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Annotation of 43BSD/usr.lib/libm/cosh.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[] = "@(#)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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