2 * Copyright (c) 2012 Stephen Montgomery-Smith <stephen@FreeBSD.ORG>
3 * Copyright (c) 2017 Mahdi Mokhtari <mmokhi@FreeBSD.org>
6 * Redistribution and use in source and binary forms, with or without
7 * modification, are permitted provided that the following conditions
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.
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
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.
35 * See catrig.c for complete comments.
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.
42 #include <sys/cdefs.h>
43 __FBSDID("$FreeBSD$");
50 #include "math_private.h"
53 #define isinf(x) (fabsl(x) == INFINITY)
55 #define isnan(x) ((x) != (x))
56 #define raise_inexact() do { volatile float junk __unused = 1 + tiny; } while(0)
58 #define signbit(x) (__builtin_signbitl(x))
60 static const long double
63 FOUR_SQRT_MIN = 0x1p-8189L,
64 QUARTER_SQRT_MAX = 0x1p8189L,
65 RECIP_EPSILON = 1 / LDBL_EPSILON,
66 SQRT_MIN = 0x1p-8191L;
68 #if LDBL_MANT_DIG == 64
69 static const union IEEEl2bits
70 um_e = LD80C(0xadf85458a2bb4a9b, 1, 2.71828182845904523536e+0L),
71 um_ln2 = LD80C(0xb17217f7d1cf79ac, -1, 6.93147180559945309417e-1L);
73 #define m_ln2 um_ln2.e
74 static const long double
75 /* The next 2 literals for non-i386. Misrounding them on i386 is harmless. */
76 SQRT_3_EPSILON = 5.70316273435758915310e-10, /* 0x9cc470a0490973e8.0p-94 */
77 SQRT_6_EPSILON = 8.06549008734932771664e-10; /* 0xddb3d742c265539e.0p-94 */
78 #elif LDBL_MANT_DIG == 113
79 static const long double
80 m_e = 2.71828182845904523536028747135266250e0L, /* 0x15bf0a8b1457695355fb8ac404e7a.0p-111 */
81 m_ln2 = 6.93147180559945309417232121458176568e-1L, /* 0x162e42fefa39ef35793c7673007e6.0p-113 */
82 SQRT_3_EPSILON = 2.40370335797945490975336727199878124e-17, /* 0x1bb67ae8584caa73b25742d7078b8.0p-168 */
83 SQRT_6_EPSILON = 3.39934988877629587239082586223300391e-17; /* 0x13988e1409212e7d0321914321a55.0p-167 */
85 #error "Unsupported long double format"
88 static const volatile float
91 static long double complex clog_for_large_values(long double complex z);
93 static inline long double
94 f(long double a, long double b, long double hypot_a_b)
97 return ((hypot_a_b - b) / 2);
100 return (a * a / (hypot_a_b + b) / 2);
104 do_hard_work(long double x, long double y, long double *rx, int *B_is_usable,
105 long double *B, long double *sqrt_A2my2, long double *new_y)
108 long double Am1, Amy;
110 R = hypotl(x, y + 1);
111 S = hypotl(x, y - 1);
117 if (A < A_crossover) {
118 if (y == 1 && x < LDBL_EPSILON * LDBL_EPSILON / 128) {
120 } else if (x >= LDBL_EPSILON * fabsl(y - 1)) {
121 Am1 = f(x, 1 + y, R) + f(x, 1 - y, S);
122 *rx = log1pl(Am1 + sqrtl(Am1 * (A + 1)));
124 *rx = x / sqrtl((1 - y) * (1 + y));
126 *rx = log1pl((y - 1) + sqrtl((y - 1) * (y + 1)));
129 *rx = logl(A + sqrtl(A * A - 1));
134 if (y < FOUR_SQRT_MIN) {
136 *sqrt_A2my2 = A * (2 / LDBL_EPSILON);
137 *new_y = y * (2 / LDBL_EPSILON);
144 if (*B > B_crossover) {
146 if (y == 1 && x < LDBL_EPSILON / 128) {
147 *sqrt_A2my2 = sqrtl(x) * sqrtl((A + y) / 2);
148 } else if (x >= LDBL_EPSILON * fabsl(y - 1)) {
149 Amy = f(x, y + 1, R) + f(x, y - 1, S);
150 *sqrt_A2my2 = sqrtl(Amy * (A + y));
152 *sqrt_A2my2 = x * (4 / LDBL_EPSILON / LDBL_EPSILON) * y /
153 sqrtl((y + 1) * (y - 1));
154 *new_y = y * (4 / LDBL_EPSILON / LDBL_EPSILON);
156 *sqrt_A2my2 = sqrtl((1 - y) * (1 + y));
162 casinhl(long double complex z)
164 long double x, y, ax, ay, rx, ry, B, sqrt_A2my2, new_y;
166 long double complex w;
173 if (isnan(x) || isnan(y)) {
175 return (CMPLXL(x, y + y));
177 return (CMPLXL(y, x + x));
179 return (CMPLXL(x + x, y));
180 return (CMPLXL(x + 0.0L + (y + 0), x + 0.0L + (y + 0)));
