sysdeps/generic/exp.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[] = "@(#)exp.c       8.1 (Berkeley) 6/4/93";
  36 #endif /* not lint */
  37
  38 /* EXP(X)
  39  * RETURN THE EXPONENTIAL OF X
  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, 2/15/85, 3/7/85, 3/24/85, 4/16/85, 6/14/86.
  43  *
  44  * Required system supported functions:
  45  *      scalb(x,n)
  46  *      copysign(x,y)
  47  *      finite(x)
  48  *
  49  * Method:
  50  *      1. Argument Reduction: given the input x, find r and integer k such
  51  *         that
  52  *                         x = k*ln2 + r,  |r| <= 0.5*ln2 .
  53  *         r will be represented as r := z+c for better accuracy.
  54  *
  55  *      2. Compute exp(r) by
  56  *
  57  *              exp(r) = 1 + r + r*R1/(2-R1),
  58  *         where
  59  *              R1 = x - x^2*(p1+x^2*(p2+x^2*(p3+x^2*(p4+p5*x^2)))).
  60  *
  61  *      3. exp(x) = 2^k * exp(r) .
  62  *
  63  * Special cases:
  64  *      exp(INF) is INF, exp(NaN) is NaN;
  65  *      exp(-INF)=  0;
  66  *      for finite argument, only exp(0)=1 is exact.
  67  *
  68  * Accuracy:
  69  *      exp(x) returns the exponential of x nearly rounded. In a test run
  70  *      with 1,156,000 random arguments on a VAX, the maximum observed
  71  *      error was 0.869 ulps (units in the last place).
  72  *
  73  * Constants:
  74  * The hexadecimal values are the intended ones for the following constants.
  75  * The decimal values may be used, provided that the compiler will convert
  76  * from decimal to binary accurately enough to produce the hexadecimal values
  77  * shown.
  78  */
  79
  80 #include "mathimpl.h"
  81
  82 vc(ln2hi,  6.9314718055829871446E-1  ,7217,4031,0000,f7d0,   0, .B17217F7D00000)
  83 vc(ln2lo,  1.6465949582897081279E-12 ,bcd5,2ce7,d9cc,e4f1, -39, .E7BCD5E4F1D9CC)
  84 vc(lnhuge, 9.4961163736712506989E1   ,ec1d,43bd,9010,a73e,   7, .BDEC1DA73E9010)
  85 vc(lntiny,-9.5654310917272452386E1   ,4f01,c3bf,33af,d72e,   7,-.BF4F01D72E33AF)
  86 vc(invln2, 1.4426950408889634148E0   ,aa3b,40b8,17f1,295c,   1, .B8AA3B295C17F1)
  87 vc(p1,     1.6666666666666602251E-1  ,aaaa,3f2a,a9f1,aaaa,  -2, .AAAAAAAAAAA9F1)
  88 vc(p2,    -2.7777777777015591216E-3  ,0b60,bc36,ec94,b5f5,  -8,-.B60B60B5F5EC94)
  89 vc(p3,     6.6137563214379341918E-5  ,b355,398a,f15f,792e, -13, .8AB355792EF15F)
  90 vc(p4,    -1.6533902205465250480E-6  ,ea0e,b6dd,5f84,2e93, -19,-.DDEA0E2E935F84)
  91 vc(p5,     4.1381367970572387085E-8  ,bb4b,3431,2683,95f5, -24, .B1BB4B95F52683)
  92
  93 #ifdef vccast
  94 #define    ln2hi    vccast(ln2hi)
  95 #define    ln2lo    vccast(ln2lo)
  96 #define   lnhuge    vccast(lnhuge)
  97 #define   lntiny    vccast(lntiny)
  98 #define   invln2    vccast(invln2)
  99 #define       p1    vccast(p1)
 100 #define       p2    vccast(p2)
 101 #define       p3    vccast(p3)
 102 #define       p4    vccast(p4)
 103 #define       p5    vccast(p5)
 104 #endif
 105
 106 ic(p1,     1.6666666666666601904E-1,  -3,  1.555555555553E)
 107 ic(p2,    -2.7777777777015593384E-3,  -9, -1.6C16C16BEBD93)
 108 ic(p3,     6.6137563214379343612E-5, -14,  1.1566AAF25DE2C)
 109 ic(p4,    -1.6533902205465251539E-6, -20, -1.BBD41C5D26BF1)
 110 ic(p5,     4.1381367970572384604E-8, -25,  1.6376972BEA4D0)
 111 ic(ln2hi,  6.9314718036912381649E-1,  -1,  1.62E42FEE00000)
