modules/fdlibm/src/e_exp.cpp

   1
   2 /* @(#)e_exp.c 1.6 04/04/22 */
   3 /*
   4  * ====================================================
   5  * Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
   6  *
   7  * Permission to use, copy, modify, and distribute this
   8  * software is freely granted, provided that this notice
   9  * is preserved.
  10  * ====================================================
  11  */
  12
  13 //#include <sys/cdefs.h>
  14 //__FBSDID("$FreeBSD$");
  15
  16 /* __ieee754_exp(x)
  17  * Returns the exponential of x.
  18  *
  19  * Method
  20  *   1. Argument reduction:
  21  *      Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
  22  *      Given x, find r and integer k such that
  23  *
  24  *               x = k*ln2 + r,  |r| <= 0.5*ln2.
  25  *
  26  *      Here r will be represented as r = hi-lo for better
  27  *      accuracy.
  28  *
  29  *   2. Approximation of exp(r) by a special rational function on
  30  *      the interval [0,0.34658]:
  31  *      Write
  32  *          R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
  33  *      We use a special Remes algorithm on [0,0.34658] to generate
  34  *      a polynomial of degree 5 to approximate R. The maximum error
  35  *      of this polynomial approximation is bounded by 2**-59. In
  36  *      other words,
  37  *          R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
  38  *      (where z=r*r, and the values of P1 to P5 are listed below)
  39  *      and
  40  *          |                  5          |     -59
  41  *          | 2.0+P1*z+...+P5*z   -  R(z) | <= 2
  42  *          |                             |
  43  *      The computation of exp(r) thus becomes
  44  *                             2*r
  45  *              exp(r) = 1 + -------
  46  *                            R - r
  47  *                                 r*R1(r)
  48  *                     = 1 + r + ----------- (for better accuracy)
  49  *                                2 - R1(r)
  50  *      where
  51  *                               2       4             10
  52  *              R1(r) = r - (P1*r  + P2*r  + ... + P5*r   ).
  53  *
  54  *   3. Scale back to obtain exp(x):
  55  *      From step 1, we have
  56  *         exp(x) = 2^k * exp(r)
  57  *
  58  * Special cases:
  59  *      exp(INF) is INF, exp(NaN) is NaN;
  60  *      exp(-INF) is 0, and
  61  *      for finite argument, only exp(0)=1 is exact.
  62  *
  63  * Accuracy:
  64  *      according to an error analysis, the error is always less than
  65  *      1 ulp (unit in the last place).
  66  *
  67  * Misc. info.
  68  *      For IEEE double
  69  *          if x >  7.09782712893383973096e+02 then exp(x) overflow
  70  *          if x < -7.45133219101941108420e+02 then exp(x) underflow
  71  *
  72  * Constants:
  73  * The hexadecimal values are the intended ones for the following
  74  * constants. The decimal values may be used, provided that the
  75  * compiler will convert from decimal to binary accurately enough
  76  * to produce the hexadecimal values shown.
  77  */
  78
  79 #include <float.h>
  80
  81 #include "math_private.h"
  82
  83 static const double
  84 one     = 1.0,
  85 halF[2] = {0.5,-0.5,},
  86 o_threshold=  7.09782712893383973096e+02,  /* 0x40862E42, 0xFEFA39EF */
  87 u_threshold= -7.45133219101941108420e+02,  /* 0xc0874910, 0xD52D3051 */
  88 ln2HI[2]   ={ 6.93147180369123816490e-01,  /* 0x3fe62e42, 0xfee00000 */
  89              -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */
  90 ln2LO[2]   ={ 1.90821492927058770002e-10,  /* 0x3dea39ef, 0x35793c76 */
  91              -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */
  92 invln2 =  1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
  93 P1   =  1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
  94 P2   = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
  95 P3   =  6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
  96 P4   = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
  97 P5   =  4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */
  98
  99 static const double E = 2.7182818284590452354;  /* e */
 100
 101 static volatile double
 102 huge    = 1.0e+300,
 103 twom1000= 9.33263618503218878990e-302;     /* 2**-1000=0x01700000,0*/
 104
 105 double
 106 __ieee754_exp(double x) /* default IEEE double exp */
 107 {
 108         double y,hi=0.0,lo=0.0,c,t,twopk;
 109         int32_t k=0,xsb;
 110         u_int32_t hx;
 111
 112         GET_HIGH_WORD(hx,x);
 113         xsb = (hx>>31)&1;               /* sign bit of x */
 114         hx &= 0x7fffffff;               /* high word of |x| */
 115
 116     /* filter out non-finite argument */
 117         if(hx >= 0x40862E42) {                  /* if |x|>=709.78... */
 118             if(hx>=0x7ff00000) {
 119                 u_int32_t lx;
 120                 GET_LOW_WORD(lx,x);
 121                 if(((hx&0xfffff)|lx)!=0)
 122                      return x+x;                /* NaN */
 123                 else return (xsb==0)? x:0.0;    /* exp(+-inf)={inf,0} */
 124             }
 125             if(x > o_threshold) return huge*huge; /* overflow */
 126             if(x < u_threshold) return twom1000*twom1000; /* underflow */
 127         }
 128
 129     /* argument reduction */
 130         if(hx > 0x3fd62e42) {           /* if  |x| > 0.5 ln2 */
 131             if(hx < 0x3FF0A2B2) {       /* and |x| < 1.5 ln2 */
 132                 if (x == 1.0) return E;
 133                 hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb;
 134             } else {
 135                 k  = (int)(invln2*x+halF[xsb]);
 136                 t  = k;
 137                 hi = x - t*ln2HI[0];    /* t*ln2HI is exact here */
 138                 lo = t*ln2LO[0];
 139             }
 140             STRICT_ASSIGN(double, x, hi - lo);
 141         }
 142         else if(hx < 0x3e300000)  {     /* when |x|<2**-28 */
 143             if(huge+x>one) return one+x;/* trigger inexact */
 144         }
 145         else k = 0;
 146
 147     /* x is now in primary range */
 148         t  = x*x;
 149         if(k >= -1021)
 150             INSERT_WORDS(twopk,((u_int32_t)(0x3ff+k))<<20, 0);
 151         else
 152             INSERT_WORDS(twopk,((u_int32_t)(0x3ff+(k+1000)))<<20, 0);
 153         c  = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
 154         if(k==0)        return one-((x*c)/(c-2.0)-x);
 155         else            y = one-((lo-(x*c)/(2.0-c))-hi);
 156         if(k >= -1021) {
 157             if (k==1024) {
 158                 double const_0x1p1023 = pow(2, 1023);
 159                 return y*2.0*const_0x1p1023;
 160             }
 161             return y*twopk;
 162         } else {
 163             return y*twopk*twom1000;
 164         }
 165 }