5 * Common Development and Distribution License (the "License").
6 * You may not use this file except in compliance with the License.
7 *
8 * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
9 * or http://www.opensolaris.org/os/licensing.
10 * See the License for the specific language governing permissions
11 * and limitations under the License.
12 *
13 * When distributing Covered Code, include this CDDL HEADER in each
14 * file and include the License file at usr/src/OPENSOLARIS.LICENSE.
15 * If applicable, add the following below this CDDL HEADER, with the
16 * fields enclosed by brackets "[]" replaced with your own identifying
17 * information: Portions Copyright [yyyy] [name of copyright owner]
18 *
19 * CDDL HEADER END
20 */
21
22 /*
23 * Copyright 2011 Nexenta Systems, Inc. All rights reserved.
24 */
25 /*
26 * Copyright 2006 Sun Microsystems, Inc. All rights reserved.
27 * Use is subject to license terms.
28 */
29
30 #pragma weak __cacos = cacos
31
32 /* INDENT OFF */
33 /*
34 * dcomplex cacos(dcomplex z);
35 *
36 * Alogrithm
37 * (based on T.E.Hull, Thomas F. Fairgrieve and Ping Tak Peter Tang's
38 * paper "Implementing the Complex Arcsine and Arccosine Functins Using
39 * Exception Handling", ACM TOMS, Vol 23, pp 299-335)
40 *
41 * The principal value of complex inverse cosine function cacos(z),
42 * where z = x+iy, can be defined by
43 *
44 * cacos(z) = acos(B) - i sign(y) log (A + sqrt(A*A-1)),
45 *
46 * where the log function is the natural log, and
47 * ____________ ____________
48 * 1 / 2 2 1 / 2 2
49 * A = --- / (x+1) + y + --- / (x-1) + y
50 * 2 \/ 2 \/
51 * ____________ ____________
52 * 1 / 2 2 1 / 2 2
173 * A ~ sqrt(x*x+y*y)
174 * B ~ x/sqrt(x*x+y*y).
175 * Thus
176 * real part = acos(B) = atan(y/x),
177 * imag part = log(A+sqrt(A*A-1)) ~ log(2A)
178 * = log(2) + 0.5*log(x*x+y*y)
179 * = log(2) + log(y) + 0.5*log(1+(x/y)^2)
180 *
181 * case 6. x < 4 sqrt(u). In this case, we have
182 * A ~ sqrt(1+y*y), B = x/sqrt(1+y*y).
183 * Since B is tiny, we have
184 * real part = acos(B) ~ pi/2
185 * imag part = log(A+sqrt(A*A-1)) = log (A+sqrt(y*y))
186 * = log(y+sqrt(1+y*y))
187 * = 0.5*log(y^2+2ysqrt(1+y^2)+1+y^2)
188 * = 0.5*log(1+2y(y+sqrt(1+y^2)));
189 * = 0.5*log1p(2y(y+A));
190 *
191 * cacos(z) = acos(B) - i sign(y) log (A + sqrt(A*A-1)),
192 */
193 /* INDENT ON */
194
195 #include "libm.h"
196 #include "complex_wrapper.h"
197
198 /* INDENT OFF */
199 static const double
200 zero = 0.0,
201 one = 1.0,
202 E = 1.11022302462515654042e-16, /* 2**-53 */
203 ln2 = 6.93147180559945286227e-01,
204 pi = 3.1415926535897931159979634685,
205 pi_l = 1.224646799147353177e-16,
206 pi_2 = 1.570796326794896558e+00,
207 pi_2_l = 6.123233995736765886e-17,
208 pi_4 = 0.78539816339744827899949,
209 pi_4_l = 3.061616997868382943e-17,
210 pi3_4 = 2.356194490192344836998,
211 pi3_4_l = 9.184850993605148829195e-17,
212 Foursqrtu = 5.96667258496016539463e-154, /* 2**(-509) */
213 Acrossover = 1.5,
214 Bcrossover = 0.6417,
215 half = 0.5;
216 /* INDENT ON */
217
218 dcomplex
219 cacos(dcomplex z) {
220 double x, y, t, R, S, A, Am1, B, y2, xm1, xp1, Apx;
221 int ix, iy, hx, hy;
222 unsigned lx, ly;
223 dcomplex ans;
224
225 x = D_RE(z);
226 y = D_IM(z);
227 hx = HI_WORD(x);
228 lx = LO_WORD(x);
