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11210 libm should be cstyle(1ONBLD) clean
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--- old/usr/src/lib/libm/common/complex/cacos.c
+++ new/usr/src/lib/libm/common/complex/cacos.c
1 1 /*
2 2 * CDDL HEADER START
3 3 *
4 4 * The contents of this file are subject to the terms of the
5 5 * Common Development and Distribution License (the "License").
6 6 * You may not use this file except in compliance with the License.
7 7 *
8 8 * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
9 9 * or http://www.opensolaris.org/os/licensing.
10 10 * See the License for the specific language governing permissions
11 11 * and limitations under the License.
12 12 *
13 13 * When distributing Covered Code, include this CDDL HEADER in each
14 14 * file and include the License file at usr/src/OPENSOLARIS.LICENSE.
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15 15 * If applicable, add the following below this CDDL HEADER, with the
16 16 * fields enclosed by brackets "[]" replaced with your own identifying
17 17 * information: Portions Copyright [yyyy] [name of copyright owner]
18 18 *
19 19 * CDDL HEADER END
20 20 */
21 21
22 22 /*
23 23 * Copyright 2011 Nexenta Systems, Inc. All rights reserved.
24 24 */
25 +
25 26 /*
26 27 * Copyright 2006 Sun Microsystems, Inc. All rights reserved.
27 28 * Use is subject to license terms.
28 29 */
29 30
30 31 #pragma weak __cacos = cacos
31 32
32 -/* INDENT OFF */
33 +
33 34 /*
34 35 * dcomplex cacos(dcomplex z);
35 36 *
36 37 * Alogrithm
37 38 * (based on T.E.Hull, Thomas F. Fairgrieve and Ping Tak Peter Tang's
38 39 * paper "Implementing the Complex Arcsine and Arccosine Functins Using
39 40 * Exception Handling", ACM TOMS, Vol 23, pp 299-335)
40 41 *
41 42 * The principal value of complex inverse cosine function cacos(z),
42 43 * where z = x+iy, can be defined by
43 44 *
44 - * cacos(z) = acos(B) - i sign(y) log (A + sqrt(A*A-1)),
45 + * cacos(z) = acos(B) - i sign(y) log (A + sqrt(A*A-1)),
45 46 *
46 47 * where the log function is the natural log, and
47 48 * ____________ ____________
48 49 * 1 / 2 2 1 / 2 2
49 50 * A = --- / (x+1) + y + --- / (x-1) + y
50 51 * 2 \/ 2 \/
51 52 * ____________ ____________
52 53 * 1 / 2 2 1 / 2 2
53 54 * B = --- / (x+1) + y - --- / (x-1) + y .
54 55 * 2 \/ 2 \/
55 56 *
56 57 * The Branch cuts are on the real line from -inf to -1 and from 1 to inf.
57 58 * The real and imaginary parts are based on Abramowitz and Stegun
58 59 * [Handbook of Mathematic Functions, 1972]. The sign of the imaginary
59 60 * part is chosen to be the generally considered the principal value of
60 61 * this function.
61 62 *
62 63 * Notes:1. A is the average of the distances from z to the points (1,0)
63 64 * and (-1,0) in the complex z-plane, and in particular A>=1.
