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Annotation of 43BSD/usr.lib/libm/asinh.c, revision 1.1.1.1

1.1       root        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) 8/21/85";
                     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: }

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