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 *
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 catanl = __catanl
31
32 /* INDENT OFF */
33 /*
34 * ldcomplex catanl(ldcomplex z);
35 *
36 * Atan(z) return A + Bi where,
37 * 1
38 * A = --- * atan2(2x, 1-x*x-y*y)
39 * 2
40 *
41 * 1 [ x*x + (y+1)*(y+1) ] 1 4y
42 * B = --- log [ ----------------- ] = - log (1+ -----------------)
43 * 4 [ x*x + (y-1)*(y-1) ] 4 x*x + (y-1)*(y-1)
44 *
45 * 2 16 3 y
46 * = t - 2t + -- t - ..., where t = -----------------
47 * 3 x*x + (y-1)*(y-1)
48 * Proof:
49 * Let w = atan(z=x+yi) = A + B i. Then tan(w) = z.
50 * Since sin(w) = (exp(iw)-exp(-iw))/(2i), cos(w)=(exp(iw)+exp(-iw))/(2),
51 * Let p = exp(iw), then z = tan(w) = ((p-1/p)/(p+1/p))/i, or
52 * iz = (p*p-1)/(p*p+1), or, after simplification,
53 * p*p = (1+iz)/(1-iz) ... (1)
54 * LHS of (1) = exp(2iw) = exp(2i(A+Bi)) = exp(-2B)*exp(2iA)
55 * = exp(-2B)*(cos(2A)+i*sin(2A)) ... (2)
56 * 1-y+ix (1-y+ix)*(1+y+ix) 1-x*x-y*y + 2xi
57 * RHS of (1) = ------ = ----------------- = --------------- ... (3)
58 * 1+y-ix (1+y)**2 + x**2 (1+y)**2 + x**2
59 *
60 * Comparing the real and imaginary parts of (2) and (3), we have:
61 * cos(2A) : 1-x*x-y*y = sin(2A) : 2x
62 * and hence
63 * tan(2A) = 2x/(1-x*x-y*y), or
64 * A = 0.5 * atan2(2x, 1-x*x-y*y) ... (4)
65 *
66 * For the imaginary part B, Note that |p*p| = exp(-2B), and
67 * |1+iz| |i-z| hypot(x,(y-1))
68 * |----| = |---| = --------------
69 * |1-iz| |i+z| hypot(x,(y+1))
70 * Thus
71 * x*x + (y+1)*(y+1)
72 * exp(4B) = -----------------, or
73 * x*x + (y-1)*(y-1)
74 *
75 * 1 [x^2+(y+1)^2] 1 4y
76 * B = - log [-----------] = - log(1+ -------------) ... (5)
77 * 4 [x^2+(y-1)^2] 4 x^2+(y-1)^2
78 *
79 * QED.
80 *
81 * Note that: if catan( x, y) = ( u, v), then
82 * catan(-x, y) = (-u, v)
83 * catan( x,-y) = ( u,-v)
84 *
85 * Also, catan(x,y) = -i*catanh(-y,x), or
86 * catanh(x,y) = i*catan(-y,x)
87 * So, if catanh(y,x) = (v,u), then catan(x,y) = -i*(-v,u) = (u,v), i.e.,
88 * catan(x,y) = (u,v)
89 *
90 * EXCEPTION CASES (conform to ISO/IEC 9899:1999(E)):
91 * catan( 0 , 0 ) = (0 , 0 )
92 * catan( NaN, 0 ) = (NaN , 0 )
93 * catan( 0 , 1 ) = (0 , +inf) with divide-by-zero
94 * catan( inf, y ) = (pi/2 , 0 ) for finite +y
95 * catan( NaN, y ) = (NaN , NaN ) with invalid for finite y != 0
96 * catan( x , inf ) = (pi/2 , 0 ) for finite +x
97 * catan( inf, inf ) = (pi/2 , 0 )
98 * catan( NaN, inf ) = (NaN , 0 )
99 * catan( x , NaN ) = (NaN , NaN ) with invalid for finite x
100 * catan( inf, NaN ) = (pi/2 , +-0 )
101 */
102 /* INDENT ON */
103
104 #include "libm.h" /* atan2l/atanl/fabsl/isinfl/iszerol/log1pl/logl */
105 #include "complex_wrapper.h"
106 #include "longdouble.h"
107
108 /* INDENT OFF */
109 static const long double
110 zero = 0.0L,
111 one = 1.0L,
112 two = 2.0L,
113 half = 0.5L,
