catrigf.c revision 331722 
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/*
28 * The algorithm is very close to that in "Implementing the complex arcsine
29 * and arccosine functions using exception handling" by T. E. Hull, Thomas F.
30 * Fairgrieve, and Ping Tak Peter Tang, published in ACM Transactions on
31 * Mathematical Software, Volume 23 Issue 3, 1997, Pages 299-335,
32 * http://dl.acm.org/citation.cfm?id=275324.
33 *
34 * See catrig.c for complete comments.
35 *
36 * XXX comments were removed automatically, and even short ones on the right
37 * of statements were removed (all of them), contrary to normal style.  Only
38 * a few comments on the right of declarations remain.
39 */
40
41#include <sys/cdefs.h>
42__FBSDID("$FreeBSD: stable/11/lib/msun/src/catrigf.c 331722 2018-03-29 02:50:57Z eadler $");
43
44#include <complex.h>
45#include <float.h>
46
47#include "math.h"
48#include "math_private.h"
49
50#undef isinf
51#define isinf(x)	(fabsf(x) == INFINITY)
52#undef isnan
53#define isnan(x)	((x) != (x))
54#define	raise_inexact()	do { volatile float junk __unused = 1 + tiny; } while(0)
55#undef signbit
56#define signbit(x)	(__builtin_signbitf(x))
57
58static const float
59A_crossover =		10,
60B_crossover =		0.6417,
61FOUR_SQRT_MIN =		0x1p-61,
62QUARTER_SQRT_MAX =	0x1p61,
63m_e =			2.7182818285e0,		/*  0xadf854.0p-22 */
64m_ln2 =			6.9314718056e-1,	/*  0xb17218.0p-24 */
65pio2_hi =		1.5707962513e0,		/*  0xc90fda.0p-23 */
66RECIP_EPSILON =		1 / FLT_EPSILON,
67SQRT_3_EPSILON =	5.9801995673e-4,	/*  0x9cc471.0p-34 */
68SQRT_6_EPSILON =	8.4572793338e-4,	/*  0xddb3d7.0p-34 */
69SQRT_MIN =		0x1p-63;
70
71static const volatile float
72pio2_lo =		7.5497899549e-8,	/*  0xa22169.0p-47 */
73tiny =			0x1p-100;
74
75static float complex clog_for_large_values(float complex z);
76
77static inline float
78f(float a, float b, float hypot_a_b)
79{
80	if (b < 0)
81		return ((hypot_a_b - b) / 2);
82	if (b == 0)
83		return (a / 2);
84	return (a * a / (hypot_a_b + b) / 2);
85}
86
87static inline void
88do_hard_work(float x, float y, float *rx, int *B_is_usable, float *B,
89    float *sqrt_A2my2, float *new_y)
90{
91	float R, S, A;
92	float Am1, Amy;
93
94	R = hypotf(x, y + 1);
95	S = hypotf(x, y - 1);
96
97	A = (R + S) / 2;
98	if (A < 1)
99		A = 1;
100
101	if (A < A_crossover) {
102		if (y == 1 && x < FLT_EPSILON * FLT_EPSILON / 128) {
103			*rx = sqrtf(x);
104		} else if (x >= FLT_EPSILON * fabsf(y - 1)) {
105			Am1 = f(x, 1 + y, R) + f(x, 1 - y, S);
106			*rx = log1pf(Am1 + sqrtf(Am1 * (A + 1)));
107		} else if (y < 1) {
108			*rx = x / sqrtf((1 - y) * (1 + y));
109		} else {
110			*rx = log1pf((y - 1) + sqrtf((y - 1) * (y + 1)));
111		}
112	} else {
113		*rx = logf(A + sqrtf(A * A - 1));
114	}
115
116	*new_y = y;
117
118	if (y < FOUR_SQRT_MIN) {
119		*B_is_usable = 0;
120		*sqrt_A2my2 = A * (2 / FLT_EPSILON);
121		*new_y = y * (2 / FLT_EPSILON);
122		return;
123	}
124
125	*B = y / A;
126	*B_is_usable = 1;
127
128	if (*B > B_crossover) {
129		*B_is_usable = 0;
130		if (y == 1 && x < FLT_EPSILON / 128) {
131			*sqrt_A2my2 = sqrtf(x) * sqrtf((A + y) / 2);