64 65 * 2. B is in [-1,1], and A*B = x
65 66 *
66 67 * Basic relations
67 68 * cacos(conj(z)) = conj(cacos(z))
68 69 * cacos(-z) = pi - cacos(z)
69 70 * cacos( z) = pi/2 - casin(z)
70 71 *
71 72 * Special cases (conform to ISO/IEC 9899:1999(E)):
72 73 * cacos(+-0 + i y ) = pi/2 - i y for y is +-0, +-inf, NaN
73 74 * cacos( x + i inf) = pi/2 - i inf for all x
74 75 * cacos( x + i NaN) = NaN + i NaN with invalid for non-zero finite x
75 76 * cacos(-inf + i y ) = pi - i inf for finite +y
76 77 * cacos( inf + i y ) = 0 - i inf for finite +y
77 78 * cacos(-inf + i inf) = 3pi/4- i inf
78 79 * cacos( inf + i inf) = pi/4 - i inf
79 80 * cacos(+-inf+ i NaN) = NaN - i inf (sign of imaginary is unspecified)
80 81 * cacos(NaN + i y ) = NaN + i NaN with invalid for finite y
81 82 * cacos(NaN + i inf) = NaN - i inf
82 83 * cacos(NaN + i NaN) = NaN + i NaN
83 84 *
84 85 * Special Regions (better formula for accuracy and for avoiding spurious
85 86 * overflow or underflow) (all x and y are assumed nonnegative):
86 87 * case 1: y = 0
87 88 * case 2: tiny y relative to x-1: y <= ulp(0.5)*|x-1|
88 89 * case 3: tiny y: y < 4 sqrt(u), where u = minimum normal number
89 90 * case 4: huge y relative to x+1: y >= (1+x)/ulp(0.5)
90 91 * case 5: huge x and y: x and y >= sqrt(M)/8, where M = maximum normal number
91 92 * case 6: tiny x: x < 4 sqrt(u)
92 93 * --------
93 94 * case 1 & 2. y=0 or y/|x-1| is tiny. We have
94 95 * ____________ _____________
95 96 * / 2 2 / y 2
96 97 * / (x+-1) + y = |x+-1| / 1 + (------)
97 98 * \/ \/ |x+-1|
98 99 *
99 100 * 1 y 2
100 101 * ~ |x+-1| ( 1 + --- (------) )
101 102 * 2 |x+-1|
102 103 *
103 104 * 2
104 105 * y
105 106 * = |x+-1| + --------.
106 107 * 2|x+-1|
107 108 *
108 109 * Consequently, it is not difficult to see that
109 110 * 2
110 111 * y
111 112 * [ 1 + ------------ , if x < 1,
112 113 * [ 2(1+x)(1-x)
113 114 * [
114 115 * [
115 116 * [ x, if x = 1 (y = 0),
116 117 * [
117 118 * A ~= [ 2
118 119 * [ x * y
119 120 * [ x + ------------ ~ x, if x > 1
120 121 * [ 2(x+1)(x-1)
121 122 *
122 123 * and hence
123 124 * ______ 2
124 125 * / 2 y y
125 126 * A + \/ A - 1 ~ 1 + ---------------- + -----------, if x < 1,
126 127 * sqrt((x+1)(1-x)) 2(x+1)(1-x)
127 128 *
128 129 *
129 130 * ~ x + sqrt((x-1)*(x+1)), if x >= 1.
130 131 *
131 132 * 2
132 133 * y
133 134 * [ x(1 - -----------) ~ x, if x < 1,
134 135 * [ 2(1+x)(1-x)
135 136 * B = x/A ~ [
136 137 * [ 1, if x = 1,
137 138 * [
138 139 * [ 2
139 140 * [ y
140 141 * [ 1 - ------------ , if x > 1,
141 142 * [ 2(x+1)(x-1)
142 143 * Thus
143 144 * [ acos(x) - i y/sqrt((x-1)*(x+1)), if x < 1,
144 145 * [
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145 146 * cacos(x+i*y)~ [ 0 - i 0, if x = 1,
146 147 * [
147 148 * [ y/sqrt(x*x-1) - i log(x+sqrt(x*x-1)), if x > 1.
148 149 *
149 150 * Note: y/sqrt(x*x-1) ~ y/x when x >= 2**26.
150 151 * case 3. y < 4 sqrt(u), where u = minimum normal x.
151 152 * After case 1 and 2, this will only occurs when x=1. When x=1, we have
152 153 * A = (sqrt(4+y*y)+y)/2 ~ 1 + y/2 + y^2/8 + ...
153 154 * and
154 155 * B = 1/A = 1 - y/2 + y^2/8 + ...
155 - * Since
156 + * Since
156 157 * cos(sqrt(y)) ~ 1 - y/2 + ...
157 158 * we have, for the real part,
158 159 * acos(B) ~ acos(1 - y/2) ~ sqrt(y)
159 160 * For the imaginary part,
160 161 * log(A+sqrt(A*A-1)) ~ log(1+y/2+sqrt(2*y/2))
161 162 * = log(1+y/2+sqrt(y))
162 163 * = (y/2+sqrt(y)) - (y/2+sqrt(y))^2/2 + ...
