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 __casin = casin
31
32 /* INDENT OFF */
33 /*
34 * dcomplex casin(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 sine function casin(z),
42 * where z = x+iy, can be defined by
43 *
44 * casin(z) = asin(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
174 * A ~ sqrt(x*x+y*y)
175 * B ~ x/sqrt(x*x+y*y).
176 * Thus
177 * real part = asin(B) = atan(x/y),
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 = asin(B) ~ B = x/sqrt(1+y*y)
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 * casin(z) = asin(B) + i sign(y) log (A + sqrt(A*A-1)),
193 */
194 /* INDENT ON */
195
196 #include "libm.h" /* asin/atan/fabs/log/log1p/sqrt */
197 #include "complex_wrapper.h"
198
199 /* INDENT OFF */
200 static const double
201 zero = 0.0,
202 one = 1.0,
203 E = 1.11022302462515654042e-16, /* 2**-53 */
204 ln2 = 6.93147180559945286227e-01,
205 pi_2 = 1.570796326794896558e+00,
206 pi_2_l = 6.123233995736765886e-17,
207 pi_4 = 7.85398163397448278999e-01,
208 Foursqrtu = 5.96667258496016539463e-154, /* 2**(-509) */
209 Acrossover = 1.5,
210 Bcrossover = 0.6417,
211 half = 0.5;
212 /* INDENT ON */
213
214 dcomplex
215 casin(dcomplex z) {
216 double x, y, t, R, S, A, Am1, B, y2, xm1, xp1, Apx;
217 int ix, iy, hx, hy;
218 unsigned lx, ly;
219 dcomplex ans;
220
221 x = D_RE(z);
222 y = D_IM(z);
223 hx = HI_WORD(x);
224 lx = LO_WORD(x);
225 hy = HI_WORD(y);
226 ly = LO_WORD(y);
227 ix = hx & 0x7fffffff;
228 iy = hy & 0x7fffffff;
229 x = fabs(x);
230 y = fabs(y);
231
232 /* special cases */
233
234 /* x is inf or NaN */
235 if (ix >= 0x7ff00000) { /* x is inf or NaN */
236 if (ISINF(ix, lx)) { /* x is INF */
237 D_IM(ans) = x;
238 if (iy >= 0x7ff00000) {
239 if (ISINF(iy, ly))
240 /* casin(inf + i inf) = pi/4 + i inf */
241 D_RE(ans) = pi_4;
242 else /* casin(inf + i NaN) = NaN + i inf */
243 D_RE(ans) = y + y;
244 } else /* casin(inf + iy) = pi/2 + i inf */
245 D_RE(ans) = pi_2;
246 } else { /* x is NaN */
247 if (iy >= 0x7ff00000) {
248 /* INDENT OFF */
249 /*
250 * casin(NaN + i inf) = NaN + i inf
251 * casin(NaN + i NaN) = NaN + i NaN
252 */
253 /* INDENT ON */
254 D_IM(ans) = y + y;
255 D_RE(ans) = x + x;
256 } else {
257 /* casin(NaN + i y ) = NaN + i NaN */
258 D_IM(ans) = D_RE(ans) = x + y;
259 }
260 }
261 if (hx < 0)
262 D_RE(ans) = -D_RE(ans);
263 if (hy < 0)
264 D_IM(ans) = -D_IM(ans);
265 return (ans);
266 }
267
268 /* casin(+0 + i 0 ) = 0 + i 0. */
269 if ((ix | lx | iy | ly) == 0)
270 return (z);
271
272 if (iy >= 0x7ff00000) { /* y is inf or NaN */
273 if (ISINF(iy, ly)) { /* casin(x + i inf) = 0 + i inf */
274 D_IM(ans) = y;
275 D_RE(ans) = zero;
276 } else { /* casin(x + i NaN) = NaN + i NaN */
277 D_IM(ans) = x + y;
278 if ((ix | lx) == 0)
279 D_RE(ans) = x;
280 else
281 D_RE(ans) = y;
282 }
283 if (hx < 0)
284 D_RE(ans) = -D_RE(ans);
285 if (hy < 0)
286 D_IM(ans) = -D_IM(ans);
287 return (ans);
288 }
289
290 if ((iy | ly) == 0) { /* region 1: y=0 */
