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[] = "@(#)asinh.c 1.2 (Berkeley) 08/21/85"; 16 #endif not lint 17 18 /* ASINH(X) 19 * RETURN THE INVERSE HYPERBOLIC SINE OF X 20 * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS) 21 * CODED IN C BY K.C. NG, 2/16/85; 22 * REVISED BY K.C. NG on 3/7/85, 3/24/85, 4/16/85. 23 * 24 * Required system supported functions : 25 * copysign(x,y) 26 * sqrt(x) 27 * 28 * Required kernel function: 29 * log1p(x) ...return log(1+x) 30 * 31 * Method : 32 * Based on 33 * asinh(x) = sign(x) * log [ |x| + sqrt(x*x+1) ] 34 * we have 35 * asinh(x) := x if 1+x*x=1, 36 * := sign(x)*(log1p(x)+ln2)) if sqrt(1+x*x)=x, else 37 * := sign(x)*log1p(|x| + |x|/(1/|x| + sqrt(1+(1/|x|)^2)) ) 38 * 39 * Accuracy: 40 * asinh(x) returns the exact inverse hyperbolic sine of x nearly rounded. 41 * In a test run with 52,000 random arguments on a VAX, the maximum 42 * observed error was 1.58 ulps (units in the last place). 43 * 44 * Constants: 45 * The hexadecimal values are the intended ones for the following constants. 46 * The decimal values may be used, provided that the compiler will convert 47 * from decimal to binary accurately enough to produce the hexadecimal values 48 * shown. 49 */ 50 51 #ifdef VAX /* VAX D format */ 52 /* static double */ 53 /* ln2hi = 6.9314718055829871446E-1 , Hex 2^ 0 * .B17217F7D00000 */ 54 /* ln2lo = 1.6465949582897081279E-12 ; Hex 2^-39 * .E7BCD5E4F1D9CC */ 55 static long ln2hix[] = { 0x72174031, 0x0000f7d0}; 56 static long ln2lox[] = { 0xbcd52ce7, 0xd9cce4f1}; 57 #define ln2hi (*(double*)ln2hix) 58 #define ln2lo (*(double*)ln2lox) 59 #else /* IEEE double */ 60 static double 61 ln2hi = 6.9314718036912381649E-1 , /*Hex 2^ -1 * 1.62E42FEE00000 */ 62 ln2lo = 1.9082149292705877000E-10 ; /*Hex 2^-33 * 1.A39EF35793C76 */ 63 #endif 64 65 double asinh(x) 66 double x; 67 { 68 double copysign(),log1p(),sqrt(),t,s; 69 static double small=1.0E-10, /* fl(1+small*small) == 1 */ 70 big =1.0E20, /* fl(1+big) == big */ 71 one =1.0 ; 72 73 #ifndef VAX 74 if(x!=x) return(x); /* x is NaN */ 75 #endif 76 if((t=copysign(x,one))>small) 77 if(t<big) { 78 s=one/t; return(copysign(log1p(t+t/(s+sqrt(one+s*s))),x)); } 79 else /* if |x| > big */ 80 {s=log1p(t)+ln2lo; return(copysign(s+ln2hi,x));} 81 else /* if |x| < small */ 82 return(x); 83 } 84