1 /* 2 * Copyright (c) 1985 Regents of the University of California. 3 * All rights reserved. 4 * 5 * Redistribution and use in source and binary forms are permitted 6 * provided that the above copyright notice and this paragraph are 7 * duplicated in all such forms and that any documentation, 8 * advertising materials, and other materials related to such 9 * distribution and use acknowledge that the software was developed 10 * by the University of California, Berkeley. The name of the 11 * University may not be used to endorse or promote products derived 12 * from this software without specific prior written permission. 13 * THIS SOFTWARE IS PROVIDED ``AS IS'' AND WITHOUT ANY EXPRESS OR 14 * IMPLIED WARRANTIES, INCLUDING, WITHOUT LIMITATION, THE IMPLIED 15 * WARRANTIES OF MERCHANTIBILITY AND FITNESS FOR A PARTICULAR PURPOSE. 16 * 17 * All recipients should regard themselves as participants in an ongoing 18 * research project and hence should feel obligated to report their 19 * experiences (good or bad) with these elementary function codes, using 20 * the sendbug(8) program, to the authors. 21 */ 22 23 #ifndef lint 24 static char sccsid[] = "@(#)acosh.c 5.3 (Berkeley) 06/30/88"; 25 #endif /* not lint */ 26 27 /* ACOSH(X) 28 * RETURN THE INVERSE HYPERBOLIC COSINE OF X 29 * DOUBLE PRECISION (VAX D FORMAT 56 BITS, IEEE DOUBLE 53 BITS) 30 * CODED IN C BY K.C. NG, 2/16/85; 31 * REVISED BY K.C. NG on 3/6/85, 3/24/85, 4/16/85, 8/17/85. 32 * 33 * Required system supported functions : 34 * sqrt(x) 35 * 36 * Required kernel function: 37 * log1p(x) ...return log(1+x) 38 * 39 * Method : 40 * Based on 41 * acosh(x) = log [ x + sqrt(x*x-1) ] 42 * we have 43 * acosh(x) := log1p(x)+ln2, if (x > 1.0E20); else 44 * acosh(x) := log1p( sqrt(x-1) * (sqrt(x-1) + sqrt(x+1)) ) . 45 * These formulae avoid the over/underflow complication. 46 * 47 * Special cases: 48 * acosh(x) is NaN with signal if x<1. 49 * acosh(NaN) is NaN without signal. 50 * 51 * Accuracy: 52 * acosh(x) returns the exact inverse hyperbolic cosine of x nearly 53 * rounded. In a test run with 512,000 random arguments on a VAX, the 54 * maximum observed error was 3.30 ulps (units of the last place) at 55 * x=1.0070493753568216 . 56 * 57 * Constants: 58 * The hexadecimal values are the intended ones for the following constants. 59 * The decimal values may be used, provided that the compiler will convert 60 * from decimal to binary accurately enough to produce the hexadecimal values 61 * shown. 62 */ 63 64 #if defined(vax)||defined(tahoe) /* VAX D format */ 65 #ifdef vax 66 #define _0x(A,B) 0x/**/A/**/B 67 #else /* vax */ 68 #define _0x(A,B) 0x/**/B/**/A 69 #endif /* vax */ 70 /* static double */ 71 /* ln2hi = 6.9314718055829871446E-1 , Hex 2^ 0 * .B17217F7D00000 */ 72 /* ln2lo = 1.6465949582897081279E-12 ; Hex 2^-39 * .E7BCD5E4F1D9CC */ 73 static long ln2hix[] = { _0x(7217,4031), _0x(0000,f7d0)}; 74 static long ln2lox[] = { _0x(bcd5,2ce7), _0x(d9cc,e4f1)}; 75 #define ln2hi (*(double*)ln2hix) 76 #define ln2lo (*(double*)ln2lox) 77 #else /* defined(vax)||defined(tahoe) */ 78 static double 79 ln2hi = 6.9314718036912381649E-1 , /*Hex 2^ -1 * 1.62E42FEE00000 */ 80 ln2lo = 1.9082149292705877000E-10 ; /*Hex 2^-33 * 1.A39EF35793C76 */ 81 #endif /* defined(vax)||defined(tahoe) */ 82 83 double acosh(x) 84 double x; 85 { 86 double log1p(),sqrt(),t,big=1.E20; /* big+1==big */ 87 88 #if !defined(vax)&&!defined(tahoe) 89 if(x!=x) return(x); /* x is NaN */ 90 #endif /* !defined(vax)&&!defined(tahoe) */ 91 92 /* return log1p(x) + log(2) if x is large */ 93 if(x>big) {t=log1p(x)+ln2lo; return(t+ln2hi);} 94 95 t=sqrt(x-1.0); 96 return(log1p(t*(t+sqrt(x+1.0)))); 97 } 98