, sy;
360 int B_is_usable;
361 double complex w;
362
363 x = creal(z);
364 y = cimag(z);
365 sx = signbit(x);
366 sy = signbit(y);
367 ax = fabs(x);
368 ay = fabs(y);
369
370 if (isnan(x) || isnan(y)) {
371 /* cacos(+-Inf + I*NaN) = NaN + I*opt(-)Inf */
372 if (isinf(x))
373 return (CMPLX(y + y, -INFINITY));
374 /* cacos(NaN + I*+-Inf) = NaN + I*-+Inf */
375 if (isinf(y))
376 return (CMPLX(x + x, -y));
377 /* cacos(0 + I*NaN) = PI/2 + I*NaN with inexact */
378 if (x == 0)
379 return (CMPLX(pio2_hi + pio2_lo, y + y));
380 /*
381 * All other cases involving NaN return NaN + I*NaN.
382 * C99 leaves it optional whether to raise invalid if one of
383 * the arguments is not NaN, so we opt not to raise it.
384 */
385 return (CMPLX(x + 0.0L + (y + 0), x + 0.0L + (y + 0)));
386 }
387
388 if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) {
389 /* clog...() will raise inexact unless x or y is infinite. */
390 w = clog_for_large_values(z);
391 rx = fabs(cimag(w));
392 ry = creal(w) + m_ln2;
393 if (sy == 0)
394 ry = -ry;
395 return (CMPLX(rx, ry));
396 }
397
398 /* Avoid spuriously raising inexact for z = 1. */
399 if (x == 1 && y == 0)
400 return (CMPLX(0, -y));
401
402 /* All remaining cases are inexact. */
403 raise_inexact();
404
405 if (ax < SQRT_6_EPSILON / 4 && ay < SQRT_6_EPSILON / 4)
406 return (CMPLX(pio2_hi - (x - pio2_lo), -y));
407
408 do_hard_work(ay, ax, &ry, &B_is_usable, &B, &sqrt_A2mx2, &new_x);
409 if (B_is_usable) {
410 if (sx == 0)
411 rx = acos(B);
412 else
413 rx = acos(-B);
414 } else {
415 if (sx == 0)
416 rx = atan2(sqrt_A2mx2, new_x);
417 else
418 rx = atan2(sqrt_A2mx2, -new_x);
419 }
420 if (sy == 0)
421 ry = -ry;
422 return (CMPLX(rx, ry));
423 }
424
425 /*
426 * cacosh(z) = I*cacos(z) or -I*cacos(z)
427 * where the sign is chosen so Re(cacosh(z)) >= 0.
428 */
429 double complex
cacosh(double complex z)430 cacosh(double complex z)
431 {
432 double complex w;
433 double rx, ry;
434
435 w = cacos(z);
436 rx = creal(w);
437 ry = cimag(w);
438 /* cacosh(NaN + I*NaN) = NaN + I*NaN */
439 if (isnan(rx) && isnan(ry))
440 return (CMPLX(ry, rx));
441 /* cacosh(NaN + I*+-Inf) = +Inf + I*NaN */
442 /* cacosh(+-Inf + I*NaN) = +Inf + I*NaN */
443 if (isnan(rx))
444 return (CMPLX(fabs(ry), rx));
445 /* cacosh(0 + I*NaN) = NaN + I*NaN */
446 if (isnan(ry))
447 return (CMPLX(ry, ry));
448 return (CMPLX(fabs(ry), copysign(rx, cimag(z))));
449 }
450
451 /*
452 * Optimized version of clog() for |z| finite and larger than ~RECIP_EPSILON.
453 */
454 static double complex
clog_for_large_values(double complex z)455 clog_for_large_values(double complex z)
456 {
457 double x, y;
458 double ax, ay, t;
459
460 x = creal(z);
461 y = cimag(z);
462 ax = fabs(x);
463 ay = fabs(y);
464 if (ax < ay) {
465 t = ax;
466 ax = ay;
467 ay = t;
468 }
469
470 /*
471 * Avoid overflow in hypot() when x and y are both very large.
472 * Divide x and y by E, and then add 1 to the logarithm. This depends
473 * on E being larger than sqrt(2).
474 * Dividing by E causes an insignificant loss of accuracy; however
475 * this method is still poor since it is uneccessarily slow.
476 */
477 if (ax > DBL_MAX / 2)
478 return (CMPLX(log(hypot(x / m_e, y / m_e)) + 1, atan2(y, x)));
479
480 /*
481 * Avoid overflow when x or y is large. Avoid underflow when x or
482 * y is small.
483 */
484 if (ax > QUARTER_SQRT_MAX || ay < SQRT_MIN)
485 return (CMPLX(log(hypot(x, y)), atan2(y, x)));
486
487 return (CMPLX(log(ax * ax + ay * ay) / 2, atan2(y, x)));
488 }
489
490 /*
491 * =================
492 * | catanh, catan |
493 * =================
494 */
495
496 /*
497 * sum_squares(x,y) = x*x + y*y (or just x*x if y*y would underflow).
498 * Assumes x*x and y*y will not overflow.
499 * Assumes x and y are finite.
500 * Assumes y is non-negative.
501 * Assumes fabs(x) >= DBL_EPSILON.
502 */
503 static inline double
sum_squares(double x,double y)504 sum_squares(double x, double y)
505 {
506
507 /* Avoid underflow when y is small. */
508 if (y < SQRT_MIN)
509 return (x * x);
510
511 return (x * x + y * y);
512 }
513
514 /*
515 * real_part_reciprocal(x, y) = Re(1/(x+I*y)) = x/(x*x + y*y).
