xref: /freebsd/lib/msun/src/catrigl.c (revision dd41de95a84d979615a2ef11df6850622bf6184e)
1 /*-
2  * Copyright (c) 2012 Stephen Montgomery-Smith <stephen@FreeBSD.ORG>
3  * Copyright (c) 2017 Mahdi Mokhtari <mmokhi@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 __FBSDID("$FreeBSD$");
44 
45 #include <complex.h>
46 #include <float.h>
47 
48 #include "invtrig.h"
49 #include "math.h"
50 #include "math_private.h"
51 
52 #undef isinf
53 #define isinf(x)	(fabsl(x) == INFINITY)
54 #undef isnan
55 #define isnan(x)	((x) != (x))
56 #define	raise_inexact()	do { volatile float junk __unused = 1 + tiny; } while(0)
57 #undef signbit
58 #define signbit(x)	(__builtin_signbitl(x))
59 
60 #if LDBL_MAX_EXP != 0x4000
61 #error "Unsupported long double format"
62 #endif
63 
64 static const long double
65 A_crossover =		10,
66 B_crossover =		0.6417,
67 FOUR_SQRT_MIN =		0x1p-8189L,
68 HALF_MAX =		0x1p16383L,
69 QUARTER_SQRT_MAX =	0x1p8189L,
70 RECIP_EPSILON =		1 / LDBL_EPSILON,
71 SQRT_MIN =		0x1p-8191L;
72 
73 #if LDBL_MANT_DIG == 64
74 static const union IEEEl2bits
75 um_e =		LD80C(0xadf85458a2bb4a9b,  1, 2.71828182845904523536e+0L),
76 um_ln2 =	LD80C(0xb17217f7d1cf79ac, -1, 6.93147180559945309417e-1L);
77 #define		m_e	um_e.e
78 #define		m_ln2	um_ln2.e
79 static const long double
80 /* The next 2 literals for non-i386.  Misrounding them on i386 is harmless. */
81 SQRT_3_EPSILON = 5.70316273435758915310e-10,	/*  0x9cc470a0490973e8.0p-94 */
82 SQRT_6_EPSILON = 8.06549008734932771664e-10;	/*  0xddb3d742c265539e.0p-94 */
83 #elif LDBL_MANT_DIG == 113
84 static const long double
85 m_e =		2.71828182845904523536028747135266250e0L,	/* 0x15bf0a8b1457695355fb8ac404e7a.0p-111 */
86 m_ln2 =		6.93147180559945309417232121458176568e-1L,	/* 0x162e42fefa39ef35793c7673007e6.0p-113 */
87 SQRT_3_EPSILON = 2.40370335797945490975336727199878124e-17,	/*  0x1bb67ae8584caa73b25742d7078b8.0p-168 */
88 SQRT_6_EPSILON = 3.39934988877629587239082586223300391e-17;	/*  0x13988e1409212e7d0321914321a55.0p-167 */
89 #else
90 #error "Unsupported long double format"
91 #endif
92 
93 static const volatile float
94 tiny =			0x1p-100;
95 
96 static long double complex clog_for_large_values(long double complex z);
97 
98 static inline long double
99 f(long double a, long double b, long double hypot_a_b)
100 {
101 	if (b < 0)
102 		return ((hypot_a_b - b) / 2);
103 	if (b == 0)
104 		return (a / 2);
105 	return (a * a / (hypot_a_b + b) / 2);
106 }
107 
108 static inline void
109 do_hard_work(long double x, long double y, long double *rx, int *B_is_usable,
110     long double *B, long double *sqrt_A2my2, long double *new_y)
111 {
112 	long double R, S, A;
113 	long double Am1, Amy;
114 
115 	R = hypotl(x, y + 1);
116 	S = hypotl(x, y - 1);
117 
118 	A = (R + S) / 2;
119 	if (A < 1)
120 		A = 1;
121 
122 	if (A < A_crossover) {
123 		if (y == 1 && x < LDBL_EPSILON * LDBL_EPSILON / 128) {
124 			*rx = sqrtl(x);
125 		} else if (x >= LDBL_EPSILON * fabsl(y - 1)) {
126 			Am1 = f(x, 1 + y, R) + f(x, 1 - y, S);
127 			*rx = log1pl(Am1 + sqrtl(Am1 * (A + 1)));
128 		} else if (y < 1) {
129 			*rx = x / sqrtl((1 - y) * (1 + y));
130 		} else {
131 			*rx = log1pl((y - 1) + sqrtl((y - 1) * (y + 1)));
132 		}
133 	} else {
134 		*rx = logl(A + sqrtl(A * A - 1));
135 	}
136 
137 	*new_y = y;
138 
139 	if (y < FOUR_SQRT_MIN) {
140 		*B_is_usable = 0;
141 		*sqrt_A2my2 = A * (2 / LDBL_EPSILON);
142 		*new_y = y * (2 / LDBL_EPSILON);
