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Annotation of 43BSD/usr.lib/libm/log.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[] = "@(#)log.c      4.5 (Berkeley) 8/21/85";
        !            16: #endif not lint
        !            17: 
        !            18: /* LOG(X)
        !            19:  * RETURN THE LOGARITHM OF x 
        !            20:  * DOUBLE PRECISION (VAX D FORMAT 56 bits or IEEE DOUBLE 53 BITS)
        !            21:  * CODED IN C BY K.C. NG, 1/19/85;
        !            22:  * REVISED BY K.C. NG on 2/7/85, 3/7/85, 3/24/85, 4/16/85.
        !            23:  *
        !            24:  * Required system supported functions:
        !            25:  *     scalb(x,n)
        !            26:  *     copysign(x,y)
        !            27:  *     logb(x) 
        !            28:  *     finite(x)
        !            29:  *
        !            30:  * Required kernel function:
        !            31:  *     log__L(z) 
        !            32:  *
        !            33:  * Method :
        !            34:  *     1. Argument Reduction: find k and f such that 
        !            35:  *                     x = 2^k * (1+f), 
        !            36:  *        where  sqrt(2)/2 < 1+f < sqrt(2) .
        !            37:  *
        !            38:  *     2. Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
        !            39:  *              = 2s + 2/3 s**3 + 2/5 s**5 + .....,
        !            40:  *        log(1+f) is computed by
        !            41:  *
        !            42:  *                     log(1+f) = 2s + s*log__L(s*s)
        !            43:  *        where
        !            44:  *             log__L(z) = z*(L1 + z*(L2 + z*(... (L6 + z*L7)...)))
        !            45:  *
        !            46:  *        See log__L() for the values of the coefficients.
        !            47:  *
        !            48:  *     3. Finally,  log(x) = k*ln2 + log(1+f).  (Here n*ln2 will be stored
        !            49:  *        in two floating point number: n*ln2hi + n*ln2lo, n*ln2hi is exact
        !            50:  *        since the last 20 bits of ln2hi is 0.)
        !            51:  *
        !            52:  * Special cases:
        !            53:  *     log(x) is NaN with signal if x < 0 (including -INF) ; 
        !            54:  *     log(+INF) is +INF; log(0) is -INF with signal;
        !            55:  *     log(NaN) is that NaN with no signal.
        !            56:  *
        !            57:  * Accuracy:
        !            58:  *     log(x) returns the exact log(x) nearly rounded. In a test run with
        !            59:  *     1,536,000 random arguments on a VAX, the maximum observed error was
        !            60:  *     .826 ulps (units in the last place).
        !            61:  *
        !            62:  * Constants:
        !            63:  * The hexadecimal values are the intended ones for the following constants.
        !            64:  * The decimal values may be used, provided that the compiler will convert
        !            65:  * from decimal to binary accurately enough to produce the hexadecimal values
        !            66:  * shown.
        !            67:  */
        !            68: 
        !            69: #ifdef VAX     /* VAX D format */
        !            70: #include <errno.h>
        !            71: 
        !            72: /* double static */
        !            73: /* ln2hi  =  6.9314718055829871446E-1    , Hex  2^  0   *  .B17217F7D00000 */
        !            74: /* ln2lo  =  1.6465949582897081279E-12   , Hex  2^-39   *  .E7BCD5E4F1D9CC */
        !            75: /* sqrt2  =  1.4142135623730950622E0     ; Hex  2^  1   *  .B504F333F9DE65 */
        !            76: static long     ln2hix[] = { 0x72174031, 0x0000f7d0};
        !            77: static long     ln2lox[] = { 0xbcd52ce7, 0xd9cce4f1};
        !            78: static long     sqrt2x[] = { 0x04f340b5, 0xde6533f9};
        !            79: #define    ln2hi    (*(double*)ln2hix)
        !            80: #define    ln2lo    (*(double*)ln2lox)
        !            81: #define    sqrt2    (*(double*)sqrt2x)
        !            82: #else  /* IEEE double */
        !            83: double static
        !            84: ln2hi  =  6.9314718036912381649E-1    , /*Hex  2^ -1   *  1.62E42FEE00000 */
        !            85: ln2lo  =  1.9082149292705877000E-10   , /*Hex  2^-33   *  1.A39EF35793C76 */
        !            86: sqrt2  =  1.4142135623730951455E0     ; /*Hex  2^  0   *  1.6A09E667F3BCD */
        !            87: #endif
        !            88: 
        !            89: double log(x)
        !            90: double x;
        !            91: {
        !            92:        static double zero=0.0, negone= -1.0, half=1.0/2.0;
        !            93:        double logb(),scalb(),copysign(),log__L(),s,z,t;
        !            94:        int k,n,finite();
        !            95: 
        !            96: #ifndef VAX
        !            97:        if(x!=x) return(x);     /* x is NaN */
        !            98: #endif
        !            99:        if(finite(x)) {
        !           100:           if( x > zero ) {
        !           101: 
        !           102:           /* argument reduction */
        !           103:              k=logb(x);   x=scalb(x,-k);
        !           104:              if(k == -1022) /* subnormal no. */
        !           105:                   {n=logb(x); x=scalb(x,-n); k+=n;} 
        !           106:              if(x >= sqrt2 ) {k += 1; x *= half;}
        !           107:              x += negone ;
        !           108: 
        !           109:           /* compute log(1+x)  */
        !           110:               s=x/(2+x); t=x*x*half;
        !           111:              z=k*ln2lo+s*(t+log__L(s*s));
        !           112:              x += (z - t) ;
        !           113: 
        !           114:              return(k*ln2hi+x);
        !           115:           }
        !           116:        /* end of if (x > zero) */
        !           117: 
        !           118:           else {
        !           119: #ifdef VAX
        !           120:                extern double infnan();
        !           121:                if ( x == zero )
        !           122:                    return (infnan(-ERANGE));   /* -INF */
        !           123:                else
        !           124:                    return (infnan(EDOM));      /* NaN */
        !           125: #else  /* IEEE double */
        !           126:                /* zero argument, return -INF with signal */
        !           127:                if ( x == zero )
        !           128:                    return( negone/zero );
        !           129: 
        !           130:                /* negative argument, return NaN with signal */
        !           131:                else 
        !           132:                    return ( zero / zero );
        !           133: #endif
        !           134:            }
        !           135:        }
        !           136:     /* end of if (finite(x)) */
        !           137:     /* NOT REACHED ifdef VAX */
        !           138: 
        !           139:     /* log(-INF) is NaN with signal */
        !           140:        else if (x<0) 
        !           141:            return(zero/zero);      
        !           142: 
        !           143:     /* log(+INF) is +INF */
        !           144:        else return(x);      
        !           145: 
        !           146: }

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