xref: /netbsd-src/lib/libm/complex/catrigf.c (revision 388550b026d49b7f7b7480b1113bf82bb8d6a480)
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 
71 static const float
72 A_crossover =		10,
73 B_crossover =		0.6417,
74 FOUR_SQRT_MIN =		0x1p-61,
75 QUARTER_SQRT_MAX =	0x1p61,
76 m_e =			2.7182818285e0,		/*  0xadf854.0p-22 */
77 m_ln2 =			6.9314718056e-1,	/*  0xb17218.0p-24 */
78 pio2_hi =		1.5707962513e0,		/*  0xc90fda.0p-23 */
79 RECIP_EPSILON =		1 / FLT_EPSILON,
80 SQRT_3_EPSILON =	5.9801995673e-4,	/*  0x9cc471.0p-34 */
81 SQRT_6_EPSILON =	8.4572793338e-4,	/*  0xddb3d7.0p-34 */
82 SQRT_MIN =		0x1p-63;
83 
84 static const volatile float
85 pio2_lo =		7.5497899549e-8,	/*  0xa22169.0p-47 */
86 tiny =			0x1p-100;
87 
88 static float complex clog_for_large_values(float complex z);
89 
90 static inline float
f(float a,float b,float hypot_a_b)91 f(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 
100 static inline void
do_hard_work(float x,float y,float * rx,int * B_is_usable,float * B,float * sqrt_A2my2,float * new_y)101 do_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 
158 float complex
casinhf(float complex z)159 casinhf(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 
205 float complex
casinf(float complex z)206 casinf(float complex z)
207 {
208 	float complex w = casinhf(CMPLXF(cimagf(z), crealf(z)));
209 
210 	return (CMPLXF(cimagf(w), crealf(w)));
211 }
212 
213 float complex
cacosf(float complex z)214 cacosf(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 
272 float complex
cacoshf(float complex z)273 cacoshf(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 
290 static float complex
clog_for_large_values(float complex z)291 clog_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 
316 static inline float
sum_squares(float x,float y)317 sum_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 
326 static inline float
real_part_reciprocal(float x,float y)327 real_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 
351 float complex
catanhf(float complex z)352 catanhf(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 
400 float complex
catanf(float complex z)401 catanf(float complex z)
402 {
403 	float complex w = catanhf(CMPLXF(cimagf(z), crealf(z)));
404 
405 	return (CMPLXF(cimagf(w), crealf(w)));
406 }
407