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").
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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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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 casin = __casin
31
32 /* INDENT OFF */
33 /*
34 * dcomplex casin(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 sine function casin(z),
42 * where z = x+iy, can be defined by
43 *
44 * casin(z) = asin(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 * Special notes: if casin( x, y) = ( u, v), then
67 * casin(-x, y) = (-u, v),
68 * casin( x,-y) = ( u,-v),
69 * in general, we have casin(conj(z)) = conj(casin(z))
70 * casin(-z) = -casin(z)
71 * casin(z) = pi/2 - cacos(z)
72 *
73 * EXCEPTION CASES (conform to ISO/IEC 9899:1999(E)):
74 * casin( 0 + i 0 ) = 0 + i 0
75 * casin( 0 + i NaN ) = 0 + i NaN
76 * casin( x + i inf ) = 0 + i inf for finite x
77 * casin( x + i NaN ) = NaN + i NaN with invalid for finite x!=0
78 * casin(inf + iy ) = pi/2 + i inf finite y
79 * casin(inf + i inf) = pi/4 + i inf
80 * casin(inf + i NaN) = NaN + i inf
81 * casin(NaN + i y ) = NaN + i NaN for finite y
82 * casin(NaN + i inf) = NaN + i inf
83 * casin(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 + ------------ , if x > 1
121 * [ 2(1+x)(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 - ------------), 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(1+x)(1-x)
143 * Thus
144 * [ asin(x) + i y/sqrt((x-1)*(x+1)), if x < 1
145 * casin(x+i*y)=[
146 * [ pi/2 + i log(x+sqrt(x*x-1)), if x >= 1
147 *
148 * case 3. y < 4 sqrt(u), where u = minimum normal x.
149 * After case 1 and 2, this will only occurs when x=1. When x=1, we have
150 * A = (sqrt(4+y*y)+y)/2 ~ 1 + y/2 + y^2/8 + ...
151 * and
152 * B = 1/A = 1 - y/2 + y^2/8 + ...
153 * Since
154 * asin(x) = pi/2-2*asin(sqrt((1-x)/2))
155 * asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
156 * we have, for the real part asin(B),
157 * asin(1-y/2) ~ pi/2 - 2 asin(sqrt(y/4))
158 * ~ pi/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 = asin(B) ~ x/y (be careful, x/y may underflow)
168 * and
169 * imag part = log(y+sqrt(y*y-one))
170 *
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 = asin(B) = atan(x/y),
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 = asin(B) ~ B = x/sqrt(1+y*y)
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 * casin(z) = asin(B) + i sign(y) log (A + sqrt(A*A-1)),
193 */
194 /* INDENT ON */
195
196 #include "libm.h" /* asin/atan/fabs/log/log1p/sqrt */
197 #include "complex_wrapper.h"
198
199 /* INDENT OFF */
200 static const double
201 zero = 0.0,
202 one = 1.0,
203 E = 1.11022302462515654042e-16, /* 2**-53 */
204 ln2 = 6.93147180559945286227e-01,
205 pi_2 = 1.570796326794896558e+00,
206 pi_2_l = 6.123233995736765886e-17,
207 pi_4 = 7.85398163397448278999e-01,
208 Foursqrtu = 5.96667258496016539463e-154, /* 2**(-509) */
209 Acrossover = 1.5,
210 Bcrossover = 0.6417,
211 half = 0.5;
212 /* INDENT ON */
213
214 dcomplex
215 casin(dcomplex z) {
216 double x, y, t, R, S, A, Am1, B, y2, xm1, xp1, Apx;
217 int ix, iy, hx, hy;
218 unsigned lx, ly;
219 dcomplex ans;
220
221 x = D_RE(z);
222 y = D_IM(z);
223 hx = HI_WORD(x);
224 lx = LO_WORD(x);
225 hy = HI_WORD(y);
226 ly = LO_WORD(y);
227 ix = hx & 0x7fffffff;
228 iy = hy & 0x7fffffff;
229 x = fabs(x);
230 y = fabs(y);
231
232 /* special cases */
233
234 /* x is inf or NaN */
235 if (ix >= 0x7ff00000) { /* x is inf or NaN */
236 if (ISINF(ix, lx)) { /* x is INF */
237 D_IM(ans) = x;
238 if (iy >= 0x7ff00000) {
239 if (ISINF(iy, ly))
240 /* casin(inf + i inf) = pi/4 + i inf */
241 D_RE(ans) = pi_4;
242 else /* casin(inf + i NaN) = NaN + i inf */
