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11210 libm should be cstyle(1ONBLD) clean
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--- old/usr/src/lib/libm/common/complex/casin.c
+++ new/usr/src/lib/libm/common/complex/casin.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 __casin = casin
31 32
32 -/* INDENT OFF */
33 +
33 34 /*
34 35 * dcomplex casin(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 sine function casin(z),
42 43 * where z = x+iy, can be defined by
43 44 *
44 - * casin(z) = asin(B) + i sign(y) log (A + sqrt(A*A-1)),
45 + * casin(z) = asin(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 * Special notes: if casin( x, y) = ( u, v), then
67 68 * casin(-x, y) = (-u, v),
68 69 * casin( x,-y) = ( u,-v),
69 70 * in general, we have casin(conj(z)) = conj(casin(z))
70 71 * casin(-z) = -casin(z)
71 72 * casin(z) = pi/2 - cacos(z)
72 73 *
73 74 * EXCEPTION CASES (conform to ISO/IEC 9899:1999(E)):
74 75 * casin( 0 + i 0 ) = 0 + i 0
75 76 * casin( 0 + i NaN ) = 0 + i NaN
76 77 * casin( x + i inf ) = 0 + i inf for finite x
77 78 * casin( x + i NaN ) = NaN + i NaN with invalid for finite x != 0
78 79 * casin(inf + iy ) = pi/2 + i inf finite y
79 80 * casin(inf + i inf) = pi/4 + i inf
80 81 * casin(inf + i NaN) = NaN + i inf
81 82 * casin(NaN + i y ) = NaN + i NaN for finite y
82 83 * casin(NaN + i inf) = NaN + i inf
83 84 * casin(NaN + i NaN) = NaN + i NaN
84 85 *
85 86 * Special Regions (better formula for accuracy and for avoiding spurious
86 87 * overflow or underflow) (all x and y are assumed nonnegative):
87 88 * case 1: y = 0
88 89 * case 2: tiny y relative to x-1: y <= ulp(0.5)*|x-1|
89 90 * case 3: tiny y: y < 4 sqrt(u), where u = minimum normal number
90 91 * case 4: huge y relative to x+1: y >= (1+x)/ulp(0.5)
91 92 * case 5: huge x and y: x and y >= sqrt(M)/8, where M = maximum normal number
92 93 * case 6: tiny x: x < 4 sqrt(u)
93 94 * --------
94 95 * case 1 & 2. y=0 or y/|x-1| is tiny. We have
95 96 * ____________ _____________
96 97 * / 2 2 / y 2
97 98 * / (x+-1) + y = |x+-1| / 1 + (------)
98 99 * \/ \/ |x+-1|
99 100 *
100 101 * 1 y 2
101 102 * ~ |x+-1| ( 1 + --- (------) )
102 103 * 2 |x+-1|
103 104 *
104 105 * 2
105 106 * y
106 107 * = |x+-1| + --------.
107 108 * 2|x+-1|
108 109 *
109 110 * Consequently, it is not difficult to see that
110 111 * 2
111 112 * y
112 113 * [ 1 + ------------ , if x < 1,
113 114 * [ 2(1+x)(1-x)
114 115 * [
115 116 * [
116 117 * [ x, if x = 1 (y = 0),
117 118 * [
118 119 * A ~= [ 2
119 120 * [ x * y
120 121 * [ x + ------------ , if x > 1
121 122 * [ 2(1+x)(x-1)
122 123 *
123 124 * and hence
124 125 * ______ 2
125 126 * / 2 y y
126 127 * A + \/ A - 1 ~ 1 + ---------------- + -----------, if x < 1,
127 128 * sqrt((x+1)(1-x)) 2(x+1)(1-x)
128 129 *
129 130 *
130 131 * ~ x + sqrt((x-1)*(x+1)), if x >= 1.
131 132 *
132 133 * 2
133 134 * y
134 135 * [ x(1 - ------------), if x < 1,
135 136 * [ 2(1+x)(1-x)
136 137 * B = x/A ~ [
137 138 * [ 1, if x = 1,
138 139 * [
139 140 * [ 2
140 141 * [ y
141 142 * [ 1 - ------------ , if x > 1,
142 143 * [ 2(1+x)(1-x)
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143 144 * Thus
144 145 * [ asin(x) + i y/sqrt((x-1)*(x+1)), if x < 1
145 146 * casin(x+i*y)=[
146 147 * [ pi/2 + i log(x+sqrt(x*x-1)), if x >= 1
147 148 *
148 149 * case 3. y < 4 sqrt(u), where u = minimum normal x.
