js/src/fdlibm/e_exp.c

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  16  * The Original Code is Mozilla Communicator client code, released
  17  * March 31, 1998.
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  21  * Portions created by the Initial Developer are Copyright (C) 1998
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  39
  40 /* @(#)e_exp.c 1.3 95/01/18 */
  41 /*
  42  * ====================================================
  43  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
  44  *
  45  * Developed at SunSoft, a Sun Microsystems, Inc. business.
  46  * Permission to use, copy, modify, and distribute this
  47  * software is freely granted, provided that this notice
  48  * is preserved.
  49  * ====================================================
  50  */
  51
  52 /* __ieee754_exp(x)
  53  * Returns the exponential of x.
  54  *
  55  * Method
  56  *   1. Argument reduction:
  57  *      Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
  58  *      Given x, find r and integer k such that
  59  *
  60  *               x = k*ln2 + r,  |r| <= 0.5*ln2.
  61  *
  62  *      Here r will be represented as r = hi-lo for better
  63  *      accuracy.
  64  *
  65  *   2. Approximation of exp(r) by a special rational function on
  66  *      the interval [0,0.34658]:
  67  *      Write
  68  *          R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
  69  *      We use a special Reme algorithm on [0,0.34658] to generate
  70  *      a polynomial of degree 5 to approximate R. The maximum error
  71  *      of this polynomial approximation is bounded by 2**-59. In
  72  *      other words,
  73  *          R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
  74  *      (where z=r*r, and the values of P1 to P5 are listed below)
  75  *      and
  76  *          |                  5          |     -59
  77  *          | 2.0+P1*z+...+P5*z   -  R(z) | <= 2
  78  *          |                             |
  79  *      The computation of exp(r) thus becomes
  80  *                             2*r
  81  *              exp(r) = 1 + -------
  82  *                            R - r
  83  *                                 r*R1(r)
  84  *                     = 1 + r + ----------- (for better accuracy)
  85  *                                2 - R1(r)
  86  *      where
  87  *                               2       4             10
  88  *              R1(r) = r - (P1*r  + P2*r  + ... + P5*r   ).
  89  *
  90  *   3. Scale back to obtain exp(x):
  91  *      From step 1, we have
  92  *         exp(x) = 2^k * exp(r)
  93  *
  94  * Special cases:
  95  *      exp(INF) is INF, exp(NaN) is NaN;
  96  *      exp(-INF) is 0, and
  97  *      for finite argument, only exp(0)=1 is exact.
  98  *
  99  * Accuracy:
 100  *      according to an error analysis, the error is always less than
 101  *      1 ulp (unit in the last place).
 102  *
 103  * Misc. info.
 104  *      For IEEE double
 105  *          if x >  7.09782712893383973096e+02 then exp(x) overflow
 106  *          if x < -7.45133219101941108420e+02 then exp(x) underflow
 107  *
 108  * Constants:
 109  * The hexadecimal values are the intended ones for the following
 110  * constants. The decimal values may be used, provided that the
 111  * compiler will convert from decimal to binary accurately enough
 112  * to produce the hexadecimal values shown.
 113  */
 114
 115 #include "fdlibm.h"
 116
 117 #ifdef __STDC__
 118 static const double
 119 #else
 120 static double
 121 #endif
 122 one     = 1.0,
 123 halF[2] = {0.5,-0.5,},
 124 really_big      = 1.0e+300,
 125 twom1000= 9.33263618503218878990e-302,     /* 2**-1000=0x01700000,0*/
 126 o_threshold=  7.09782712893383973096e+02,  /* 0x40862E42, 0xFEFA39EF */
 127 u_threshold= -7.45133219101941108420e+02,  /* 0xc0874910, 0xD52D3051 */
 128 ln2HI[2]   ={ 6.93147180369123816490e-01,  /* 0x3fe62e42, 0xfee00000 */
 129              -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */
 130 ln2LO[2]   ={ 1.90821492927058770002e-10,  /* 0x3dea39ef, 0x35793c76 */
 131              -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */
 132 invln2 =  1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
 133 P1   =  1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
 134 P2   = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
 135 P3   =  6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
 136 P4   = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
 137 P5   =  4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */
 138
 139
 140 #ifdef __STDC__
 141         double __ieee754_exp(double x)  /* default IEEE double exp */
 142 #else
 143         double __ieee754_exp(x) /* default IEEE double exp */
 144         double x;
 145 #endif
 146 {
 147         fd_twoints u;
 148         double y,hi,lo,c,t;
 149         int k, xsb;
 150         unsigned hx;
 151
 152         u.d = x;
 153         hx  = __HI(u);  /* high word of x */
 154         xsb = (hx>>31)&1;               /* sign bit of x */
 155         hx &= 0x7fffffff;               /* high word of |x| */
 156
 157     /* filter out non-finite argument */
 158         if(hx >= 0x40862E42) {                  /* if |x|>=709.78... */
 159             if(hx>=0x7ff00000) {
 160                 u.d = x;
 161                 if(((hx&0xfffff)|__LO(u))!=0)
 162                      return x+x;                /* NaN */
 163                 else return (xsb==0)? x:0.0;    /* exp(+-inf)={inf,0} */
 164             }
 165             if(x > o_threshold) return really_big*really_big; /* overflow */
 166             if(x < u_threshold) return twom1000*twom1000; /* underflow */
 167         }
 168
 169     /* argument reduction */
 170         if(hx > 0x3fd62e42) {           /* if  |x| > 0.5 ln2 */
 171             if(hx < 0x3FF0A2B2) {       /* and |x| < 1.5 ln2 */
 172                 hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb;
 173             } else {
 174                 k  = (int)(invln2*x+halF[xsb]);
 175                 t  = k;
 176                 hi = x - t*ln2HI[0];    /* t*ln2HI is exact here */
 177                 lo = t*ln2LO[0];
 178             }
 179             x  = hi - lo;
 180         }
 181         else if(hx < 0x3e300000)  {     /* when |x|<2**-28 */
 182             if(really_big+x>one) return one+x;/* trigger inexact */
 183         }
 184         else k = 0;
 185
 186     /* x is now in primary range */
 187         t  = x*x;
 188         c  = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
 189         if(k==0)        return one-((x*c)/(c-2.0)-x);
 190         else            y = one-((lo-(x*c)/(2.0-c))-hi);
 191         if(k >= -1021) {
 192             u.d = y;
 193             __HI(u) += (k<<20); /* add k to y's exponent */
 194             y = u.d;
 195             return y;
 196         } else {
 197             u.d = y;
 198             __HI(u) += ((k+1000)<<20);/* add k to y's exponent */
 199             y = u.d;
 200             return y*twom1000;
 201         }
 202 }