1 /*-
2 * SPDX-License-Identifier: BSD-2-Clause
3 *
4 * Copyright (c) 2012 Stephen Montgomery-Smith <stephen@FreeBSD.ORG>
5 * All rights reserved.
6 *
7 * Redistribution and use in source and binary forms, with or without
8 * modification, are permitted provided that the following conditions
9 * are met:
10 * 1. Redistributions of source code must retain the above copyright
11 * notice, this list of conditions and the following disclaimer.
12 * 2. Redistributions in binary form must reproduce the above copyright
13 * notice, this list of conditions and the following disclaimer in the
14 * documentation and/or other materials provided with the distribution.
15 *
16 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND
17 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
18 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
19 * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
20 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
21 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
22 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
23 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
24 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
25 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
26 * SUCH DAMAGE.
27 */
28
29 /*
30 * The algorithm is very close to that in "Implementing the complex arcsine
31 * and arccosine functions using exception handling" by T. E. Hull, Thomas F.
32 * Fairgrieve, and Ping Tak Peter Tang, published in ACM Transactions on
33 * Mathematical Software, Volume 23 Issue 3, 1997, Pages 299-335,
34 * http://dl.acm.org/citation.cfm?id=275324.
35 *
36 * See catrig.c for complete comments.
37 *
38 * XXX comments were removed automatically, and even short ones on the right
39 * of statements were removed (all of them), contrary to normal style. Only
40 * a few comments on the right of declarations remain.
41 */
42
43 #include <sys/cdefs.h>
44 __FBSDID("$FreeBSD$");
45
46 #include <complex.h>
47 #include <float.h>
48
49 #include "math.h"
50 #include "math_private.h"
51
52 #undef isinf
53 #define isinf(x) (fabsf(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_signbitf(x))
59
60 static const float
61 A_crossover = 10,
62 B_crossover = 0.6417,
63 FOUR_SQRT_MIN = 0x1p-61,
64 QUARTER_SQRT_MAX = 0x1p61,
65 m_e = 2.7182818285e0, /* 0xadf854.0p-22 */
66 m_ln2 = 6.9314718056e-1, /* 0xb17218.0p-24 */
67 pio2_hi = 1.5707962513e0, /* 0xc90fda.0p-23 */
68 RECIP_EPSILON = 1 / FLT_EPSILON,
69 SQRT_3_EPSILON = 5.9801995673e-4, /* 0x9cc471.0p-34 */
70 SQRT_6_EPSILON = 8.4572793338e-4, /* 0xddb3d7.0p-34 */
71 SQRT_MIN = 0x1p-63;
72
73 static const volatile float
74 pio2_lo = 7.5497899549e-8, /* 0xa22169.0p-47 */
75 tiny = 0x1p-100;
76
77 static float complex clog_for_large_values(float complex z);
78
79 static inline float
f(float a,float b,float hypot_a_b)80 f(float a, float b, float hypot_a_b)
81 {
82 if (b < 0)
83 return ((hypot_a_b - b) / 2);
84 if (b == 0)
85 return (a / 2);
86 return (a * a / (hypot_a_b + b) / 2);
87 }
88
89 static inline void
do_hard_work(float x,float y,float * rx,int * B_is_usable,float * B,float * sqrt_A2my2,float * new_y)90 do_hard_work(float x, float y, float *rx, int *B_is_usable, float *B,
91 float *sqrt_A2my2, float *new_y)
92 {
93 float R, S, A;
94 float Am1, Amy;
95
96 R = hypotf(x, y + 1);
97 S = hypotf(x, y - 1);
98
99 A = (R + S) / 2;
100 if (A < 1)
101 A = 1;
102
103 if (A < A_crossover) {
104 if (y == 1 && x < FLT_EPSILON * FLT_EPSILON / 128) {
105 *rx = sqrtf(x);