 112 ic(ln2lo,  1.9082149292705877000E-10,-33,  1.A39EF35793C76)
 113 ic(lnhuge, 7.1602103751842355450E2,    9,  1.6602B15B7ECF2)
 114 ic(lntiny,-7.5137154372698068983E2,    9, -1.77AF8EBEAE354)
 115 ic(invln2, 1.4426950408889633870E0,    0,  1.71547652B82FE)
 116
 117 double exp(x)
 118 double x;
 119 {
 120         double  z,hi,lo,c;
 121         int k;
 122
 123 #if !defined(vax)&&!defined(tahoe)
 124         if(x!=x) return(x);     /* x is NaN */
 125 #endif  /* !defined(vax)&&!defined(tahoe) */
 126         if( x <= lnhuge ) {
 127                 if( x >= lntiny ) {
 128
 129                     /* argument reduction : x --> x - k*ln2 */
 130
 131                         k=invln2*x+copysign(0.5,x);     /* k=NINT(x/ln2) */
 132
 133                     /* express x-k*ln2 as hi-lo and let x=hi-lo rounded */
 134
 135                         hi=x-k*ln2hi;
 136                         x=hi-(lo=k*ln2lo);
 137
 138                     /* return 2^k*[1+x+x*c/(2+c)]  */
 139                         z=x*x;
 140                         c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5))));
 141                         return  scalb(1.0+(hi-(lo-(x*c)/(2.0-c))),k);
 142
 143                 }
 144                 /* end of x > lntiny */
 145
 146                 else
 147                      /* exp(-big#) underflows to zero */
 148                      if(finite(x))  return(scalb(1.0,-5000));
 149
 150                      /* exp(-INF) is zero */
 151                      else return(0.0);
 152         }
 153         /* end of x < lnhuge */
 154
 155         else
 156         /* exp(INF) is INF, exp(+big#) overflows to INF */
 157             return( finite(x) ?  scalb(1.0,5000)  : x);
 158 }
 159
 160 /* returns exp(r = x + c) for |c| < |x| with no overlap.  */
 161
 162 double __exp__D(x, c)
 163 double x, c;
 164 {
 165         double  z,hi,lo, t;
 166         int k;
 167
 168 #if !defined(vax)&&!defined(tahoe)
 169         if (x!=x) return(x);    /* x is NaN */
 170 #endif  /* !defined(vax)&&!defined(tahoe) */
 171         if ( x <= lnhuge ) {
 172                 if ( x >= lntiny ) {
 173
 174                     /* argument reduction : x --> x - k*ln2 */
 175                         z = invln2*x;
 176                         k = z + copysign(.5, x);
 177
 178                     /* express (x+c)-k*ln2 as hi-lo and let x=hi-lo rounded */
 179
 180                         hi=(x-k*ln2hi);                 /* Exact. */
 181                         x= hi - (lo = k*ln2lo-c);
 182                     /* return 2^k*[1+x+x*c/(2+c)]  */
 183                         z=x*x;
 184                         c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5))));
 185                         c = (x*c)/(2.0-c);
 186
 187                         return  scalb(1.+(hi-(lo - c)), k);
 188                 }
 189                 /* end of x > lntiny */
 190
 191                 else
 192                      /* exp(-big#) underflows to zero */
 193                      if(finite(x))  return(scalb(1.0,-5000));
 194
 195                      /* exp(-INF) is zero */
 196                      else return(0.0);
 197         }
 198         /* end of x < lnhuge */
 199
 200         else
 201         /* exp(INF) is INF, exp(+big#) overflows to INF */
 202             return( finite(x) ?  scalb(1.0,5000)  : x);
 203 }