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 this notice is preserved and that due credit is given 7 * to the University of California at Berkeley. The name of the University 8 * may not be used to endorse or promote products derived from this 9 * software without specific prior written permission. This software 10 * is provided ``as is'' without express or implied warranty. 11 * 12 * All recipients should regard themselves as participants in an ongoing 13 * research project and hence should feel obligated to report their 14 * experiences (good or bad) with these elementary function codes, using 15 * the sendbug(8) program, to the authors. 16 */ 17 18 #ifndef lint 19 static char sccsid[] = "@(#)acosh.c 5.2 (Berkeley) 04/29/88"; 20 #endif /* not lint */ 21 22 /* ACOSH(X) 23 * RETURN THE INVERSE HYPERBOLIC COSINE OF X 24 * DOUBLE PRECISION (VAX D FORMAT 56 BITS, IEEE DOUBLE 53 BITS) 25 * CODED IN C BY K.C. NG, 2/16/85; 26 * REVISED BY K.C. NG on 3/6/85, 3/24/85, 4/16/85, 8/17/85. 27 * 28 * Required system supported functions : 29 * sqrt(x) 30 * 31 * Required kernel function: 32 * log1p(x) ...return log(1+x) 33 * 34 * Method : 35 * Based on 36 * acosh(x) = log [ x + sqrt(x*x-1) ] 37 * we have 38 * acosh(x) := log1p(x)+ln2, if (x > 1.0E20); else 39 * acosh(x) := log1p( sqrt(x-1) * (sqrt(x-1) + sqrt(x+1)) ) . 40 * These formulae avoid the over/underflow complication. 41 * 42 * Special cases: 43 * acosh(x) is NaN with signal if x<1. 44 * acosh(NaN) is NaN without signal. 45 * 46 * Accuracy: 47 * acosh(x) returns the exact inverse hyperbolic cosine of x nearly 48 * rounded. In a test run with 512,000 random arguments on a VAX, the 49 * maximum observed error was 3.30 ulps (units of the last place) at 50 * x=1.0070493753568216 . 51 * 52 * Constants: 53 * The hexadecimal values are the intended ones for the following constants. 54 * The decimal values may be used, provided that the compiler will convert 55 * from decimal to binary accurately enough to produce the hexadecimal values 56 * shown. 57 */ 58 59 #if defined(vax)||defined(tahoe) /* VAX D format */ 60 #ifdef vax 61 #define _0x(A,B) 0x/**/A/**/B 62 #else /* vax */ 63 #define _0x(A,B) 0x/**/B/**/A 64 #endif /* vax */ 65 /* static double */ 66 /* ln2hi = 6.9314718055829871446E-1 , Hex 2^ 0 * .B17217F7D00000 */ 67 /* ln2lo = 1.6465949582897081279E-12 ; Hex 2^-39 * .E7BCD5E4F1D9CC */ 68 static long ln2hix[] = { _0x(7217,4031), _0x(0000,f7d0)}; 69 static long ln2lox[] = { _0x(bcd5,2ce7), _0x(d9cc,e4f1)}; 70 #define ln2hi (*(double*)ln2hix) 71 #define ln2lo (*(double*)ln2lox) 72 #else /* defined(vax)||defined(tahoe) */ 73 static double 74 ln2hi = 6.9314718036912381649E-1 , /*Hex 2^ -1 * 1.62E42FEE00000 */ 75 ln2lo = 1.9082149292705877000E-10 ; /*Hex 2^-33 * 1.A39EF35793C76 */ 76 #endif /* defined(vax)||defined(tahoe) */ 77 78 double acosh(x) 79 double x; 80 { 81 double log1p(),sqrt(),t,big=1.E20; /* big+1==big */ 82 83 #if !defined(vax)&&!defined(tahoe) 84 if(x!=x) return(x); /* x is NaN */ 85 #endif /* !defined(vax)&&!defined(tahoe) */ 86 87 /* return log1p(x) + log(2) if x is large */ 88 if(x>big) {t=log1p(x)+ln2lo; return(t+ln2hi);} 89 90 t=sqrt(x-1.0); 91 return(log1p(t*(t+sqrt(x+1.0)))); 92 } 93