229 hy = HI_WORD(y);
230 ly = LO_WORD(y);
231 ix = hx & 0x7fffffff;
232 iy = hy & 0x7fffffff;
233
234 /* x is 0 */
235 if ((ix | lx) == 0) {
236 if (((iy | ly) == 0) || (iy >= 0x7ff00000)) {
237 D_RE(ans) = pi_2;
238 D_IM(ans) = -y;
239 return (ans);
240 }
241 }
242
243 /* |y| is inf or NaN */
244 if (iy >= 0x7ff00000) {
245 if (ISINF(iy, ly)) { /* cacos(x + i inf) = pi/2 - i inf */
246 D_IM(ans) = -y;
247 if (ix < 0x7ff00000) {
248 D_RE(ans) = pi_2 + pi_2_l;
249 } else if (ISINF(ix, lx)) {
250 if (hx >= 0)
251 D_RE(ans) = pi_4 + pi_4_l;
252 else
253 D_RE(ans) = pi3_4 + pi3_4_l;
254 } else {
255 D_RE(ans) = x;
256 }
257 } else { /* cacos(x + i NaN) = NaN + i NaN */
258 D_RE(ans) = y + x;
259 if (ISINF(ix, lx))
260 D_IM(ans) = -fabs(x);
261 else
262 D_IM(ans) = y;
263 }
264 return (ans);
265 }
266
267 x = fabs(x);
268 y = fabs(y);
269
270 /* x is inf or NaN */
271 if (ix >= 0x7ff00000) { /* x is inf or NaN */
272 if (ISINF(ix, lx)) { /* x is INF */
273 D_IM(ans) = -x;
274 if (iy >= 0x7ff00000) {
275 if (ISINF(iy, ly)) {
276 /* INDENT OFF */
277 /* cacos(inf + i inf) = pi/4 - i inf */
278 /* cacos(-inf+ i inf) =3pi/4 - i inf */
279 /* INDENT ON */
280 if (hx >= 0)
281 D_RE(ans) = pi_4 + pi_4_l;
282 else
283 D_RE(ans) = pi3_4 + pi3_4_l;
284 } else
285 /* INDENT OFF */
286 /* cacos(inf + i NaN) = NaN - i inf */
287 /* INDENT ON */
288 D_RE(ans) = y + y;
289 } else
290 /* INDENT OFF */
291 /* cacos(inf + iy ) = 0 - i inf */
292 /* cacos(-inf+ iy ) = pi - i inf */
293 /* INDENT ON */
294 if (hx >= 0)
295 D_RE(ans) = zero;
296 else
297 D_RE(ans) = pi + pi_l;
298 } else { /* x is NaN */
299 /* INDENT OFF */
300 /*
301 * cacos(NaN + i inf) = NaN - i inf
302 * cacos(NaN + i y ) = NaN + i NaN
303 * cacos(NaN + i NaN) = NaN + i NaN
304 */
305 /* INDENT ON */
306 D_RE(ans) = x + y;
307 if (iy >= 0x7ff00000) {
308 D_IM(ans) = -y;
309 } else {
310 D_IM(ans) = x;
311 }
312 }
313 if (hy < 0)
314 D_IM(ans) = -D_IM(ans);
315 return (ans);
316 }
317
318 if ((iy | ly) == 0) { /* region 1: y=0 */
319 if (ix < 0x3ff00000) { /* |x| < 1 */
320 D_RE(ans) = acos(x);
321 D_IM(ans) = zero;
322 } else {
323 D_RE(ans) = zero;
324 if (ix >= 0x43500000) /* |x| >= 2**54 */
325 D_IM(ans) = ln2 + log(x);
326 else if (ix >= 0x3ff80000) /* x > Acrossover */
327 D_IM(ans) = log(x + sqrt((x - one) * (x +
328 one)));
329 else {
330 xm1 = x - one;
331 D_IM(ans) = log1p(xm1 + sqrt(xm1 * (x + one)));
332 }
333 }
334 } else if (y <= E * fabs(x - one)) { /* region 2: y < tiny*|x-1| */
335 if (ix < 0x3ff00000) { /* x < 1 */
336 D_RE(ans) = acos(x);
337 D_IM(ans) = y / sqrt((one + x) * (one - x));
338 } else if (ix >= 0x43500000) { /* |x| >= 2**54 */
339 D_RE(ans) = y / x;
340 D_IM(ans) = ln2 + log(x);
341 } else {
342 t = sqrt((x - one) * (x + one));
343 D_RE(ans) = y / t;
344 if (ix >= 0x3ff80000) /* x > Acrossover */
345 D_IM(ans) = log(x + t);
346 else
347 D_IM(ans) = log1p((x - one) + t);
348 }
349 } else if (y < Foursqrtu) { /* region 3 */
350 t = sqrt(y);
351 D_RE(ans) = t;
352 D_IM(ans) = t;
353 } else if (E * y - one >= x) { /* region 4 */
354 D_RE(ans) = pi_2;
355 D_IM(ans) = ln2 + log(y);
356 } else if (ix >= 0x5fc00000 || iy >= 0x5fc00000) { /* x,y>2**509 */
357 /* region 5: x+1 or y is very large (>= sqrt(max)/8) */
358 t = x / y;
