1*24588Szliu /* 2*24588Szliu * Copyright (c) 1985 Regents of the University of California. 3*24588Szliu * 4*24588Szliu * Use and reproduction of this software are granted in accordance with 5*24588Szliu * the terms and conditions specified in the Berkeley Software License 6*24588Szliu * Agreement (in particular, this entails acknowledgement of the programs' 7*24588Szliu * source, and inclusion of this notice) with the additional understanding 8*24588Szliu * that all recipients should regard themselves as participants in an 9*24588Szliu * ongoing research project and hence should feel obligated to report 10*24588Szliu * their experiences (good or bad) with these elementary function codes, 11*24588Szliu * using "sendbug 4bsd-bugs@BERKELEY", to the authors. 12*24588Szliu */ 13*24588Szliu 14*24588Szliu #ifndef lint 15*24588Szliu static char sccsid[] = "@(#)asinh.c 1.1 (ELEFUNT) 09/06/85"; 16*24588Szliu #endif not lint 17*24588Szliu 18*24588Szliu /* ASINH(X) 19*24588Szliu * RETURN THE INVERSE HYPERBOLIC SINE OF X 20*24588Szliu * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS) 21*24588Szliu * CODED IN C BY K.C. NG, 2/16/85; 22*24588Szliu * REVISED BY K.C. NG on 3/7/85, 3/24/85, 4/16/85. 23*24588Szliu * 24*24588Szliu * Required system supported functions : 25*24588Szliu * copysign(x,y) 26*24588Szliu * sqrt(x) 27*24588Szliu * 28*24588Szliu * Required kernel function: 29*24588Szliu * log1p(x) ...return log(1+x) 30*24588Szliu * 31*24588Szliu * Method : 32*24588Szliu * Based on 33*24588Szliu * asinh(x) = sign(x) * log [ |x| + sqrt(x*x+1) ] 34*24588Szliu * we have 35*24588Szliu * asinh(x) := x if 1+x*x=1, 36*24588Szliu * := sign(x)*(log1p(x)+ln2)) if sqrt(1+x*x)=x, else 37*24588Szliu * := sign(x)*log1p(|x| + |x|/(1/|x| + sqrt(1+(1/|x|)^2)) ) 38*24588Szliu * 39*24588Szliu * Accuracy: 40*24588Szliu * asinh(x) returns the exact inverse hyperbolic sine of x nearly rounded. 41*24588Szliu * In a test run with 52,000 random arguments on a VAX, the maximum 42*24588Szliu * observed error was 1.58 ulps (units in the last place). 43*24588Szliu * 44*24588Szliu * Constants: 45*24588Szliu * The hexadecimal values are the intended ones for the following constants. 46*24588Szliu * The decimal values may be used, provided that the compiler will convert 47*24588Szliu * from decimal to binary accurately enough to produce the hexadecimal values 48*24588Szliu * shown. 49*24588Szliu */ 50*24588Szliu 51*24588Szliu #ifdef VAX /* VAX D format */ 52*24588Szliu /* static double */ 53*24588Szliu /* ln2hi = 6.9314718055829871446E-1 , Hex 2^ 0 * .B17217F7D00000 */ 54*24588Szliu /* ln2lo = 1.6465949582897081279E-12 ; Hex 2^-39 * .E7BCD5E4F1D9CC */ 55*24588Szliu static long ln2hix[] = { 0x72174031, 0x0000f7d0}; 56*24588Szliu static long ln2lox[] = { 0xbcd52ce7, 0xd9cce4f1}; 57*24588Szliu #define ln2hi (*(double*)ln2hix) 58*24588Szliu #define ln2lo (*(double*)ln2lox) 59*24588Szliu #else /* IEEE double */ 60*24588Szliu static double 61*24588Szliu ln2hi = 6.9314718036912381649E-1 , /*Hex 2^ -1 * 1.62E42FEE00000 */ 62*24588Szliu ln2lo = 1.9082149292705877000E-10 ; /*Hex 2^-33 * 1.A39EF35793C76 */ 63*24588Szliu #endif 64*24588Szliu 65*24588Szliu double asinh(x) 66*24588Szliu double x; 67*24588Szliu { 68*24588Szliu double copysign(),log1p(),sqrt(),t,s; 69*24588Szliu static double small=1.0E-10, /* fl(1+small*small) == 1 */ 70*24588Szliu big =1.0E20, /* fl(1+big) == big */ 71*24588Szliu one =1.0 ; 72*24588Szliu 73*24588Szliu #ifndef VAX 74*24588Szliu if(x!=x) return(x); /* x is NaN */ 75*24588Szliu #endif 76*24588Szliu if((t=copysign(x,one))>small) 77*24588Szliu if(t<big) { 78*24588Szliu s=one/t; return(copysign(log1p(t+t/(s+sqrt(one+s*s))),x)); } 79*24588Szliu else /* if |x| > big */ 80*24588Szliu {s=log1p(t)+ln2lo; return(copysign(s+ln2hi,x));} 81*24588Szliu else /* if |x| < small */ 82*24588Szliu return(x); 83*24588Szliu } 84