143 		return;
144 	}
145 
146 	*B = y / A;
147 	*B_is_usable = 1;
148 
149 	if (*B > B_crossover) {
150 		*B_is_usable = 0;
151 		if (y == 1 && x < LDBL_EPSILON / 128) {
152 			*sqrt_A2my2 = sqrtl(x) * sqrtl((A + y) / 2);
153 		} else if (x >= LDBL_EPSILON * fabsl(y - 1)) {
154 			Amy = f(x, y + 1, R) + f(x, y - 1, S);
155 			*sqrt_A2my2 = sqrtl(Amy * (A + y));
156 		} else if (y > 1) {
157 			*sqrt_A2my2 = x * (4 / LDBL_EPSILON / LDBL_EPSILON) * y /
158 			    sqrtl((y + 1) * (y - 1));
159 			*new_y = y * (4 / LDBL_EPSILON / LDBL_EPSILON);
160 		} else {
161 			*sqrt_A2my2 = sqrtl((1 - y) * (1 + y));
162 		}
163 	}
164 }
165 
166 long double complex
167 casinhl(long double complex z)
168 {
169 	long double x, y, ax, ay, rx, ry, B, sqrt_A2my2, new_y;
170 	int B_is_usable;
171 	long double complex w;
172 
173 	x = creall(z);
174 	y = cimagl(z);
175 	ax = fabsl(x);
176 	ay = fabsl(y);
177 
178 	if (isnan(x) || isnan(y)) {
179 		if (isinf(x))
180 			return (CMPLXL(x, y + y));
181 		if (isinf(y))
182 			return (CMPLXL(y, x + x));
183 		if (y == 0)
184 			return (CMPLXL(x + x, y));
185 		return (CMPLXL(nan_mix(x, y), nan_mix(x, y)));
186 	}
187 
188 	if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) {
189 		if (signbit(x) == 0)
190 			w = clog_for_large_values(z) + m_ln2;
191 		else
192 			w = clog_for_large_values(-z) + m_ln2;
193 		return (CMPLXL(copysignl(creall(w), x),
194 		    copysignl(cimagl(w), y)));
195 	}
196 
197 	if (x == 0 && y == 0)
198 		return (z);
199 
200 	raise_inexact();
201 
202 	if (ax < SQRT_6_EPSILON / 4 && ay < SQRT_6_EPSILON / 4)
203 		return (z);
204 
205 	do_hard_work(ax, ay, &rx, &B_is_usable, &B, &sqrt_A2my2, &new_y);
206 	if (B_is_usable)
207 		ry = asinl(B);
208 	else
209 		ry = atan2l(new_y, sqrt_A2my2);
210 	return (CMPLXL(copysignl(rx, x), copysignl(ry, y)));
211 }
212 
213 long double complex
214 casinl(long double complex z)
215 {
216 	long double complex w;
217 
218 	w = casinhl(CMPLXL(cimagl(z), creall(z)));
219 	return (CMPLXL(cimagl(w), creall(w)));
220 }
221 
222 long double complex
223 cacosl(long double complex z)
224 {
225 	long double x, y, ax, ay, rx, ry, B, sqrt_A2mx2, new_x;
226 	int sx, sy;
227 	int B_is_usable;
228 	long double complex w;
229 
230 	x = creall(z);
231 	y = cimagl(z);
232 	sx = signbit(x);
233 	sy = signbit(y);
234 	ax = fabsl(x);
235 	ay = fabsl(y);
236 
237 	if (isnan(x) || isnan(y)) {
238 		if (isinf(x))
239 			return (CMPLXL(y + y, -INFINITY));
240 		if (isinf(y))
241 			return (CMPLXL(x + x, -y));
242 		if (x == 0)
243 			return (CMPLXL(pio2_hi + pio2_lo, y + y));
244 		return (CMPLXL(nan_mix(x, y), nan_mix(x, y)));
245 	}
246 
247 	if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) {
248 		w = clog_for_large_values(z);
249 		rx = fabsl(cimagl(w));
250 		ry = creall(w) + m_ln2;
251 		if (sy == 0)
252 			ry = -ry;
253 		return (CMPLXL(rx, ry));
254 	}
255 
256 	if (x == 1 && y == 0)
257 		return (CMPLXL(0, -y));
258 
259 	raise_inexact();
260 
261 	if (ax < SQRT_6_EPSILON / 4 && ay < SQRT_6_EPSILON / 4)
262 		return (CMPLXL(pio2_hi - (x - pio2_lo), -y));
263 
264 	do_hard_work(ay, ax, &ry, &B_is_usable, &B, &sqrt_A2mx2, &new_x);
265 	if (B_is_usable) {
266 		if (sx == 0)
267 			rx = acosl(B);
268 		else
269 			rx = acosl(-B);
270 	} else {
271 		if (sx == 0)
272 			rx = atan2l(sqrt_A2mx2, new_x);
273 		else
274 			rx = atan2l(sqrt_A2mx2, -new_x);
275 	}
276 	if (sy == 0)
277 		ry = -ry;
