62 static double complex clog_for_large_values(double complex z);
90  * Throughout we use the convention z = x + I*y.
92  * casinh(z) = sign(x)*log(A+sqrt(A*A-1)) + I*asin(B)
94  * A = (|z+I| + |z-I|) / 2
95  * B = (|z+I| - |z-I|) / 2 = y/A
98  *   (a) for Re(casinh(z)) when z is close to the line segment [-I, I] (that
99  *       is, Re(casinh(z)) is close to 0);
100  *   (b) for Im(casinh(z)) when z is close to either of the intervals
101  *       [I, I*infinity) or (-I*infinity, -I] (that is, |Im(casinh(z))| is
145  * rx = Re(casinh(z)) = -Im(cacos(y + I*x)).
158 	R = hypot(x, y + 1);		/* |z+I| */  in do_hard_work()
159 	S = hypot(x, y - 1);		/* |z-I| */  in do_hard_work()
161 	/* A = (|z+I| + |z-I|) / 2 */  in do_hard_work()
219 	/* B = (|z+I| - |z-I|) / 2 = y/A */  in do_hard_work()
266  * casinh(z) = z + O(z^3)   as z -> 0
268  * casinh(z) = sign(x)*clog(sign(x)*z) + O(1/z^2)   as z -> infinity
270  * Im(casinh(z)) = sign(x)*atan2(sign(x)*y, fabs(x)) + O(y/z^3)
271  *    as z -> infinity, uniformly in y
274 casinh(double complex z)  in casinh()  argument
280 	x = creal(z);  in casinh()
281 	y = cimag(z);  in casinh()
306 			w = clog_for_large_values(z) + m_ln2;  in casinh()
308 			w = clog_for_large_values(-z) + m_ln2;  in casinh()
312 	/* Avoid spuriously raising inexact for z = 0. */  in casinh()
314 		return (z);  in casinh()
320 		return (z);  in casinh()
331  * casin(z) = reverse(casinh(reverse(z)))
332  * where reverse(x + I*y) = y + I*x = I*conj(z).
335 casin(double complex z)  in casin()  argument
337 	double complex w = casinh(CMPLX(cimag(z), creal(z)));  in casin()
343  * cacos(z) = PI/2 - casin(z)
344  * but do the computation carefully so cacos(z) is accurate when z is
347  * cacos(z) = PI/2 - z + O(z^3)   as z -> 0
349  * cacos(z) = -sign(y)*I*clog(z) + O(1/z^2)   as z -> infinity
351  * Re(cacos(z)) = atan2(fabs(y), x) + O(y/z^3)
352  *    as z -> infinity, uniformly in y
355 cacos(double complex z)  in cacos()  argument
362 	x = creal(z);  in cacos()
363 	y = cimag(z);  in cacos()
389 		w = clog_for_large_values(z);  in cacos()
397 	/* Avoid spuriously raising inexact for z = 1. */  in cacos()
425  * cacosh(z) = I*cacos(z) or -I*cacos(z)
426  * where the sign is chosen so Re(cacosh(z)) >= 0.
429 cacosh(double complex z)  in cacosh()  argument
434 	w = cacos(z);  in cacosh()
447 	return (CMPLX(fabs(ry), copysign(rx, cimag(z))));  in cacosh()
451  * Optimized version of clog() for |z| finite and larger than ~RECIP_EPSILON.
454 clog_for_large_values(double complex z)  in clog_for_large_values()  argument
459 	x = creal(z);  in clog_for_large_values()
460 	y = cimag(z);  in clog_for_large_values()
522  * the code creal(1/z), because the imaginary part may produce an unwanted
559  * catanh(z) = log((1+z)/(1-z)) / 2
560  *           = log1p(4*x / |z-1|^2) / 4
563  * catanh(z) = z + O(z^3)   as z -> 0
565  * catanh(z) = 1/z + sign(y)*I*PI/2 + O(1/z^3)   as z -> infinity
567  * Re(catanh(z)) = x/|z|^2 + O(x/z^4)
568  *    as z -> infinity, uniformly in x
571 catanh(double complex z)  in catanh()  argument
575 	x = creal(z);  in catanh()
576 	y = cimag(z);  in catanh()
584 	/* To ensure the same accuracy as atan(), and to filter out z = 0. */  in catanh()
610 		 * z = 0 was filtered out above.  All other cases must raise  in catanh()
615 		return (z);  in catanh()
634  * catan(z) = reverse(catanh(reverse(z)))
635  * where reverse(x + I*y) = y + I*x = I*conj(z).
638 catan(double complex z)  in catan()  argument
640 	double complex w = catanh(CMPLX(cimag(z), creal(z)));  in catan()