1 /*
2 * CDDL HEADER START
3 *
4 * The contents of this file are subject to the terms of the
5 * Common Development and Distribution License (the "License").
6 * You may not use this file except in compliance with the License.
7 *
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10 * See the License for the specific language governing permissions
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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
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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
54 * B = --- / (x+1) + y - --- / (x-1) + y .
55 * 2 \/ 2 \/
56 *
57 * The Branch cuts are on the real line from -inf to -1 and from 1 to inf.
58 * The real and imaginary parts are based on Abramowitz and Stegun
59 * [Handbook of Mathematic Functions, 1972]. The sign of the imaginary
60 * part is chosen to be the generally considered the principal value of
61 * this function.
62 *
63 * Notes:1. A is the average of the distances from z to the points (1,0)
64 * and (-1,0) in the complex z-plane, and in particular A>=1.
65 * 2. B is in [-1,1], and A*B = x
66 *
67 * Basic relations
68 * cacos(conj(z)) = conj(cacos(z))
69 * cacos(-z) = pi - cacos(z)
70 * cacos( z) = pi/2 - casin(z)
71 *
72 * Special cases (conform to ISO/IEC 9899:1999(E)):
73 * cacos(+-0 + i y ) = pi/2 - i y for y is +-0, +-inf, NaN
74 * cacos( x + i inf) = pi/2 - i inf for all x
75 * cacos( x + i NaN) = NaN + i NaN with invalid for non-zero finite x
76 * cacos(-inf + i y ) = pi - i inf for finite +y
77 * cacos( inf + i y ) = 0 - i inf for finite +y
78 * cacos(-inf + i inf) = 3pi/4- i inf
79 * cacos( inf + i inf) = pi/4 - i inf
80 * cacos(+-inf+ i NaN) = NaN - i inf (sign of imaginary is unspecified)
81 * cacos(NaN + i y ) = NaN + i NaN with invalid for finite y
82 * cacos(NaN + i inf) = NaN - i inf
83 * cacos(NaN + i NaN) = NaN + i NaN
84 *
85 * Special Regions (better formula for accuracy and for avoiding spurious
86 * overflow or underflow) (all x and y are assumed nonnegative):
87 * case 1: y = 0
88 * case 2: tiny y relative to x-1: y <= ulp(0.5)*|x-1|
89 * case 3: tiny y: y < 4 sqrt(u), where u = minimum normal number
90 * case 4: huge y relative to x+1: y >= (1+x)/ulp(0.5)
91 * case 5: huge x and y: x and y >= sqrt(M)/8, where M = maximum normal number
92 * case 6: tiny x: x < 4 sqrt(u)
93 * --------
94 * case 1 & 2. y=0 or y/|x-1| is tiny. We have
95 * ____________ _____________
96 * / 2 2 / y 2
97 * / (x+-1) + y = |x+-1| / 1 + (------)
98 * \/ \/ |x+-1|
99 *
100 * 1 y 2
101 * ~ |x+-1| ( 1 + --- (------) )
102 * 2 |x+-1|
103 *
104 * 2
105 * y
106 * = |x+-1| + --------.
107 * 2|x+-1|
108 *
109 * Consequently, it is not difficult to see that
110 * 2
111 * y
112 * [ 1 + ------------ , if x < 1,
113 * [ 2(1+x)(1-x)
114 * [
115 * [
116 * [ x, if x = 1 (y = 0),
117 * [
118 * A ~= [ 2
119 * [ x * y
120 * [ x + ------------ ~ x, if x > 1
121 * [ 2(x+1)(x-1)
122 *
123 * and hence
124 * ______ 2
125 * / 2 y y
126 * A + \/ A - 1 ~ 1 + ---------------- + -----------, if x < 1,
127 * sqrt((x+1)(1-x)) 2(x+1)(1-x)
128 *
129 *
130 * ~ x + sqrt((x-1)*(x+1)), if x >= 1.
131 *
132 * 2
133 * y
134 * [ x(1 - -----------) ~ x, if x < 1,
135 * [ 2(1+x)(1-x)
136 * B = x/A ~ [
137 * [ 1, if x = 1,
138 * [
139 * [ 2
140 * [ y
141 * [ 1 - ------------ , if x > 1,
142 * [ 2(x+1)(x-1)
143 * Thus
144 * [ acos(x) - i y/sqrt((x-1)*(x+1)), if x < 1,
145 * [
146 * cacos(x+i*y)~ [ 0 - i 0, if x = 1,
147 * [
148 * [ y/sqrt(x*x-1) - i log(x+sqrt(x*x-1)), if x > 1.
149 *
150 * Note: y/sqrt(x*x-1) ~ y/x when x >= 2**26.
151 * case 3. y < 4 sqrt(u), where u = minimum normal x.
152 * After case 1 and 2, this will only occurs when x=1. When x=1, we have
153 * A = (sqrt(4+y*y)+y)/2 ~ 1 + y/2 + y^2/8 + ...
154 * and
155 * B = 1/A = 1 - y/2 + y^2/8 + ...
156 * Since
157 * cos(sqrt(y)) ~ 1 - y/2 + ...
158 * we have, for the real part,
159 * acos(B) ~ acos(1 - y/2) ~ sqrt(y)
160 * For the imaginary part,
161 * log(A+sqrt(A*A-1)) ~ log(1+y/2+sqrt(2*y/2))
162 * = log(1+y/2+sqrt(y))
163 * = (y/2+sqrt(y)) - (y/2+sqrt(y))^2/2 + ...
164 * ~ sqrt(y) - y*(sqrt(y)+y/2)/2
165 * ~ sqrt(y)
166 *
167 * case 4. y >= (x+1)/ulp(0.5). In this case, A ~ y and B ~ x/y. Thus
168 * real part = acos(B) ~ pi/2
169 * and
170 * imag part = log(y+sqrt(y*y-one))
171 *
172 * case 5. Both x and y are large: x and y > sqrt(M)/8, where M = maximum x
173 * In this case,
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 }