359 D_RE(ans) = atan(y / x);
360 D_IM(ans) = ln2 + log(y) + half * log1p(t * t);
361 } else if (x < Foursqrtu) {
362 /* region 6: x is very small, < 4sqrt(min) */
363 D_RE(ans) = pi_2;
364 A = sqrt(one + y * y);
365 if (iy >= 0x3ff80000) /* if y > Acrossover */
366 D_IM(ans) = log(y + A);
367 else
368 D_IM(ans) = half * log1p((y + y) * (y + A));
369 } else { /* safe region */
370 y2 = y * y;
371 xp1 = x + one;
372 xm1 = x - one;
373 R = sqrt(xp1 * xp1 + y2);
374 S = sqrt(xm1 * xm1 + y2);
375 A = half * (R + S);
376 B = x / A;
377 if (B <= Bcrossover)
378 D_RE(ans) = acos(B);
379 else { /* use atan and an accurate approx to a-x */
380 Apx = A + x;
381 if (x <= one)
382 D_RE(ans) = atan(sqrt(half * Apx * (y2 / (R +
383 xp1) + (S - xm1))) / x);
384 else
385 D_RE(ans) = atan((y * sqrt(half * (Apx / (R +
386 xp1) + Apx / (S + xm1)))) / x);
387 }
388 if (A <= Acrossover) {
389 /* use log1p and an accurate approx to A-1 */
390 if (x < one)
391 Am1 = half * (y2 / (R + xp1) + y2 / (S - xm1));
392 else
393 Am1 = half * (y2 / (R + xp1) + (S + xm1));
394 D_IM(ans) = log1p(Am1 + sqrt(Am1 * (A + one)));
395 } else {
396 D_IM(ans) = log(A + sqrt(A * A - one));
397 }
398 }
399 if (hx < 0)
400 D_RE(ans) = pi - D_RE(ans);
401 if (hy >= 0)
402 D_IM(ans) = -D_IM(ans);
403 return (ans);
404 }
|
5 * Common Development and Distribution License (the "License").
6 * You may not use this file except in compliance with the License.
7 *
8 * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
9 * or http://www.opensolaris.org/os/licensing.
10 * See the License for the specific language governing permissions
11 * and limitations under the License.
12 *
13 * When distributing Covered Code, include this CDDL HEADER in each
14 * file and include the License file at usr/src/OPENSOLARIS.LICENSE.
15 * If applicable, add the following below this CDDL HEADER, with the
16 * fields enclosed by brackets "[]" replaced with your own identifying
17 * information: Portions Copyright [yyyy] [name of copyright owner]
18 *
19 * CDDL HEADER END
20 */
21
22 /*
23 * Copyright 2011 Nexenta Systems, Inc. All rights reserved.
24 */
25
26 /*
27 * Copyright 2006 Sun Microsystems, Inc. All rights reserved.
28 * Use is subject to license terms.
29 */
30
31 #pragma weak __cacos = cacos
32
33
34 /*
35 * dcomplex cacos(dcomplex z);
36 *
37 * Alogrithm
38 * (based on T.E.Hull, Thomas F. Fairgrieve and Ping Tak Peter Tang's
39 * paper "Implementing the Complex Arcsine and Arccosine Functins Using
40 * Exception Handling", ACM TOMS, Vol 23, pp 299-335)
41 *
42 * The principal value of complex inverse cosine function cacos(z),
43 * where z = x+iy, can be defined by
44 *
45 * cacos(z) = acos(B) - i sign(y) log (A + sqrt(A*A-1)),
46 *
47 * where the log function is the natural log, and
48 * ____________ ____________
49 * 1 / 2 2 1 / 2 2
50 * A = --- / (x+1) + y + --- / (x-1) + y
51 * 2 \/ 2 \/
52 * ____________ ____________
53 * 1 / 2 2 1 / 2 2
174 * A ~ sqrt(x*x+y*y)
175 * B ~ x/sqrt(x*x+y*y).
176 * Thus
177 * real part = acos(B) = atan(y/x),
178 * imag part = log(A+sqrt(A*A-1)) ~ log(2A)
179 * = log(2) + 0.5*log(x*x+y*y)
180 * = log(2) + log(y) + 0.5*log(1+(x/y)^2)
181 *
182 * case 6. x < 4 sqrt(u). In this case, we have
183 * A ~ sqrt(1+y*y), B = x/sqrt(1+y*y).
