sysdeps/generic/log1p.c

   1 /*
   2  * Copyright (c) 1985, 1993
   3  *      The Regents of the University of California.  All rights reserved.
   4  *
   5  * Redistribution and use in source and binary forms, with or without
   6  * modification, are permitted provided that the following conditions
   7  * are met:
   8  * 1. Redistributions of source code must retain the above copyright
   9  *    notice, this list of conditions and the following disclaimer.
  10  * 2. Redistributions in binary form must reproduce the above copyright
  11  *    notice, this list of conditions and the following disclaimer in the
  12  *    documentation and/or other materials provided with the distribution.
  13  * 3. All advertising materials mentioning features or use of this software
  14  *    must display the following acknowledgement:
  15  *      This product includes software developed by the University of
  16  *      California, Berkeley and its contributors.
  17  * 4. Neither the name of the University nor the names of its contributors
  18  *    may be used to endorse or promote products derived from this software
  19  *    without specific prior written permission.
  20  *
  21  * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
  22  * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
  23  * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
  24  * ARE DISCLAIMED.  IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
  25  * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
  26  * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
  27  * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
  28  * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
  29  * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
  30  * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
  31  * SUCH DAMAGE.
  32  */
  33
  34 #ifndef lint
  35 static char sccsid[] = "@(#)log1p.c     8.1 (Berkeley) 6/4/93";
  36 #endif /* not lint */
  37
  38 /* LOG1P(x)
  39  * RETURN THE LOGARITHM OF 1+x
  40  * DOUBLE PRECISION (VAX D FORMAT 56 bits, IEEE DOUBLE 53 BITS)
  41  * CODED IN C BY K.C. NG, 1/19/85;
  42  * REVISED BY K.C. NG on 2/6/85, 3/7/85, 3/24/85, 4/16/85.
  43  *
  44  * Required system supported functions:
  45  *      scalb(x,n)
  46  *      copysign(x,y)
  47  *      logb(x)
  48  *      finite(x)
  49  *
  50  * Required kernel function:
  51  *      log__L(z)
  52  *
  53  * Method :
  54  *      1. Argument Reduction: find k and f such that
  55  *                      1+x  = 2^k * (1+f),
  56  *         where  sqrt(2)/2 < 1+f < sqrt(2) .
  57  *
  58  *      2. Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
  59  *               = 2s + 2/3 s**3 + 2/5 s**5 + .....,
  60  *         log(1+f) is computed by
  61  *
  62  *                      log(1+f) = 2s + s*log__L(s*s)
  63  *         where
  64  *              log__L(z) = z*(L1 + z*(L2 + z*(... (L6 + z*L7)...)))
  65  *
  66  *         See log__L() for the values of the coefficients.
  67  *
  68  *      3. Finally,  log(1+x) = k*ln2 + log(1+f).
  69  *
  70  *      Remarks 1. In step 3 n*ln2 will be stored in two floating point numbers
  71  *                 n*ln2hi + n*ln2lo, where ln2hi is chosen such that the last
  72  *                 20 bits (for VAX D format), or the last 21 bits ( for IEEE
  73  *                 double) is 0. This ensures n*ln2hi is exactly representable.
  74  *              2. In step 1, f may not be representable. A correction term c
  75  *                 for f is computed. It follows that the correction term for
  76  *                 f - t (the leading term of log(1+f) in step 2) is c-c*x. We
  77  *                 add this correction term to n*ln2lo to attenuate the error.
  78  *
  79  *
  80  * Special cases:
  81  *      log1p(x) is NaN with signal if x < -1; log1p(NaN) is NaN with no signal;
  82  *      log1p(INF) is +INF; log1p(-1) is -INF with signal;
  83  *      only log1p(0)=0 is exact for finite argument.
  84  *
  85  * Accuracy:
  86  *      log1p(x) returns the exact log(1+x) nearly rounded. In a test run
  87  *      with 1,536,000 random arguments on a VAX, the maximum observed
  88  *      error was .846 ulps (units in the last place).
  89  *
  90  * Constants:
  91  * The hexadecimal values are the intended ones for the following constants.
  92  * The decimal values may be used, provided that the compiler will convert
  93  * from decimal to binary accurately enough to produce the hexadecimal values
  94  * shown.
  95  */
  96
  97 #include <errno.h>
  98 #include "mathimpl.h"
  99
 100 vc(ln2hi, 6.9314718055829871446E-1  ,7217,4031,0000,f7d0,   0, .B17217F7D00000)
 101 vc(ln2lo, 1.6465949582897081279E-12 ,bcd5,2ce7,d9cc,e4f1, -39, .E7BCD5E4F1D9CC)
 102 vc(sqrt2, 1.4142135623730950622E0   ,04f3,40b5,de65,33f9,   1, .B504F333F9DE65)
 103
 104 ic(ln2hi, 6.9314718036912381649E-1,   -1, 1.62E42FEE00000)
 105 ic(ln2lo, 1.9082149292705877000E-10, -33, 1.A39EF35793C76)
 106 ic(sqrt2, 1.4142135623730951455E0,     0, 1.6A09E667F3BCD)
 107
 108 #ifdef vccast
 109 #define ln2hi   vccast(ln2hi)
 110 #define ln2lo   vccast(ln2lo)
 111 #define sqrt2   vccast(sqrt2)
 112 #endif
 113
 114 double log1p(x)
 115 double x;
 116 {
 117         const static double zero=0.0, negone= -1.0, one=1.0,
 118                       half=1.0/2.0, small=1.0E-20;   /* 1+small == 1 */
 119         double z,s,t,c;
 120         int k;
 121
 122 #if !defined(vax)&&!defined(tahoe)
 123         if(x!=x) return(x);     /* x is NaN */
 124 #endif  /* !defined(vax)&&!defined(tahoe) */
 125
 126         if(finite(x)) {
 127            if( x > negone ) {
 128
 129            /* argument reduction */
 130               if(copysign(x,one)<small) return(x);
 131               k=logb(one+x); z=scalb(x,-k); t=scalb(one,-k);
 132               if(z+t >= sqrt2 )
 133                   { k += 1 ; z *= half; t *= half; }
 134               t += negone; x = z + t;
 135               c = (t-x)+z ;             /* correction term for x */
 136
 137            /* compute log(1+x)  */
 138               s = x/(2+x); t = x*x*half;
 139               c += (k*ln2lo-c*x);
 140               z = c+s*(t+__log__L(s*s));
 141               x += (z - t) ;
 142
 143               return(k*ln2hi+x);
 144            }
 145         /* end of if (x > negone) */
 146
 147             else {
 148 #if defined(vax)||defined(tahoe)
 149                 if ( x == negone )
 150                     return (infnan(-ERANGE));   /* -INF */
 151                 else
 152                     return (infnan(EDOM));      /* NaN */
 153 #else   /* defined(vax)||defined(tahoe) */
 154                 /* x = -1, return -INF with signal */
 155                 if ( x == negone ) return( negone/zero );
 156
 157                 /* negative argument for log, return NaN with signal */
 158                 else return ( zero / zero );
 159 #endif  /* defined(vax)||defined(tahoe) */
 160             }
 161         }
 162     /* end of if (finite(x)) */
 163
 164     /* log(-INF) is NaN */
 165         else if(x<0)
 166              return(zero/zero);
 167
 168     /* log(+INF) is INF */
 169         else return(x);
 170 }