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Annotation of 43BSD/usr.lib/libm/log1p.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[] = "@(#)log1p.c    1.3 (Berkeley) 8/21/85";
        !            16: #endif not lint
        !            17: 
        !            18: /* LOG1P(x) 
        !            19:  * RETURN THE LOGARITHM OF 1+x
        !            20:  * DOUBLE PRECISION (VAX D FORMAT 56 bits, IEEE DOUBLE 53 BITS)
        !            21:  * CODED IN C BY K.C. NG, 1/19/85; 
        !            22:  * REVISED BY K.C. NG on 2/6/85, 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:  *                     1+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(1+x) = k*ln2 + log(1+f).  
        !            49:  *
        !            50:  *     Remarks 1. In step 3 n*ln2 will be stored in two floating point numbers
        !            51:  *                n*ln2hi + n*ln2lo, where ln2hi is chosen such that the last 
        !            52:  *                20 bits (for VAX D format), or the last 21 bits ( for IEEE 
        !            53:  *                double) is 0. This ensures n*ln2hi is exactly representable.
        !            54:  *             2. In step 1, f may not be representable. A correction term c
        !            55:  *                for f is computed. It follows that the correction term for
        !            56:  *                f - t (the leading term of log(1+f) in step 2) is c-c*x. We
        !            57:  *                add this correction term to n*ln2lo to attenuate the error.
        !            58:  *
        !            59:  *
        !            60:  * Special cases:
        !            61:  *     log1p(x) is NaN with signal if x < -1; log1p(NaN) is NaN with no signal;
        !            62:  *     log1p(INF) is +INF; log1p(-1) is -INF with signal;
        !            63:  *     only log1p(0)=0 is exact for finite argument.
        !            64:  *
        !            65:  * Accuracy:
        !            66:  *     log1p(x) returns the exact log(1+x) nearly rounded. In a test run 
        !            67:  *     with 1,536,000 random arguments on a VAX, the maximum observed
        !            68:  *     error was .846 ulps (units in the last place).
        !            69:  *
        !            70:  * Constants:
        !            71:  * The hexadecimal values are the intended ones for the following constants.
        !            72:  * The decimal values may be used, provided that the compiler will convert
        !            73:  * from decimal to binary accurately enough to produce the hexadecimal values
        !            74:  * shown.
        !            75:  */
        !            76: 
        !            77: #ifdef VAX     /* VAX D format */
        !            78: #include <errno.h>
        !            79: 
        !            80: /* double static */
        !            81: /* ln2hi  =  6.9314718055829871446E-1    , Hex  2^  0   *  .B17217F7D00000 */
        !            82: /* ln2lo  =  1.6465949582897081279E-12   , Hex  2^-39   *  .E7BCD5E4F1D9CC */
        !            83: /* sqrt2  =  1.4142135623730950622E0     ; Hex  2^  1   *  .B504F333F9DE65 */
        !            84: static long     ln2hix[] = { 0x72174031, 0x0000f7d0};
        !            85: static long     ln2lox[] = { 0xbcd52ce7, 0xd9cce4f1};
        !            86: static long     sqrt2x[] = { 0x04f340b5, 0xde6533f9};
        !            87: #define    ln2hi    (*(double*)ln2hix)
        !            88: #define    ln2lo    (*(double*)ln2lox)
        !            89: #define    sqrt2    (*(double*)sqrt2x)
        !            90: #else  /* IEEE double */
        !            91: double static
        !            92: ln2hi  =  6.9314718036912381649E-1    , /*Hex  2^ -1   *  1.62E42FEE00000 */
        !            93: ln2lo  =  1.9082149292705877000E-10   , /*Hex  2^-33   *  1.A39EF35793C76 */
        !            94: sqrt2  =  1.4142135623730951455E0     ; /*Hex  2^  0   *  1.6A09E667F3BCD */
        !            95: #endif
        !            96: 
        !            97: double log1p(x)
        !            98: double x;
        !            99: {
        !           100:        static double zero=0.0, negone= -1.0, one=1.0, 
        !           101:                      half=1.0/2.0, small=1.0E-20;   /* 1+small == 1 */
        !           102:        double logb(),copysign(),scalb(),log__L(),z,s,t,c;
        !           103:        int k,finite();
        !           104: 
        !           105: #ifndef VAX
        !           106:        if(x!=x) return(x);     /* x is NaN */
        !           107: #endif
        !           108: 
        !           109:        if(finite(x)) {
        !           110:           if( x > negone ) {
        !           111: 
        !           112:           /* argument reduction */
        !           113:              if(copysign(x,one)<small) return(x);
        !           114:              k=logb(one+x); z=scalb(x,-k); t=scalb(one,-k);
        !           115:              if(z+t >= sqrt2 ) 
        !           116:                  { k += 1 ; z *= half; t *= half; }
        !           117:              t += negone; x = z + t;
        !           118:              c = (t-x)+z ;             /* correction term for x */
        !           119: 
        !           120:           /* compute log(1+x)  */
        !           121:               s = x/(2+x); t = x*x*half;
        !           122:              c += (k*ln2lo-c*x);
        !           123:              z = c+s*(t+log__L(s*s));
        !           124:              x += (z - t) ;
        !           125: 
        !           126:              return(k*ln2hi+x);
        !           127:           }
        !           128:        /* end of if (x > negone) */
        !           129: 
        !           130:            else {
        !           131: #ifdef VAX
        !           132:                extern double infnan();
        !           133:                if ( x == negone )
        !           134:                    return (infnan(-ERANGE));   /* -INF */
        !           135:                else
        !           136:                    return (infnan(EDOM));      /* NaN */
        !           137: #else  /* IEEE double */
        !           138:                /* x = -1, return -INF with signal */
        !           139:                if ( x == negone ) return( negone/zero );
        !           140: 
        !           141:                /* negative argument for log, return NaN with signal */
        !           142:                else return ( zero / zero );
        !           143: #endif
        !           144:            }
        !           145:        }
        !           146:     /* end of if (finite(x)) */
        !           147: 
        !           148:     /* log(-INF) is NaN */
        !           149:        else if(x<0) 
        !           150:             return(zero/zero);
        !           151: 
        !           152:     /* log(+INF) is INF */
        !           153:        else return(x);      
        !           154: }

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