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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7 *
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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 /*
27 * Copyright 2006 Sun Microsystems, Inc. All rights reserved.
28 * Use is subject to license terms.
29 */
30
31 #pragma weak __casin = casin
32
33
34 /*
35 * dcomplex casin(dcomplex z);
36 *
37 * Alogrithm
38 * (based on T.E.Hull, Thomas F. Fairgrieve and Ping Tak Peter Tang's
39 * paper "Implementing the Complex Arcsine and Arccosine Functins Using
40 * Exception Handling", ACM TOMS, Vol 23, pp 299-335)
41 *
42 * The principal value of complex inverse sine function casin(z),
43 * where z = x+iy, can be defined by
44 *
45 * casin(z) = asin(B) + i sign(y) log (A + sqrt(A*A-1)),
46 *
47 * where the log function is the natural log, and
48 * ____________ ____________
49 * 1 / 2 2 1 / 2 2
50 * A = --- / (x+1) + y + --- / (x-1) + y
51 * 2 \/ 2 \/
52 * ____________ ____________
53 * 1 / 2 2 1 / 2 2
54 * B = --- / (x+1) + y - --- / (x-1) + y .
55 * 2 \/ 2 \/
56 *
57 * The Branch cuts are on the real line from -inf to -1 and from 1 to inf.
58 * The real and imaginary parts are based on Abramowitz and Stegun
59 * [Handbook of Mathematic Functions, 1972]. The sign of the imaginary
60 * part is chosen to be the generally considered the principal value of
61 * this function.
62 *
63 * Notes:1. A is the average of the distances from z to the points (1,0)
64 * and (-1,0) in the complex z-plane, and in particular A>=1.
65 * 2. B is in [-1,1], and A*B = x.
66 *
67 * Special notes: if casin( x, y) = ( u, v), then
68 * casin(-x, y) = (-u, v),
69 * casin( x,-y) = ( u,-v),
70 * in general, we have casin(conj(z)) = conj(casin(z))
71 * casin(-z) = -casin(z)
72 * casin(z) = pi/2 - cacos(z)
73 *
74 * EXCEPTION CASES (conform to ISO/IEC 9899:1999(E)):
75 * casin( 0 + i 0 ) = 0 + i 0
76 * casin( 0 + i NaN ) = 0 + i NaN
77 * casin( x + i inf ) = 0 + i inf for finite x
78 * casin( x + i NaN ) = NaN + i NaN with invalid for finite x != 0
79 * casin(inf + iy ) = pi/2 + i inf finite y
80 * casin(inf + i inf) = pi/4 + i inf
81 * casin(inf + i NaN) = NaN + i inf
82 * casin(NaN + i y ) = NaN + i NaN for finite y
83 * casin(NaN + i inf) = NaN + i inf
84 * casin(NaN + i NaN) = NaN + i NaN
85 *
86 * Special Regions (better formula for accuracy and for avoiding spurious
87 * overflow or underflow) (all x and y are assumed nonnegative):
88 * case 1: y = 0
89 * case 2: tiny y relative to x-1: y <= ulp(0.5)*|x-1|
90 * case 3: tiny y: y < 4 sqrt(u), where u = minimum normal number
91 * case 4: huge y relative to x+1: y >= (1+x)/ulp(0.5)
92 * case 5: huge x and y: x and y >= sqrt(M)/8, where M = maximum normal number
93 * case 6: tiny x: x < 4 sqrt(u)
94 * --------
95 * case 1 & 2. y=0 or y/|x-1| is tiny. We have
96 * ____________ _____________
97 * / 2 2 / y 2
98 * / (x+-1) + y = |x+-1| / 1 + (------)
99 * \/ \/ |x+-1|
100 *
101 * 1 y 2
102 * ~ |x+-1| ( 1 + --- (------) )
103 * 2 |x+-1|
104 *
105 * 2
106 * y
107 * = |x+-1| + --------.
108 * 2|x+-1|
109 *
110 * Consequently, it is not difficult to see that
111 * 2
112 * y
113 * [ 1 + ------------ , if x < 1,
114 * [ 2(1+x)(1-x)
115 * [
116 * [
117 * [ x, if x = 1 (y = 0),
118 * [
119 * A ~= [ 2
120 * [ x * y
121 * [ x + ------------ , if x > 1
122 * [ 2(1+x)(x-1)
123 *
124 * and hence
125 * ______ 2
126 * / 2 y y
127 * A + \/ A - 1 ~ 1 + ---------------- + -----------, if x < 1,
128 * sqrt((x+1)(1-x)) 2(x+1)(1-x)
129 *
130 *
131 * ~ x + sqrt((x-1)*(x+1)), if x >= 1.