106 } else if (x >= FLT_EPSILON * fabsf(y - 1)) {
107 Am1 = f(x, 1 + y, R) + f(x, 1 - y, S);
108 *rx = log1pf(Am1 + sqrtf(Am1 * (A + 1)));
109 } else if (y < 1) {
110 *rx = x / sqrtf((1 - y) * (1 + y));
111 } else {
112 *rx = log1pf((y - 1) + sqrtf((y - 1) * (y + 1)));
113 }
114 } else {
115 *rx = logf(A + sqrtf(A * A - 1));
116 }
117
118 *new_y = y;
119
120 if (y < FOUR_SQRT_MIN) {
121 *B_is_usable = 0;
122 *sqrt_A2my2 = A * (2 / FLT_EPSILON);
123 *new_y = y * (2 / FLT_EPSILON);
124 return;
125 }
126
127 *B = y / A;
128 *B_is_usable = 1;
129
130 if (*B > B_crossover) {
131 *B_is_usable = 0;
132 if (y == 1 && x < FLT_EPSILON / 128) {
133 *sqrt_A2my2 = sqrtf(x) * sqrtf((A + y) / 2);
134 } else if (x >= FLT_EPSILON * fabsf(y - 1)) {
135 Amy = f(x, y + 1, R) + f(x, y - 1, S);
136 *sqrt_A2my2 = sqrtf(Amy * (A + y));
137 } else if (y > 1) {
138 *sqrt_A2my2 = x * (4 / FLT_EPSILON / FLT_EPSILON) * y /
139 sqrtf((y + 1) * (y - 1));
140 *new_y = y * (4 / FLT_EPSILON / FLT_EPSILON);
141 } else {
142 *sqrt_A2my2 = sqrtf((1 - y) * (1 + y));
143 }
144 }
145 }
146
147 float complex
casinhf(float complex z)148 casinhf(float complex z)
149 {
150 float x, y, ax, ay, rx, ry, B, sqrt_A2my2, new_y;
151 int B_is_usable;
152 float complex w;
153
154 x = crealf(z);
155 y = cimagf(z);
156 ax = fabsf(x);
157 ay = fabsf(y);
158
159 if (isnan(x) || isnan(y)) {
160 if (isinf(x))
161 return (CMPLXF(x, y + y));
162 if (isinf(y))
163 return (CMPLXF(y, x + x));
164 if (y == 0)
165 return (CMPLXF(x + x, y));
166 return (CMPLXF(nan_mix(x, y), nan_mix(x, y)));
167 }
168
169 if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) {
170 if (signbit(x) == 0)
171 w = clog_for_large_values(z) + m_ln2;
172 else
173 w = clog_for_large_values(-z) + m_ln2;
174 return (CMPLXF(copysignf(crealf(w), x),
175 copysignf(cimagf(w), y)));
176 }
177
178 if (x == 0 && y == 0)
179 return (z);
180
181 raise_inexact();
182
183 if (ax < SQRT_6_EPSILON / 4 && ay < SQRT_6_EPSILON / 4)
184 return (z);
185
186 do_hard_work(ax, ay, &rx, &B_is_usable, &B, &sqrt_A2my2, &new_y);
187 if (B_is_usable)
188 ry = asinf(B);
189 else
190 ry = atan2f(new_y, sqrt_A2my2);
191 return (CMPLXF(copysignf(rx, x), copysignf(ry, y)));
192 }
193
194 float complex
casinf(float complex z)195 casinf(float complex z)
196 {
197 float complex w = casinhf(CMPLXF(cimagf(z), crealf(z)));
198
199 return (CMPLXF(cimagf(w), crealf(w)));
200 }
201
202 float complex
cacosf(float complex z)203 cacosf(float complex z)
204 {
205 float x, y, ax, ay, rx, ry, B, sqrt_A2mx2, new_x;
206 int sx, sy;
207 int B_is_usable;
208 float complex w;
209
210 x = crealf(z);
211 y = cimagf(z);
212 sx = signbit(x);
213 sy = signbit(y);
214 ax = fabsf(x);
215 ay = fabsf(y);
216
217 if (isnan(x) || isnan(y)) {
218 if (isinf(x))
219 return (CMPLXF(y + y, -INFINITY));
220 if (isinf(y))
221 return (CMPLXF(x + x, -y));
222 if (x == 0)
223 return (CMPLXF(pio2_hi + pio2_lo, y + y));
224 return (CMPLXF(nan_mix(x, y), nan_mix(x, y)));
225 }
226
227 if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) {
228 w = clog_for_large_values(z);
229 rx = fabsf(cimagf(w));
230 ry = crealf(w) + m_ln2;
231 if (sy == 0)
232 ry = -ry;
233 return (CMPLXF(rx, ry));
234 }
235
236 if (x == 1 && y == 0)
237 return (CMPLXF(0, -y));
238
239 raise_inexact();
240
241 if (ax < SQRT_6_EPSILON / 4 && ay < SQRT_6_EPSILON / 4)
242 return (CMPLXF(pio2_hi - (x - pio2_lo), -y));
243
244 do_hard_work(ay, ax, &ry, &B_is_usable, &B, &sqrt_A2mx2, &new_x);
245 if (B_is_usable) {
246 if (sx == 0)