184 * Since B is tiny, we have
185 * real part = acos(B) ~ pi/2
186 * imag part = log(A+sqrt(A*A-1)) = log (A+sqrt(y*y))
187 * = log(y+sqrt(1+y*y))
188 * = 0.5*log(y^2+2ysqrt(1+y^2)+1+y^2)
189 * = 0.5*log(1+2y(y+sqrt(1+y^2)));
190 * = 0.5*log1p(2y(y+A));
191 *
192 * cacos(z) = acos(B) - i sign(y) log (A + sqrt(A*A-1)),
193 */
194
195 #include "libm.h"
196 #include "complex_wrapper.h"
197
198 static const double zero = 0.0,
199 one = 1.0,
200 E = 1.11022302462515654042e-16, /* 2**-53 */
201 ln2 = 6.93147180559945286227e-01,
202 pi = 3.1415926535897931159979634685,
203 pi_l = 1.224646799147353177e-16,
204 pi_2 = 1.570796326794896558e+00,
205 pi_2_l = 6.123233995736765886e-17,
206 pi_4 = 0.78539816339744827899949,
207 pi_4_l = 3.061616997868382943e-17,
208 pi3_4 = 2.356194490192344836998,
209 pi3_4_l = 9.184850993605148829195e-17,
210 Foursqrtu = 5.96667258496016539463e-154, /* 2**(-509) */
211 Acrossover = 1.5,
212 Bcrossover = 0.6417,
213 half = 0.5;
214
215
216 dcomplex
217 cacos(dcomplex z)
218 {
219 double x, y, t, R, S, A, Am1, B, y2, xm1, xp1, Apx;
220 int ix, iy, hx, hy;
221 unsigned lx, ly;
222 dcomplex ans;
223
224 x = D_RE(z);
225 y = D_IM(z);
226 hx = HI_WORD(x);
227 lx = LO_WORD(x);
228 hy = HI_WORD(y);
229 ly = LO_WORD(y);
230 ix = hx & 0x7fffffff;
231 iy = hy & 0x7fffffff;
232
233 /* x is 0 */
234 if ((ix | lx) == 0) {
235 if (((iy | ly) == 0) || (iy >= 0x7ff00000)) {
236 D_RE(ans) = pi_2;
237 D_IM(ans) = -y;
238 return (ans);
239 }
240 }
241
242 /* |y| is inf or NaN */
243 if (iy >= 0x7ff00000) {
244 if (ISINF(iy, ly)) { /* cacos(x + i inf) = pi/2 - i inf */
245 D_IM(ans) = -y;
246
247 if (ix < 0x7ff00000) {
248 D_RE(ans) = pi_2 + pi_2_l;
249 } else if (ISINF(ix, lx)) {
250 if (hx >= 0)
251 D_RE(ans) = pi_4 + pi_4_l;
252 else
253 D_RE(ans) = pi3_4 + pi3_4_l;
254 } else {
255 D_RE(ans) = x;
256 }
257 } else { /* cacos(x + i NaN) = NaN + i NaN */
258 D_RE(ans) = y + x;
259
260 if (ISINF(ix, lx))
261 D_IM(ans) = -fabs(x);
262 else
263 D_IM(ans) = y;
264 }
265
266 return (ans);
267 }
268
269 x = fabs(x);
270 y = fabs(y);
271
272 /* x is inf or NaN */
273 if (ix >= 0x7ff00000) { /* x is inf or NaN */
274 if (ISINF(ix, lx)) { /* x is INF */
275 D_IM(ans) = -x;
276
277 if (iy >= 0x7ff00000) {
278 if (ISINF(iy, ly)) {
279 /*
280 * cacos(inf + i inf) = pi/4 - i inf
281 * cacos(-inf+ i inf) =3pi/4 - i inf
282 */
283 if (hx >= 0)
284 D_RE(ans) = pi_4 + pi_4_l;
285 else
286 D_RE(ans) = pi3_4 + pi3_4_l;
287 } else {
288 /*
289 * cacos(inf + i NaN) = NaN - i inf
290 */
291 D_RE(ans) = y + y;
292 }
293 } else
294 /*
295 * cacos(inf + iy ) = 0 - i inf
296 * cacos(-inf+ iy ) = pi - i inf
297 */
298 if (hx >= 0) {
299 D_RE(ans) = zero;
300 } else {
301 D_RE(ans) = pi + pi_l;
302 }
303 } else { /* x is NaN */
304
305 /*
306 * cacos(NaN + i inf) = NaN - i inf
307 * cacos(NaN + i y ) = NaN + i NaN