132 *
133 * 2
134 * y
135 * [ x(1 - ------------), if x < 1,
136 * [ 2(1+x)(1-x)
137 * B = x/A ~ [
138 * [ 1, if x = 1,
139 * [
140 * [ 2
141 * [ y
142 * [ 1 - ------------ , if x > 1,
143 * [ 2(1+x)(1-x)
144 * Thus
145 * [ asin(x) + i y/sqrt((x-1)*(x+1)), if x < 1
146 * casin(x+i*y)=[
147 * [ pi/2 + i log(x+sqrt(x*x-1)), if x >= 1
148 *
149 * case 3. y < 4 sqrt(u), where u = minimum normal x.
150 * After case 1 and 2, this will only occurs when x=1. When x=1, we have
151 * A = (sqrt(4+y*y)+y)/2 ~ 1 + y/2 + y^2/8 + ...
152 * and
153 * B = 1/A = 1 - y/2 + y^2/8 + ...
154 * Since
155 * asin(x) = pi/2-2*asin(sqrt((1-x)/2))
156 * asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
157 * we have, for the real part asin(B),
158 * asin(1-y/2) ~ pi/2 - 2 asin(sqrt(y/4))
159 * ~ pi/2 - sqrt(y)
160 * For the imaginary part,
161 * log(A+sqrt(A*A-1)) ~ log(1+y/2+sqrt(2*y/2))
162 * = log(1+y/2+sqrt(y))
163 * = (y/2+sqrt(y)) - (y/2+sqrt(y))^2/2 + ...
164 * ~ sqrt(y) - y*(sqrt(y)+y/2)/2
165 * ~ sqrt(y)
166 *
167 * case 4. y >= (x+1)ulp(0.5). In this case, A ~ y and B ~ x/y. Thus
168 * real part = asin(B) ~ x/y (be careful, x/y may underflow)
169 * and
170 * imag part = log(y+sqrt(y*y-one))
171 *
172 *
173 * case 5. Both x and y are large: x and y > sqrt(M)/8, where M = maximum x
174 * In this case,
175 * A ~ sqrt(x*x+y*y)
176 * B ~ x/sqrt(x*x+y*y).
177 * Thus
178 * real part = asin(B) = atan(x/y),
179 * imag part = log(A+sqrt(A*A-1)) ~ log(2A)
180 * = log(2) + 0.5*log(x*x+y*y)
181 * = log(2) + log(y) + 0.5*log(1+(x/y)^2)
182 *
183 * case 6. x < 4 sqrt(u). In this case, we have
184 * A ~ sqrt(1+y*y), B = x/sqrt(1+y*y).
185 * Since B is tiny, we have
186 * real part = asin(B) ~ B = x/sqrt(1+y*y)
187 * imag part = log(A+sqrt(A*A-1)) = log (A+sqrt(y*y))
188 * = log(y+sqrt(1+y*y))
189 * = 0.5*log(y^2+2ysqrt(1+y^2)+1+y^2)
190 * = 0.5*log(1+2y(y+sqrt(1+y^2)));
191 * = 0.5*log1p(2y(y+A));
192 *
193 * casin(z) = asin(B) + i sign(y) log (A + sqrt(A*A-1)),
194 */
195
196 #include "libm.h" /* asin/atan/fabs/log/log1p/sqrt */
197 #include "complex_wrapper.h"
198
199 static const double zero = 0.0,
200 one = 1.0,
201 E = 1.11022302462515654042e-16, /* 2**-53 */
202 ln2 = 6.93147180559945286227e-01,
203 pi_2 = 1.570796326794896558e+00,
204 pi_2_l = 6.123233995736765886e-17,
205 pi_4 = 7.85398163397448278999e-01,
206 Foursqrtu = 5.96667258496016539463e-154, /* 2**(-509) */
207 Acrossover = 1.5,
208 Bcrossover = 0.6417,
209 half = 0.5;
210
211
212 dcomplex
213 casin(dcomplex z)
214 {
215 double x, y, t, R, S, A, Am1, B, y2, xm1, xp1, Apx;
216 int ix, iy, hx, hy;
217 unsigned lx, ly;
218 dcomplex ans;
219
220 x = D_RE(z);
221 y = D_IM(z);
222 hx = HI_WORD(x);
223 lx = LO_WORD(x);
224 hy = HI_WORD(y);
225 ly = LO_WORD(y);
226 ix = hx & 0x7fffffff;
227 iy = hy & 0x7fffffff;
228 x = fabs(x);
229 y = fabs(y);
230
231 /* special cases */
232
233 /* x is inf or NaN */
234 if (ix >= 0x7ff00000) { /* x is inf or NaN */
235 if (ISINF(ix, lx)) { /* x is INF */
236 D_IM(ans) = x;
237
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 }
247 } else { /* x is NaN */
248 if (iy >= 0x7ff00000) {
249
250 /*
251 * casin(NaN + i inf) = NaN + i inf
252 * casin(NaN + i NaN) = NaN + i NaN
253 */
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
262 if (hx < 0)
263 D_RE(ans) = -D_RE(ans);
