catrigf.c revision 251121 
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 * The code for catrig.c contains complete comments.
35 */
36
37#include <sys/cdefs.h>
38__FBSDID("$FreeBSD: head/lib/msun/src/catrigf.c 251121 2013-05-30 04:49:26Z das $");
39
40#include <complex.h>
41#include <float.h>
42
43#include "math.h"
44#include "math_private.h"
45
46#undef isinf
47#define isinf(x)	(fabsf(x) == INFINITY)
48#undef isnan
49#define isnan(x)	((x) != (x))
50#define	raise_inexact()	do { volatile float junk = 1 + tiny; } while(0)
51#undef signbit
52#define signbit(x)	(__builtin_signbitf(x))
53
54static const float
55A_crossover =		10,
56B_crossover =		0.6417,
57FOUR_SQRT_MIN =		0x1p-61,
58QUARTER_SQRT_MAX =	0x1p61,
59m_e =			2.7182818285e0,		/*  0xadf854.0p-22 */
60m_ln2 =			6.9314718056e-1,	/*  0xb17218.0p-24 */
61pio2_hi =		1.5707962513e0,		/*  0xc90fda.0p-23 */
62RECIP_EPSILON =		1 / FLT_EPSILON,
63SQRT_3_EPSILON =	5.9801995673e-4,	/*  0x9cc471.0p-34 */
64SQRT_6_EPSILON =	8.4572793338e-4,	/*  0xddb3d7.0p-34 */
65SQRT_MIN =		0x1p-63;
66
67static const volatile float
68pio2_lo =		7.5497899549e-8,	/*  0xa22169.0p-47 */
69tiny =			0x1p-100;
70
71static float complex clog_for_large_values(float complex z);
72
73static inline float
74f(float a, float b, float hypot_a_b)
75{
76	if (b < 0)
77		return ((hypot_a_b - b) / 2);
78	if (b == 0)
79		return (a / 2);
80	return (a * a / (hypot_a_b + b) / 2);
81}
82
83static inline void
84do_hard_work(float x, float y, float *rx, int *B_is_usable, float *B,
85	     float *sqrt_A2my2, float *new_y)
86{
87	float R, S, A;
88	float Am1, Amy;
89
90	R = hypotf(x, y + 1);
91	S = hypotf(x, y - 1);
92
93	A = (R + S) / 2;
94	if (A < 1)
95		A = 1;
96
97	if (A < A_crossover) {
98		if (y == 1 && x < FLT_EPSILON * FLT_EPSILON / 128) {
99			*rx = sqrtf(x);
100		} else if (x >= FLT_EPSILON * fabsf(y - 1)) {
101			Am1 = f(x, 1 + y, R) + f(x, 1 - y, S);
102			*rx = log1pf(Am1 + sqrtf(Am1 * (A + 1)));
103		} else if (y < 1) {
104			*rx = x / sqrtf((1 - y)*(1 + y));
105		} else {
106			*rx = log1pf((y - 1) + sqrtf((y - 1) * (y + 1)));
107		}
108	} else {
109		*rx = logf(A + sqrtf(A * A - 1));
110	}
111
112	*new_y = y;
113
114	if (y < FOUR_SQRT_MIN) {
115		*B_is_usable = 0;
116		*sqrt_A2my2 = A * (2 / FLT_EPSILON);
117		*new_y = y * (2 / FLT_EPSILON);
118		return;
119	}
120
121	*B = y / A;
122	*B_is_usable = 1;
123
124	if (*B > B_crossover) {
125		*B_is_usable = 0;
126		if (y == 1 && x < FLT_EPSILON / 128) {
127			*sqrt_A2my2 = sqrtf(x) * sqrtf((A + y) / 2);
128		} else if (x >= FLT_EPSILON * fabsf(y - 1)) {
129			Amy = f(x, y + 1, R) + f(x, y - 1, S);
130			*sqrt_A2my2 = sqrtf(Amy * (A + y));
131		} else if (y > 1) {
132			*sqrt_A2my2 = x * (4 / FLT_EPSILON / FLT_EPSILON) * y /
133				sqrtf((y + 1) * (y - 1));
134			*new_y = y * (4 / FLT_EPSILON / FLT_EPSILON);
135		} else {
136			*sqrt_A2my2 = sqrtf((1 - y) * (1 + y));
137		}
138	}
139}
140