149 150 * After case 1 and 2, this will only occurs when x=1. When x=1, we have
150 151 * A = (sqrt(4+y*y)+y)/2 ~ 1 + y/2 + y^2/8 + ...
151 152 * and
152 153 * B = 1/A = 1 - y/2 + y^2/8 + ...
153 - * Since
154 + * Since
154 155 * asin(x) = pi/2-2*asin(sqrt((1-x)/2))
155 156 * asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
156 157 * we have, for the real part asin(B),
157 158 * asin(1-y/2) ~ pi/2 - 2 asin(sqrt(y/4))
158 159 * ~ pi/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 = asin(B) ~ x/y (be careful, x/y may underflow)
168 - * and
169 + * and
169 170 * imag part = log(y+sqrt(y*y-one))
170 171 *
171 172 *
172 173 * case 5. Both x and y are large: x and y > sqrt(M)/8, where M = maximum x
173 174 * In this case,
174 175 * A ~ sqrt(x*x+y*y)
175 176 * B ~ x/sqrt(x*x+y*y).
176 177 * Thus
177 178 * real part = asin(B) = atan(x/y),
178 179 * imag part = log(A+sqrt(A*A-1)) ~ log(2A)
179 180 * = log(2) + 0.5*log(x*x+y*y)
180 181 * = log(2) + log(y) + 0.5*log(1+(x/y)^2)
181 182 *
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182 183 * case 6. x < 4 sqrt(u). In this case, we have
183 184 * A ~ sqrt(1+y*y), B = x/sqrt(1+y*y).
184 185 * Since B is tiny, we have
185 186 * real part = asin(B) ~ B = x/sqrt(1+y*y)
186 187 * imag part = log(A+sqrt(A*A-1)) = log (A+sqrt(y*y))
187 188 * = log(y+sqrt(1+y*y))
188 189 * = 0.5*log(y^2+2ysqrt(1+y^2)+1+y^2)
189 190 * = 0.5*log(1+2y(y+sqrt(1+y^2)));
190 191 * = 0.5*log1p(2y(y+A));
191 192 *
192 - * casin(z) = asin(B) + i sign(y) log (A + sqrt(A*A-1)),
193 + * casin(z) = asin(B) + i sign(y) log (A + sqrt(A*A-1)),
193 194 */
194 -/* INDENT ON */
195 195
196 -#include "libm.h" /* asin/atan/fabs/log/log1p/sqrt */
196 +#include "libm.h" /* asin/atan/fabs/log/log1p/sqrt */
197 197 #include "complex_wrapper.h"
198 198
199 -/* INDENT OFF */
200 -static const double
201 - zero = 0.0,
199 +static const double zero = 0.0,
202 200 one = 1.0,
203 - E = 1.11022302462515654042e-16, /* 2**-53 */
201 + E = 1.11022302462515654042e-16, /* 2**-53 */
204 202 ln2 = 6.93147180559945286227e-01,
205 203 pi_2 = 1.570796326794896558e+00,
206 204 pi_2_l = 6.123233995736765886e-17,
207 205 pi_4 = 7.85398163397448278999e-01,
208 - Foursqrtu = 5.96667258496016539463e-154, /* 2**(-509) */
206 + Foursqrtu = 5.96667258496016539463e-154, /* 2**(-509) */
209 207 Acrossover = 1.5,
210 208 Bcrossover = 0.6417,
211 209 half = 0.5;
212 -/* INDENT ON */
210 +
213 211
214 212 dcomplex
215 -casin(dcomplex z) {
213 +casin(dcomplex z)
214 +{
216 215 double x, y, t, R, S, A, Am1, B, y2, xm1, xp1, Apx;
217 216 int ix, iy, hx, hy;
218 217 unsigned lx, ly;
219 218 dcomplex ans;
220 219
221 220 x = D_RE(z);
222 221 y = D_IM(z);
223 222 hx = HI_WORD(x);
224 223 lx = LO_WORD(x);
225 224 hy = HI_WORD(y);
226 225 ly = LO_WORD(y);
227 226 ix = hx & 0x7fffffff;
