usr/src/lib/libm/common_source/log1p.c

/*
 * Copyright (c) 1985 Regents of the University of California.
 * All rights reserved.
 *
 * Redistribution and use in source and binary forms are permitted
 * provided that this notice is preserved and that due credit is given
 * to the University of California at Berkeley. The name of the University
 * may not be used to endorse or promote products derived from this
 * software without specific prior written permission. This software
 * is provided ``as is'' without express or implied warranty.
 *
 * All recipients should regard themselves as participants in an ongoing
 * research project and hence should feel obligated to report their
 * experiences (good or bad) with these elementary function codes, using
 * the sendbug(8) program, to the authors.
 */

#ifndef lint
static char sccsid[] = "@(#)log1p.c     5.2 (Berkeley) %G%";
#endif /* not lint */

/* LOG1P(x)
 * RETURN THE LOGARITHM OF 1+x
 * DOUBLE PRECISION (VAX D FORMAT 56 bits, IEEE DOUBLE 53 BITS)
 * CODED IN C BY K.C. NG, 1/19/85;
 * REVISED BY K.C. NG on 2/6/85, 3/7/85, 3/24/85, 4/16/85.
 *
 * Required system supported functions:
 *      scalb(x,n)
 *      copysign(x,y)
 *      logb(x)
 *      finite(x)
 *
 * Required kernel function:
 *      log__L(z)
 *
 * Method :
 *      1. Argument Reduction: find k and f such that
 *                      1+x  = 2^k * (1+f),
 *         where  sqrt(2)/2 < 1+f < sqrt(2) .
 *
 *      2. Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
 *               = 2s + 2/3 s**3 + 2/5 s**5 + .....,
 *         log(1+f) is computed by
 *
 *                      log(1+f) = 2s + s*log__L(s*s)
 *         where
 *              log__L(z) = z*(L1 + z*(L2 + z*(... (L6 + z*L7)...)))
 *
 *         See log__L() for the values of the coefficients.
 *
 *      3. Finally,  log(1+x) = k*ln2 + log(1+f).
 *
 *      Remarks 1. In step 3 n*ln2 will be stored in two floating point numbers
 *                 n*ln2hi + n*ln2lo, where ln2hi is chosen such that the last
 *                 20 bits (for VAX D format), or the last 21 bits ( for IEEE
 *                 double) is 0. This ensures n*ln2hi is exactly representable.
 *              2. In step 1, f may not be representable. A correction term c
 *                 for f is computed. It follows that the correction term for
 *                 f - t (the leading term of log(1+f) in step 2) is c-c*x. We
 *                 add this correction term to n*ln2lo to attenuate the error.
 *
 *
 * Special cases:
 *      log1p(x) is NaN with signal if x < -1; log1p(NaN) is NaN with no signal;
 *      log1p(INF) is +INF; log1p(-1) is -INF with signal;
 *      only log1p(0)=0 is exact for finite argument.
 *
 * Accuracy:
 *      log1p(x) returns the exact log(1+x) nearly rounded. In a test run
 *      with 1,536,000 random arguments on a VAX, the maximum observed
 *      error was .846 ulps (units in the last place).
 *
 * Constants:
 * The hexadecimal values are the intended ones for the following constants.
 * The decimal values may be used, provided that the compiler will convert
 * from decimal to binary accurately enough to produce the hexadecimal values
 * shown.
 */

#if defined(vax)||defined(tahoe)        /* VAX D format */
#include <errno.h>
#ifdef vax
#define _0x(A,B)        0x/**/A/**/B
#else   /* vax */
#define _0x(A,B)        0x/**/B/**/A
#endif  /* vax */
/* static double */
/* ln2hi  =  6.9314718055829871446E-1    , Hex  2^  0   *  .B17217F7D00000 */
/* ln2lo  =  1.6465949582897081279E-12   , Hex  2^-39   *  .E7BCD5E4F1D9CC */
/* sqrt2  =  1.4142135623730950622E0     ; Hex  2^  1   *  .B504F333F9DE65 */
static long     ln2hix[] = { _0x(7217,4031), _0x(0000,f7d0)};
static long     ln2lox[] = { _0x(bcd5,2ce7), _0x(d9cc,e4f1)};
static long     sqrt2x[] = { _0x(04f3,40b5), _0x(de65,33f9)};
#define    ln2hi    (*(double*)ln2hix)
#define    ln2lo    (*(double*)ln2lox)
#define    sqrt2    (*(double*)sqrt2x)
#else   /* defined(vax)||defined(tahoe) */
static double
ln2hi  =  6.9314718036912381649E-1    , /*Hex  2^ -1   *  1.62E42FEE00000 */
ln2lo  =  1.9082149292705877000E-10   , /*Hex  2^-33   *  1.A39EF35793C76 */
sqrt2  =  1.4142135623730951455E0     ; /*Hex  2^  0   *  1.6A09E667F3BCD */
#endif  /* defined(vax)||defined(tahoe) */

double log1p(x)
double x;
{
        static double zero=0.0, negone= -1.0, one=1.0,
                      half=1.0/2.0, small=1.0E-20;   /* 1+small == 1 */
        double logb(),copysign(),scalb(),log__L(),z,s,t,c;
        int k,finite();

#if !defined(vax)&&!defined(tahoe)
        if(x!=x) return(x);     /* x is NaN */
#endif  /* !defined(vax)&&!defined(tahoe) */

        if(finite(x)) {
           if( x > negone ) {

           /* argument reduction */
              if(copysign(x,one)<small) return(x);
              k=logb(one+x); z=scalb(x,-k); t=scalb(one,-k);
              if(z+t >= sqrt2 )
                  { k += 1 ; z *= half; t *= half; }
              t += negone; x = z + t;
              c = (t-x)+z ;             /* correction term for x */

           /* compute log(1+x)  */
              s = x/(2+x); t = x*x*half;
              c += (k*ln2lo-c*x);
              z = c+s*(t+log__L(s*s));
              x += (z - t) ;

              return(k*ln2hi+x);
           }
        /* end of if (x > negone) */

            else {
#if defined(vax)||defined(tahoe)
                extern double infnan();
                if ( x == negone )
                    return (infnan(-ERANGE));   /* -INF */
                else
                    return (infnan(EDOM));      /* NaN */
#else   /* defined(vax)||defined(tahoe) */
                /* x = -1, return -INF with signal */
                if ( x == negone ) return( negone/zero );

                /* negative argument for log, return NaN with signal */
                else return ( zero / zero );
#endif  /* defined(vax)||defined(tahoe) */
            }
        }
    /* end of if (finite(x)) */

    /* log(-INF) is NaN */
        else if(x<0)
             return(zero/zero);

    /* log(+INF) is INF */
        else return(x);
}