1/*	$NetBSD: catrigf.c,v 1.2 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/*
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#if 0
44__FBSDID("$FreeBSD: head/lib/msun/src/catrigf.c 275819 2014-12-16 09:21:56Z ed $");
45#endif
46__RCSID("$NetBSD: catrigf.c,v 1.2 2022/04/19 20:32:16 rillig Exp $");
47
48#include "namespace.h"
49#ifdef __weak_alias
50__weak_alias(casinf, _casinf)
51#endif
52#ifdef __weak_alias
53__weak_alias(catanf, _catanf)
54#endif
55
56
57#include <complex.h>
58#include <float.h>
59
60#include "math.h"
61#include "math_private.h"
62
63#undef isinf
64#define isinf(x)	(fabsf(x) == INFINITY)
65#undef isnan
66#define isnan(x)	((x) != (x))
67#define	raise_inexact()	do { volatile float junk __unused = /*LINTED*/1 + tiny; } while (0)
68#undef signbit
69#define signbit(x)	(__builtin_signbitf(x))
70
71static const float
72A_crossover =		10,
73B_crossover =		0.6417,
74FOUR_SQRT_MIN =		0x1p-61,
75QUARTER_SQRT_MAX =	0x1p61,
76m_e =			2.7182818285e0,		/*  0xadf854.0p-22 */
77m_ln2 =			6.9314718056e-1,	/*  0xb17218.0p-24 */
78pio2_hi =		1.5707962513e0,		/*  0xc90fda.0p-23 */
79RECIP_EPSILON =		1 / FLT_EPSILON,
80SQRT_3_EPSILON =	5.9801995673e-4,	/*  0x9cc471.0p-34 */
81SQRT_6_EPSILON =	8.4572793338e-4,	/*  0xddb3d7.0p-34 */
82SQRT_MIN =		0x1p-63;
83
84static const volatile float
85pio2_lo =		7.5497899549e-8,	/*  0xa22169.0p-47 */
86tiny =			0x1p-100;
87
88static float complex clog_for_large_values(float complex z);
89
90static inline float
91f(float a, float b, float hypot_a_b)
92{
93	if (b < 0)
94		return ((hypot_a_b - b) / 2);
95	if (b == 0)
96		return (a / 2);
97	return (a * a / (hypot_a_b + b) / 2);
98}
99
100static inline void
101do_hard_work(float x, float y, float *rx, int *B_is_usable, float *B,
102    float *sqrt_A2my2, float *new_y)
103{
104	float R, S, A;
105	float Am1, Amy;
106
107	R = hypotf(x, y + 1);
108	S = hypotf(x, y - 1);
109
110	A = (R + S) / 2;
111	if (A < 1)
112		A = 1;
113
114	if (A < A_crossover) {
115		if (y == 1 && x < FLT_EPSILON * FLT_EPSILON / 128) {
116			*rx = sqrtf(x);
117		} else if (x >= FLT_EPSILON * fabsf(y - 1)) {
118			Am1 = f(x, 1 + y, R) + f(x, 1 - y, S);
119			*rx = log1pf(Am1 + sqrtf(Am1 * (A + 1)));
120		} else if (y < 1) {
121			*rx = x / sqrtf((1 - y) * (1 + y));
122		} else {
123			*rx = log1pf((y - 1) + sqrtf((y - 1) * (y + 1)));
124		}
125	} else {
126		*rx = logf(A + sqrtf(A * A - 1));
127	}
128
129	*new_y = y;
130
131	if (y < FOUR_SQRT_MIN) {
132		*B_is_usable = 0;
133		*sqrt_A2my2 = A * (2 / FLT_EPSILON);
134		*new_y = y * (2 / FLT_EPSILON);
135		return;
136	}
137
138	*B = y / A;
139	*B_is_usable = 1;
140
141	if (*B > B_crossover) {
142		*B_is_usable = 0;
143		if (y == 1 && x < FLT_EPSILON / 128) {
144			*sqrt_A2my2 = sqrtf(x) * sqrtf((A + y) / 2);
145		} else if (x >= FLT_EPSILON * fabsf(y - 1)) {
146			Amy = f(x, y + 1, R) + f(x, y - 1, S);
147			*sqrt_A2my2 = sqrtf(Amy * (A + y));
148		} else if (y > 1) {
149			*sqrt_A2my2 = x * (4 / FLT_EPSILON / FLT_EPSILON) * y /
