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 * See catrig.c for complete comments.
35 *
36 * XXX comments were removed automatically, and even short ones on the right
37 * of statements were removed (all of them), contrary to normal style. Only
38 * a few comments on the right of declarations remain.
39 */
40
41 #include <sys/cdefs.h>
42 __FBSDID("$FreeBSD: head/lib/msun/src/catrigf.c 275819 2014-12-16 09:21:56Z ed $");
43
44 #include <complex.h>
45 #include <float.h>
46
47 #include "math.h"
48 #include "math_private.h"
49
50 #undef isinf
51 #define isinf(x) (fabsf(x) == INFINITY)
52 #undef isnan
53 #define isnan(x) ((x) != (x))
54 #define raise_inexact() do { volatile float junk = 1 + tiny; } while(0)
55 #undef signbit
56 #define signbit(x) (__builtin_signbitf(x))
57
58 static const float
59 A_crossover = 10,
60 B_crossover = 0.6417,
61 FOUR_SQRT_MIN = 0x1p-61,
62 QUARTER_SQRT_MAX = 0x1p61,
63 m_e = 2.7182818285e0, /* 0xadf854.0p-22 */
64 m_ln2 = 6.9314718056e-1, /* 0xb17218.0p-24 */
65 pio2_hi = 1.5707962513e0, /* 0xc90fda.0p-23 */
66 RECIP_EPSILON = 1 / FLT_EPSILON,
67 SQRT_3_EPSILON = 5.9801995673e-4, /* 0x9cc471.0p-34 */
68 SQRT_6_EPSILON = 8.4572793338e-4, /* 0xddb3d7.0p-34 */
69 SQRT_MIN = 0x1p-63;
70
71 static const volatile float
72 pio2_lo = 7.5497899549e-8, /* 0xa22169.0p-47 */
73 tiny = 0x1p-100;
74
75 static float complex clog_for_large_values(float complex z);
76
77 static inline float
f(float a,float b,float hypot_a_b)78 f(float a, float b, float hypot_a_b)
79 {
80 if (b < 0)
81 return ((hypot_a_b - b) / 2);
82 if (b == 0)
83 return (a / 2);
84 return (a * a / (hypot_a_b + b) / 2);
85 }
86
87 static inline void
do_hard_work(float x,float y,float * rx,int * B_is_usable,float * B,float * sqrt_A2my2,float * new_y)88 do_hard_work(float x, float y, float *rx, int *B_is_usable, float *B,
89 float *sqrt_A2my2, float *new_y)
90 {
91 float R, S, A;
92 float Am1, Amy;
93
94 R = hypotf(x, y + 1);
95 S = hypotf(x, y - 1);
96
97 A = (R + S) / 2;
98 if (A < 1)
99 A = 1;
100
101 if (A < A_crossover) {
102 if (y == 1 && x < FLT_EPSILON * FLT_EPSILON / 128) {
103 *rx = sqrtf(x);
104 } else if (x >= FLT_EPSILON * fabsf(y - 1)) {
105 Am1 = f(x, 1 + y, R) + f(x, 1 - y, S);
106 *rx = log1pf(Am1 + sqrtf(Am1 * (A + 1)));
107 } else if (y < 1) {
108 *rx = x / sqrtf((1 - y) * (1 + y));
109 } else {
110 *rx = log1pf((y - 1) + sqrtf((y - 1) * (y + 1)));
111 }
112 } else {
113 *rx = logf(A + sqrtf(A * A - 1));
114 }
115
116 *new_y = y;
117
118 if (y < FOUR_SQRT_MIN) {
119 *B_is_usable = 0;
120 *sqrt_A2my2 = A * (2 / FLT_EPSILON);
121 *new_y = y * (2 / FLT_EPSILON);
122 return;
123 }
124
125 *B = y / A;
126 *B_is_usable = 1;
127
128 if (*B > B_crossover) {
129 *B_is_usable = 0;
130 if (y == 1 && x < FLT_EPSILON / 128) {
131 *sqrt_A2my2 = sqrtf(x) * sqrtf((A + y) / 2);
132 } else if (x >= FLT_EPSILON * fabsf(y - 1)) {
133 Amy = f(x, y + 1, R) + f(x, y - 1, S);
134 *sqrt_A2my2 = sqrtf(Amy * (A + y));
135 } else if (y > 1) {
136 *sqrt_A2my2 = x * (4 / FLT_EPSILON / FLT_EPSILON) * y /
137 sqrtf((y + 1) * (y - 1));