291 if (ix < 0x3ff00000) { /* |x| < 1 */
292 D_RE(ans) = asin(x);
293 D_IM(ans) = zero;
294 } else {
295 D_RE(ans) = pi_2;
296 if (ix >= 0x43500000) /* |x| >= 2**54 */
297 D_IM(ans) = ln2 + log(x);
298 else if (ix >= 0x3ff80000) /* x > Acrossover */
299 D_IM(ans) = log(x + sqrt((x - one) * (x +
300 one)));
301 else {
302 xm1 = x - one;
303 D_IM(ans) = log1p(xm1 + sqrt(xm1 * (x + one)));
304 }
305 }
306 } else if (y <= E * fabs(x - one)) { /* region 2: y < tiny*|x-1| */
307 if (ix < 0x3ff00000) { /* x < 1 */
308 D_RE(ans) = asin(x);
309 D_IM(ans) = y / sqrt((one + x) * (one - x));
310 } else {
311 D_RE(ans) = pi_2;
312 if (ix >= 0x43500000) { /* |x| >= 2**54 */
313 D_IM(ans) = ln2 + log(x);
314 } else if (ix >= 0x3ff80000) /* x > Acrossover */
315 D_IM(ans) = log(x + sqrt((x - one) * (x +
316 one)));
317 else
318 D_IM(ans) = log1p((x - one) + sqrt((x - one) *
319 (x + one)));
320 }
321 } else if (y < Foursqrtu) { /* region 3 */
322 t = sqrt(y);
323 D_RE(ans) = pi_2 - (t - pi_2_l);
324 D_IM(ans) = t;
325 } else if (E * y - one >= x) { /* region 4 */
326 D_RE(ans) = x / y; /* need to fix underflow cases */
327 D_IM(ans) = ln2 + log(y);
328 } else if (ix >= 0x5fc00000 || iy >= 0x5fc00000) { /* x,y>2**509 */
329 /* region 5: x+1 or y is very large (>= sqrt(max)/8) */
330 t = x / y;
331 D_RE(ans) = atan(t);
332 D_IM(ans) = ln2 + log(y) + half * log1p(t * t);
333 } else if (x < Foursqrtu) {
334 /* region 6: x is very small, < 4sqrt(min) */
335 A = sqrt(one + y * y);
336 D_RE(ans) = x / A; /* may underflow */
337 if (iy >= 0x3ff80000) /* if y > Acrossover */
338 D_IM(ans) = log(y + A);
339 else
340 D_IM(ans) = half * log1p((y + y) * (y + A));
341 } else { /* safe region */
342 y2 = y * y;
343 xp1 = x + one;
344 xm1 = x - one;
345 R = sqrt(xp1 * xp1 + y2);
346 S = sqrt(xm1 * xm1 + y2);
347 A = half * (R + S);
348 B = x / A;
349
350 if (B <= Bcrossover)
351 D_RE(ans) = asin(B);
352 else { /* use atan and an accurate approx to a-x */
353 Apx = A + x;
354 if (x <= one)
355 D_RE(ans) = atan(x / sqrt(half * Apx * (y2 /
356 (R + xp1) + (S - xm1))));
357 else
358 D_RE(ans) = atan(x / (y * sqrt(half * (Apx /
359 (R + xp1) + Apx / (S + xm1)))));
360 }
361 if (A <= Acrossover) {
362 /* use log1p and an accurate approx to A-1 */
363 if (x < one)
364 Am1 = half * (y2 / (R + xp1) + y2 / (S - xm1));
365 else
366 Am1 = half * (y2 / (R + xp1) + (S + xm1));
367 D_IM(ans) = log1p(Am1 + sqrt(Am1 * (A + one)));
368 } else {
369 D_IM(ans) = log(A + sqrt(A * A - one));
370 }
371 }
372
373 if (hx < 0)
374 D_RE(ans) = -D_RE(ans);
375 if (hy < 0)
376 D_IM(ans) = -D_IM(ans);
377
378 return (ans);
379 }
|
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 __casin = casin
32
33
34 /*
35 * dcomplex casin(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 sine function casin(z),
43 * where z = x+iy, can be defined by
44 *
45 * casin(z) = asin(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
175 * A ~ sqrt(x*x+y*y)
176 * B ~ x/sqrt(x*x+y*y).