163 164 * ~ sqrt(y) - y*(sqrt(y)+y/2)/2
164 165 * ~ sqrt(y)
165 166 *
166 167 * case 4. y >= (x+1)/ulp(0.5). In this case, A ~ y and B ~ x/y. Thus
167 168 * real part = acos(B) ~ pi/2
168 - * and
169 + * and
169 170 * imag part = log(y+sqrt(y*y-one))
170 171 *
171 172 * case 5. Both x and y are large: x and y > sqrt(M)/8, where M = maximum x
172 173 * In this case,
173 174 * A ~ sqrt(x*x+y*y)
174 175 * B ~ x/sqrt(x*x+y*y).
175 176 * Thus
176 177 * real part = acos(B) = atan(y/x),
177 178 * imag part = log(A+sqrt(A*A-1)) ~ log(2A)
178 179 * = log(2) + 0.5*log(x*x+y*y)
179 180 * = log(2) + log(y) + 0.5*log(1+(x/y)^2)
180 181 *
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181 182 * case 6. x < 4 sqrt(u). In this case, we have
182 183 * A ~ sqrt(1+y*y), B = x/sqrt(1+y*y).
183 184 * Since B is tiny, we have
184 185 * real part = acos(B) ~ pi/2
185 186 * imag part = log(A+sqrt(A*A-1)) = log (A+sqrt(y*y))
186 187 * = log(y+sqrt(1+y*y))
187 188 * = 0.5*log(y^2+2ysqrt(1+y^2)+1+y^2)
188 189 * = 0.5*log(1+2y(y+sqrt(1+y^2)));
189 190 * = 0.5*log1p(2y(y+A));
190 191 *
191 - * cacos(z) = acos(B) - i sign(y) log (A + sqrt(A*A-1)),
192 + * cacos(z) = acos(B) - i sign(y) log (A + sqrt(A*A-1)),
192 193 */
193 -/* INDENT ON */
194 194
195 195 #include "libm.h"
196 196 #include "complex_wrapper.h"
197 197
198 -/* INDENT OFF */
199 -static const double
200 - zero = 0.0,
198 +static const double zero = 0.0,
201 199 one = 1.0,
202 - E = 1.11022302462515654042e-16, /* 2**-53 */
200 + E = 1.11022302462515654042e-16, /* 2**-53 */
203 201 ln2 = 6.93147180559945286227e-01,
204 202 pi = 3.1415926535897931159979634685,
205 203 pi_l = 1.224646799147353177e-16,
206 204 pi_2 = 1.570796326794896558e+00,
207 205 pi_2_l = 6.123233995736765886e-17,
208 206 pi_4 = 0.78539816339744827899949,
209 207 pi_4_l = 3.061616997868382943e-17,
210 208 pi3_4 = 2.356194490192344836998,
211 209 pi3_4_l = 9.184850993605148829195e-17,
212 210 Foursqrtu = 5.96667258496016539463e-154, /* 2**(-509) */
213 211 Acrossover = 1.5,
214 212 Bcrossover = 0.6417,
215 213 half = 0.5;
216 -/* INDENT ON */
214 +
217 215
218 216 dcomplex
219 -cacos(dcomplex z) {
217 +cacos(dcomplex z)
218 +{
220 219 double x, y, t, R, S, A, Am1, B, y2, xm1, xp1, Apx;
221 220 int ix, iy, hx, hy;
222 221 unsigned lx, ly;
223 222 dcomplex ans;
224 223
225 224 x = D_RE(z);
226 225 y = D_IM(z);
227 226 hx = HI_WORD(x);
228 227 lx = LO_WORD(x);
229 228 hy = HI_WORD(y);
230 229 ly = LO_WORD(y);
231 230 ix = hx & 0x7fffffff;
232 231 iy = hy & 0x7fffffff;
233 232
234 233 /* x is 0 */
235 234 if ((ix | lx) == 0) {
236 235 if (((iy | ly) == 0) || (iy >= 0x7ff00000)) {
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237 236 D_RE(ans) = pi_2;
238 237 D_IM(ans) = -y;
239 238 return (ans);
240 239 }
241 240 }
242 241
243 242 /* |y| is inf or NaN */
244 243 if (iy >= 0x7ff00000) {