138 *new_y = y * (4 / FLT_EPSILON / FLT_EPSILON);
139 } else {
140 *sqrt_A2my2 = sqrtf((1 - y) * (1 + y));
141 }
142 }
143 }
144
145 float complex
casinhf(float complex z)146 casinhf(float complex z)
147 {
148 float x, y, ax, ay, rx, ry, B, sqrt_A2my2, new_y;
149 int B_is_usable;
150 float complex w;
151
152 x = crealf(z);
153 y = cimagf(z);
154 ax = fabsf(x);
155 ay = fabsf(y);
156
157 if (isnan(x) || isnan(y)) {
158 if (isinf(x))
159 return (CMPLXF(x, y + y));
160 if (isinf(y))
161 return (CMPLXF(y, x + x));
162 if (y == 0)
163 return (CMPLXF(x + x, y));
164 return (CMPLXF(x + 0.0L + (y + 0), x + 0.0L + (y + 0)));
165 }
166
167 if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) {
168 if (signbit(x) == 0)
169 w = clog_for_large_values(z) + m_ln2;
170 else
171 w = clog_for_large_values(-z) + m_ln2;
172 return (CMPLXF(copysignf(crealf(w), x),
173 copysignf(cimagf(w), y)));
174 }
175
176 if (x == 0 && y == 0)
177 return (z);
178
179 raise_inexact();
180
181 if (ax < SQRT_6_EPSILON / 4 && ay < SQRT_6_EPSILON / 4)
182 return (z);
183
184 do_hard_work(ax, ay, &rx, &B_is_usable, &B, &sqrt_A2my2, &new_y);
185 if (B_is_usable)
186 ry = asinf(B);
187 else
188 ry = atan2f(new_y, sqrt_A2my2);
189 return (CMPLXF(copysignf(rx, x), copysignf(ry, y)));
190 }
191
192 float complex
casinf(float complex z)193 casinf(float complex z)
194 {
195 float complex w = casinhf(CMPLXF(cimagf(z), crealf(z)));
196
197 return (CMPLXF(cimagf(w), crealf(w)));
198 }
199
200 float complex
cacosf(float complex z)201 cacosf(float complex z)
202 {
203 float x, y, ax, ay, rx, ry, B, sqrt_A2mx2, new_x;
204 int sx, sy;
205 int B_is_usable;
206 float complex w;
207
208 x = crealf(z);
209 y = cimagf(z);
210 sx = signbit(x);
211 sy = signbit(y);
212 ax = fabsf(x);
213 ay = fabsf(y);
214
215 if (isnan(x) || isnan(y)) {
216 if (isinf(x))
217 return (CMPLXF(y + y, -INFINITY));
218 if (isinf(y))
219 return (CMPLXF(x + x, -y));
220 if (x == 0)
221 return (CMPLXF(pio2_hi + pio2_lo, y + y));
222 return (CMPLXF(x + 0.0L + (y + 0), x + 0.0L + (y + 0)));
223 }
224
225 if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) {
226 w = clog_for_large_values(z);
227 rx = fabsf(cimagf(w));
228 ry = crealf(w) + m_ln2;
229 if (sy == 0)
230 ry = -ry;
231 return (CMPLXF(rx, ry));
232 }
233
234 if (x == 1 && y == 0)
235 return (CMPLXF(0, -y));
236
237 raise_inexact();
238
239 if (ax < SQRT_6_EPSILON / 4 && ay < SQRT_6_EPSILON / 4)
240 return (CMPLXF(pio2_hi - (x - pio2_lo), -y));
241
242 do_hard_work(ay, ax, &ry, &B_is_usable, &B, &sqrt_A2mx2, &new_x);
243 if (B_is_usable) {
244 if (sx == 0)
245 rx = acosf(B);
246 else
247 rx = acosf(-B);
248 } else {
249 if (sx == 0)
250 rx = atan2f(sqrt_A2mx2, new_x);
251 else
252 rx = atan2f(sqrt_A2mx2, -new_x);
253 }
254 if (sy == 0)
255 ry = -ry;
256 return (CMPLXF(rx, ry));
257 }
258
259 float complex
cacoshf(float complex z)260 cacoshf(float complex z)
261 {
262 float complex w;
263 float rx, ry;
264
265 w = cacosf(z);
266 rx = crealf(w);
267 ry = cimagf(w);
268 if (isnan(rx) && isnan(ry))
269 return (CMPLXF(ry, rx));