141float complex
142casinhf(float complex z)
143{
144	float x, y, ax, ay, rx, ry, B, sqrt_A2my2, new_y;
145	int B_is_usable;
146	float complex w;
147
148	x = crealf(z);
149	y = cimagf(z);
150	ax = fabsf(x);
151	ay = fabsf(y);
152
153	if (isnan(x) || isnan(y)) {
154		if (isinf(x))
155			return (cpackf(x, y + y));
156		if (isinf(y))
157			return (cpackf(y, x + x));
158		if (y == 0)
159			return (cpackf(x + x, y));
160		return (cpackf(x + 0.0L + (y + 0), x + 0.0L + (y + 0)));
161	}
162
163	if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) {
164		if (signbit(x) == 0)
165			w = clog_for_large_values(z) + m_ln2;
166		else
167			w = clog_for_large_values(-z) + m_ln2;
168		return (cpackf(copysignf(crealf(w), x),
169			       copysignf(cimagf(w), y)));
170	}
171
172	if (x == 0 && y == 0)
173		return (z);
174
175	raise_inexact();
176
177	if (ax < SQRT_6_EPSILON / 4 && ay < SQRT_6_EPSILON / 4)
178		return (z);
179
180	do_hard_work(ax, ay, &rx, &B_is_usable, &B, &sqrt_A2my2, &new_y);
181	if (B_is_usable)
182		ry = asinf(B);
183	else
184		ry = atan2f(new_y, sqrt_A2my2);
185	return (cpackf(copysignf(rx, x), copysignf(ry, y)));
186}
187
188float complex
189casinf(float complex z)
190{
191	float complex w = casinhf(cpackf(cimagf(z), crealf(z)));
192	return (cpackf(cimagf(w), crealf(w)));
193}
194
195float complex
196cacosf(float complex z)
197{
198	float x, y, ax, ay, rx, ry, B, sqrt_A2mx2, new_x;
199	int sx, sy;
200	int B_is_usable;
201	float complex w;
202
203	x = crealf(z);
204	y = cimagf(z);
205	sx = signbit(x);
206	sy = signbit(y);
207	ax = fabsf(x);
208	ay = fabsf(y);
209
210	if (isnan(x) || isnan(y)) {
211		if (isinf(x))
212			return (cpackf(y + y, -INFINITY));
213		if (isinf(y))
214			return (cpackf(x + x, -y));
215		if (x == 0) return (cpackf(pio2_hi + pio2_lo, y + y));
216		return (cpackf(x + 0.0L + (y + 0), x + 0.0L + (y + 0)));
217	}
218
219	if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) {
220		w = clog_for_large_values(z);
221		rx = fabsf(cimagf(w));
222		ry = crealf(w) + m_ln2;
223		if (sy == 0)
224			ry = -ry;
225		return (cpackf(rx, ry));
226	}
227
228	if (x == 1 && y == 0)
229		return (cpackf(0, -y));
230
231	raise_inexact();
232
233	if (ax < SQRT_6_EPSILON / 4 && ay < SQRT_6_EPSILON / 4)
234		return (cpackf(pio2_hi - (x - pio2_lo), -y));
235
236	do_hard_work(ay, ax, &ry, &B_is_usable, &B, &sqrt_A2mx2, &new_x);
237	if (B_is_usable) {
238		if (sx==0)
239			rx = acosf(B);
240		else
241			rx = acosf(-B);
242	} else {
243		if (sx==0)
244			rx = atan2f(sqrt_A2mx2, new_x);
245		else
246			rx = atan2f(sqrt_A2mx2, -new_x);
247	}
248	if (sy==0)
249		ry = -ry;
250	return (cpackf(rx, ry));
251}
252
253float complex
254cacoshf(float complex z)
255{
256	float complex w;
257	float rx, ry;
258
259	w = cacosf(z);
260	rx = crealf(w);
261	ry = cimagf(w);
262	if (isnan(rx) && isnan(ry))
263		return (cpackf(ry, rx));
264	if (isnan(rx))
