sysdeps/generic/exp__E.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__E.c    8.1 (Berkeley) 6/4/93";
  36 #endif /* not lint */
  37
  38 /* exp__E(x,c)
  39  * ASSUMPTION: c << x  SO THAT  fl(x+c)=x.
  40  * (c is the correction term for x)
  41  * exp__E RETURNS
  42  *
  43  *                       /  exp(x+c) - 1 - x ,  1E-19 < |x| < .3465736
  44  *       exp__E(x,c) =  |
  45  *                       \  0 ,  |x| < 1E-19.
  46  *
  47  * DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS)
  48  * KERNEL FUNCTION OF EXP, EXPM1, POW FUNCTIONS
  49  * CODED IN C BY K.C. NG, 1/31/85;
  50  * REVISED BY K.C. NG on 3/16/85, 4/16/85.
  51  *
  52  * Required system supported function:
  53  *      copysign(x,y)
  54  *
  55  * Method:
  56  *      1. Rational approximation. Let r=x+c.
  57  *         Based on
  58  *                                   2 * sinh(r/2)
  59  *                exp(r) - 1 =   ----------------------   ,
  60  *                               cosh(r/2) - sinh(r/2)
  61  *         exp__E(r) is computed using
  62  *                   x*x            (x/2)*W - ( Q - ( 2*P  + x*P ) )
  63  *                   --- + (c + x*[---------------------------------- + c ])
  64  *                    2                          1 - W
  65  *         where  P := p1*x^2 + p2*x^4,
  66  *                Q := q1*x^2 + q2*x^4 (for 56 bits precision, add q3*x^6)
  67  *                W := x/2-(Q-x*P),
  68  *
  69  *         (See the listing below for the values of p1,p2,q1,q2,q3. The poly-
  70  *          nomials P and Q may be regarded as the approximations to sinh
  71  *          and cosh :
  72  *              sinh(r/2) =  r/2 + r * P  ,  cosh(r/2) =  1 + Q . )
  73  *
  74  *         The coefficients were obtained by a special Remez algorithm.
  75  *
  76  * Approximation error:
  77  *
  78  *   |  exp(x) - 1                         |        2**(-57),  (IEEE double)
  79  *   | ------------  -  (exp__E(x,0)+x)/x  |  <=
  80  *   |       x                             |        2**(-69).  (VAX D)
  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(p1, 1.5150724356786683059E-2 ,3abe,3d78,066a,67e1,  -6, .F83ABE67E1066A)
  92 vc(p2, 6.3112487873718332688E-5 ,5b42,3984,0173,48cd, -13, .845B4248CD0173)
  93 vc(q1, 1.1363478204690669916E-1 ,b95a,3ee8,ec45,44a2,  -3, .E8B95A44A2EC45)
  94 vc(q2, 1.2624568129896839182E-3 ,7905,3ba5,f5e7,72e4,  -9, .A5790572E4F5E7)
  95 vc(q3, 1.5021856115869022674E-6 ,9eb4,36c9,c395,604a, -19, .C99EB4604AC395)
  96
  97 ic(p1, 1.3887401997267371720E-2,  -7, 1.C70FF8B3CC2CF)
  98 ic(p2, 3.3044019718331897649E-5, -15, 1.15317DF4526C4)
  99 ic(q1, 1.1110813732786649355E-1,  -4, 1.C719538248597)
 100 ic(q2, 9.9176615021572857300E-4, -10, 1.03FC4CB8C98E8)
 101
 102 #ifdef vccast
 103 #define       p1    vccast(p1)
 104 #define       p2    vccast(p2)
 105 #define       q1    vccast(q1)
 106 #define       q2    vccast(q2)
 107 #define       q3    vccast(q3)
 108 #endif
 109
 110 double __exp__E(x,c)
 111 double x,c;
 112 {
 113         const static double zero=0.0, one=1.0, half=1.0/2.0, small=1.0E-19;
 114         double z,p,q,xp,xh,w;
 115         if(copysign(x,one)>small) {
 116            z = x*x  ;
 117            p = z*( p1 +z* p2 );
 118 #if defined(vax)||defined(tahoe)
 119            q = z*( q1 +z*( q2 +z* q3 ));
 120 #else   /* defined(vax)||defined(tahoe) */
 121            q = z*( q1 +z*  q2 );
 122 #endif  /* defined(vax)||defined(tahoe) */
 123            xp= x*p     ;
 124            xh= x*half  ;
 125            w = xh-(q-xp)  ;
 126            p = p+p;
 127            c += x*((xh*w-(q-(p+xp)))/(one-w)+c);
 128            return(z*half+c);
 129         }
 130         /* end of |x| > small */
 131
 132         else {
 133             if(x!=zero) one+small;      /* raise the inexact flag */
 134             return(copysign(zero,x));
 135         }
 136 }