1 /* 2 * Copyright (c) 1985, 1993 3 * The Regents of the University of California. All rights reserved. 4 * 5 * %sccs.include.redist.c% 6 */ 7 8 #ifndef lint 9 static char sccsid[] = "@(#)acosh.c 8.1 (Berkeley) 06/04/93"; 10 #endif /* not lint */ 11 12 /* ACOSH(X) 13 * RETURN THE INVERSE HYPERBOLIC COSINE OF X 14 * DOUBLE PRECISION (VAX D FORMAT 56 BITS, IEEE DOUBLE 53 BITS) 15 * CODED IN C BY K.C. NG, 2/16/85; 16 * REVISED BY K.C. NG on 3/6/85, 3/24/85, 4/16/85, 8/17/85. 17 * 18 * Required system supported functions : 19 * sqrt(x) 20 * 21 * Required kernel function: 22 * log1p(x) ...return log(1+x) 23 * 24 * Method : 25 * Based on 26 * acosh(x) = log [ x + sqrt(x*x-1) ] 27 * we have 28 * acosh(x) := log1p(x)+ln2, if (x > 1.0E20); else 29 * acosh(x) := log1p( sqrt(x-1) * (sqrt(x-1) + sqrt(x+1)) ) . 30 * These formulae avoid the over/underflow complication. 31 * 32 * Special cases: 33 * acosh(x) is NaN with signal if x<1. 34 * acosh(NaN) is NaN without signal. 35 * 36 * Accuracy: 37 * acosh(x) returns the exact inverse hyperbolic cosine of x nearly 38 * rounded. In a test run with 512,000 random arguments on a VAX, the 39 * maximum observed error was 3.30 ulps (units of the last place) at 40 * x=1.0070493753568216 . 41 * 42 * Constants: 43 * The hexadecimal values are the intended ones for the following constants. 44 * The decimal values may be used, provided that the compiler will convert 45 * from decimal to binary accurately enough to produce the hexadecimal values 46 * shown. 47 */ 48 49 #include "mathimpl.h" 50 51 vc(ln2hi, 6.9314718055829871446E-1 ,7217,4031,0000,f7d0, 0, .B17217F7D00000) 52 vc(ln2lo, 1.6465949582897081279E-12 ,bcd5,2ce7,d9cc,e4f1, -39, .E7BCD5E4F1D9CC) 53 54 ic(ln2hi, 6.9314718036912381649E-1, -1, 1.62E42FEE00000) 55 ic(ln2lo, 1.9082149292705877000E-10,-33, 1.A39EF35793C76) 56 57 #ifdef vccast 58 #define ln2hi vccast(ln2hi) 59 #define ln2lo vccast(ln2lo) 60 #endif 61 62 double acosh(x) 63 double x; 64 { 65 double t,big=1.E20; /* big+1==big */ 66 67 #if !defined(vax)&&!defined(tahoe) 68 if(x!=x) return(x); /* x is NaN */ 69 #endif /* !defined(vax)&&!defined(tahoe) */ 70 71 /* return log1p(x) + log(2) if x is large */ 72 if(x>big) {t=log1p(x)+ln2lo; return(t+ln2hi);} 73 74 t=sqrt(x-1.0); 75 return(log1p(t*(t+sqrt(x+1.0)))); 76 } 77