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[] = 16 "@(#)acosh.c 1.2 (Berkeley) 8/21/85; 1.2 (ucb.elefunt) 09/11/85"; 17 #endif not lint 18 19 /* ACOSH(X) 20 * RETURN THE INVERSE HYPERBOLIC COSINE OF X 21 * DOUBLE PRECISION (VAX D FORMAT 56 BITS, IEEE DOUBLE 53 BITS) 22 * CODED IN C BY K.C. NG, 2/16/85; 23 * REVISED BY K.C. NG on 3/6/85, 3/24/85, 4/16/85, 8/17/85. 24 * 25 * Required system supported functions : 26 * sqrt(x) 27 * 28 * Required kernel function: 29 * log1p(x) ...return log(1+x) 30 * 31 * Method : 32 * Based on 33 * acosh(x) = log [ x + sqrt(x*x-1) ] 34 * we have 35 * acosh(x) := log1p(x)+ln2, if (x > 1.0E20); else 36 * acosh(x) := log1p( sqrt(x-1) * (sqrt(x-1) + sqrt(x+1)) ) . 37 * These formulae avoid the over/underflow complication. 38 * 39 * Special cases: 40 * acosh(x) is NaN with signal if x<1. 41 * acosh(NaN) is NaN without signal. 42 * 43 * Accuracy: 44 * acosh(x) returns the exact inverse hyperbolic cosine of x nearly 45 * rounded. In a test run with 512,000 random arguments on a VAX, the 46 * maximum observed error was 3.30 ulps (units of the last place) at 47 * x=1.0070493753568216 . 48 * 49 * Constants: 50 * The hexadecimal values are the intended ones for the following constants. 51 * The decimal values may be used, provided that the compiler will convert 52 * from decimal to binary accurately enough to produce the hexadecimal values 53 * shown. 54 */ 55 56 #ifdef VAX /* VAX D format */ 57 /* static double */ 58 /* ln2hi = 6.9314718055829871446E-1 , Hex 2^ 0 * .B17217F7D00000 */ 59 /* ln2lo = 1.6465949582897081279E-12 ; Hex 2^-39 * .E7BCD5E4F1D9CC */ 60 static long ln2hix[] = { 0x72174031, 0x0000f7d0}; 61 static long ln2lox[] = { 0xbcd52ce7, 0xd9cce4f1}; 62 #define ln2hi (*(double*)ln2hix) 63 #define ln2lo (*(double*)ln2lox) 64 #else /* IEEE double */ 65 static double 66 ln2hi = 6.9314718036912381649E-1 , /*Hex 2^ -1 * 1.62E42FEE00000 */ 67 ln2lo = 1.9082149292705877000E-10 ; /*Hex 2^-33 * 1.A39EF35793C76 */ 68 #endif 69 70 double acosh(x) 71 double x; 72 { 73 double log1p(),sqrt(),t,big=1.E20; /* big+1==big */ 74 75 #ifndef VAX 76 if(x!=x) return(x); /* x is NaN */ 77 #endif 78 79 /* return log1p(x) + log(2) if x is large */ 80 if(x>big) {t=log1p(x)+ln2lo; return(t+ln2hi);} 81 82 t=sqrt(x-1.0); 83 return(log1p(t*(t+sqrt(x+1.0)))); 84 } 85