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. */
45static const double
46A_crossover =		10, /* Hull et al suggest 1.5, but 10 works better */
47B_crossover =		0.6417,			/* suggested by Hull et al */
48FOUR_SQRT_MIN =		0x1p-509,		/* >= 4 * sqrt(DBL_MIN) */
49QUARTER_SQRT_MAX =	0x1p509,		/* <= sqrt(DBL_MAX) / 4 */
50m_e =			2.7182818284590452e0,	/*  0x15bf0a8b145769.0p-51 */
51m_ln2 =			6.9314718055994531e-1,	/*  0x162e42fefa39ef.0p-53 */
52pio2_hi =		1.5707963267948966e0,	/*  0x1921fb54442d18.0p-52 */
53RECIP_EPSILON =		1 / DBL_EPSILON,
54SQRT_3_EPSILON =	2.5809568279517849e-8,	/*  0x1bb67ae8584caa.0p-78 */
55SQRT_6_EPSILON =	3.6500241499888571e-8,	/*  0x13988e1409212e.0p-77 */
56SQRT_MIN =		0x1p-511;		/* >= sqrt(DBL_MIN) */
57
58static const volatile double
59pio2_lo =		6.1232339957367659e-17;	/*  0x11a62633145c07.0p-106 */
60static const volatile float
61tiny =			0x1p-100;
62
63static 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 */
132static inline double
133f(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 */
152static inline void
153do_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 */
274double complex
275casinh(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 */
335double complex
336casin(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 */
355double complex
356cacos(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 */
429double complex
430cacosh(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 */
454static double complex
455clog_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 */
503static inline double
504sum_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 */
523static inline double
524real_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 */
566double complex
567catanh(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 */
633double complex
634catan(double complex z)
635{
636	double complex w = catanh(CMPLX(cimag(z), creal(z)));
637
638	return (CMPLX(cimag(w), creal(w)));
639}
640