132		} else if (x >= FLT_EPSILON * fabsf(y - 1)) {
133			Amy = f(x, y + 1, R) + f(x, y - 1, S);
134			*sqrt_A2my2 = sqrtf(Amy * (A + y));
135		} else if (y > 1) {
136			*sqrt_A2my2 = x * (4 / FLT_EPSILON / FLT_EPSILON) * y /
137			    sqrtf((y + 1) * (y - 1));
138			*new_y = y * (4 / FLT_EPSILON / FLT_EPSILON);
139		} else {
140			*sqrt_A2my2 = sqrtf((1 - y) * (1 + y));
141		}
142	}
143}
144
145float complex
146casinhf(float complex z)
147{
148	float x, y, ax, ay, rx, ry, B, sqrt_A2my2, new_y;
149	int B_is_usable;
150	float complex w;
151
152	x = crealf(z);
153	y = cimagf(z);
154	ax = fabsf(x);
155	ay = fabsf(y);
156
157	if (isnan(x) || isnan(y)) {
158		if (isinf(x))
159			return (CMPLXF(x, y + y));
160		if (isinf(y))
161			return (CMPLXF(y, x + x));
162		if (y == 0)
163			return (CMPLXF(x + x, y));
164		return (CMPLXF(x + 0.0L + (y + 0), x + 0.0L + (y + 0)));
165	}
166
167	if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) {
168		if (signbit(x) == 0)
169			w = clog_for_large_values(z) + m_ln2;
170		else
171			w = clog_for_large_values(-z) + m_ln2;
172		return (CMPLXF(copysignf(crealf(w), x),
173		    copysignf(cimagf(w), y)));
174	}
175
176	if (x == 0 && y == 0)
177		return (z);
178
179	raise_inexact();
180
181	if (ax < SQRT_6_EPSILON / 4 && ay < SQRT_6_EPSILON / 4)
182		return (z);
183
184	do_hard_work(ax, ay, &rx, &B_is_usable, &B, &sqrt_A2my2, &new_y);
185	if (B_is_usable)
186		ry = asinf(B);
187	else
188		ry = atan2f(new_y, sqrt_A2my2);
189	return (CMPLXF(copysignf(rx, x), copysignf(ry, y)));
190}
191
192float complex
193casinf(float complex z)
194{
195	float complex w = casinhf(CMPLXF(cimagf(z), crealf(z)));
196
197	return (CMPLXF(cimagf(w), crealf(w)));
198}
199
200float complex
201cacosf(float complex z)
202{
203	float x, y, ax, ay, rx, ry, B, sqrt_A2mx2, new_x;
204	int sx, sy;
205	int B_is_usable;
206	float complex w;
207
208	x = crealf(z);
209	y = cimagf(z);
210	sx = signbit(x);
211	sy = signbit(y);
212	ax = fabsf(x);
213	ay = fabsf(y);
214
215	if (isnan(x) || isnan(y)) {
216		if (isinf(x))
217			return (CMPLXF(y + y, -INFINITY));
218		if (isinf(y))
219			return (CMPLXF(x + x, -y));
220		if (x == 0)
221			return (CMPLXF(pio2_hi + pio2_lo, y + y));
222		return (CMPLXF(x + 0.0L + (y + 0), x + 0.0L + (y + 0)));
223	}
224
225	if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) {
226		w = clog_for_large_values(z);
227		rx = fabsf(cimagf(w));
228		ry = crealf(w) + m_ln2;
229		if (sy == 0)
230			ry = -ry;
231		return (CMPLXF(rx, ry));
232	}
233
234	if (x == 1 && y == 0)
235		return (CMPLXF(0, -y));
236
237	raise_inexact();
238
239	if (ax < SQRT_6_EPSILON / 4 && ay < SQRT_6_EPSILON / 4)
240		return (CMPLXF(pio2_hi - (x - pio2_lo), -y));
241
242	do_hard_work(ay, ax, &ry, &B_is_usable, &B, &sqrt_A2mx2, &new_x);
243	if (B_is_usable) {
244		if (sx == 0)
245			rx = acosf(B);
246		else
247			rx = acosf(-B);
248	} else {
249		if (sx == 0)
250			rx = atan2f(sqrt_A2mx2, new_x);
251		else
252			rx = atan2f(sqrt_A2mx2, -new_x);
253	}
254	if (sy == 0)
255		ry = -ry;
256	return (CMPLXF(rx, ry));
257}
258
259float complex
260cacoshf(float complex z)
