134124Sbostic /* 224586Szliu * Copyright (c) 1985 Regents of the University of California. 334124Sbostic * All rights reserved. 434124Sbostic * 534124Sbostic * Redistribution and use in source and binary forms are permitted 634931Sbostic * provided that the above copyright notice and this paragraph are 734931Sbostic * duplicated in all such forms and that any documentation, 834931Sbostic * advertising materials, and other materials related to such 934931Sbostic * distribution and use acknowledge that the software was developed 1034931Sbostic * by the University of California, Berkeley. The name of the 1134931Sbostic * University may not be used to endorse or promote products derived 1234931Sbostic * from this software without specific prior written permission. 1334931Sbostic * THIS SOFTWARE IS PROVIDED ``AS IS'' AND WITHOUT ANY EXPRESS OR 1434931Sbostic * IMPLIED WARRANTIES, INCLUDING, WITHOUT LIMITATION, THE IMPLIED 1534931Sbostic * WARRANTIES OF MERCHANTIBILITY AND FITNESS FOR A PARTICULAR PURPOSE. 1634124Sbostic * 1734124Sbostic * All recipients should regard themselves as participants in an ongoing 1834124Sbostic * research project and hence should feel obligated to report their 1934124Sbostic * experiences (good or bad) with these elementary function codes, using 2034124Sbostic * the sendbug(8) program, to the authors. 2124586Szliu */ 2224586Szliu 2324586Szliu #ifndef lint 24*35679Sbostic static char sccsid[] = "@(#)acosh.c 5.4 (Berkeley) 09/22/88"; 2534124Sbostic #endif /* not lint */ 2634124Sbostic 2724586Szliu /* ACOSH(X) 2824586Szliu * RETURN THE INVERSE HYPERBOLIC COSINE OF X 2924586Szliu * DOUBLE PRECISION (VAX D FORMAT 56 BITS, IEEE DOUBLE 53 BITS) 3024586Szliu * CODED IN C BY K.C. NG, 2/16/85; 3124586Szliu * REVISED BY K.C. NG on 3/6/85, 3/24/85, 4/16/85, 8/17/85. 3224586Szliu * 3324586Szliu * Required system supported functions : 3424586Szliu * sqrt(x) 3524586Szliu * 3624586Szliu * Required kernel function: 3724586Szliu * log1p(x) ...return log(1+x) 3824586Szliu * 3924586Szliu * Method : 4024586Szliu * Based on 4124586Szliu * acosh(x) = log [ x + sqrt(x*x-1) ] 4224586Szliu * we have 4324586Szliu * acosh(x) := log1p(x)+ln2, if (x > 1.0E20); else 4424586Szliu * acosh(x) := log1p( sqrt(x-1) * (sqrt(x-1) + sqrt(x+1)) ) . 4524586Szliu * These formulae avoid the over/underflow complication. 4624586Szliu * 4724586Szliu * Special cases: 4824586Szliu * acosh(x) is NaN with signal if x<1. 4924586Szliu * acosh(NaN) is NaN without signal. 5024586Szliu * 5124586Szliu * Accuracy: 5224586Szliu * acosh(x) returns the exact inverse hyperbolic cosine of x nearly 5324586Szliu * rounded. In a test run with 512,000 random arguments on a VAX, the 5424586Szliu * maximum observed error was 3.30 ulps (units of the last place) at 5524586Szliu * x=1.0070493753568216 . 5624586Szliu * 5724586Szliu * Constants: 5824586Szliu * The hexadecimal values are the intended ones for the following constants. 5924586Szliu * The decimal values may be used, provided that the compiler will convert 6024586Szliu * from decimal to binary accurately enough to produce the hexadecimal values 6124586Szliu * shown. 6224586Szliu */ 6324586Szliu 64*35679Sbostic #include "mathimpl.h" 6524586Szliu 66*35679Sbostic vc(ln2hi, 6.9314718055829871446E-1 ,7217,4031,0000,f7d0, 0, .B17217F7D00000) 67*35679Sbostic vc(ln2lo, 1.6465949582897081279E-12 ,bcd5,2ce7,d9cc,e4f1, -39, .E7BCD5E4F1D9CC) 68*35679Sbostic 69*35679Sbostic ic(ln2hi, 6.9314718036912381649E-1, -1, 1.62E42FEE00000) 70*35679Sbostic ic(ln2lo, 1.9082149292705877000E-10,-33, 1.A39EF35793C76) 71*35679Sbostic 72*35679Sbostic #ifdef vccast 73*35679Sbostic #define ln2hi vccast(ln2hi) 74*35679Sbostic #define ln2lo vccast(ln2lo) 75*35679Sbostic #endif 76*35679Sbostic 7724586Szliu double acosh(x) 7824586Szliu double x; 7924586Szliu { 80*35679Sbostic double t,big=1.E20; /* big+1==big */ 8124586Szliu 8231853Szliu #if !defined(vax)&&!defined(tahoe) 8324586Szliu if(x!=x) return(x); /* x is NaN */ 8431853Szliu #endif /* !defined(vax)&&!defined(tahoe) */ 8524586Szliu 8624586Szliu /* return log1p(x) + log(2) if x is large */ 8724586Szliu if(x>big) {t=log1p(x)+ln2lo; return(t+ln2hi);} 8824586Szliu 8924586Szliu t=sqrt(x-1.0); 9024586Szliu return(log1p(t*(t+sqrt(x+1.0)))); 9124586Szliu } 92