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Annotation of 43BSD/usr.lib/libm/cosh.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[] = "@(#)cosh.c     1.2 (Berkeley) 8/21/85";
        !            16: #endif not lint
        !            17: 
        !            18: /* COSH(X)
        !            19:  * RETURN THE HYPERBOLIC COSINE OF X
        !            20:  * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS)
        !            21:  * CODED IN C BY K.C. NG, 1/8/85; 
        !            22:  * REVISED BY K.C. NG on 2/8/85, 2/23/85, 3/7/85, 3/29/85, 4/16/85.
        !            23:  *
        !            24:  * Required system supported functions :
        !            25:  *     copysign(x,y)
        !            26:  *     scalb(x,N)
        !            27:  *
        !            28:  * Required kernel function:
        !            29:  *     exp(x) 
        !            30:  *     exp__E(x,c)     ...return exp(x+c)-1-x for |x|<0.3465
        !            31:  *
        !            32:  * Method :
        !            33:  *     1. Replace x by |x|. 
        !            34:  *     2. 
        !            35:  *                                                     [ exp(x) - 1 ]^2 
        !            36:  *         0        <= x <= 0.3465  :  cosh(x) := 1 + -------------------
        !            37:  *                                                        2*exp(x)
        !            38:  *
        !            39:  *                                                exp(x) +  1/exp(x)
        !            40:  *         0.3465   <= x <= 22      :  cosh(x) := -------------------
        !            41:  *                                                        2
        !            42:  *         22       <= x <= lnovfl  :  cosh(x) := exp(x)/2 
        !            43:  *         lnovfl   <= x <= lnovfl+log(2)
        !            44:  *                                  :  cosh(x) := exp(x)/2 (avoid overflow)
        !            45:  *         log(2)+lnovfl <  x <  INF:  overflow to INF
        !            46:  *
        !            47:  *     Note: .3465 is a number near one half of ln2.
        !            48:  *
        !            49:  * Special cases:
        !            50:  *     cosh(x) is x if x is +INF, -INF, or NaN.
        !            51:  *     only cosh(0)=1 is exact for finite x.
        !            52:  *
        !            53:  * Accuracy:
        !            54:  *     cosh(x) returns the exact hyperbolic cosine of x nearly rounded.
        !            55:  *     In a test run with 768,000 random arguments on a VAX, the maximum
        !            56:  *     observed error was 1.23 ulps (units in the last place).
        !            57:  *
        !            58:  * Constants:
        !            59:  * The hexadecimal values are the intended ones for the following constants.
        !            60:  * The decimal values may be used, provided that the compiler will convert
        !            61:  * from decimal to binary accurately enough to produce the hexadecimal values
        !            62:  * shown.
        !            63:  */
        !            64: 
        !            65: #ifdef VAX
        !            66: /* double static  */
        !            67: /* mln2hi =  8.8029691931113054792E1     , Hex  2^  7   *  .B00F33C7E22BDB */
        !            68: /* mln2lo = -4.9650192275318476525E-16   , Hex  2^-50   * -.8F1B60279E582A */
        !            69: /* lnovfl =  8.8029691931113053016E1     ; Hex  2^  7   *  .B00F33C7E22BDA */
        !            70: static long    mln2hix[] = { 0x0f3343b0, 0x2bdbc7e2};
        !            71: static long    mln2lox[] = { 0x1b60a70f, 0x582a279e};
        !            72: static long    lnovflx[] = { 0x0f3343b0, 0x2bdac7e2};
        !            73: #define   mln2hi    (*(double*)mln2hix)
        !            74: #define   mln2lo    (*(double*)mln2lox)
        !            75: #define   lnovfl    (*(double*)lnovflx)
        !            76: #else  /* IEEE double */
        !            77: double static 
        !            78: mln2hi =  7.0978271289338397310E2     , /*Hex  2^ 10   *  1.62E42FEFA39EF */
        !            79: mln2lo =  2.3747039373786107478E-14   , /*Hex  2^-45   *  1.ABC9E3B39803F */
        !            80: lnovfl =  7.0978271289338397310E2     ; /*Hex  2^  9   *  1.62E42FEFA39EF */
        !            81: #endif
        !            82: 
        !            83: #ifdef VAX
        !            84: static max = 126                      ;
        !            85: #else  /* IEEE double */
        !            86: static max = 1023                     ;
        !            87: #endif
        !            88: 
        !            89: double cosh(x)
        !            90: double x;
        !            91: {      
        !            92:        static double half=1.0/2.0,one=1.0, small=1.0E-18; /* fl(1+small)==1 */
        !            93:        double scalb(),copysign(),exp(),exp__E(),t;
        !            94: 
        !            95: #ifndef VAX
        !            96:        if(x!=x) return(x);     /* x is NaN */
        !            97: #endif
        !            98:        if((x=copysign(x,one)) <= 22)
        !            99:            if(x<0.3465) 
        !           100:                if(x<small) return(one+x);
        !           101:                else {t=x+exp__E(x,0.0);x=t+t; return(one+t*t/(2.0+x)); }
        !           102: 
        !           103:            else /* for x lies in [0.3465,22] */
        !           104:                { t=exp(x); return((t+one/t)*half); }
        !           105: 
        !           106:        if( lnovfl <= x && x <= (lnovfl+0.7)) 
        !           107:         /* for x lies in [lnovfl, lnovfl+ln2], decrease x by ln(2^(max+1)) 
        !           108:          * and return 2^max*exp(x) to avoid unnecessary overflow 
        !           109:          */
        !           110:            return(scalb(exp((x-mln2hi)-mln2lo), max)); 
        !           111: 
        !           112:        else 
        !           113:            return(exp(x)*half);        /* for large x,  cosh(x)=exp(x)/2 */
        !           114: }

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