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 6*34931Sbostic * provided that the above copyright notice and this paragraph are 7*34931Sbostic * duplicated in all such forms and that any documentation, 8*34931Sbostic * advertising materials, and other materials related to such 9*34931Sbostic * distribution and use acknowledge that the software was developed 10*34931Sbostic * by the University of California, Berkeley. The name of the 11*34931Sbostic * University may not be used to endorse or promote products derived 12*34931Sbostic * from this software without specific prior written permission. 13*34931Sbostic * THIS SOFTWARE IS PROVIDED ``AS IS'' AND WITHOUT ANY EXPRESS OR 14*34931Sbostic * IMPLIED WARRANTIES, INCLUDING, WITHOUT LIMITATION, THE IMPLIED 15*34931Sbostic * 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*34931Sbostic static char sccsid[] = "@(#)acosh.c 5.3 (Berkeley) 06/30/88"; 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 6431853Szliu #if defined(vax)||defined(tahoe) /* VAX D format */ 6531853Szliu #ifdef vax 6631812Szliu #define _0x(A,B) 0x/**/A/**/B 6731853Szliu #else /* vax */ 6831812Szliu #define _0x(A,B) 0x/**/B/**/A 6931853Szliu #endif /* vax */ 7024586Szliu /* static double */ 7124586Szliu /* ln2hi = 6.9314718055829871446E-1 , Hex 2^ 0 * .B17217F7D00000 */ 7224586Szliu /* ln2lo = 1.6465949582897081279E-12 ; Hex 2^-39 * .E7BCD5E4F1D9CC */ 7331812Szliu static long ln2hix[] = { _0x(7217,4031), _0x(0000,f7d0)}; 7431812Szliu static long ln2lox[] = { _0x(bcd5,2ce7), _0x(d9cc,e4f1)}; 7524586Szliu #define ln2hi (*(double*)ln2hix) 7624586Szliu #define ln2lo (*(double*)ln2lox) 7731853Szliu #else /* defined(vax)||defined(tahoe) */ 7824586Szliu static double 7924586Szliu ln2hi = 6.9314718036912381649E-1 , /*Hex 2^ -1 * 1.62E42FEE00000 */ 8024586Szliu ln2lo = 1.9082149292705877000E-10 ; /*Hex 2^-33 * 1.A39EF35793C76 */ 8131853Szliu #endif /* defined(vax)||defined(tahoe) */ 8224586Szliu 8324586Szliu double acosh(x) 8424586Szliu double x; 8524586Szliu { 8624586Szliu double log1p(),sqrt(),t,big=1.E20; /* big+1==big */ 8724586Szliu 8831853Szliu #if !defined(vax)&&!defined(tahoe) 8924586Szliu if(x!=x) return(x); /* x is NaN */ 9031853Szliu #endif /* !defined(vax)&&!defined(tahoe) */ 9124586Szliu 9224586Szliu /* return log1p(x) + log(2) if x is large */ 9324586Szliu if(x>big) {t=log1p(x)+ln2lo; return(t+ln2hi);} 9424586Szliu 9524586Szliu t=sqrt(x-1.0); 9624586Szliu return(log1p(t*(t+sqrt(x+1.0)))); 9724586Szliu } 98