1*24586Szliu /* 2*24586Szliu * Copyright (c) 1985 Regents of the University of California. 3*24586Szliu * 4*24586Szliu * Use and reproduction of this software are granted in accordance with 5*24586Szliu * the terms and conditions specified in the Berkeley Software License 6*24586Szliu * Agreement (in particular, this entails acknowledgement of the programs' 7*24586Szliu * source, and inclusion of this notice) with the additional understanding 8*24586Szliu * that all recipients should regard themselves as participants in an 9*24586Szliu * ongoing research project and hence should feel obligated to report 10*24586Szliu * their experiences (good or bad) with these elementary function codes, 11*24586Szliu * using "sendbug 4bsd-bugs@BERKELEY", to the authors. 12*24586Szliu */ 13*24586Szliu 14*24586Szliu #ifndef lint 15*24586Szliu static char sccsid[] = "@(#)acosh.c 1.1 (ELEFUNT) 09/06/85"; 16*24586Szliu #endif not lint 17*24586Szliu 18*24586Szliu /* ACOSH(X) 19*24586Szliu * RETURN THE INVERSE HYPERBOLIC COSINE OF X 20*24586Szliu * DOUBLE PRECISION (VAX D FORMAT 56 BITS, IEEE DOUBLE 53 BITS) 21*24586Szliu * CODED IN C BY K.C. NG, 2/16/85; 22*24586Szliu * REVISED BY K.C. NG on 3/6/85, 3/24/85, 4/16/85, 8/17/85. 23*24586Szliu * 24*24586Szliu * Required system supported functions : 25*24586Szliu * sqrt(x) 26*24586Szliu * 27*24586Szliu * Required kernel function: 28*24586Szliu * log1p(x) ...return log(1+x) 29*24586Szliu * 30*24586Szliu * Method : 31*24586Szliu * Based on 32*24586Szliu * acosh(x) = log [ x + sqrt(x*x-1) ] 33*24586Szliu * we have 34*24586Szliu * acosh(x) := log1p(x)+ln2, if (x > 1.0E20); else 35*24586Szliu * acosh(x) := log1p( sqrt(x-1) * (sqrt(x-1) + sqrt(x+1)) ) . 36*24586Szliu * These formulae avoid the over/underflow complication. 37*24586Szliu * 38*24586Szliu * Special cases: 39*24586Szliu * acosh(x) is NaN with signal if x<1. 40*24586Szliu * acosh(NaN) is NaN without signal. 41*24586Szliu * 42*24586Szliu * Accuracy: 43*24586Szliu * acosh(x) returns the exact inverse hyperbolic cosine of x nearly 44*24586Szliu * rounded. In a test run with 512,000 random arguments on a VAX, the 45*24586Szliu * maximum observed error was 3.30 ulps (units of the last place) at 46*24586Szliu * x=1.0070493753568216 . 47*24586Szliu * 48*24586Szliu * Constants: 49*24586Szliu * The hexadecimal values are the intended ones for the following constants. 50*24586Szliu * The decimal values may be used, provided that the compiler will convert 51*24586Szliu * from decimal to binary accurately enough to produce the hexadecimal values 52*24586Szliu * shown. 53*24586Szliu */ 54*24586Szliu 55*24586Szliu #ifdef VAX /* VAX D format */ 56*24586Szliu /* static double */ 57*24586Szliu /* ln2hi = 6.9314718055829871446E-1 , Hex 2^ 0 * .B17217F7D00000 */ 58*24586Szliu /* ln2lo = 1.6465949582897081279E-12 ; Hex 2^-39 * .E7BCD5E4F1D9CC */ 59*24586Szliu static long ln2hix[] = { 0x72174031, 0x0000f7d0}; 60*24586Szliu static long ln2lox[] = { 0xbcd52ce7, 0xd9cce4f1}; 61*24586Szliu #define ln2hi (*(double*)ln2hix) 62*24586Szliu #define ln2lo (*(double*)ln2lox) 63*24586Szliu #else /* IEEE double */ 64*24586Szliu static double 65*24586Szliu ln2hi = 6.9314718036912381649E-1 , /*Hex 2^ -1 * 1.62E42FEE00000 */ 66*24586Szliu ln2lo = 1.9082149292705877000E-10 ; /*Hex 2^-33 * 1.A39EF35793C76 */ 67*24586Szliu #endif 68*24586Szliu 69*24586Szliu double acosh(x) 70*24586Szliu double x; 71*24586Szliu { 72*24586Szliu double log1p(),sqrt(),t,big=1.E20; /* big+1==big */ 73*24586Szliu 74*24586Szliu #ifndef VAX 75*24586Szliu if(x!=x) return(x); /* x is NaN */ 76*24586Szliu #endif 77*24586Szliu 78*24586Szliu /* return log1p(x) + log(2) if x is large */ 79*24586Szliu if(x>big) {t=log1p(x)+ln2lo; return(t+ln2hi);} 80*24586Szliu 81*24586Szliu t=sqrt(x-1.0); 82*24586Szliu return(log1p(t*(t+sqrt(x+1.0)))); 83*24586Szliu } 84