308 * cacos(NaN + i NaN) = NaN + i NaN
309 */
310 D_RE(ans) = x + y;
311
312 if (iy >= 0x7ff00000)
313 D_IM(ans) = -y;
314 else
315 D_IM(ans) = x;
316 }
317
318 if (hy < 0)
319 D_IM(ans) = -D_IM(ans);
320
321 return (ans);
322 }
323
324 if ((iy | ly) == 0) { /* region 1: y=0 */
325 if (ix < 0x3ff00000) { /* |x| < 1 */
326 D_RE(ans) = acos(x);
327 D_IM(ans) = zero;
328 } else {
329 D_RE(ans) = zero;
330
331 if (ix >= 0x43500000) { /* |x| >= 2**54 */
332 D_IM(ans) = ln2 + log(x);
333 } else if (ix >= 0x3ff80000) { /* x > Acrossover */
334 D_IM(ans) = log(x + sqrt((x - one) * (x +
335 one)));
336 } else {
337 xm1 = x - one;
338 D_IM(ans) = log1p(xm1 + sqrt(xm1 * (x + one)));
339 }
340 }
341 } else if (y <= E * fabs(x - one)) { /* region 2: y < tiny*|x-1| */
342 if (ix < 0x3ff00000) { /* x < 1 */
343 D_RE(ans) = acos(x);
344 D_IM(ans) = y / sqrt((one + x) * (one - x));
345 } else if (ix >= 0x43500000) { /* |x| >= 2**54 */
346 D_RE(ans) = y / x;
347 D_IM(ans) = ln2 + log(x);
348 } else {
349 t = sqrt((x - one) * (x + one));
350 D_RE(ans) = y / t;
351
352 if (ix >= 0x3ff80000) /* x > Acrossover */
353 D_IM(ans) = log(x + t);
354 else
355 D_IM(ans) = log1p((x - one) + t);
356 }
357 } else if (y < Foursqrtu) { /* region 3 */
358 t = sqrt(y);
359 D_RE(ans) = t;
360 D_IM(ans) = t;
361 } else if (E * y - one >= x) { /* region 4 */
362 D_RE(ans) = pi_2;
363 D_IM(ans) = ln2 + log(y);
364 } else if (ix >= 0x5fc00000 || iy >= 0x5fc00000) { /* x,y>2**509 */
365 /* region 5: x+1 or y is very large (>= sqrt(max)/8) */
366 t = x / y;
367 D_RE(ans) = atan(y / x);
368 D_IM(ans) = ln2 + log(y) + half * log1p(t * t);
369 } else if (x < Foursqrtu) {
370 /* region 6: x is very small, < 4sqrt(min) */
371 D_RE(ans) = pi_2;
372 A = sqrt(one + y * y);
373
374 if (iy >= 0x3ff80000) /* if y > Acrossover */
375 D_IM(ans) = log(y + A);
376 else
377 D_IM(ans) = half * log1p((y + y) * (y + A));
378 } else { /* safe region */
379 y2 = y * y;
380 xp1 = x + one;
381 xm1 = x - one;
382 R = sqrt(xp1 * xp1 + y2);
383 S = sqrt(xm1 * xm1 + y2);
384 A = half * (R + S);
385 B = x / A;
386
387 if (B <= Bcrossover) {
388 D_RE(ans) = acos(B);
389 } else { /* use atan and an accurate approx to a-x */
390 Apx = A + x;
391
392 if (x <= one)
393 D_RE(ans) = atan(sqrt(half * Apx * (y2 / (R +
394 xp1) + (S - xm1))) / x);
395 else
396 D_RE(ans) = atan((y * sqrt(half * (Apx / (R +
397 xp1) + Apx / (S + xm1)))) / x);
398 }
399
400 if (A <= Acrossover) {
401 /* use log1p and an accurate approx to A-1 */
402 if (x < one)
403 Am1 = half * (y2 / (R + xp1) + y2 / (S - xm1));
404 else
405 Am1 = half * (y2 / (R + xp1) + (S + xm1));
406
407 D_IM(ans) = log1p(Am1 + sqrt(Am1 * (A + one)));
408 } else {
409 D_IM(ans) = log(A + sqrt(A * A - one));
410 }
411 }
412
413 if (hx < 0)
414 D_RE(ans) = pi - D_RE(ans);
415
416 if (hy >= 0)
417 D_IM(ans) = -D_IM(ans);
418
419 return (ans);
420 }
|