247 rx = acosf(B);
248 else
249 rx = acosf(-B);
250 } else {
251 if (sx == 0)
252 rx = atan2f(sqrt_A2mx2, new_x);
253 else
254 rx = atan2f(sqrt_A2mx2, -new_x);
255 }
256 if (sy == 0)
257 ry = -ry;
258 return (CMPLXF(rx, ry));
259 }
260
261 float complex
cacoshf(float complex z)262 cacoshf(float complex z)
263 {
264 float complex w;
265 float rx, ry;
266
267 w = cacosf(z);
268 rx = crealf(w);
269 ry = cimagf(w);
270 if (isnan(rx) && isnan(ry))
271 return (CMPLXF(ry, rx));
272 if (isnan(rx))
273 return (CMPLXF(fabsf(ry), rx));
274 if (isnan(ry))
275 return (CMPLXF(ry, ry));
276 return (CMPLXF(fabsf(ry), copysignf(rx, cimagf(z))));
277 }
278
279 static float complex
clog_for_large_values(float complex z)280 clog_for_large_values(float complex z)
281 {
282 float x, y;
283 float ax, ay, t;
284
285 x = crealf(z);
286 y = cimagf(z);
287 ax = fabsf(x);
288 ay = fabsf(y);
289 if (ax < ay) {
290 t = ax;
291 ax = ay;
292 ay = t;
293 }
294
295 if (ax > FLT_MAX / 2)
296 return (CMPLXF(logf(hypotf(x / m_e, y / m_e)) + 1,
297 atan2f(y, x)));
298
299 if (ax > QUARTER_SQRT_MAX || ay < SQRT_MIN)
300 return (CMPLXF(logf(hypotf(x, y)), atan2f(y, x)));
301
302 return (CMPLXF(logf(ax * ax + ay * ay) / 2, atan2f(y, x)));
303 }
304
305 static inline float
sum_squares(float x,float y)306 sum_squares(float x, float y)
307 {
308
309 if (y < SQRT_MIN)
310 return (x * x);
311
312 return (x * x + y * y);
313 }
314
315 static inline float
real_part_reciprocal(float x,float y)316 real_part_reciprocal(float x, float y)
317 {
318 float scale;
319 uint32_t hx, hy;
320 int32_t ix, iy;
321
322 GET_FLOAT_WORD(hx, x);
323 ix = hx & 0x7f800000;
324 GET_FLOAT_WORD(hy, y);
325 iy = hy & 0x7f800000;
326 #define BIAS (FLT_MAX_EXP - 1)
327 #define CUTOFF (FLT_MANT_DIG / 2 + 1)
328 if (ix - iy >= CUTOFF << 23 || isinf(x))
329 return (1 / x);
330 if (iy - ix >= CUTOFF << 23)
331 return (x / y / y);
332 if (ix <= (BIAS + FLT_MAX_EXP / 2 - CUTOFF) << 23)
333 return (x / (x * x + y * y));
334 SET_FLOAT_WORD(scale, 0x7f800000 - ix);
335 x *= scale;
336 y *= scale;
337 return (x / (x * x + y * y) * scale);
338 }
339
340 float complex
catanhf(float complex z)341 catanhf(float complex z)
342 {
343 float x, y, ax, ay, rx, ry;
344
345 x = crealf(z);
346 y = cimagf(z);
347 ax = fabsf(x);
348 ay = fabsf(y);
349
350 if (y == 0 && ax <= 1)
351 return (CMPLXF(atanhf(x), y));
352
353 if (x == 0)
354 return (CMPLXF(x, atanf(y)));
355
356 if (isnan(x) || isnan(y)) {
357 if (isinf(x))
358 return (CMPLXF(copysignf(0, x), y + y));
359 if (isinf(y))
360 return (CMPLXF(copysignf(0, x),
361 copysignf(pio2_hi + pio2_lo, y)));
362 return (CMPLXF(nan_mix(x, y), nan_mix(x, y)));
363 }
364
365 if (ax > RECIP_EPSILON || ay > RECIP_EPSILON)
366 return (CMPLXF(real_part_reciprocal(x, y),
367 copysignf(pio2_hi + pio2_lo, y)));
368
369 if (ax < SQRT_3_EPSILON / 2 && ay < SQRT_3_EPSILON / 2) {
370 raise_inexact();
371 return (z);
372 }
373
374 if (ax == 1 && ay < FLT_EPSILON)
375 rx = (m_ln2 - logf(ay)) / 2;
376 else
377 rx = log1pf(4 * ax / sum_squares(ax - 1, ay)) / 4;
378
379 if (ax == 1)
380 ry = atan2f(2, -ay) / 2;
381 else if (ay < FLT_EPSILON)
382 ry = atan2f(2 * ay, (1 - ax) * (1 + ax)) / 2;
383 else
384 ry = atan2f(2 * ay, (1 - ax) * (1 + ax) - ay * ay) / 2;
385
386 return (CMPLXF(copysignf(rx, x), copysignf(ry, y)));
387 }
388
389 float complex
catanf(float complex z)390 catanf(float complex z)
391 {
392 float complex w = catanhf(CMPLXF(cimagf(z), crealf(z)));
393
394 return (CMPLXF(cimagf(w), crealf(w)));
395 }
396