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