1 /* 2 * Copyright (c) 1985 Regents of the University of California. 3 * All rights reserved. 4 * 5 * %sccs.include.redist.c% 6 * 7 * All recipients should regard themselves as participants in an ongoing 8 * research project and hence should feel obligated to report their 9 * experiences (good or bad) with these elementary function codes, using 10 * the sendbug(8) program, to the authors. 11 */ 12 13 #ifndef lint 14 static char sccsid[] = "@(#)acosh.c 5.5 (Berkeley) 06/01/90"; 15 #endif /* not lint */ 16 17 /* ACOSH(X) 18 * RETURN THE INVERSE HYPERBOLIC COSINE OF X 19 * DOUBLE PRECISION (VAX D FORMAT 56 BITS, IEEE DOUBLE 53 BITS) 20 * CODED IN C BY K.C. NG, 2/16/85; 21 * REVISED BY K.C. NG on 3/6/85, 3/24/85, 4/16/85, 8/17/85. 22 * 23 * Required system supported functions : 24 * sqrt(x) 25 * 26 * Required kernel function: 27 * log1p(x) ...return log(1+x) 28 * 29 * Method : 30 * Based on 31 * acosh(x) = log [ x + sqrt(x*x-1) ] 32 * we have 33 * acosh(x) := log1p(x)+ln2, if (x > 1.0E20); else 34 * acosh(x) := log1p( sqrt(x-1) * (sqrt(x-1) + sqrt(x+1)) ) . 35 * These formulae avoid the over/underflow complication. 36 * 37 * Special cases: 38 * acosh(x) is NaN with signal if x<1. 39 * acosh(NaN) is NaN without signal. 40 * 41 * Accuracy: 42 * acosh(x) returns the exact inverse hyperbolic cosine of x nearly 43 * rounded. In a test run with 512,000 random arguments on a VAX, the 44 * maximum observed error was 3.30 ulps (units of the last place) at 45 * x=1.0070493753568216 . 46 * 47 * Constants: 48 * The hexadecimal values are the intended ones for the following constants. 49 * The decimal values may be used, provided that the compiler will convert 50 * from decimal to binary accurately enough to produce the hexadecimal values 51 * shown. 52 */ 53 54 #include "mathimpl.h" 55 56 vc(ln2hi, 6.9314718055829871446E-1 ,7217,4031,0000,f7d0, 0, .B17217F7D00000) 57 vc(ln2lo, 1.6465949582897081279E-12 ,bcd5,2ce7,d9cc,e4f1, -39, .E7BCD5E4F1D9CC) 58 59 ic(ln2hi, 6.9314718036912381649E-1, -1, 1.62E42FEE00000) 60 ic(ln2lo, 1.9082149292705877000E-10,-33, 1.A39EF35793C76) 61 62 #ifdef vccast 63 #define ln2hi vccast(ln2hi) 64 #define ln2lo vccast(ln2lo) 65 #endif 66 67 double acosh(x) 68 double x; 69 { 70 double t,big=1.E20; /* big+1==big */ 71 72 #if !defined(vax)&&!defined(tahoe) 73 if(x!=x) return(x); /* x is NaN */ 74 #endif /* !defined(vax)&&!defined(tahoe) */ 75 76 /* return log1p(x) + log(2) if x is large */ 77 if(x>big) {t=log1p(x)+ln2lo; return(t+ln2hi);} 78 79 t=sqrt(x-1.0); 80 return(log1p(t*(t+sqrt(x+1.0)))); 81 } 82