245 244 if (ISINF(iy, ly)) { /* cacos(x + i inf) = pi/2 - i inf */
246 245 D_IM(ans) = -y;
246 +
247 247 if (ix < 0x7ff00000) {
248 248 D_RE(ans) = pi_2 + pi_2_l;
249 249 } else if (ISINF(ix, lx)) {
250 250 if (hx >= 0)
251 251 D_RE(ans) = pi_4 + pi_4_l;
252 252 else
253 253 D_RE(ans) = pi3_4 + pi3_4_l;
254 254 } else {
255 255 D_RE(ans) = x;
256 256 }
257 257 } else { /* cacos(x + i NaN) = NaN + i NaN */
258 258 D_RE(ans) = y + x;
259 +
259 260 if (ISINF(ix, lx))
260 261 D_IM(ans) = -fabs(x);
261 262 else
262 263 D_IM(ans) = y;
263 264 }
265 +
264 266 return (ans);
265 267 }
266 268
267 269 x = fabs(x);
268 270 y = fabs(y);
269 271
270 272 /* x is inf or NaN */
271 - if (ix >= 0x7ff00000) { /* x is inf or NaN */
273 + if (ix >= 0x7ff00000) { /* x is inf or NaN */
272 274 if (ISINF(ix, lx)) { /* x is INF */
273 275 D_IM(ans) = -x;
276 +
274 277 if (iy >= 0x7ff00000) {
275 278 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 */
279 + /*
280 + * cacos(inf + i inf) = pi/4 - i inf
281 + * cacos(-inf+ i inf) =3pi/4 - i inf
282 + */
280 283 if (hx >= 0)
281 284 D_RE(ans) = pi_4 + pi_4_l;
282 285 else
283 286 D_RE(ans) = pi3_4 + pi3_4_l;
284 - } else
285 - /* INDENT OFF */
286 - /* cacos(inf + i NaN) = NaN - i inf */
287 - /* INDENT ON */
287 + } else {
288 + /*
289 + * cacos(inf + i NaN) = NaN - i inf
290 + */
288 291 D_RE(ans) = y + y;
292 + }
289 293 } 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)
294 + /*
295 + * cacos(inf + iy ) = 0 - i inf
296 + * cacos(-inf+ iy ) = pi - i inf
297 + */
298 + if (hx >= 0) {
295 299 D_RE(ans) = zero;
296 - else
300 + } else {
297 301 D_RE(ans) = pi + pi_l;
302 + }
298 303 } else { /* x is NaN */
299 - /* INDENT OFF */
304 +
300 305 /*
301 306 * cacos(NaN + i inf) = NaN - i inf
302 307 * cacos(NaN + i y ) = NaN + i NaN
303 308 * cacos(NaN + i NaN) = NaN + i NaN
304 309 */
305 - /* INDENT ON */
306 310 D_RE(ans) = x + y;
307 - if (iy >= 0x7ff00000) {
311 +
312 + if (iy >= 0x7ff00000)
308 313 D_IM(ans) = -y;
309 - } else {
314 + else
310 315 D_IM(ans) = x;
311 - }
312 316 }
317 +
313 318 if (hy < 0)
314 319 D_IM(ans) = -D_IM(ans);
320 +
315 321 return (ans);
316 322 }
317 323
318 - if ((iy | ly) == 0) { /* region 1: y=0 */
324 + if ((iy | ly) == 0) { /* region 1: y=0 */
319 325 if (ix < 0x3ff00000) { /* |x| < 1 */
320 326 D_RE(ans) = acos(x);
321 327 D_IM(ans) = zero;
322 328 } else {
323 329 D_RE(ans) = zero;
324 - if (ix >= 0x43500000) /* |x| >= 2**54 */
330 +
331 + if (ix >= 0x43500000) { /* |x| >= 2**54 */
325 332 D_IM(ans) = ln2 + log(x);
326 - else if (ix >= 0x3ff80000) /* x > Acrossover */
333 + } else if (ix >= 0x3ff80000) { /* x > Acrossover */
327 334 D_IM(ans) = log(x + sqrt((x - one) * (x +
328 - one)));
329 - else {
335 + one)));
336 + } else {
330 337 xm1 = x - one;
331 338 D_IM(ans) = log1p(xm1 + sqrt(xm1 * (x + one)));