150			    sqrtf((y + 1) * (y - 1));
151			*new_y = y * (4 / FLT_EPSILON / FLT_EPSILON);
152		} else {
153			*sqrt_A2my2 = sqrtf((1 - y) * (1 + y));
154		}
155	}
156}
157
158float complex
159casinhf(float complex z)
160{
161	float x, y, ax, ay, rx, ry, B, sqrt_A2my2, new_y;
162	int B_is_usable;
163	float complex w;
164
165	x = crealf(z);
166	y = cimagf(z);
167	ax = fabsf(x);
168	ay = fabsf(y);
169
170	if (isnan(x) || isnan(y)) {
171		if (isinf(x))
172			return (CMPLXF(x, y + y));
173		if (isinf(y))
174			return (CMPLXF(y, x + x));
175		if (y == 0)
176			return (CMPLXF(x + x, y));
177		return (CMPLXF(x + 0.0L + (y + 0), x + 0.0L + (y + 0)));
178	}
179
180	if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) {
181		if (signbit(x) == 0)
182			w = clog_for_large_values(z) + m_ln2;
183		else
184			w = clog_for_large_values(-z) + m_ln2;
185		return (CMPLXF(copysignf(crealf(w), x),
186		    copysignf(cimagf(w), y)));
187	}
188
189	if (x == 0 && y == 0)
190		return (z);
191
192	raise_inexact();
193
194	if (ax < SQRT_6_EPSILON / 4 && ay < SQRT_6_EPSILON / 4)
195		return (z);
196
197	do_hard_work(ax, ay, &rx, &B_is_usable, &B, &sqrt_A2my2, &new_y);
198	if (B_is_usable)
199		ry = asinf(B);
200	else
201		ry = atan2f(new_y, sqrt_A2my2);
202	return (CMPLXF(copysignf(rx, x), copysignf(ry, y)));
203}
204
205float complex
206casinf(float complex z)
207{
208	float complex w = casinhf(CMPLXF(cimagf(z), crealf(z)));
209
210	return (CMPLXF(cimagf(w), crealf(w)));
211}
212
213float complex
214cacosf(float complex z)
215{
216	float x, y, ax, ay, rx, ry, B, sqrt_A2mx2, new_x;
217	int sx, sy;
218	int B_is_usable;
219	float complex w;
220
221	x = crealf(z);
222	y = cimagf(z);
223	sx = signbit(x);
224	sy = signbit(y);
225	ax = fabsf(x);
226	ay = fabsf(y);
227
228	if (isnan(x) || isnan(y)) {
229		if (isinf(x))
230			return (CMPLXF(y + y, -INFINITY));
231		if (isinf(y))
232			return (CMPLXF(x + x, -y));
233		if (x == 0)
234			return (CMPLXF(pio2_hi + pio2_lo, y + y));
235		return (CMPLXF(x + 0.0L + (y + 0), x + 0.0L + (y + 0)));
236	}
237
238	if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) {
239		w = clog_for_large_values(z);
240		rx = fabsf(cimagf(w));
241		ry = crealf(w) + m_ln2;
242		if (sy == 0)
243			ry = -ry;
244		return (CMPLXF(rx, ry));
245	}
246
247	if (x == 1 && y == 0)
248		return (CMPLXF(0, -y));
249
250	raise_inexact();
251
252	if (ax < SQRT_6_EPSILON / 4 && ay < SQRT_6_EPSILON / 4)
253		return (CMPLXF(pio2_hi - (x - pio2_lo), -y));
254
255	do_hard_work(ay, ax, &ry, &B_is_usable, &B, &sqrt_A2mx2, &new_x);
256	if (B_is_usable) {
257		if (sx == 0)
258			rx = acosf(B);
259		else
260			rx = acosf(-B);
261	} else {
262		if (sx == 0)
263			rx = atan2f(sqrt_A2mx2, new_x);
264		else
265			rx = atan2f(sqrt_A2mx2, -new_x);
266	}
267	if (sy == 0)
268		ry = -ry;
269	return (CMPLXF(rx, ry));
270}
271
272float complex
273cacoshf(float complex z)
274{
275	float complex w;
276	float rx, ry;
277
278	w = cacosf(z);
279	rx = crealf(w);
280	ry = cimagf(w);