264
265 if (hy < 0)
266 D_IM(ans) = -D_IM(ans);
267
268 return (ans);
269 }
270
271 /* casin(+0 + i 0 ) = 0 + i 0. */
272 if ((ix | lx | iy | ly) == 0)
273 return (z);
274
275 if (iy >= 0x7ff00000) { /* y is inf or NaN */
276 if (ISINF(iy, ly)) { /* casin(x + i inf) = 0 + i inf */
277 D_IM(ans) = y;
278 D_RE(ans) = zero;
279 } else { /* casin(x + i NaN) = NaN + i NaN */
280 D_IM(ans) = x + y;
281
282 if ((ix | lx) == 0)
283 D_RE(ans) = x;
284 else
285 D_RE(ans) = y;
286 }
287
288 if (hx < 0)
289 D_RE(ans) = -D_RE(ans);
290
291 if (hy < 0)
292 D_IM(ans) = -D_IM(ans);
293
294 return (ans);
295 }
296
297 if ((iy | ly) == 0) { /* region 1: y=0 */
298 if (ix < 0x3ff00000) { /* |x| < 1 */
299 D_RE(ans) = asin(x);
300 D_IM(ans) = zero;
301 } else {
302 D_RE(ans) = pi_2;
303
304 if (ix >= 0x43500000) { /* |x| >= 2**54 */
305 D_IM(ans) = ln2 + log(x);
306 } else if (ix >= 0x3ff80000) { /* x > Acrossover */
307 D_IM(ans) = log(x + sqrt((x - one) * (x +
308 one)));
309 } else {
310 xm1 = x - one;
311 D_IM(ans) = log1p(xm1 + sqrt(xm1 * (x + one)));
312 }
313 }
314 } else if (y <= E * fabs(x - one)) { /* region 2: y < tiny*|x-1| */
315 if (ix < 0x3ff00000) { /* x < 1 */
316 D_RE(ans) = asin(x);
317 D_IM(ans) = y / sqrt((one + x) * (one - x));
318 } else {
319 D_RE(ans) = pi_2;
320
321 if (ix >= 0x43500000) /* |x| >= 2**54 */
322 D_IM(ans) = ln2 + log(x);
323 else if (ix >= 0x3ff80000) /* x > Acrossover */
324 D_IM(ans) = log(x + sqrt((x - one) * (x +
325 one)));
326 else
327 D_IM(ans) = log1p((x - one) + sqrt((x - one) *
328 (x + one)));
329 }
330 } else if (y < Foursqrtu) { /* region 3 */
331 t = sqrt(y);
332 D_RE(ans) = pi_2 - (t - pi_2_l);
333 D_IM(ans) = t;
334 } else if (E * y - one >= x) { /* region 4 */
335 D_RE(ans) = x / y; /* need to fix underflow cases */
336 D_IM(ans) = ln2 + log(y);
337 } else if (ix >= 0x5fc00000 || iy >= 0x5fc00000) { /* x,y>2**509 */
338 /* region 5: x+1 or y is very large (>= sqrt(max)/8) */
339 t = x / y;
340 D_RE(ans) = atan(t);
341 D_IM(ans) = ln2 + log(y) + half * log1p(t * t);
342 } else if (x < Foursqrtu) {
343 /* region 6: x is very small, < 4sqrt(min) */
344 A = sqrt(one + y * y);
345 D_RE(ans) = x / A; /* may underflow */
346
347 if (iy >= 0x3ff80000) /* if y > Acrossover */
348 D_IM(ans) = log(y + A);
349 else
350 D_IM(ans) = half * log1p((y + y) * (y + A));
351 } else { /* safe region */
352 y2 = y * y;
353 xp1 = x + one;
354 xm1 = x - one;
355 R = sqrt(xp1 * xp1 + y2);
356 S = sqrt(xm1 * xm1 + y2);
357 A = half * (R + S);
358 B = x / A;
359
360 if (B <= Bcrossover) {
361 D_RE(ans) = asin(B);
362 } else { /* use atan and an accurate approx to a-x */
363 Apx = A + x;
364
365 if (x <= one)
366 D_RE(ans) = atan(x / sqrt(half * Apx * (y2 /
367 (R + xp1) + (S - xm1))));
368 else
369 D_RE(ans) = atan(x / (y * sqrt(half * (Apx /
370 (R + xp1) + Apx / (S + xm1)))));
371 }
372
373 if (A <= Acrossover) {
374 /* use log1p and an accurate approx to A-1 */
375 if (x < one)
376 Am1 = half * (y2 / (R + xp1) + y2 / (S - xm1));
377 else
378 Am1 = half * (y2 / (R + xp1) + (S + xm1));
379
380 D_IM(ans) = log1p(Am1 + sqrt(Am1 * (A + one)));
381 } else {
382 D_IM(ans) = log(A + sqrt(A * A - one));
383 }
384 }
385
386 if (hx < 0)
387 D_RE(ans) = -D_RE(ans);
388
389 if (hy < 0)
390 D_IM(ans) = -D_IM(ans);
391
392 return (ans);
393 }