332 339 }
333 340 }
334 341 } else if (y <= E * fabs(x - one)) { /* region 2: y < tiny*|x-1| */
335 - if (ix < 0x3ff00000) { /* x < 1 */
342 + if (ix < 0x3ff00000) { /* x < 1 */
336 343 D_RE(ans) = acos(x);
337 344 D_IM(ans) = y / sqrt((one + x) * (one - x));
338 345 } else if (ix >= 0x43500000) { /* |x| >= 2**54 */
339 346 D_RE(ans) = y / x;
340 347 D_IM(ans) = ln2 + log(x);
341 348 } else {
342 349 t = sqrt((x - one) * (x + one));
343 350 D_RE(ans) = y / t;
351 +
344 352 if (ix >= 0x3ff80000) /* x > Acrossover */
345 353 D_IM(ans) = log(x + t);
346 354 else
347 355 D_IM(ans) = log1p((x - one) + t);
348 356 }
349 357 } else if (y < Foursqrtu) { /* region 3 */
350 358 t = sqrt(y);
351 359 D_RE(ans) = t;
352 360 D_IM(ans) = t;
353 - } else if (E * y - one >= x) { /* region 4 */
361 + } else if (E * y - one >= x) { /* region 4 */
354 362 D_RE(ans) = pi_2;
355 363 D_IM(ans) = ln2 + log(y);
356 364 } else if (ix >= 0x5fc00000 || iy >= 0x5fc00000) { /* x,y>2**509 */
357 365 /* region 5: x+1 or y is very large (>= sqrt(max)/8) */
358 366 t = x / y;
359 367 D_RE(ans) = atan(y / x);
360 368 D_IM(ans) = ln2 + log(y) + half * log1p(t * t);
361 369 } else if (x < Foursqrtu) {
362 370 /* region 6: x is very small, < 4sqrt(min) */
363 371 D_RE(ans) = pi_2;
364 372 A = sqrt(one + y * y);
373 +
365 374 if (iy >= 0x3ff80000) /* if y > Acrossover */
366 375 D_IM(ans) = log(y + A);
367 376 else
368 377 D_IM(ans) = half * log1p((y + y) * (y + A));
369 - } else { /* safe region */
378 + } else { /* safe region */
370 379 y2 = y * y;
371 380 xp1 = x + one;
372 381 xm1 = x - one;
373 382 R = sqrt(xp1 * xp1 + y2);
374 383 S = sqrt(xm1 * xm1 + y2);
375 384 A = half * (R + S);
376 385 B = x / A;
377 - if (B <= Bcrossover)
386 +
387 + if (B <= Bcrossover) {
378 388 D_RE(ans) = acos(B);
379 - else { /* use atan and an accurate approx to a-x */
389 + } else { /* use atan and an accurate approx to a-x */
380 390 Apx = A + x;
391 +
381 392 if (x <= one)
382 393 D_RE(ans) = atan(sqrt(half * Apx * (y2 / (R +
383 - xp1) + (S - xm1))) / x);
394 + xp1) + (S - xm1))) / x);
384 395 else
385 396 D_RE(ans) = atan((y * sqrt(half * (Apx / (R +
386 - xp1) + Apx / (S + xm1)))) / x);
397 + xp1) + Apx / (S + xm1)))) / x);
387 398 }
399 +
388 400 if (A <= Acrossover) {
389 401 /* use log1p and an accurate approx to A-1 */
390 402 if (x < one)
391 403 Am1 = half * (y2 / (R + xp1) + y2 / (S - xm1));
392 404 else
393 405 Am1 = half * (y2 / (R + xp1) + (S + xm1));
406 +
394 407 D_IM(ans) = log1p(Am1 + sqrt(Am1 * (A + one)));
395 408 } else {
396 409 D_IM(ans) = log(A + sqrt(A * A - one));
397 410 }
398 411 }
412 +
399 413 if (hx < 0)
400 414 D_RE(ans) = pi - D_RE(ans);
415 +
401 416 if (hy >= 0)
402 417 D_IM(ans) = -D_IM(ans);
418 +
403 419 return (ans);
404 420 }
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