177 * Thus
178 * real part = asin(B) = atan(x/y),
179 * imag part = log(A+sqrt(A*A-1)) ~ log(2A)
180 * = log(2) + 0.5*log(x*x+y*y)
181 * = log(2) + log(y) + 0.5*log(1+(x/y)^2)
182 *
183 * case 6. x < 4 sqrt(u). In this case, we have
184 * A ~ sqrt(1+y*y), B = x/sqrt(1+y*y).
185 * Since B is tiny, we have
186 * real part = asin(B) ~ B = x/sqrt(1+y*y)
187 * imag part = log(A+sqrt(A*A-1)) = log (A+sqrt(y*y))
188 * = log(y+sqrt(1+y*y))
189 * = 0.5*log(y^2+2ysqrt(1+y^2)+1+y^2)
190 * = 0.5*log(1+2y(y+sqrt(1+y^2)));
191 * = 0.5*log1p(2y(y+A));
192 *
193 * casin(z) = asin(B) + i sign(y) log (A + sqrt(A*A-1)),
194 */
195
196 #include "libm.h" /* asin/atan/fabs/log/log1p/sqrt */
197 #include "complex_wrapper.h"
198
199 static const double zero = 0.0,
200 one = 1.0,
201 E = 1.11022302462515654042e-16, /* 2**-53 */
202 ln2 = 6.93147180559945286227e-01,
203 pi_2 = 1.570796326794896558e+00,
204 pi_2_l = 6.123233995736765886e-17,
205 pi_4 = 7.85398163397448278999e-01,
206 Foursqrtu = 5.96667258496016539463e-154, /* 2**(-509) */
207 Acrossover = 1.5,
208 Bcrossover = 0.6417,
209 half = 0.5;
210
211
212 dcomplex
213 casin(dcomplex z)
214 {
215 double x, y, t, R, S, A, Am1, B, y2, xm1, xp1, Apx;
216 int ix, iy, hx, hy;
217 unsigned lx, ly;
218 dcomplex ans;
219
220 x = D_RE(z);
221 y = D_IM(z);
222 hx = HI_WORD(x);
223 lx = LO_WORD(x);
224 hy = HI_WORD(y);
225 ly = LO_WORD(y);
226 ix = hx & 0x7fffffff;
227 iy = hy & 0x7fffffff;
228 x = fabs(x);
229 y = fabs(y);
230
231 /* special cases */
232
233 /* x is inf or NaN */
234 if (ix >= 0x7ff00000) { /* x is inf or NaN */
235 if (ISINF(ix, lx)) { /* x is INF */
236 D_IM(ans) = x;
237
238 if (iy >= 0x7ff00000) {
239 if (ISINF(iy, ly))
240 /* casin(inf + i inf) = pi/4 + i inf */
241 D_RE(ans) = pi_4;
242 else /* casin(inf + i NaN) = NaN + i inf */
243 D_RE(ans) = y + y;
244 } else { /* casin(inf + iy) = pi/2 + i inf */
245 D_RE(ans) = pi_2;
246 }
247 } else { /* x is NaN */
248 if (iy >= 0x7ff00000) {
249
250 /*
251 * casin(NaN + i inf) = NaN + i inf
252 * casin(NaN + i NaN) = NaN + i NaN
253 */
254 D_IM(ans) = y + y;
255 D_RE(ans) = x + x;
256 } else {
257 /* casin(NaN + i y ) = NaN + i NaN */
258 D_IM(ans) = D_RE(ans) = x + y;
259 }
260 }
261
262 if (hx < 0)
263 D_RE(ans) = -D_RE(ans);
264
265 if (hy < 0)
266 D_IM(ans) = -D_IM(ans);
267
268 return (ans);
269 }
270
271 /* casin(+0 + i 0 ) = 0 + i 0. */
272 if ((ix | lx | iy | ly) == 0)
273 return (z);
274
275 if (iy >= 0x7ff00000) { /* y is inf or NaN */
276 if (ISINF(iy, ly)) { /* casin(x + i inf) = 0 + i inf */
277 D_IM(ans) = y;
278 D_RE(ans) = zero;
279 } else { /* casin(x + i NaN) = NaN + i NaN */
280 D_IM(ans) = x + y;
281
282 if ((ix | lx) == 0)
283 D_RE(ans) = x;
284 else
285 D_RE(ans) = y;
286 }
287
288 if (hx < 0)
289 D_RE(ans) = -D_RE(ans);