243 D_RE(ans) = y + y;
244 } else /* casin(inf + iy) = pi/2 + i inf */
245 D_RE(ans) = pi_2;
246 } else { /* x is NaN */
247 if (iy >= 0x7ff00000) {
248 /* INDENT OFF */
249 /*
250 * casin(NaN + i inf) = NaN + i inf
251 * casin(NaN + i NaN) = NaN + i NaN
252 */
253 /* INDENT ON */
254 D_IM(ans) = y + y;
255 D_RE(ans) = x + x;
256 } else {
257 /* casin(NaN + i y ) = NaN + i NaN */
258 D_IM(ans) = D_RE(ans) = x + y;
259 }
260 }
261 if (hx < 0)
262 D_RE(ans) = -D_RE(ans);
263 if (hy < 0)
264 D_IM(ans) = -D_IM(ans);
265 return (ans);
266 }
267
268 /* casin(+0 + i 0 ) = 0 + i 0. */
269 if ((ix | lx | iy | ly) == 0)
270 return (z);
271
272 if (iy >= 0x7ff00000) { /* y is inf or NaN */
273 if (ISINF(iy, ly)) { /* casin( x + i inf ) = 0 + i inf */
274 D_IM(ans) = y;
275 D_RE(ans) = zero;
276 } else { /* casin( x + i NaN ) = NaN + i NaN */
277 D_IM(ans) = x + y;
278 if ((ix | lx) == 0)
279 D_RE(ans) = x;
280 else
281 D_RE(ans) = y;
282 }
283 if (hx < 0)
284 D_RE(ans) = -D_RE(ans);
285 if (hy < 0)
286 D_IM(ans) = -D_IM(ans);
287 return (ans);
288 }
289
290 if ((iy | ly) == 0) { /* region 1: y=0 */
291 if (ix < 0x3ff00000) { /* |x| < 1 */
292 D_RE(ans) = asin(x);
293 D_IM(ans) = zero;
294 } else {
295 D_RE(ans) = pi_2;
296 if (ix >= 0x43500000) /* |x| >= 2**54 */
297 D_IM(ans) = ln2 + log(x);
298 else if (ix >= 0x3ff80000) /* x > Acrossover */
299 D_IM(ans) = log(x + sqrt((x - one) * (x +
300 one)));
301 else {
302 xm1 = x - one;
303 D_IM(ans) = log1p(xm1 + sqrt(xm1 * (x + one)));
304 }
305 }
306 } else if (y <= E * fabs(x - one)) { /* region 2: y < tiny*|x-1| */
307 if (ix < 0x3ff00000) { /* x < 1 */
308 D_RE(ans) = asin(x);
309 D_IM(ans) = y / sqrt((one + x) * (one - x));
310 } else {
311 D_RE(ans) = pi_2;
312 if (ix >= 0x43500000) { /* |x| >= 2**54 */
313 D_IM(ans) = ln2 + log(x);
314 } else if (ix >= 0x3ff80000) /* x > Acrossover */
315 D_IM(ans) = log(x + sqrt((x - one) * (x +
316 one)));
317 else
318 D_IM(ans) = log1p((x - one) + sqrt((x - one) *
319 (x + one)));
320 }
321 } else if (y < Foursqrtu) { /* region 3 */
322 t = sqrt(y);
323 D_RE(ans) = pi_2 - (t - pi_2_l);
324 D_IM(ans) = t;
325 } else if (E * y - one >= x) { /* region 4 */
326 D_RE(ans) = x / y; /* need to fix underflow cases */
327 D_IM(ans) = ln2 + log(y);
328 } else if (ix >= 0x5fc00000 || iy >= 0x5fc00000) { /* x,y>2**509 */
329 /* region 5: x+1 or y is very large (>= sqrt(max)/8) */
330 t = x / y;
331 D_RE(ans) = atan(t);
332 D_IM(ans) = ln2 + log(y) + half * log1p(t * t);
333 } else if (x < Foursqrtu) {
334 /* region 6: x is very small, < 4sqrt(min) */
335 A = sqrt(one + y * y);
336 D_RE(ans) = x / A; /* may underflow */
337 if (iy >= 0x3ff80000) /* if y > Acrossover */
338 D_IM(ans) = log(y + A);
339 else
340 D_IM(ans) = half * log1p((y + y) * (y + A));
341 } else { /* safe region */
342 y2 = y * y;
343 xp1 = x + one;
344 xm1 = x - one;
345 R = sqrt(xp1 * xp1 + y2);
346 S = sqrt(xm1 * xm1 + y2);
347 A = half * (R + S);
348 B = x / A;
349
350 if (B <= Bcrossover)
351 D_RE(ans) = asin(B);
352 else { /* use atan and an accurate approx to a-x */
353 Apx = A + x;
354 if (x <= one)
355 D_RE(ans) = atan(x / sqrt(half * Apx * (y2 /
356 (R + xp1) + (S - xm1))));
357 else
358 D_RE(ans) = atan(x / (y * sqrt(half * (Apx /
359 (R + xp1) + Apx / (S + xm1)))));
360 }
361 if (A <= Acrossover) {
362 /* use log1p and an accurate approx to A-1 */
363 if (x < one)
364 Am1 = half * (y2 / (R + xp1) + y2 / (S - xm1));
365 else
366 Am1 = half * (y2 / (R + xp1) + (S + xm1));
367 D_IM(ans) = log1p(Am1 + sqrt(Am1 * (A + one)));
368 } else {
369 D_IM(ans) = log(A + sqrt(A * A - one));
370 }
371 }
372
373 if (hx < 0)
374 D_RE(ans) = -D_RE(ans);
375 if (hy < 0)
376 D_IM(ans) = -D_IM(ans);
377
378 return (ans);
379 }