270 if (isnan(rx))
271 return (CMPLXF(fabsf(ry), rx));
272 if (isnan(ry))
273 return (CMPLXF(ry, ry));
274 return (CMPLXF(fabsf(ry), copysignf(rx, cimagf(z))));
275 }
276
277 static float complex
clog_for_large_values(float complex z)278 clog_for_large_values(float complex z)
279 {
280 float x, y;
281 float ax, ay, t;
282
283 x = crealf(z);
284 y = cimagf(z);
285 ax = fabsf(x);
286 ay = fabsf(y);
287 if (ax < ay) {
288 t = ax;
289 ax = ay;
290 ay = t;
291 }
292
293 if (ax > FLT_MAX / 2)
294 return (CMPLXF(logf(hypotf(x / m_e, y / m_e)) + 1,
295 atan2f(y, x)));
296
297 if (ax > QUARTER_SQRT_MAX || ay < SQRT_MIN)
298 return (CMPLXF(logf(hypotf(x, y)), atan2f(y, x)));
299
300 return (CMPLXF(logf(ax * ax + ay * ay) / 2, atan2f(y, x)));
301 }
302
303 static inline float
sum_squares(float x,float y)304 sum_squares(float x, float y)
305 {
306
307 if (y < SQRT_MIN)
308 return (x * x);
309
310 return (x * x + y * y);
311 }
312
313 static inline float
real_part_reciprocal(float x,float y)314 real_part_reciprocal(float x, float y)
315 {
316 float scale;
317 uint32_t hx, hy;
318 int32_t ix, iy;
319
320 GET_FLOAT_WORD(hx, x);
321 ix = hx & 0x7f800000;
322 GET_FLOAT_WORD(hy, y);
323 iy = hy & 0x7f800000;
324 #define BIAS (FLT_MAX_EXP - 1)
325 #define CUTOFF (FLT_MANT_DIG / 2 + 1)
326 if (ix - iy >= CUTOFF << 23 || isinf(x))
327 return (1 / x);
328 if (iy - ix >= CUTOFF << 23)
329 return (x / y / y);
330 if (ix <= (BIAS + FLT_MAX_EXP / 2 - CUTOFF) << 23)
331 return (x / (x * x + y * y));
332 SET_FLOAT_WORD(scale, 0x7f800000 - ix);
333 x *= scale;
334 y *= scale;
335 return (x / (x * x + y * y) * scale);
336 }
337
338 float complex
catanhf(float complex z)339 catanhf(float complex z)
340 {
341 float x, y, ax, ay, rx, ry;
342
343 x = crealf(z);
344 y = cimagf(z);
345 ax = fabsf(x);
346 ay = fabsf(y);
347
348 if (y == 0 && ax <= 1)
349 return (CMPLXF(atanhf(x), y));
350
351 if (x == 0)
352 return (CMPLXF(x, atanf(y)));
353
354 if (isnan(x) || isnan(y)) {
355 if (isinf(x))
356 return (CMPLXF(copysignf(0, x), y + y));
357 if (isinf(y))
358 return (CMPLXF(copysignf(0, x),
359 copysignf(pio2_hi + pio2_lo, y)));
360 return (CMPLXF(x + 0.0L + (y + 0), x + 0.0L + (y + 0)));
361 }
362
363 if (ax > RECIP_EPSILON || ay > RECIP_EPSILON)
364 return (CMPLXF(real_part_reciprocal(x, y),
365 copysignf(pio2_hi + pio2_lo, y)));
366
367 if (ax < SQRT_3_EPSILON / 2 && ay < SQRT_3_EPSILON / 2) {
368 raise_inexact();
369 return (z);
370 }
371
372 if (ax == 1 && ay < FLT_EPSILON)
373 rx = (m_ln2 - logf(ay)) / 2;
374 else
375 rx = log1pf(4 * ax / sum_squares(ax - 1, ay)) / 4;
376
377 if (ax == 1)
378 ry = atan2f(2, -ay) / 2;
379 else if (ay < FLT_EPSILON)
380 ry = atan2f(2 * ay, (1 - ax) * (1 + ax)) / 2;
381 else
382 ry = atan2f(2 * ay, (1 - ax) * (1 + ax) - ay * ay) / 2;
383
384 return (CMPLXF(copysignf(rx, x), copysignf(ry, y)));
385 }
386
387 float complex
catanf(float complex z)388 catanf(float complex z)
389 {
390 float complex w = catanhf(CMPLXF(cimagf(z), crealf(z)));
391
392 return (CMPLXF(cimagf(w), crealf(w)));
393 }
394