1*34124Sbostic /* 224586Szliu * Copyright (c) 1985 Regents of the University of California. 3*34124Sbostic * All rights reserved. 4*34124Sbostic * 5*34124Sbostic * Redistribution and use in source and binary forms are permitted 6*34124Sbostic * provided that this notice is preserved and that due credit is given 7*34124Sbostic * to the University of California at Berkeley. The name of the University 8*34124Sbostic * may not be used to endorse or promote products derived from this 9*34124Sbostic * software without specific prior written permission. This software 10*34124Sbostic * is provided ``as is'' without express or implied warranty. 11*34124Sbostic * 12*34124Sbostic * All recipients should regard themselves as participants in an ongoing 13*34124Sbostic * research project and hence should feel obligated to report their 14*34124Sbostic * experiences (good or bad) with these elementary function codes, using 15*34124Sbostic * the sendbug(8) program, to the authors. 1624586Szliu */ 1724586Szliu 1824586Szliu #ifndef lint 19*34124Sbostic static char sccsid[] = "@(#)acosh.c 5.2 (Berkeley) 04/29/88"; 20*34124Sbostic #endif /* not lint */ 21*34124Sbostic 2224586Szliu /* ACOSH(X) 2324586Szliu * RETURN THE INVERSE HYPERBOLIC COSINE OF X 2424586Szliu * DOUBLE PRECISION (VAX D FORMAT 56 BITS, IEEE DOUBLE 53 BITS) 2524586Szliu * CODED IN C BY K.C. NG, 2/16/85; 2624586Szliu * REVISED BY K.C. NG on 3/6/85, 3/24/85, 4/16/85, 8/17/85. 2724586Szliu * 2824586Szliu * Required system supported functions : 2924586Szliu * sqrt(x) 3024586Szliu * 3124586Szliu * Required kernel function: 3224586Szliu * log1p(x) ...return log(1+x) 3324586Szliu * 3424586Szliu * Method : 3524586Szliu * Based on 3624586Szliu * acosh(x) = log [ x + sqrt(x*x-1) ] 3724586Szliu * we have 3824586Szliu * acosh(x) := log1p(x)+ln2, if (x > 1.0E20); else 3924586Szliu * acosh(x) := log1p( sqrt(x-1) * (sqrt(x-1) + sqrt(x+1)) ) . 4024586Szliu * These formulae avoid the over/underflow complication. 4124586Szliu * 4224586Szliu * Special cases: 4324586Szliu * acosh(x) is NaN with signal if x<1. 4424586Szliu * acosh(NaN) is NaN without signal. 4524586Szliu * 4624586Szliu * Accuracy: 4724586Szliu * acosh(x) returns the exact inverse hyperbolic cosine of x nearly 4824586Szliu * rounded. In a test run with 512,000 random arguments on a VAX, the 4924586Szliu * maximum observed error was 3.30 ulps (units of the last place) at 5024586Szliu * x=1.0070493753568216 . 5124586Szliu * 5224586Szliu * Constants: 5324586Szliu * The hexadecimal values are the intended ones for the following constants. 5424586Szliu * The decimal values may be used, provided that the compiler will convert 5524586Szliu * from decimal to binary accurately enough to produce the hexadecimal values 5624586Szliu * shown. 5724586Szliu */ 5824586Szliu 5931853Szliu #if defined(vax)||defined(tahoe) /* VAX D format */ 6031853Szliu #ifdef vax 6131812Szliu #define _0x(A,B) 0x/**/A/**/B 6231853Szliu #else /* vax */ 6331812Szliu #define _0x(A,B) 0x/**/B/**/A 6431853Szliu #endif /* vax */ 6524586Szliu /* static double */ 6624586Szliu /* ln2hi = 6.9314718055829871446E-1 , Hex 2^ 0 * .B17217F7D00000 */ 6724586Szliu /* ln2lo = 1.6465949582897081279E-12 ; Hex 2^-39 * .E7BCD5E4F1D9CC */ 6831812Szliu static long ln2hix[] = { _0x(7217,4031), _0x(0000,f7d0)}; 6931812Szliu static long ln2lox[] = { _0x(bcd5,2ce7), _0x(d9cc,e4f1)}; 7024586Szliu #define ln2hi (*(double*)ln2hix) 7124586Szliu #define ln2lo (*(double*)ln2lox) 7231853Szliu #else /* defined(vax)||defined(tahoe) */ 7324586Szliu static double 7424586Szliu ln2hi = 6.9314718036912381649E-1 , /*Hex 2^ -1 * 1.62E42FEE00000 */ 7524586Szliu ln2lo = 1.9082149292705877000E-10 ; /*Hex 2^-33 * 1.A39EF35793C76 */ 7631853Szliu #endif /* defined(vax)||defined(tahoe) */ 7724586Szliu 7824586Szliu double acosh(x) 7924586Szliu double x; 8024586Szliu { 8124586Szliu double log1p(),sqrt(),t,big=1.E20; /* big+1==big */ 8224586Szliu 8331853Szliu #if !defined(vax)&&!defined(tahoe) 8424586Szliu if(x!=x) return(x); /* x is NaN */ 8531853Szliu #endif /* !defined(vax)&&!defined(tahoe) */ 8624586Szliu 8724586Szliu /* return log1p(x) + log(2) if x is large */ 8824586Szliu if(x>big) {t=log1p(x)+ln2lo; return(t+ln2hi);} 8924586Szliu 9024586Szliu t=sqrt(x-1.0); 9124586Szliu return(log1p(t*(t+sqrt(x+1.0)))); 9224586Szliu } 93