278 	return (CMPLXL(rx, ry));
279 }
280 
281 long double complex
282 cacoshl(long double complex z)
283 {
284 	long double complex w;
285 	long double rx, ry;
286 
287 	w = cacosl(z);
288 	rx = creall(w);
289 	ry = cimagl(w);
290 	if (isnan(rx) && isnan(ry))
291 		return (CMPLXL(ry, rx));
292 	if (isnan(rx))
293 		return (CMPLXL(fabsl(ry), rx));
294 	if (isnan(ry))
295 		return (CMPLXL(ry, ry));
296 	return (CMPLXL(fabsl(ry), copysignl(rx, cimagl(z))));
297 }
298 
299 static long double complex
300 clog_for_large_values(long double complex z)
301 {
302 	long double x, y;
303 	long double ax, ay, t;
304 
305 	x = creall(z);
306 	y = cimagl(z);
307 	ax = fabsl(x);
308 	ay = fabsl(y);
309 	if (ax < ay) {
310 		t = ax;
311 		ax = ay;
312 		ay = t;
313 	}
314 
315 	if (ax > HALF_MAX)
316 		return (CMPLXL(logl(hypotl(x / m_e, y / m_e)) + 1,
317 		    atan2l(y, x)));
318 
319 	if (ax > QUARTER_SQRT_MAX || ay < SQRT_MIN)
320 		return (CMPLXL(logl(hypotl(x, y)), atan2l(y, x)));
321 
322 	return (CMPLXL(logl(ax * ax + ay * ay) / 2, atan2l(y, x)));
323 }
324 
325 static inline long double
326 sum_squares(long double x, long double y)
327 {
328 
329 	if (y < SQRT_MIN)
330 		return (x * x);
331 
332 	return (x * x + y * y);
333 }
334 
335 static inline long double
336 real_part_reciprocal(long double x, long double y)
337 {
338 	long double scale;
339 	uint16_t hx, hy;
340 	int16_t ix, iy;
341 
342 	GET_LDBL_EXPSIGN(hx, x);
343 	ix = hx & 0x7fff;
344 	GET_LDBL_EXPSIGN(hy, y);
345 	iy = hy & 0x7fff;
346 #define	BIAS	(LDBL_MAX_EXP - 1)
347 #define	CUTOFF	(LDBL_MANT_DIG / 2 + 1)
348 	if (ix - iy >= CUTOFF || isinf(x))
349 		return (1 / x);
350 	if (iy - ix >= CUTOFF)
351 		return (x / y / y);
352 	if (ix <= BIAS + LDBL_MAX_EXP / 2 - CUTOFF)
353 		return (x / (x * x + y * y));
354 	scale = 1;
355 	SET_LDBL_EXPSIGN(scale, 0x7fff - ix);
356 	x *= scale;
357 	y *= scale;
358 	return (x / (x * x + y * y) * scale);
359 }
360 
361 long double complex
362 catanhl(long double complex z)
363 {
364 	long double x, y, ax, ay, rx, ry;
365 
366 	x = creall(z);
367 	y = cimagl(z);
368 	ax = fabsl(x);
369 	ay = fabsl(y);
370 
371 	if (y == 0 && ax <= 1)
372 		return (CMPLXL(atanhl(x), y));
373 
374 	if (x == 0)
375 		return (CMPLXL(x, atanl(y)));
376 
377 	if (isnan(x) || isnan(y)) {
378 		if (isinf(x))
379 			return (CMPLXL(copysignl(0, x), y + y));
380 		if (isinf(y))
381 			return (CMPLXL(copysignl(0, x),
382 			    copysignl(pio2_hi + pio2_lo, y)));
383 		return (CMPLXL(nan_mix(x, y), nan_mix(x, y)));
384 	}
385 
386 	if (ax > RECIP_EPSILON || ay > RECIP_EPSILON)
387 		return (CMPLXL(real_part_reciprocal(x, y),
388 		    copysignl(pio2_hi + pio2_lo, y)));
389 
390 	if (ax < SQRT_3_EPSILON / 2 && ay < SQRT_3_EPSILON / 2) {
391 		raise_inexact();
392 		return (z);
393 	}
394 
395 	if (ax == 1 && ay < LDBL_EPSILON)
396 		rx = (m_ln2 - logl(ay)) / 2;
397 	else
398 		rx = log1pl(4 * ax / sum_squares(ax - 1, ay)) / 4;
399 
400 	if (ax == 1)
401 		ry = atan2l(2, -ay) / 2;
402 	else if (ay < LDBL_EPSILON)
403 		ry = atan2l(2 * ay, (1 - ax) * (1 + ax)) / 2;
404 	else
405 		ry = atan2l(2 * ay, (1 - ax) * (1 + ax) - ay * ay) / 2;
406 
407 	return (CMPLXL(copysignl(rx, x), copysignl(ry, y)));
408 }
409 
410 long double complex
411 catanl(long double complex z)
412 {
413 	long double complex w;
414 
415 	w = catanhl(CMPLXL(cimagl(z), creall(z)));
416 	return (CMPLXL(cimagl(w), creall(w)));
417 }
418