114 ln2 = 6.931471805599453094172321214581765680755e-0001L,
115 pi_2 = 1.570796326794896619231321691639751442098584699687552910487472L,
116 #if defined(__x86)
117 E = 2.910383045673370361328125000000000000000e-11L, /* 2**-35 */
118 Einv = 3.435973836800000000000000000000000000000e+10L; /* 2**+35 */
119 #else
120 E = 8.673617379884035472059622406959533691406e-19L, /* 2**-60 */
121 Einv = 1.152921504606846976000000000000000000000e18L; /* 2**+60 */
122 #endif
123 /* INDENT ON */
124
125 ldcomplex
126 catanl(ldcomplex z) {
127 ldcomplex ans;
128 long double x, y, t1, ax, ay, t;
129 int hx, hy, ix, iy;
130
131 x = LD_RE(z);
132 y = LD_IM(z);
133 ax = fabsl(x);
134 ay = fabsl(y);
135 hx = HI_XWORD(x);
136 hy = HI_XWORD(y);
137 ix = hx & 0x7fffffff;
138 iy = hy & 0x7fffffff;
139
140 /* x is inf or NaN */
141 if (ix >= 0x7fff0000) {
142 if (isinfl(x)) {
143 LD_RE(ans) = pi_2;
144 LD_IM(ans) = zero;
145 } else {
146 LD_RE(ans) = x + x;
147 if (iszerol(y) || (isinfl(y)))
148 LD_IM(ans) = zero;
149 else
150 LD_IM(ans) = (fabsl(y) - ay) / (fabsl(y) - ay);
151 }
152 } else if (iy >= 0x7fff0000) {
153 /* y is inf or NaN */
154 if (isinfl(y)) {
155 LD_RE(ans) = pi_2;
156 LD_IM(ans) = zero;
157 } else {
158 LD_RE(ans) = (fabsl(x) - ax) / (fabsl(x) - ax);
159 LD_IM(ans) = y;
160 }
161 } else if (iszerol(x)) {
162 /* INDENT OFF */
163 /*
164 * x = 0
165 * 1 1
166 * A = --- * atan2(2x, 1-x*x-y*y) = --- atan2(0,1-|y|)
167 * 2 2
168 *
169 * 1 [ (y+1)*(y+1) ] 1 2 1 2y
170 * B = - log [ ----------- ] = - log (1+ ---) or - log(1+ ----)
171 * 4 [ (y-1)*(y-1) ] 2 y-1 2 1-y
172 */
173 /* INDENT ON */
174 t = one - ay;
175 if (ay == one) {
176 /* y=1: catan(0,1)=(0,+inf) with 1/0 signal */
177 LD_IM(ans) = ay / ax;
178 LD_RE(ans) = zero;
179 } else if (ay > one) { /* y>1 */
180 LD_IM(ans) = half * log1pl(two / (-t));
181 LD_RE(ans) = pi_2;
182 } else { /* y<1 */
183 LD_IM(ans) = half * log1pl((ay + ay) / t);
184 LD_RE(ans) = zero;
185 }
186 } else if (ay < E * (one + ax)) {
187 /* INDENT OFF */
188 /*
189 * Tiny y (relative to 1+|x|)
190 * |y| < E*(1+|x|)
191 * where E=2**-29, -35, -60 for double, extended, quad precision
192 *
193 * 1 [x<=1: atan(x)
194 * A = - * atan2(2x,1-x*x-y*y) ~ [ 1 1+x
195 * 2 [x>=1: - atan2(2,(1-x)*(-----))
196 * 2 x
197 *
198 * y/x
199 * B ~ t*(1-2t), where t = ----------------- is tiny
200 * x + (y-1)*(y-1)/x
201 *
202 * y
203 * (when x < 2**-60, t = ----------- )
204 * (y-1)*(y-1)
205 */
206 /* INDENT ON */
207 if (ay == zero)
208 LD_IM(ans) = ay;
209 else {
210 t1 = ay - one;
211 if (ix < 0x3fc30000)
212 t = ay / (t1 * t1);
213 else if (ix > 0x403b0000)
214 t = (ay / ax) / ax;
215 else
216 t = ay / (ax * ax + t1 * t1);
217 LD_IM(ans) = t * (one - two * t);
218 }
219 if (ix < 0x3fff0000)
220 LD_RE(ans) = atanl(ax);
221 else
222 LD_RE(ans) = half * atan2l(two, (one - ax) * (one +
223 one / ax));
224
