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 "@(#)asinh.c 1.2 (Berkeley) 8/21/85; 1.2 (ucb.elefunt) 09/11/85"; 17 #endif not lint 18 19 /* ASINH(X) 20 * RETURN THE INVERSE HYPERBOLIC SINE 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 3/7/85, 3/24/85, 4/16/85. 24 * 25 * Required system supported functions : 26 * copysign(x,y) 27 * sqrt(x) 28 * 29 * Required kernel function: 30 * log1p(x) ...return log(1+x) 31 * 32 * Method : 33 * Based on 34 * asinh(x) = sign(x) * log [ |x| + sqrt(x*x+1) ] 35 * we have 36 * asinh(x) := x if 1+x*x=1, 37 * := sign(x)*(log1p(x)+ln2)) if sqrt(1+x*x)=x, else 38 * := sign(x)*log1p(|x| + |x|/(1/|x| + sqrt(1+(1/|x|)^2)) ) 39 * 40 * Accuracy: 41 * asinh(x) returns the exact inverse hyperbolic sine of x nearly rounded. 42 * In a test run with 52,000 random arguments on a VAX, the maximum 43 * observed error was 1.58 ulps (units in the last place). 44 * 45 * Constants: 46 * The hexadecimal values are the intended ones for the following constants. 47 * The decimal values may be used, provided that the compiler will convert 48 * from decimal to binary accurately enough to produce the hexadecimal values 49 * shown. 50 */ 51 52 #ifdef VAX /* VAX D format */ 53 /* static double */ 54 /* ln2hi = 6.9314718055829871446E-1 , Hex 2^ 0 * .B17217F7D00000 */ 55 /* ln2lo = 1.6465949582897081279E-12 ; Hex 2^-39 * .E7BCD5E4F1D9CC */ 56 static long ln2hix[] = { 0x72174031, 0x0000f7d0}; 57 static long ln2lox[] = { 0xbcd52ce7, 0xd9cce4f1}; 58 #define ln2hi (*(double*)ln2hix) 59 #define ln2lo (*(double*)ln2lox) 60 #else /* IEEE double */ 61 static double 62 ln2hi = 6.9314718036912381649E-1 , /*Hex 2^ -1 * 1.62E42FEE00000 */ 63 ln2lo = 1.9082149292705877000E-10 ; /*Hex 2^-33 * 1.A39EF35793C76 */ 64 #endif 65 66 double asinh(x) 67 double x; 68 { 69 double copysign(),log1p(),sqrt(),t,s; 70 static double small=1.0E-10, /* fl(1+small*small) == 1 */ 71 big =1.0E20, /* fl(1+big) == big */ 72 one =1.0 ; 73 74 #ifndef VAX 75 if(x!=x) return(x); /* x is NaN */ 76 #endif 77 if((t=copysign(x,one))>small) 78 if(t<big) { 79 s=one/t; return(copysign(log1p(t+t/(s+sqrt(one+s*s))),x)); } 80 else /* if |x| > big */ 81 {s=log1p(t)+ln2lo; return(copysign(s+ln2hi,x));} 82 else /* if |x| < small */ 83 return(x); 84 } 85