1 /* 2 * Copyright (c) 1985 Regents of the University of California. 3 * All rights reserved. 4 * 5 * Redistribution and use in source and binary forms are permitted 6 * provided that this notice is preserved and that due credit is given 7 * to the University of California at Berkeley. The name of the University 8 * may not be used to endorse or promote products derived from this 9 * software without specific prior written permission. This software 10 * is provided ``as is'' without express or implied warranty. 11 * 12 * All recipients should regard themselves as participants in an ongoing 13 * research project and hence should feel obligated to report their 14 * experiences (good or bad) with these elementary function codes, using 15 * the sendbug(8) program, to the authors. 16 */ 17 18 #ifndef lint 19 static char sccsid[] = "@(#)asinh.c 5.2 (Berkeley) 04/29/88"; 20 #endif /* not lint */ 21 22 /* ASINH(X) 23 * RETURN THE INVERSE HYPERBOLIC SINE OF X 24 * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS) 25 * CODED IN C BY K.C. NG, 2/16/85; 26 * REVISED BY K.C. NG on 3/7/85, 3/24/85, 4/16/85. 27 * 28 * Required system supported functions : 29 * copysign(x,y) 30 * sqrt(x) 31 * 32 * Required kernel function: 33 * log1p(x) ...return log(1+x) 34 * 35 * Method : 36 * Based on 37 * asinh(x) = sign(x) * log [ |x| + sqrt(x*x+1) ] 38 * we have 39 * asinh(x) := x if 1+x*x=1, 40 * := sign(x)*(log1p(x)+ln2)) if sqrt(1+x*x)=x, else 41 * := sign(x)*log1p(|x| + |x|/(1/|x| + sqrt(1+(1/|x|)^2)) ) 42 * 43 * Accuracy: 44 * asinh(x) returns the exact inverse hyperbolic sine of x nearly rounded. 45 * In a test run with 52,000 random arguments on a VAX, the maximum 46 * observed error was 1.58 ulps (units in the last place). 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 #if defined(vax)||defined(tahoe) /* VAX/TAHOE D format */ 56 #ifdef vax 57 #define _0x(A,B) 0x/**/A/**/B 58 #else /* vax */ 59 #define _0x(A,B) 0x/**/B/**/A 60 #endif /* vax */ 61 /* static double */ 62 /* ln2hi = 6.9314718055829871446E-1 , Hex 2^ 0 * .B17217F7D00000 */ 63 /* ln2lo = 1.6465949582897081279E-12 ; Hex 2^-39 * .E7BCD5E4F1D9CC */ 64 static long ln2hix[] = { _0x(7217,4031), _0x(0000,f7d0)}; 65 static long ln2lox[] = { _0x(bcd5,2ce7), _0x(d9cc,e4f1)}; 66 #define ln2hi (*(double*)ln2hix) 67 #define ln2lo (*(double*)ln2lox) 68 #else /* defined(vax)||defined(tahoe) */ 69 static double 70 ln2hi = 6.9314718036912381649E-1 , /*Hex 2^ -1 * 1.62E42FEE00000 */ 71 ln2lo = 1.9082149292705877000E-10 ; /*Hex 2^-33 * 1.A39EF35793C76 */ 72 #endif /* defined(vax)||defined(tahoe) */ 73 74 double asinh(x) 75 double x; 76 { 77 double copysign(),log1p(),sqrt(),t,s; 78 static double small=1.0E-10, /* fl(1+small*small) == 1 */ 79 big =1.0E20, /* fl(1+big) == big */ 80 one =1.0 ; 81 82 #if !defined(vax)&&!defined(tahoe) 83 if(x!=x) return(x); /* x is NaN */ 84 #endif /* !defined(vax)&&!defined(tahoe) */ 85 if((t=copysign(x,one))>small) 86 if(t<big) { 87 s=one/t; return(copysign(log1p(t+t/(s+sqrt(one+s*s))),x)); } 88 else /* if |x| > big */ 89 {s=log1p(t)+ln2lo; return(copysign(s+ln2hi,x));} 90 else /* if |x| < small */ 91 return(x); 92 } 93