lib/libm/noieee_src/n_expm1.c

   1 /*      $NetBSD: n_expm1.c,v 1.7 2008/04/29 15:10:02 uwe Exp $ */
   2 /*
   3  * Copyright (c) 1985, 1993
   4  *      The Regents of the University of California.  All rights reserved.
   5  *
   6  * Redistribution and use in source and binary forms, with or without
   7  * modification, are permitted provided that the following conditions
   8  * are met:
   9  * 1. Redistributions of source code must retain the above copyright
  10  *    notice, this list of conditions and the following disclaimer.
  11  * 2. Redistributions in binary form must reproduce the above copyright
  12  *    notice, this list of conditions and the following disclaimer in the
  13  *    documentation and/or other materials provided with the distribution.
  14  * 3. Neither the name of the University nor the names of its contributors
  15  *    may be used to endorse or promote products derived from this software
  16  *    without specific prior written permission.
  17  *
  18  * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
  19  * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
  20  * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
  21  * ARE DISCLAIMED.  IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
  22  * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
  23  * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
  24  * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
  25  * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
  26  * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
  27  * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
  28  * SUCH DAMAGE.
  29  */
  30
  31 #ifndef lint
  32 #if 0
  33 static char sccsid[] = "@(#)expm1.c     8.1 (Berkeley) 6/4/93";
  34 #endif
  35 #endif /* not lint */
  36
  37 /* EXPM1(X)
  38  * RETURN THE EXPONENTIAL OF X MINUS ONE
  39  * DOUBLE PRECISION (IEEE 53 BITS, VAX D FORMAT 56 BITS)
  40  * CODED IN C BY K.C. NG, 1/19/85;
  41  * REVISED BY K.C. NG on 2/6/85, 3/7/85, 3/21/85, 4/16/85.
  42  *
  43  * Required system supported functions:
  44  *      scalb(x,n)
  45  *      copysign(x,y)
  46  *      finite(x)
  47  *
  48  * Kernel function:
  49  *      exp__E(x,c)
  50  *
  51  * Method:
  52  *      1. Argument Reduction: given the input x, find r and integer k such
  53  *         that
  54  *                         x = k*ln2 + r,  |r| <= 0.5*ln2 .
  55  *         r will be represented as r := z+c for better accuracy.
  56  *
  57  *      2. Compute EXPM1(r)=exp(r)-1 by
  58  *
  59  *                      EXPM1(r=z+c) := z + exp__E(z,c)
  60  *
  61  *      3. EXPM1(x) =  2^k * ( EXPM1(r) + 1-2^-k ).
  62  *
  63  *      Remarks:
  64  *         1. When k=1 and z < -0.25, we use the following formula for
  65  *            better accuracy:
  66  *                      EXPM1(x) = 2 * ( (z+0.5) + exp__E(z,c) )
  67  *         2. To avoid rounding error in 1-2^-k where k is large, we use
  68  *                      EXPM1(x) = 2^k * { [z+(exp__E(z,c)-2^-k )] + 1 }
  69  *            when k>56.
  70  *
  71  * Special cases:
  72  *      EXPM1(INF) is INF, EXPM1(NaN) is NaN;
  73  *      EXPM1(-INF)= -1;
  74  *      for finite argument, only EXPM1(0)=0 is exact.
  75  *
  76  * Accuracy:
  77  *      EXPM1(x) returns the exact (exp(x)-1) nearly rounded. In a test run with
  78  *      1,166,000 random arguments on a VAX, the maximum observed error was
  79  *      .872 ulps (units of the last place).
  80  *
  81  * Constants:
  82  * The hexadecimal values are the intended ones for the following constants.
  83  * The decimal values may be used, provided that the compiler will convert
  84  * from decimal to binary accurately enough to produce the hexadecimal values
  85  * shown.
  86  */
  87
  88 #define _LIBM_STATIC
  89 #include "mathimpl.h"
  90
  91 vc(ln2hi,  6.9314718055829871446E-1  ,7217,4031,0000,f7d0,   0, .B17217F7D00000)
  92 vc(ln2lo,  1.6465949582897081279E-12 ,bcd5,2ce7,d9cc,e4f1, -39, .E7BCD5E4F1D9CC)
  93 vc(lnhuge, 9.4961163736712506989E1   ,ec1d,43bd,9010,a73e,   7, .BDEC1DA73E9010)
  94 vc(invln2, 1.4426950408889634148E0   ,aa3b,40b8,17f1,295c,   1, .B8AA3B295C17F1)
  95
  96 ic(ln2hi,  6.9314718036912381649E-1,   -1, 1.62E42FEE00000)
  97 ic(ln2lo,  1.9082149292705877000E-10, -33, 1.A39EF35793C76)
  98 ic(lnhuge, 7.1602103751842355450E2,     9, 1.6602B15B7ECF2)
  99 ic(invln2, 1.4426950408889633870E0,     0, 1.71547652B82FE)
 100
 101 #ifdef vccast
 102 #define ln2hi   vccast(ln2hi)
 103 #define ln2lo   vccast(ln2lo)
 104 #define lnhuge  vccast(lnhuge)
 105 #define invln2  vccast(invln2)
 106 #endif
 107
 108 #if defined(__vax__)||defined(tahoe)
 109 #define PREC    56
 110 #else   /* defined(__vax__)||defined(tahoe) */
 111 #define PREC    53
 112 #endif  /* defined(__vax__)||defined(tahoe) */
 113
 114 double
 115 expm1(double x)
 116 {
 117         static const double one=1.0, half=1.0/2.0;
 118         double  z,hi,lo,c;
 119         int k;
 120
 121 #if !defined(__vax__)&&!defined(tahoe)
 122         if(x!=x) return(x);     /* x is NaN */
 123 #endif  /* !defined(__vax__)&&!defined(tahoe) */
 124
 125         if( x <= lnhuge ) {
 126                 if( x >= -40.0 ) {
 127
 128                     /* argument reduction : x - k*ln2 */
 129                         k= invln2 *x+copysign(0.5,x);   /* k=NINT(x/ln2) */
 130                         hi=x-k*ln2hi ;
 131                         z=hi-(lo=k*ln2lo);
 132                         c=(hi-z)-lo;
 133
 134                         if(k==0) return(z+__exp__E(z,c));
 135                         if(k==1)
 136                             if(z< -0.25)
 137                                 {x=z+half;x +=__exp__E(z,c); return(x+x);}
 138                             else
 139                                 {z+=__exp__E(z,c); x=half+z; return(x+x);}
 140                     /* end of k=1 */
 141
 142                         else {
 143                             if(k<=PREC)
 144                               { x=one-scalb(one,-k); z += __exp__E(z,c);}
 145                             else if(k<100)
 146                               { x = __exp__E(z,c)-scalb(one,-k); x+=z; z=one;}
 147                             else
 148                               { x = __exp__E(z,c)+z; z=one;}
 149
 150                             return (scalb(x+z,k));
 151                         }
 152                 }
 153                 /* end of x > lnunfl */
 154
 155                 else
 156                      /* expm1(-big#) rounded to -1 (inexact) */
 157                      if(finite(x))
 158                          { c=ln2hi+ln2lo; return(-one);} /* ??? -ragge */
 159
 160                      /* expm1(-INF) is -1 */
 161                      else return(-one);
 162         }
 163         /* end of x < lnhuge */
 164
 165         else
 166         /*  expm1(INF) is INF, expm1(+big#) overflows to INF */
 167             return( finite(x) ?  scalb(one,5000) : x);
 168 }