265		return (cpackf(fabsf(ry), rx));
266	if (isnan(ry))
267		return (cpackf(ry, ry));
268	return (cpackf(fabsf(ry), copysignf(rx, cimagf(z))));
269}
270
271static float complex
272clog_for_large_values(float complex z)
273{
274	float x, y;
275	float ax, ay, t;
276
277	x = crealf(z);
278	y = cimagf(z);
279	ax = fabsf(x);
280	ay = fabsf(y);
281	if (ax < ay) {
282		t = ax;
283		ax = ay;
284		ay = t;
285	}
286
287	if (ax > FLT_MAX / 2) {
288		return (cpackf(logf(hypotf(x / m_e, y / m_e)) + 1,
289			       atan2f(y, x)));
290	}
291
292	if (ax > QUARTER_SQRT_MAX || ay < SQRT_MIN)
293		return (cpackf(logf(hypotf(x, y)), atan2f(y, x)));
294
295	return (cpackf(logf(ax * ax + ay * ay) / 2, atan2f(y, x)));
296}
297
298static inline float
299sum_squares(float x, float y)
300{
301
302	if (y < SQRT_MIN)
303		return (x*x);
304	return (x*x + y*y);
305}
306
307static inline float
308real_part_reciprocal(float x, float y)
309{
310	float scale;
311	uint32_t hx, hy;
312	int32_t ix, iy;
313
314	GET_FLOAT_WORD(hx, x);
315	ix = hx & 0x7f800000;
316	GET_FLOAT_WORD(hy, y);
317	iy = hy & 0x7f800000;
318#define	BIAS	(FLT_MAX_EXP - 1)
319#define	CUTOFF	(FLT_MANT_DIG / 2 + 1)
320	if (ix - iy >= CUTOFF << 23 || isinf(x))
321		return (1/x);
322	if (iy - ix >= CUTOFF << 23)
323		return (x/y/y);
324	if (ix <= (BIAS + FLT_MAX_EXP / 2 - CUTOFF) << 23)
325		return (x / (x * x + y * y));
326	SET_FLOAT_WORD(scale, 0x7f800000 - ix);
327	x *= scale;
328	y *= scale;
329	return (x / (x * x + y * y) * scale);
330}
331
332float complex
333catanhf(float complex z)
334{
335	float x, y, ax, ay, rx, ry;
336
337	x = crealf(z);
338	y = cimagf(z);
339	ax = fabsf(x);
340	ay = fabsf(y);
341
342	if (y == 0 && ax <= 1)
343		return (cpackf(atanhf(x), y));
344
345	if (x == 0)
346		return (cpackf(x, atanf(y)));
347
348	if (isnan(x) || isnan(y)) {
349		if (isinf(x))
350			return (cpackf(copysignf(0, x), y+y));
351		if (isinf(y)) {
352			return (cpackf(copysignf(0, x),
353				       copysignf(pio2_hi + pio2_lo, y)));
354		}
355		return (cpackf(x + 0.0L + (y + 0), x + 0.0L + (y + 0)));
356	}
357
358	if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) {
359		return (cpackf(real_part_reciprocal(x, y),
360			       copysignf(pio2_hi + pio2_lo, y)));
361	}
362
363	if (ax < SQRT_3_EPSILON / 2 && ay < SQRT_3_EPSILON / 2) {
364		raise_inexact();
365		return (z);
366	}
367
368	if (ax == 1 && ay < FLT_EPSILON)
369		rx = (logf(ay) - m_ln2) / -2;
370	else
371		rx = log1pf(4 * ax / sum_squares(ax - 1, ay)) / 4;
372
373	if (ax == 1)
374		ry = atan2f(2, -ay) / 2;
375	else if (ay < FLT_EPSILON)
376		ry = atan2f(2 * ay, (1 - ax) * (1 + ax)) / 2;
377	else
378		ry = atan2f(2 * ay, (1 - ax) * (1 + ax) - ay * ay) / 2;
379
380	return (cpackf(copysignf(rx, x), copysignf(ry, y)));
381}
382
383float complex
384catanf(float complex z)
385{
386	float complex w = catanhf(cpackf(cimagf(z), crealf(z)));
387	return (cpackf(cimagf(w), crealf(w)));
388}
389