225 } else if (ay > Einv * (one + ax)) {
226 /* INDENT OFF */
227 /*
228 * Huge y relative to 1+|x|
229 * |y| > Einv*(1+|x|), where Einv~2**(prec/2+3),
230 * 1
231 * A ~ --- * atan2(2x, -y*y) ~ pi/2
232 * 2
233 * y
234 * B ~ t*(1-2t), where t = --------------- is tiny
235 * (y-1)*(y-1)
236 */
237 /* INDENT ON */
238 LD_RE(ans) = pi_2;
239 t = (ay / (ay - one)) / (ay - one);
240 LD_IM(ans) = t * (one - (t + t));
241 } else if (ay == one) {
242 /* INDENT OFF */
243 /*
244 * y=1
245 * 1 1
246 * A = - * atan2(2x, -x*x) = --- atan2(2,-x)
247 * 2 2
248 *
249 * 1 [ x*x+4] 1 4 [ 0.5(log2-logx) if
250 * B = - log [ -----] = - log (1+ ---) = [ |x|<E, else 0.25*
251 * 4 [ x*x ] 4 x*x [ log1p((2/x)*(2/x))
252 */
253 /* INDENT ON */
254 LD_RE(ans) = half * atan2l(two, -ax);
255 if (ax < E)
256 LD_IM(ans) = half * (ln2 - logl(ax));
257 else {
258 t = two / ax;
259 LD_IM(ans) = 0.25L * log1pl(t * t);
260 }
261 } else if (ax > Einv * Einv) {
262 /* INDENT OFF */
263 /*
264 * Huge x:
265 * when |x| > 1/E^2,
266 * 1 pi
267 * A ~ --- * atan2(2x, -x*x-y*y) ~ ---
268 * 2 2
269 * y y/x
270 * B ~ t*(1-2t), where t = --------------- = (-------------- )/x
271 * x*x+(y-1)*(y-1) 1+((y-1)/x)^2
272 */
273 /* INDENT ON */
274 LD_RE(ans) = pi_2;
275 t = ((ay / ax) / (one + ((ay - one) / ax) * ((ay - one) /
276 ax))) / ax;
277 LD_IM(ans) = t * (one - (t + t));
278 } else if (ax < E * E * E * E) {
279 /* INDENT OFF */
280 /*
281 * Tiny x:
282 * when |x| < E^4, (note that y != 1)
283 * 1 1
284 * A = --- * atan2(2x, 1-x*x-y*y) ~ --- * atan2(2x,1-y*y)
285 * 2 2
286 *
287 * 1 [ (y+1)*(y+1) ] 1 2 1 2y
288 * B = - log [ ----------- ] = - log (1+ ---) or - log(1+ ----)
289 * 4 [ (y-1)*(y-1) ] 2 y-1 2 1-y
290 */
291 /* INDENT ON */
292 LD_RE(ans) = half * atan2l(ax + ax, (one - ay) * (one + ay));
293 if (ay > one) /* y>1 */
294 LD_IM(ans) = half * log1pl(two / (ay - one));
295 else /* y<1 */
296 LD_IM(ans) = half * log1pl((ay + ay) / (one - ay));
297 } else {
298 /* INDENT OFF */
299 /*
300 * normal x,y
301 * 1
302 * A = --- * atan2(2x, 1-x*x-y*y)
303 * 2
304 *
305 * 1 [ x*x+(y+1)*(y+1) ] 1 4y
306 * B = - log [ --------------- ] = - log (1+ -----------------)
307 * 4 [ x*x+(y-1)*(y-1) ] 4 x*x + (y-1)*(y-1)
308 */
309 /* INDENT ON */
310 t = one - ay;
311 if (iy >= 0x3ffe0000 && iy < 0x40000000) {
312 /* y close to 1 */
313 LD_RE(ans) = half * (atan2l((ax + ax), (t * (one +
314 ay) - ax * ax)));
315 } else if (ix >= 0x3ffe0000 && ix < 0x40000000) {
316 /* x close to 1 */
317 LD_RE(ans) = half * atan2l((ax + ax), ((one - ax) *
318 (one + ax) - ay * ay));
319 } else
320 LD_RE(ans) = half * atan2l((ax + ax), ((one - ax *
321 ax) - ay * ay));
322 LD_IM(ans) = 0.25L * log1pl((4.0L * ay) / (ax * ax + t * t));
323 }
324 if (hx < 0)
325 LD_RE(ans) = -LD_RE(ans);
326 if (hy < 0)
327 LD_IM(ans) = -LD_IM(ans);
328 return (ans);
329 }