516 * Assumes x and y are not NaN, and one of x and y is larger than
517 * RECIP_EPSILON. We avoid unwarranted underflow. It is important to not use
518 * the code creal(1/z), because the imaginary part may produce an unwanted
519 * underflow.
520 * This is only called in a context where inexact is always raised before
521 * the call, so no effort is made to avoid or force inexact.
522 */
523 static inline double
real_part_reciprocal(double x,double y)524 real_part_reciprocal(double x, double y)
525 {
526 double scale;
527 uint32_t hx, hy;
528 int32_t ix, iy;
529
530 /*
531 * This code is inspired by the C99 document n1124.pdf, Section G.5.1,
532 * example 2.
533 */
534 GET_HIGH_WORD(hx, x);
535 ix = hx & 0x7ff00000;
536 GET_HIGH_WORD(hy, y);
537 iy = hy & 0x7ff00000;
538 #define BIAS (DBL_MAX_EXP - 1)
539 /* XXX more guard digits are useful iff there is extra precision. */
540 #define CUTOFF (DBL_MANT_DIG / 2 + 1) /* just half or 1 guard digit */
541 if (ix - iy >= CUTOFF << 20 || isinf(x))
542 return (1 / x); /* +-Inf -> +-0 is special */
543 if (iy - ix >= CUTOFF << 20)
544 return (x / y / y); /* should avoid double div, but hard */
545 if (ix <= (BIAS + DBL_MAX_EXP / 2 - CUTOFF) << 20)
546 return (x / (x * x + y * y));
547 scale = 1;
548 SET_HIGH_WORD(scale, 0x7ff00000 - ix); /* 2**(1-ilogb(x)) */
549 x *= scale;
550 y *= scale;
551 return (x / (x * x + y * y) * scale);
552 }
553
554 /*
555 * catanh(z) = log((1+z)/(1-z)) / 2
556 * = log1p(4*x / |z-1|^2) / 4
557 * + I * atan2(2*y, (1-x)*(1+x)-y*y) / 2
558 *
559 * catanh(z) = z + O(z^3) as z -> 0
560 *
561 * catanh(z) = 1/z + sign(y)*I*PI/2 + O(1/z^3) as z -> infinity
562 * The above formula works for the real part as well, because
563 * Re(catanh(z)) = x/|z|^2 + O(x/z^4)
564 * as z -> infinity, uniformly in x
565 */
566 double complex
catanh(double complex z)567 catanh(double complex z)
568 {
569 double x, y, ax, ay, rx, ry;
570
571 x = creal(z);
572 y = cimag(z);
573 ax = fabs(x);
574 ay = fabs(y);
575
576 /* This helps handle many cases. */
577 if (y == 0 && ax <= 1)
578 return (CMPLX(atanh(x), y));
579
580 /* To ensure the same accuracy as atan(), and to filter out z = 0. */
581 if (x == 0)
582 return (CMPLX(x, atan(y)));
583
584 if (isnan(x) || isnan(y)) {
585 /* catanh(+-Inf + I*NaN) = +-0 + I*NaN */
586 if (isinf(x))
587 return (CMPLX(copysign(0, x), y + y));
588 /* catanh(NaN + I*+-Inf) = sign(NaN)0 + I*+-PI/2 */
589 if (isinf(y))
590 return (CMPLX(copysign(0, x),
591 copysign(pio2_hi + pio2_lo, y)));
592 /*
593 * All other cases involving NaN return NaN + I*NaN.
594 * C99 leaves it optional whether to raise invalid if one of
595 * the arguments is not NaN, so we opt not to raise it.
596 */
597 return (CMPLX(x + 0.0L + (y + 0), x + 0.0L + (y + 0)));
598 }
599
600 if (ax > RECIP_EPSILON || ay > RECIP_EPSILON)
601 return (CMPLX(real_part_reciprocal(x, y),
602 copysign(pio2_hi + pio2_lo, y)));
603
604 if (ax < SQRT_3_EPSILON / 2 && ay < SQRT_3_EPSILON / 2) {
605 /*
606 * z = 0 was filtered out above. All other cases must raise
607 * inexact, but this is the only only that needs to do it
608 * explicitly.
609 */
610 raise_inexact();
611 return (z);
612 }
613
614 if (ax == 1 && ay < DBL_EPSILON)
615 rx = (m_ln2 - log(ay)) / 2;
616 else
617 rx = log1p(4 * ax / sum_squares(ax - 1, ay)) / 4;
618
619 if (ax == 1)
620 ry = atan2(2, -ay) / 2;
621 else if (ay < DBL_EPSILON)
622 ry = atan2(2 * ay, (1 - ax) * (1 + ax)) / 2;
623 else
624 ry = atan2(2 * ay, (1 - ax) * (1 + ax) - ay * ay) / 2;
625
626 return (CMPLX(copysign(rx, x), copysign(ry, y)));
627 }
628
629 /*
630 * catan(z) = reverse(catanh(reverse(z)))
631 * where reverse(x + I*y) = y + I*x = I*conj(z).
632 */
633 double complex
catan(double complex z)634 catan(double complex z)
635 {
636 double complex w = catanh(CMPLX(cimag(z), creal(z)));
637
638 return (CMPLX(cimag(w), creal(w)));
639 }
640