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