261{
262	float complex w;
263	float rx, ry;
264
265	w = cacosf(z);
266	rx = crealf(w);
267	ry = cimagf(w);
268	if (isnan(rx) && isnan(ry))
269		return (CMPLXF(ry, rx));
270	if (isnan(rx))
271		return (CMPLXF(fabsf(ry), rx));
272	if (isnan(ry))
273		return (CMPLXF(ry, ry));
274	return (CMPLXF(fabsf(ry), copysignf(rx, cimagf(z))));
275}
276
277static float complex
278clog_for_large_values(float complex z)
279{
280	float x, y;
281	float ax, ay, t;
282
283	x = crealf(z);
284	y = cimagf(z);
285	ax = fabsf(x);
286	ay = fabsf(y);
287	if (ax < ay) {
288		t = ax;
289		ax = ay;
290		ay = t;
291	}
292
293	if (ax > FLT_MAX / 2)
294		return (CMPLXF(logf(hypotf(x / m_e, y / m_e)) + 1,
295		    atan2f(y, x)));
296
297	if (ax > QUARTER_SQRT_MAX || ay < SQRT_MIN)
298		return (CMPLXF(logf(hypotf(x, y)), atan2f(y, x)));
299
300	return (CMPLXF(logf(ax * ax + ay * ay) / 2, atan2f(y, x)));
301}
302
303static inline float
304sum_squares(float x, float y)
305{
306
307	if (y < SQRT_MIN)
308		return (x * x);
309
310	return (x * x + y * y);
311}
312
313static inline float
314real_part_reciprocal(float x, float y)
315{
316	float scale;
317	uint32_t hx, hy;
318	int32_t ix, iy;
319
320	GET_FLOAT_WORD(hx, x);
321	ix = hx & 0x7f800000;
322	GET_FLOAT_WORD(hy, y);
323	iy = hy & 0x7f800000;
324#define	BIAS	(FLT_MAX_EXP - 1)
325#define	CUTOFF	(FLT_MANT_DIG / 2 + 1)
326	if (ix - iy >= CUTOFF << 23 || isinf(x))
327		return (1 / x);
328	if (iy - ix >= CUTOFF << 23)
329		return (x / y / y);
330	if (ix <= (BIAS + FLT_MAX_EXP / 2 - CUTOFF) << 23)
331		return (x / (x * x + y * y));
332	SET_FLOAT_WORD(scale, 0x7f800000 - ix);
333	x *= scale;
334	y *= scale;
335	return (x / (x * x + y * y) * scale);
336}
337
338float complex
339catanhf(float complex z)
340{
341	float x, y, ax, ay, rx, ry;
342
343	x = crealf(z);
344	y = cimagf(z);
345	ax = fabsf(x);
346	ay = fabsf(y);
347
348	if (y == 0 && ax <= 1)
349		return (CMPLXF(atanhf(x), y));
350
351	if (x == 0)
352		return (CMPLXF(x, atanf(y)));
353
354	if (isnan(x) || isnan(y)) {
355		if (isinf(x))
356			return (CMPLXF(copysignf(0, x), y + y));
357		if (isinf(y))
358			return (CMPLXF(copysignf(0, x),
359			    copysignf(pio2_hi + pio2_lo, y)));
360		return (CMPLXF(x + 0.0L + (y + 0), x + 0.0L + (y + 0)));
361	}
362
363	if (ax > RECIP_EPSILON || ay > RECIP_EPSILON)
364		return (CMPLXF(real_part_reciprocal(x, y),
365		    copysignf(pio2_hi + pio2_lo, y)));
366
367	if (ax < SQRT_3_EPSILON / 2 && ay < SQRT_3_EPSILON / 2) {
368		raise_inexact();
369		return (z);
370	}
371
372	if (ax == 1 && ay < FLT_EPSILON)
373		rx = (m_ln2 - logf(ay)) / 2;
374	else
375		rx = log1pf(4 * ax / sum_squares(ax - 1, ay)) / 4;
376
377	if (ax == 1)
378		ry = atan2f(2, -ay) / 2;
379	else if (ay < FLT_EPSILON)
380		ry = atan2f(2 * ay, (1 - ax) * (1 + ax)) / 2;
381	else
382		ry = atan2f(2 * ay, (1 - ax) * (1 + ax) - ay * ay) / 2;
383
384	return (CMPLXF(copysignf(rx, x), copysignf(ry, y)));
385}
386
387float complex
388catanf(float complex z)
389{
390	float complex w = catanhf(CMPLXF(cimagf(z), crealf(z)));
391
392	return (CMPLXF(cimagf(w), crealf(w)));
393}
394