228 227 iy = hy & 0x7fffffff;
229 228 x = fabs(x);
230 229 y = fabs(y);
231 230
232 231 /* special cases */
233 232
234 233 /* x is inf or NaN */
235 - if (ix >= 0x7ff00000) { /* x is inf or NaN */
234 + if (ix >= 0x7ff00000) { /* x is inf or NaN */
236 235 if (ISINF(ix, lx)) { /* x is INF */
237 236 D_IM(ans) = x;
237 +
238 238 if (iy >= 0x7ff00000) {
239 239 if (ISINF(iy, ly))
240 240 /* casin(inf + i inf) = pi/4 + i inf */
241 241 D_RE(ans) = pi_4;
242 242 else /* casin(inf + i NaN) = NaN + i inf */
243 243 D_RE(ans) = y + y;
244 - } else /* casin(inf + iy) = pi/2 + i inf */
244 + } else { /* casin(inf + iy) = pi/2 + i inf */
245 245 D_RE(ans) = pi_2;
246 + }
246 247 } else { /* x is NaN */
247 248 if (iy >= 0x7ff00000) {
248 - /* INDENT OFF */
249 +
249 250 /*
250 251 * casin(NaN + i inf) = NaN + i inf
251 252 * casin(NaN + i NaN) = NaN + i NaN
252 253 */
253 - /* INDENT ON */
254 254 D_IM(ans) = y + y;
255 255 D_RE(ans) = x + x;
256 256 } else {
257 257 /* casin(NaN + i y ) = NaN + i NaN */
258 258 D_IM(ans) = D_RE(ans) = x + y;
259 259 }
260 260 }
261 +
261 262 if (hx < 0)
262 263 D_RE(ans) = -D_RE(ans);
264 +
263 265 if (hy < 0)
264 266 D_IM(ans) = -D_IM(ans);
267 +
265 268 return (ans);
266 269 }
267 270
268 271 /* casin(+0 + i 0 ) = 0 + i 0. */
269 272 if ((ix | lx | iy | ly) == 0)
270 273 return (z);
271 274
272 - if (iy >= 0x7ff00000) { /* y is inf or NaN */
275 + if (iy >= 0x7ff00000) { /* y is inf or NaN */
273 276 if (ISINF(iy, ly)) { /* casin(x + i inf) = 0 + i inf */
274 277 D_IM(ans) = y;
275 278 D_RE(ans) = zero;
276 279 } else { /* casin(x + i NaN) = NaN + i NaN */
277 280 D_IM(ans) = x + y;
281 +
278 282 if ((ix | lx) == 0)
279 283 D_RE(ans) = x;
280 284 else
281 285 D_RE(ans) = y;
282 286 }
287 +
283 288 if (hx < 0)
284 289 D_RE(ans) = -D_RE(ans);
290 +
285 291 if (hy < 0)
286 292 D_IM(ans) = -D_IM(ans);
293 +
287 294 return (ans);
288 295 }
289 296
290 - if ((iy | ly) == 0) { /* region 1: y=0 */
297 + if ((iy | ly) == 0) { /* region 1: y=0 */
291 298 if (ix < 0x3ff00000) { /* |x| < 1 */
292 299 D_RE(ans) = asin(x);
293 300 D_IM(ans) = zero;
294 301 } else {
295 302 D_RE(ans) = pi_2;
296 - if (ix >= 0x43500000) /* |x| >= 2**54 */
303 +
304 + if (ix >= 0x43500000) { /* |x| >= 2**54 */
297 305 D_IM(ans) = ln2 + log(x);
298 - else if (ix >= 0x3ff80000) /* x > Acrossover */
306 + } else if (ix >= 0x3ff80000) { /* x > Acrossover */
299 307 D_IM(ans) = log(x + sqrt((x - one) * (x +
300 - one)));
301 - else {
308 + one)));
309 + } else {
302 310 xm1 = x - one;
303 311 D_IM(ans) = log1p(xm1 + sqrt(xm1 * (x + one)));
304 312 }
305 313 }
306 314 } else if (y <= E * fabs(x - one)) { /* region 2: y < tiny*|x-1| */
307 - if (ix < 0x3ff00000) { /* x < 1 */
315 + if (ix < 0x3ff00000) { /* x < 1 */
308 316 D_RE(ans) = asin(x);
309 317 D_IM(ans) = y / sqrt((one + x) * (one - x));
