1 /* 2 * Copyright (c) 1985 Regents of the University of California. 3 * 4 * Use and reproduction of this software are granted in accordance with 5 * the terms and conditions specified in the Berkeley Software License 6 * Agreement (in particular, this entails acknowledgement of the programs' 7 * source, and inclusion of this notice) with the additional understanding 8 * that all recipients should regard themselves as participants in an 9 * ongoing research project and hence should feel obligated to report 10 * their experiences (good or bad) with these elementary function codes, 11 * using "sendbug 4bsd-bugs@BERKELEY", to the authors. 12 */ 13 14 #ifndef lint 15 static char sccsid[] = "@(#)acosh.c 1.2 (Berkeley) 08/21/85"; 16 #endif not lint 17 18 /* ACOSH(X) 19 * RETURN THE INVERSE HYPERBOLIC COSINE OF X 20 * DOUBLE PRECISION (VAX D FORMAT 56 BITS, IEEE DOUBLE 53 BITS) 21 * CODED IN C BY K.C. NG, 2/16/85; 22 * REVISED BY K.C. NG on 3/6/85, 3/24/85, 4/16/85, 8/17/85. 23 * 24 * Required system supported functions : 25 * sqrt(x) 26 * 27 * Required kernel function: 28 * log1p(x) ...return log(1+x) 29 * 30 * Method : 31 * Based on 32 * acosh(x) = log [ x + sqrt(x*x-1) ] 33 * we have 34 * acosh(x) := log1p(x)+ln2, if (x > 1.0E20); else 35 * acosh(x) := log1p( sqrt(x-1) * (sqrt(x-1) + sqrt(x+1)) ) . 36 * These formulae avoid the over/underflow complication. 37 * 38 * Special cases: 39 * acosh(x) is NaN with signal if x<1. 40 * acosh(NaN) is NaN without signal. 41 * 42 * Accuracy: 43 * acosh(x) returns the exact inverse hyperbolic cosine of x nearly 44 * rounded. In a test run with 512,000 random arguments on a VAX, the 45 * maximum observed error was 3.30 ulps (units of the last place) at 46 * x=1.0070493753568216 . 47 * 48 * Constants: 49 * The hexadecimal values are the intended ones for the following constants. 50 * The decimal values may be used, provided that the compiler will convert 51 * from decimal to binary accurately enough to produce the hexadecimal values 52 * shown. 53 */ 54 55 #ifdef VAX /* VAX D format */ 56 /* static double */ 57 /* ln2hi = 6.9314718055829871446E-1 , Hex 2^ 0 * .B17217F7D00000 */ 58 /* ln2lo = 1.6465949582897081279E-12 ; Hex 2^-39 * .E7BCD5E4F1D9CC */ 59 static long ln2hix[] = { 0x72174031, 0x0000f7d0}; 60 static long ln2lox[] = { 0xbcd52ce7, 0xd9cce4f1}; 61 #define ln2hi (*(double*)ln2hix) 62 #define ln2lo (*(double*)ln2lox) 63 #else /* IEEE double */ 64 static double 65 ln2hi = 6.9314718036912381649E-1 , /*Hex 2^ -1 * 1.62E42FEE00000 */ 66 ln2lo = 1.9082149292705877000E-10 ; /*Hex 2^-33 * 1.A39EF35793C76 */ 67 #endif 68 69 double acosh(x) 70 double x; 71 { 72 double log1p(),sqrt(),t,big=1.E20; /* big+1==big */ 73 74 #ifndef VAX 75 if(x!=x) return(x); /* x is NaN */ 76 #endif 77 78 /* return log1p(x) + log(2) if x is large */ 79 if(x>big) {t=log1p(x)+ln2lo; return(t+ln2hi);} 80 81 t=sqrt(x-1.0); 82 return(log1p(t*(t+sqrt(x+1.0)))); 83 } 84