| 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 1.1 (ELEFUNT) %G%"; |
| 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 | } |