310 318 } else {
311 319 D_RE(ans) = pi_2;
312 - if (ix >= 0x43500000) { /* |x| >= 2**54 */
320 +
321 + if (ix >= 0x43500000) /* |x| >= 2**54 */
313 322 D_IM(ans) = ln2 + log(x);
314 - } else if (ix >= 0x3ff80000) /* x > Acrossover */
323 + else if (ix >= 0x3ff80000) /* x > Acrossover */
315 324 D_IM(ans) = log(x + sqrt((x - one) * (x +
316 - one)));
325 + one)));
317 326 else
318 327 D_IM(ans) = log1p((x - one) + sqrt((x - one) *
319 - (x + one)));
328 + (x + one)));
320 329 }
321 330 } else if (y < Foursqrtu) { /* region 3 */
322 331 t = sqrt(y);
323 332 D_RE(ans) = pi_2 - (t - pi_2_l);
324 333 D_IM(ans) = t;
325 - } else if (E * y - one >= x) { /* region 4 */
326 - D_RE(ans) = x / y; /* need to fix underflow cases */
334 + } else if (E * y - one >= x) { /* region 4 */
335 + D_RE(ans) = x / y; /* need to fix underflow cases */
327 336 D_IM(ans) = ln2 + log(y);
328 337 } else if (ix >= 0x5fc00000 || iy >= 0x5fc00000) { /* x,y>2**509 */
329 338 /* region 5: x+1 or y is very large (>= sqrt(max)/8) */
330 339 t = x / y;
331 340 D_RE(ans) = atan(t);
332 341 D_IM(ans) = ln2 + log(y) + half * log1p(t * t);
333 342 } else if (x < Foursqrtu) {
334 343 /* region 6: x is very small, < 4sqrt(min) */
335 344 A = sqrt(one + y * y);
336 345 D_RE(ans) = x / A; /* may underflow */
346 +
337 347 if (iy >= 0x3ff80000) /* if y > Acrossover */
338 348 D_IM(ans) = log(y + A);
339 349 else
340 350 D_IM(ans) = half * log1p((y + y) * (y + A));
341 - } else { /* safe region */
351 + } else { /* safe region */
342 352 y2 = y * y;
343 353 xp1 = x + one;
344 354 xm1 = x - one;
345 355 R = sqrt(xp1 * xp1 + y2);
346 356 S = sqrt(xm1 * xm1 + y2);
347 357 A = half * (R + S);
348 358 B = x / A;
349 359
350 - if (B <= Bcrossover)
360 + if (B <= Bcrossover) {
351 361 D_RE(ans) = asin(B);
352 - else { /* use atan and an accurate approx to a-x */
362 + } else { /* use atan and an accurate approx to a-x */
353 363 Apx = A + x;
364 +
354 365 if (x <= one)
355 366 D_RE(ans) = atan(x / sqrt(half * Apx * (y2 /
356 - (R + xp1) + (S - xm1))));
367 + (R + xp1) + (S - xm1))));
357 368 else
358 369 D_RE(ans) = atan(x / (y * sqrt(half * (Apx /
359 - (R + xp1) + Apx / (S + xm1)))));
370 + (R + xp1) + Apx / (S + xm1)))));
360 371 }
372 +
361 373 if (A <= Acrossover) {
362 374 /* use log1p and an accurate approx to A-1 */
363 375 if (x < one)
364 376 Am1 = half * (y2 / (R + xp1) + y2 / (S - xm1));
365 377 else
366 378 Am1 = half * (y2 / (R + xp1) + (S + xm1));
379 +
367 380 D_IM(ans) = log1p(Am1 + sqrt(Am1 * (A + one)));
368 381 } else {
369 382 D_IM(ans) = log(A + sqrt(A * A - one));
370 383 }
371 384 }
372 385
373 386 if (hx < 0)
374 387 D_RE(ans) = -D_RE(ans);
388 +
375 389 if (hy < 0)
376 390 D_IM(ans) = -D_IM(ans);
377 391
378 392 return (ans);
379 393 }
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