290
291 if (hy < 0)
292 D_IM(ans) = -D_IM(ans);
293
294 return (ans);
295 }
296
297 if ((iy | ly) == 0) { /* region 1: y=0 */
298 if (ix < 0x3ff00000) { /* |x| < 1 */
299 D_RE(ans) = asin(x);
300 D_IM(ans) = zero;
301 } else {
302 D_RE(ans) = pi_2;
303
304 if (ix >= 0x43500000) { /* |x| >= 2**54 */
305 D_IM(ans) = ln2 + log(x);
306 } else if (ix >= 0x3ff80000) { /* x > Acrossover */
307 D_IM(ans) = log(x + sqrt((x - one) * (x +
308 one)));
309 } else {
310 xm1 = x - one;
311 D_IM(ans) = log1p(xm1 + sqrt(xm1 * (x + one)));
312 }
313 }
314 } else if (y <= E * fabs(x - one)) { /* region 2: y < tiny*|x-1| */
315 if (ix < 0x3ff00000) { /* x < 1 */
316 D_RE(ans) = asin(x);
317 D_IM(ans) = y / sqrt((one + x) * (one - x));
318 } else {
319 D_RE(ans) = pi_2;
320
321 if (ix >= 0x43500000) /* |x| >= 2**54 */
322 D_IM(ans) = ln2 + log(x);
323 else if (ix >= 0x3ff80000) /* x > Acrossover */
324 D_IM(ans) = log(x + sqrt((x - one) * (x +
325 one)));
326 else
327 D_IM(ans) = log1p((x - one) + sqrt((x - one) *
328 (x + one)));
329 }
330 } else if (y < Foursqrtu) { /* region 3 */
331 t = sqrt(y);
332 D_RE(ans) = pi_2 - (t - pi_2_l);
333 D_IM(ans) = t;
334 } else if (E * y - one >= x) { /* region 4 */
335 D_RE(ans) = x / y; /* need to fix underflow cases */
336 D_IM(ans) = ln2 + log(y);
337 } else if (ix >= 0x5fc00000 || iy >= 0x5fc00000) { /* x,y>2**509 */
338 /* region 5: x+1 or y is very large (>= sqrt(max)/8) */
339 t = x / y;
340 D_RE(ans) = atan(t);
341 D_IM(ans) = ln2 + log(y) + half * log1p(t * t);
342 } else if (x < Foursqrtu) {
343 /* region 6: x is very small, < 4sqrt(min) */
344 A = sqrt(one + y * y);
345 D_RE(ans) = x / A; /* may underflow */
346
347 if (iy >= 0x3ff80000) /* if y > Acrossover */
348 D_IM(ans) = log(y + A);
349 else
350 D_IM(ans) = half * log1p((y + y) * (y + A));
351 } else { /* safe region */
352 y2 = y * y;
353 xp1 = x + one;
354 xm1 = x - one;
355 R = sqrt(xp1 * xp1 + y2);
356 S = sqrt(xm1 * xm1 + y2);
357 A = half * (R + S);
358 B = x / A;
359
360 if (B <= Bcrossover) {
361 D_RE(ans) = asin(B);
362 } else { /* use atan and an accurate approx to a-x */
363 Apx = A + x;
364
365 if (x <= one)
366 D_RE(ans) = atan(x / sqrt(half * Apx * (y2 /
367 (R + xp1) + (S - xm1))));
368 else
369 D_RE(ans) = atan(x / (y * sqrt(half * (Apx /
370 (R + xp1) + Apx / (S + xm1)))));
371 }
372
373 if (A <= Acrossover) {
374 /* use log1p and an accurate approx to A-1 */
375 if (x < one)
376 Am1 = half * (y2 / (R + xp1) + y2 / (S - xm1));
377 else
378 Am1 = half * (y2 / (R + xp1) + (S + xm1));
379
380 D_IM(ans) = log1p(Am1 + sqrt(Am1 * (A + one)));
381 } else {
382 D_IM(ans) = log(A + sqrt(A * A - one));
383 }
384 }
385
386 if (hx < 0)
387 D_RE(ans) = -D_RE(ans);
388
389 if (hy < 0)
390 D_IM(ans) = -D_IM(ans);
391
392 return (ans);
393 }
|