sysdeps/generic/expm1.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[] = "@(#)expm1.c     8.1 (Berkeley) 6/4/93";
  36 #endif /* not lint */
  37
  38 /* EXPM1(X)
  39  * RETURN THE EXPONENTIAL OF X MINUS ONE
  40  * DOUBLE PRECISION (IEEE 53 BITS, VAX D FORMAT 56 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, 3/21/85, 4/16/85.
  43  *
  44  * Required system supported functions:
  45  *      scalb(x,n)
  46  *      copysign(x,y)
  47  *      finite(x)
  48  *
  49  * Kernel function:
  50  *      exp__E(x,c)
  51  *
  52  * Method:
  53  *      1. Argument Reduction: given the input x, find r and integer k such
  54  *         that
  55  *                         x = k*ln2 + r,  |r| <= 0.5*ln2 .
  56  *         r will be represented as r := z+c for better accuracy.
  57  *
  58  *      2. Compute EXPM1(r)=exp(r)-1 by
  59  *
  60  *                      EXPM1(r=z+c) := z + exp__E(z,c)
  61  *
  62  *      3. EXPM1(x) =  2^k * ( EXPM1(r) + 1-2^-k ).
  63  *
  64  *      Remarks:
  65  *         1. When k=1 and z < -0.25, we use the following formula for
  66  *            better accuracy:
  67  *                      EXPM1(x) = 2 * ( (z+0.5) + exp__E(z,c) )
  68  *         2. To avoid rounding error in 1-2^-k where k is large, we use
  69  *                      EXPM1(x) = 2^k * { [z+(exp__E(z,c)-2^-k )] + 1 }
  70  *            when k>56.
  71  *
  72  * Special cases:
  73  *      EXPM1(INF) is INF, EXPM1(NaN) is NaN;
  74  *      EXPM1(-INF)= -1;
  75  *      for finite argument, only EXPM1(0)=0 is exact.
  76  *
  77  * Accuracy:
  78  *      EXPM1(x) returns the exact (exp(x)-1) nearly rounded. In a test run with
  79  *      1,166,000 random arguments on a VAX, the maximum observed error was
  80  *      .872 ulps (units of the last place).
  81  *
  82  * Constants:
  83  * The hexadecimal values are the intended ones for the following constants.
  84  * The decimal values may be used, provided that the compiler will convert
  85  * from decimal to binary accurately enough to produce the hexadecimal values
  86  * shown.
  87  */
  88
  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 double expm1(x)
 109 double x;
 110 {
 111         const static double one=1.0, half=1.0/2.0;
 112         double  z,hi,lo,c;
 113         int k;
 114 #if defined(vax)||defined(tahoe)
 115         static prec=56;
 116 #else   /* defined(vax)||defined(tahoe) */
 117         static prec=53;
 118 #endif  /* defined(vax)||defined(tahoe) */
 119
 120 #if !defined(vax)&&!defined(tahoe)
 121         if(x!=x) return(x);     /* x is NaN */
 122 #endif  /* !defined(vax)&&!defined(tahoe) */
 123
 124         if( x <= lnhuge ) {
 125                 if( x >= -40.0 ) {
 126
 127                     /* argument reduction : x - k*ln2 */
 128                         k= invln2 *x+copysign(0.5,x);   /* k=NINT(x/ln2) */
 129                         hi=x-k*ln2hi ;
 130                         z=hi-(lo=k*ln2lo);
 131                         c=(hi-z)-lo;
 132
 133                         if(k==0) return(z+__exp__E(z,c));
 134                         if(k==1)
 135                             if(z< -0.25)
 136                                 {x=z+half;x +=__exp__E(z,c); return(x+x);}
 137                             else
 138                                 {z+=__exp__E(z,c); x=half+z; return(x+x);}
 139                     /* end of k=1 */
 140
 141                         else {
 142                             if(k<=prec)
 143                               { x=one-scalb(one,-k); z += __exp__E(z,c);}
 144                             else if(k<100)
 145                               { x = __exp__E(z,c)-scalb(one,-k); x+=z; z=one;}
 146                             else
 147                               { x = __exp__E(z,c)+z; z=one;}
 148
 149                             return (scalb(x+z,k));
 150                         }
 151                 }
 152                 /* end of x > lnunfl */
 153
 154                 else
 155                      /* expm1(-big#) rounded to -1 (inexact) */
 156                      if(finite(x))
 157                          { ln2hi+ln2lo; return(-one);}
 158
 159                      /* expm1(-INF) is -1 */
 160                      else return(-one);
 161         }
 162         /* end of x < lnhuge */
 163
 164         else
 165         /*  expm1(INF) is INF, expm1(+big#) overflows to INF */
 166             return( finite(x) ?  scalb(one,5000) : x);
 167 }
 168
 169 weak_alias (__expm1, expm1)