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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12 *
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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 * 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
53 * B = --- / (x+1) + y - --- / (x-1) + y .
54 * 2 \/ 2 \/
55 *
56 * The Branch cuts are on the real line from -inf to -1 and from 1 to inf.
57 * The real and imaginary parts are based on Abramowitz and Stegun
58 * [Handbook of Mathematic Functions, 1972]. The sign of the imaginary
59 * part is chosen to be the generally considered the principal value of
60 * this function.
61 *
62 * Notes:1. A is the average of the distances from z to the points (1,0)
63 * and (-1,0) in the complex z-plane, and in particular A>=1.
64 * 2. B is in [-1,1], and A*B = x
65 *
66 * Basic relations
67 * cacos(conj(z)) = conj(cacos(z))
68 * cacos(-z) = pi - cacos(z)
69 * cacos( z) = pi/2 - casin(z)
70 *
71 * Special cases (conform to ISO/IEC 9899:1999(E)):
72 * cacos(+-0 + i y ) = pi/2 - i y for y is +-0, +-inf, NaN
73 * cacos( x + i inf) = pi/2 - i inf for all x
74 * cacos( x + i NaN) = NaN + i NaN with invalid for non-zero finite x
75 * cacos(-inf + i y ) = pi - i inf for finite +y
76 * cacos( inf + i y ) = 0 - i inf for finite +y
77 * cacos(-inf + i inf) = 3pi/4- i inf
78 * cacos( inf + i inf) = pi/4 - i inf
79 * cacos(+-inf+ i NaN) = NaN - i inf (sign of imaginary is unspecified)
80 * cacos(NaN + i y ) = NaN + i NaN with invalid for finite y
81 * cacos(NaN + i inf) = NaN - i inf
82 * cacos(NaN + i NaN) = NaN + i NaN
83 *
84 * Special Regions (better formula for accuracy and for avoiding spurious
85 * overflow or underflow) (all x and y are assumed nonnegative):
86 * case 1: y = 0
87 * case 2: tiny y relative to x-1: y <= ulp(0.5)*|x-1|
88 * case 3: tiny y: y < 4 sqrt(u), where u = minimum normal number
89 * case 4: huge y relative to x+1: y >= (1+x)/ulp(0.5)
90 * case 5: huge x and y: x and y >= sqrt(M)/8, where M = maximum normal number
91 * case 6: tiny x: x < 4 sqrt(u)
92 * --------
93 * case 1 & 2. y=0 or y/|x-1| is tiny. We have
94 * ____________ _____________
95 * / 2 2 / y 2
96 * / (x+-1) + y = |x+-1| / 1 + (------)
97 * \/ \/ |x+-1|
98 *
99 * 1 y 2
100 * ~ |x+-1| ( 1 + --- (------) )
101 * 2 |x+-1|
102 *
103 * 2
104 * y
105 * = |x+-1| + --------.
106 * 2|x+-1|
107 *
108 * Consequently, it is not difficult to see that
109 * 2
110 * y
111 * [ 1 + ------------ , if x < 1,
112 * [ 2(1+x)(1-x)
113 * [
114 * [
115 * [ x, if x = 1 (y = 0),
116 * [
117 * A ~= [ 2
118 * [ x * y
119 * [ x + ------------ ~ x, if x > 1
120 * [ 2(x+1)(x-1)
121 *
122 * and hence
123 * ______ 2
124 * / 2 y y
125 * A + \/ A - 1 ~ 1 + ---------------- + -----------, if x < 1,
126 * sqrt((x+1)(1-x)) 2(x+1)(1-x)
127 *
128 *
129 * ~ x + sqrt((x-1)*(x+1)), if x >= 1.
130 *
131 * 2
132 * y
133 * [ x(1 - -----------) ~ x, if x < 1,
134 * [ 2(1+x)(1-x)
135 * B = x/A ~ [
136 * [ 1, if x = 1,
137 * [
138 * [ 2
139 * [ y
140 * [ 1 - ------------ , if x > 1,
141 * [ 2(x+1)(x-1)
142 * Thus
143 * [ acos(x) - i y/sqrt((x-1)*(x+1)), if x < 1,
144 * [
145 * cacos(x+i*y)~ [ 0 - i 0, if x = 1,
146 * [
147 * [ y/sqrt(x*x-1) - i log(x+sqrt(x*x-1)), if x > 1.
148 *
149 * Note: y/sqrt(x*x-1) ~ y/x when x >= 2**26.
150 * case 3. y < 4 sqrt(u), where u = minimum normal x.
151 * After case 1 and 2, this will only occurs when x=1. When x=1, we have
152 * A = (sqrt(4+y*y)+y)/2 ~ 1 + y/2 + y^2/8 + ...
153 * and
154 * B = 1/A = 1 - y/2 + y^2/8 + ...
155 * Since
156 * cos(sqrt(y)) ~ 1 - y/2 + ...
157 * we have, for the real part,
158 * acos(B) ~ acos(1 - y/2) ~ sqrt(y)
159 * For the imaginary part,
160 * log(A+sqrt(A*A-1)) ~ log(1+y/2+sqrt(2*y/2))
161 * = log(1+y/2+sqrt(y))
162 * = (y/2+sqrt(y)) - (y/2+sqrt(y))^2/2 + ...
163 * ~ sqrt(y) - y*(sqrt(y)+y/2)/2
164 * ~ sqrt(y)
165 *
166 * case 4. y >= (x+1)/ulp(0.5). In this case, A ~ y and B ~ x/y. Thus
167 * real part = acos(B) ~ pi/2
168 * and
169 * imag part = log(y+sqrt(y*y-one))
170 *
171 * case 5. Both x and y are large: x and y > sqrt(M)/8, where M = maximum x
172 * In this case,
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 }