281	if (isnan(rx) && isnan(ry))
282		return (CMPLXF(ry, rx));
283	if (isnan(rx))
284		return (CMPLXF(fabsf(ry), rx));
285	if (isnan(ry))
286		return (CMPLXF(ry, ry));
287	return (CMPLXF(fabsf(ry), copysignf(rx, cimagf(z))));
288}
289
290static float complex
291clog_for_large_values(float complex z)
292{
293	float x, y;
294	float ax, ay, t;
295
296	x = crealf(z);
297	y = cimagf(z);
298	ax = fabsf(x);
299	ay = fabsf(y);
300	if (ax < ay) {
301		t = ax;
302		ax = ay;
303		ay = t;
304	}
305
306	if (ax > FLT_MAX / 2)
307		return (CMPLXF(logf(hypotf(x / m_e, y / m_e)) + 1,
308		    atan2f(y, x)));
309
310	if (ax > QUARTER_SQRT_MAX || ay < SQRT_MIN)
311		return (CMPLXF(logf(hypotf(x, y)), atan2f(y, x)));
312
313	return (CMPLXF(logf(ax * ax + ay * ay) / 2, atan2f(y, x)));
314}
315
316static inline float
317sum_squares(float x, float y)
318{
319
320	if (y < SQRT_MIN)
321		return (x * x);
322
323	return (x * x + y * y);
324}
325
326static inline float
327real_part_reciprocal(float x, float y)
328{
329	float scale;
330	uint32_t hx, hy;
331	int32_t ix, iy;
332
333	GET_FLOAT_WORD(hx, x);
334	ix = hx & 0x7f800000;
335	GET_FLOAT_WORD(hy, y);
336	iy = hy & 0x7f800000;
337#define	BIAS	(FLT_MAX_EXP - 1)
338#define	CUTOFF	(FLT_MANT_DIG / 2 + 1)
339	if (ix - iy >= CUTOFF << 23 || isinf(x))
340		return (1 / x);
341	if (iy - ix >= CUTOFF << 23)
342		return (x / y / y);
343	if (ix <= (BIAS + FLT_MAX_EXP / 2 - CUTOFF) << 23)
344		return (x / (x * x + y * y));
345	SET_FLOAT_WORD(scale, 0x7f800000 - ix);
346	x *= scale;
347	y *= scale;
348	return (x / (x * x + y * y) * scale);
349}
350
351float complex
352catanhf(float complex z)
353{
354	float x, y, ax, ay, rx, ry;
355
356	x = crealf(z);
357	y = cimagf(z);
358	ax = fabsf(x);
359	ay = fabsf(y);
360
361	if (y == 0 && ax <= 1)
362		return (CMPLXF(atanhf(x), y));
363
364	if (x == 0)
365		return (CMPLXF(x, atanf(y)));
366
367	if (isnan(x) || isnan(y)) {
368		if (isinf(x))
369			return (CMPLXF(copysignf(0, x), y + y));
370		if (isinf(y))
371			return (CMPLXF(copysignf(0, x),
372			    copysignf(pio2_hi + pio2_lo, y)));
373		return (CMPLXF(x + 0.0L + (y + 0), x + 0.0L + (y + 0)));
374	}
375
376	if (ax > RECIP_EPSILON || ay > RECIP_EPSILON)
377		return (CMPLXF(real_part_reciprocal(x, y),
378		    copysignf(pio2_hi + pio2_lo, y)));
379
380	if (ax < SQRT_3_EPSILON / 2 && ay < SQRT_3_EPSILON / 2) {
381		raise_inexact();
382		return (z);
383	}
384
385	if (ax == 1 && ay < FLT_EPSILON)
386		rx = (m_ln2 - logf(ay)) / 2;
387	else
388		rx = log1pf(4 * ax / sum_squares(ax - 1, ay)) / 4;
389
390	if (ax == 1)
391		ry = atan2f(2, -ay) / 2;
392	else if (ay < FLT_EPSILON)
393		ry = atan2f(2 * ay, (1 - ax) * (1 + ax)) / 2;
394	else
395		ry = atan2f(2 * ay, (1 - ax) * (1 + ax) - ay * ay) / 2;
396
397	return (CMPLXF(copysignf(rx, x), copysignf(ry, y)));
398}
399
400float complex
401catanf(float complex z)
402{
403	float complex w = catanhf(CMPLXF(cimagf(z), crealf(z)));
404
405	return (CMPLXF(cimagf(w), crealf(w)));
406}
407