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Annotation of 43BSD/usr.lib/libm/acosh.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[] = "@(#)acosh.c    1.2 (Berkeley) 8/21/85";
                     16: #endif not lint
                     17: 
                     18: /* ACOSH(X)
                     19:  * RETURN THE INVERSE HYPERBOLIC COSINE 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 3/6/85, 3/24/85, 4/16/85, 8/17/85.
                     23:  *
                     24:  * Required system supported functions :
                     25:  *     sqrt(x)
                     26:  *
                     27:  * Required kernel function:
                     28:  *     log1p(x)                ...return log(1+x)
                     29:  *
                     30:  * Method :
                     31:  *     Based on 
                     32:  *             acosh(x) = log [ x + sqrt(x*x-1) ]
                     33:  *     we have
                     34:  *             acosh(x) := log1p(x)+ln2,       if (x > 1.0E20); else           
                     35:  *             acosh(x) := log1p( sqrt(x-1) * (sqrt(x-1) + sqrt(x+1)) ) .
                     36:  *     These formulae avoid the over/underflow complication.
                     37:  *
                     38:  * Special cases:
                     39:  *     acosh(x) is NaN with signal if x<1.
                     40:  *     acosh(NaN) is NaN without signal.
                     41:  *
                     42:  * Accuracy:
                     43:  *     acosh(x) returns the exact inverse hyperbolic cosine of x nearly 
                     44:  *     rounded. In a test run with 512,000 random arguments on a VAX, the
                     45:  *     maximum observed error was 3.30 ulps (units of the last place) at
                     46:  *     x=1.0070493753568216 .
                     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: #ifdef VAX     /* VAX D format */
                     56: /* static double */
                     57: /* ln2hi  =  6.9314718055829871446E-1    , Hex  2^  0   *  .B17217F7D00000 */
                     58: /* ln2lo  =  1.6465949582897081279E-12   ; Hex  2^-39   *  .E7BCD5E4F1D9CC */
                     59: static long     ln2hix[] = { 0x72174031, 0x0000f7d0};
                     60: static long     ln2lox[] = { 0xbcd52ce7, 0xd9cce4f1};
                     61: #define    ln2hi    (*(double*)ln2hix)
                     62: #define    ln2lo    (*(double*)ln2lox)
                     63: #else  /* IEEE double */
                     64: static double
                     65: ln2hi  =  6.9314718036912381649E-1    , /*Hex  2^ -1   *  1.62E42FEE00000 */
                     66: ln2lo  =  1.9082149292705877000E-10   ; /*Hex  2^-33   *  1.A39EF35793C76 */
                     67: #endif
                     68: 
                     69: double acosh(x)
                     70: double x;
                     71: {      
                     72:        double log1p(),sqrt(),t,big=1.E20; /* big+1==big */
                     73: 
                     74: #ifndef VAX
                     75:        if(x!=x) return(x);     /* x is NaN */
                     76: #endif
                     77: 
                     78:     /* return log1p(x) + log(2) if x is large */
                     79:        if(x>big) {t=log1p(x)+ln2lo; return(t+ln2hi);} 
                     80: 
                     81:        t=sqrt(x-1.0);
                     82:        return(log1p(t*(t+sqrt(x+1.0))));
                     83: }

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