134124Sbostic /* 224588Szliu * Copyright (c) 1985 Regents of the University of California. 334124Sbostic * All rights reserved. 434124Sbostic * 534124Sbostic * Redistribution and use in source and binary forms are permitted 6*34931Sbostic * provided that the above copyright notice and this paragraph are 7*34931Sbostic * duplicated in all such forms and that any documentation, 8*34931Sbostic * advertising materials, and other materials related to such 9*34931Sbostic * distribution and use acknowledge that the software was developed 10*34931Sbostic * by the University of California, Berkeley. The name of the 11*34931Sbostic * University may not be used to endorse or promote products derived 12*34931Sbostic * from this software without specific prior written permission. 13*34931Sbostic * THIS SOFTWARE IS PROVIDED ``AS IS'' AND WITHOUT ANY EXPRESS OR 14*34931Sbostic * IMPLIED WARRANTIES, INCLUDING, WITHOUT LIMITATION, THE IMPLIED 15*34931Sbostic * WARRANTIES OF MERCHANTIBILITY AND FITNESS FOR A PARTICULAR PURPOSE. 1634124Sbostic * 1734124Sbostic * All recipients should regard themselves as participants in an ongoing 1834124Sbostic * research project and hence should feel obligated to report their 1934124Sbostic * experiences (good or bad) with these elementary function codes, using 2034124Sbostic * the sendbug(8) program, to the authors. 2124588Szliu */ 2224588Szliu 2324588Szliu #ifndef lint 24*34931Sbostic static char sccsid[] = "@(#)asinh.c 5.3 (Berkeley) 06/30/88"; 2534124Sbostic #endif /* not lint */ 2624588Szliu 2724588Szliu /* ASINH(X) 2824588Szliu * RETURN THE INVERSE HYPERBOLIC SINE OF X 2924588Szliu * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS) 3024588Szliu * CODED IN C BY K.C. NG, 2/16/85; 3124588Szliu * REVISED BY K.C. NG on 3/7/85, 3/24/85, 4/16/85. 3224588Szliu * 3324588Szliu * Required system supported functions : 3424588Szliu * copysign(x,y) 3524588Szliu * sqrt(x) 3624588Szliu * 3724588Szliu * Required kernel function: 3824588Szliu * log1p(x) ...return log(1+x) 3924588Szliu * 4024588Szliu * Method : 4124588Szliu * Based on 4224588Szliu * asinh(x) = sign(x) * log [ |x| + sqrt(x*x+1) ] 4324588Szliu * we have 4424588Szliu * asinh(x) := x if 1+x*x=1, 4524588Szliu * := sign(x)*(log1p(x)+ln2)) if sqrt(1+x*x)=x, else 4624588Szliu * := sign(x)*log1p(|x| + |x|/(1/|x| + sqrt(1+(1/|x|)^2)) ) 4724588Szliu * 4824588Szliu * Accuracy: 4924588Szliu * asinh(x) returns the exact inverse hyperbolic sine of x nearly rounded. 5024588Szliu * In a test run with 52,000 random arguments on a VAX, the maximum 5124588Szliu * observed error was 1.58 ulps (units in the last place). 5224588Szliu * 5324588Szliu * Constants: 5424588Szliu * The hexadecimal values are the intended ones for the following constants. 5524588Szliu * The decimal values may be used, provided that the compiler will convert 5624588Szliu * from decimal to binary accurately enough to produce the hexadecimal values 5724588Szliu * shown. 5824588Szliu */ 5924588Szliu 6031853Szliu #if defined(vax)||defined(tahoe) /* VAX/TAHOE D format */ 6131853Szliu #ifdef vax 6231812Szliu #define _0x(A,B) 0x/**/A/**/B 6331853Szliu #else /* vax */ 6431812Szliu #define _0x(A,B) 0x/**/B/**/A 6531853Szliu #endif /* vax */ 6624588Szliu /* static double */ 6724588Szliu /* ln2hi = 6.9314718055829871446E-1 , Hex 2^ 0 * .B17217F7D00000 */ 6824588Szliu /* ln2lo = 1.6465949582897081279E-12 ; Hex 2^-39 * .E7BCD5E4F1D9CC */ 6931812Szliu static long ln2hix[] = { _0x(7217,4031), _0x(0000,f7d0)}; 7031812Szliu static long ln2lox[] = { _0x(bcd5,2ce7), _0x(d9cc,e4f1)}; 7124588Szliu #define ln2hi (*(double*)ln2hix) 7224588Szliu #define ln2lo (*(double*)ln2lox) 7331853Szliu #else /* defined(vax)||defined(tahoe) */ 7424588Szliu static double 7524588Szliu ln2hi = 6.9314718036912381649E-1 , /*Hex 2^ -1 * 1.62E42FEE00000 */ 7624588Szliu ln2lo = 1.9082149292705877000E-10 ; /*Hex 2^-33 * 1.A39EF35793C76 */ 7731853Szliu #endif /* defined(vax)||defined(tahoe) */ 7824588Szliu 7924588Szliu double asinh(x) 8024588Szliu double x; 8124588Szliu { 8224588Szliu double copysign(),log1p(),sqrt(),t,s; 8324588Szliu static double small=1.0E-10, /* fl(1+small*small) == 1 */ 8424588Szliu big =1.0E20, /* fl(1+big) == big */ 8524588Szliu one =1.0 ; 8624588Szliu 8731853Szliu #if !defined(vax)&&!defined(tahoe) 8824588Szliu if(x!=x) return(x); /* x is NaN */ 8931853Szliu #endif /* !defined(vax)&&!defined(tahoe) */ 9024588Szliu if((t=copysign(x,one))>small) 9124588Szliu if(t<big) { 9224588Szliu s=one/t; return(copysign(log1p(t+t/(s+sqrt(one+s*s))),x)); } 9324588Szliu else /* if |x| > big */ 9424588Szliu {s=log1p(t)+ln2lo; return(copysign(s+ln2hi,x));} 9524588Szliu else /* if |x| < small */ 9624588Szliu return(x); 9724588Szliu } 98