183 if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) {
185 w = clog_for_large_values(z) + m_ln2;
187 w = clog_for_large_values(-z) + m_ln2;
188 return (CMPLXL(copysignl(creall(w), x),
189 copysignl(cimagl(w), y)));
192 if (x == 0 && y == 0)
197 if (ax < SQRT_6_EPSILON / 4 && ay < SQRT_6_EPSILON / 4)
200 do_hard_work(ax, ay, &rx, &B_is_usable, &B, &sqrt_A2my2, &new_y);
204 ry = atan2l(new_y, sqrt_A2my2);
205 return (CMPLXL(copysignl(rx, x), copysignl(ry, y)));
209 casinl(long double complex z)
211 long double complex w;
213 w = casinhl(CMPLXL(cimagl(z), creall(z)));
214 return (CMPLXL(cimagl(w), creall(w)));
218 cacosl(long double complex z)
220 long double x, y, ax, ay, rx, ry, B, sqrt_A2mx2, new_x;
223 long double complex w;
232 if (isnan(x) || isnan(y)) {
234 return (CMPLXL(y + y, -INFINITY));
236 return (CMPLXL(x + x, -y));
238 return (CMPLXL(pio2_hi + pio2_lo, y + y));
239 return (CMPLXL(x + 0.0L + (y + 0), x + 0.0L + (y + 0)));
242 if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) {
243 w = clog_for_large_values(z);
244 rx = fabsl(cimagl(w));
245 ry = creall(w) + m_ln2;
248 return (CMPLXL(rx, ry));
251 if (x == 1 && y == 0)
252 return (CMPLXL(0, -y));
256 if (ax < SQRT_6_EPSILON / 4 && ay < SQRT_6_EPSILON / 4)
257 return (CMPLXL(pio2_hi - (x - pio2_lo), -y));
259 do_hard_work(ay, ax, &ry, &B_is_usable, &B, &sqrt_A2mx2, &new_x);
267 rx = atan2l(sqrt_A2mx2, new_x);
269 rx = atan2l(sqrt_A2mx2, -new_x);
273 return (CMPLXL(rx, ry));
277 cacoshl(long double complex z)
279 long double complex w;
285 if (isnan(rx) && isnan(ry))
286 return (CMPLXL(ry, rx));
288 return (CMPLXL(fabsl(ry), rx));
290 return (CMPLXL(ry, ry));
291 return (CMPLXL(fabsl(ry), copysignl(rx, cimagl(z))));
294 static long double complex
295 clog_for_large_values(long double complex z)
298 long double ax, ay, t;
310 if (ax > LDBL_MAX / 2)
311 return (CMPLXL(logl(hypotl(x / m_e, y / m_e)) + 1,
314 if (ax > QUARTER_SQRT_MAX || ay < SQRT_MIN)
315 return (CMPLXL(logl(hypotl(x, y)), atan2l(y, x)));
317 return (CMPLXL(logl(ax * ax + ay * ay) / 2, atan2l(y, x)));
320 static inline long double
321 sum_squares(long double x, long double y)
327 return (x * x + y * y);
330 static inline long double
331 real_part_reciprocal(long double x, long double y)
337 GET_LDBL_EXPSIGN(hx, x);
339 GET_LDBL_EXPSIGN(hy, y);
341 #define BIAS (LDBL_MAX_EXP - 1)
342 #define CUTOFF (LDBL_MANT_DIG / 2 + 1)
343 if (ix - iy >= CUTOFF || isinf(x))
345 if (iy - ix >= CUTOFF)
347 if (ix <= BIAS + LDBL_MAX_EXP / 2 - CUTOFF)
348 return (x / (x * x + y * y));
350 SET_LDBL_EXPSIGN(scale, 0x7fff - ix);
353 return (x / (x * x + y * y) * scale);
357 catanhl(long double complex z)
359 long double x, y, ax, ay, rx, ry;
366 if (y == 0 && ax <= 1)
367 return (CMPLXL(atanhl(x), y));
370 return (CMPLXL(x, atanl(y)));
372 if (isnan(x) || isnan(y)) {
374 return (CMPLXL(copysignl(0, x), y + y));
376 return (CMPLXL(copysignl(0, x),
377 copysignl(pio2_hi + pio2_lo, y)));
378 return (CMPLXL(x + 0.0L + (y + 0), x + 0.0L + (y + 0)));
381 if (ax > RECIP_EPSILON || ay > RECIP_EPSILON)
382 return (CMPLXL(real_part_reciprocal(x, y),
383 copysignl(pio2_hi + pio2_lo, y)));
385 if (ax < SQRT_3_EPSILON / 2 && ay < SQRT_3_EPSILON / 2) {
390 if (ax == 1 && ay < LDBL_EPSILON)
391 rx = (m_ln2 - logl(ay)) / 2;
393 rx = log1pl(4 * ax / sum_squares(ax - 1, ay)) / 4;
396 ry = atan2l(2, -ay) / 2;
397 else if (ay < LDBL_EPSILON)
398 ry = atan2l(2 * ay, (1 - ax) * (1 + ax)) / 2;
400 ry = atan2l(2 * ay, (1 - ax) * (1 + ax) - ay * ay) / 2;
402 return (CMPLXL(copysignl(rx, x), copysignl(ry, y)));
406 catanl(long double complex z)
408 long double complex w;
410 w = catanhl(CMPLXL(cimagl(z), creall(z)));
411 return (CMPLXL(cimagl(w), creall(w)));