lib/libm/noieee_src/n_exp.c

   1 /*      $NetBSD: n_exp.c,v 1.9 2014/10/10 20:58:09 martin 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[] = "@(#)exp.c       8.1 (Berkeley) 6/4/93";
  34 #endif
  35 #endif /* not lint */
  36
  37 /* EXP(X)
  38  * RETURN THE EXPONENTIAL OF X
  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, 2/15/85, 3/7/85, 3/24/85, 4/16/85, 6/14/86.
  42  *
  43  * Required system supported functions:
  44  *      scalb(x,n)
  45  *      copysign(x,y)
  46  *      finite(x)
  47  *
  48  * Method:
  49  *      1. Argument Reduction: given the input x, find r and integer k such
  50  *         that
  51  *                         x = k*ln2 + r,  |r| <= 0.5*ln2 .
  52  *         r will be represented as r := z+c for better accuracy.
  53  *
  54  *      2. Compute exp(r) by
  55  *
  56  *              exp(r) = 1 + r + r*R1/(2-R1),
  57  *         where
  58  *              R1 = x - x^2*(p1+x^2*(p2+x^2*(p3+x^2*(p4+p5*x^2)))).
  59  *
  60  *      3. exp(x) = 2^k * exp(r) .
  61  *
  62  * Special cases:
  63  *      exp(INF) is INF, exp(NaN) is NaN;
  64  *      exp(-INF)=  0;
  65  *      for finite argument, only exp(0)=1 is exact.
  66  *
  67  * Accuracy:
  68  *      exp(x) returns the exponential of x nearly rounded. In a test run
  69  *      with 1,156,000 random arguments on a VAX, the maximum observed
  70  *      error was 0.869 ulps (units in the last place).
  71  *
  72  * Constants:
  73  * The hexadecimal values are the intended ones for the following constants.
  74  * The decimal values may be used, provided that the compiler will convert
  75  * from decimal to binary accurately enough to produce the hexadecimal values
  76  * shown.
  77  */
  78
  79 #define _LIBM_STATIC
  80 #include "../src/namespace.h"
  81 #include "mathimpl.h"
  82
  83 #ifdef __weak_alias
  84 __weak_alias(exp, _exp);
  85 __weak_alias(_expl, _exp);
  86 __weak_alias(expf, _expf);
  87 #endif
  88
  89 vc(ln2hi,  6.9314718055829871446E-1  ,7217,4031,0000,f7d0,   0, .B17217F7D00000)
  90 vc(ln2lo,  1.6465949582897081279E-12 ,bcd5,2ce7,d9cc,e4f1, -39, .E7BCD5E4F1D9CC)
  91 vc(lnhuge, 9.4961163736712506989E1   ,ec1d,43bd,9010,a73e,   7, .BDEC1DA73E9010)
  92 vc(lntiny,-9.5654310917272452386E1   ,4f01,c3bf,33af,d72e,   7,-.BF4F01D72E33AF)
  93 vc(invln2, 1.4426950408889634148E0   ,aa3b,40b8,17f1,295c,   1, .B8AA3B295C17F1)
  94 vc(p1,     1.6666666666666602251E-1  ,aaaa,3f2a,a9f1,aaaa,  -2, .AAAAAAAAAAA9F1)
  95 vc(p2,    -2.7777777777015591216E-3  ,0b60,bc36,ec94,b5f5,  -8,-.B60B60B5F5EC94)
  96 vc(p3,     6.6137563214379341918E-5  ,b355,398a,f15f,792e, -13, .8AB355792EF15F)
  97 vc(p4,    -1.6533902205465250480E-6  ,ea0e,b6dd,5f84,2e93, -19,-.DDEA0E2E935F84)
  98 vc(p5,     4.1381367970572387085E-8  ,bb4b,3431,2683,95f5, -24, .B1BB4B95F52683)
  99
 100 #ifdef vccast
 101 #define    ln2hi    vccast(ln2hi)
 102 #define    ln2lo    vccast(ln2lo)
 103 #define   lnhuge    vccast(lnhuge)
 104 #define   lntiny    vccast(lntiny)
 105 #define   invln2    vccast(invln2)
 106 #define       p1    vccast(p1)
 107 #define       p2    vccast(p2)
 108 #define       p3    vccast(p3)
 109 #define       p4    vccast(p4)
 110 #define       p5    vccast(p5)
 111 #endif
 112
 113 ic(p1,     1.6666666666666601904E-1,  -3,  1.555555555553E)
 114 ic(p2,    -2.7777777777015593384E-3,  -9, -1.6C16C16BEBD93)
 115 ic(p3,     6.6137563214379343612E-5, -14,  1.1566AAF25DE2C)
 116 ic(p4,    -1.6533902205465251539E-6, -20, -1.BBD41C5D26BF1)
 117 ic(p5,     4.1381367970572384604E-8, -25,  1.6376972BEA4D0)
 118 ic(ln2hi,  6.9314718036912381649E-1,  -1,  1.62E42FEE00000)
 119 ic(ln2lo,  1.9082149292705877000E-10,-33,  1.A39EF35793C76)
 120 ic(lnhuge, 7.1602103751842355450E2,    9,  1.6602B15B7ECF2)
 121 ic(lntiny,-7.5137154372698068983E2,    9, -1.77AF8EBEAE354)
 122 ic(invln2, 1.4426950408889633870E0,    0,  1.71547652B82FE)
 123
 124 double
 125 exp(double x)
 126 {
 127         double  z,hi,lo,c;
 128         int k;
 129
 130 #if !defined(__vax__)&&!defined(tahoe)
 131         if(x!=x) return(x);     /* x is NaN */
 132 #endif  /* !defined(__vax__)&&!defined(tahoe) */
 133         if( x <= lnhuge ) {
 134                 if( x >= lntiny ) {
 135
 136                     /* argument reduction : x --> x - k*ln2 */
 137
 138                         k=invln2*x+copysign(0.5,x);     /* k=NINT(x/ln2) */
 139
 140                     /* express x-k*ln2 as hi-lo and let x=hi-lo rounded */
 141
 142                         hi=x-k*ln2hi;
 143                         x=hi-(lo=k*ln2lo);
 144
 145                     /* return 2^k*[1+x+x*c/(2+c)]  */
 146                         z=x*x;
 147                         c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5))));
 148                         return  scalb(1.0+(hi-(lo-(x*c)/(2.0-c))),k);
 149
 150                 }
 151                 /* end of x > lntiny */
 152
 153                 else
 154                      /* exp(-big#) underflows to zero */
 155                      if(finite(x))  return(scalb(1.0,-5000));
 156
 157                      /* exp(-INF) is zero */
 158                      else return(0.0);
 159         }
 160         /* end of x < lnhuge */
 161
 162         else
 163         /* exp(INF) is INF, exp(+big#) overflows to INF */
 164             return( finite(x) ?  scalb(1.0,5000)  : x);
 165 }
 166
 167 float
 168 expf(float x)
 169 {
 170         return(exp((double)x));
 171 }
 172
 173 /* returns exp(r = x + c) for |c| < |x| with no overlap.  */
 174
 175 double
 176 __exp__D(double x, double c)
 177 {
 178         double  z,hi,lo;
 179         int k;
 180
 181 #if !defined(__vax__)&&!defined(tahoe)
 182         if (x!=x) return(x);    /* x is NaN */
 183 #endif  /* !defined(__vax__)&&!defined(tahoe) */
 184         if ( x <= lnhuge ) {
 185                 if ( x >= lntiny ) {
 186
 187                     /* argument reduction : x --> x - k*ln2 */
 188                         z = invln2*x;
 189                         k = z + copysign(.5, x);
 190
 191                     /* express (x+c)-k*ln2 as hi-lo and let x=hi-lo rounded */
 192
 193                         hi=(x-k*ln2hi);                 /* Exact. */
 194                         x= hi - (lo = k*ln2lo-c);
 195                     /* return 2^k*[1+x+x*c/(2+c)]  */
 196                         z=x*x;
 197                         c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5))));
 198                         c = (x*c)/(2.0-c);
 199
 200                         return  scalb(1.+(hi-(lo - c)), k);
 201                 }
 202                 /* end of x > lntiny */
 203
 204                 else
 205                      /* exp(-big#) underflows to zero */
 206                      if(finite(x))  return(scalb(1.0,-5000));
 207
 208                      /* exp(-INF) is zero */
 209                      else return(0.0);
 210         }
 211         /* end of x < lnhuge */
 212
 213         else
 214         /* exp(INF) is INF, exp(+big#) overflows to INF */
 215             return( finite(x) ?  scalb(1.0,5000)  : x);
 216 }