Quelle e_exp.cpp Sprache: C

/* @(#)e_exp.c 1.6 04/04/22 */
/*
* ====================================================
* Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
*
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/

//#include <sys/cdefs.h>
//__FBSDID("$FreeBSD$");

/* __ieee754_exp(x)
* Returns the exponential of x.
*
* Method
*   1. Argument reduction:
*      Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
* Given x, find r and integer k such that
*
*               x = k*ln2 + r,  |r| <= 0.5*ln2.
*
*      Here r will be represented as r = hi-lo for better
* accuracy.
*
*   2. Approximation of exp(r) by a special rational function on
* the interval [0,0.34658]:
* Write
*     R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
*      We use a special Remes algorithm on [0,0.34658] to generate
* a polynomial of degree 5 to approximate R. The maximum error
* of this polynomial approximation is bounded by 2**-59. In
* other words,
*     R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
*   (where z=r*r, and the values of P1 to P5 are listed below)
* and
*     |                  5          |     -59
*     | 2.0+P1*z+...+P5*z   -  R(z) | <= 2
*     |                             |
* The computation of exp(r) thus becomes
*                             2*r
* exp(r) = 1 + -------
*               R - r
*                                 r*R1(r)
*        = 1 + r + ----------- (for better accuracy)
*                   2 - R1(r)
* where
*          2       4             10
* R1(r) = r - (P1*r  + P2*r  + ... + P5*r   ).
*
*   3. Scale back to obtain exp(x):
* From step 1, we have
*    exp(x) = 2^k * exp(r)
*
* Special cases:
* exp(INF) is INF, exp(NaN) is NaN;
* exp(-INF) is 0, and
* for finite argument, only exp(0)=1 is exact.
*
* Accuracy:
* according to an error analysis, the error is always less than
* 1 ulp (unit in the last place).
*
* Misc. info.
* For IEEE double
*     if x >  7.09782712893383973096e+02 then exp(x) overflow
*     if x < -7.45133219101941108420e+02 then exp(x) underflow
*
* Constants:
* The hexadecimal values are the intended ones for the following
* constants. The decimal values may be used, provided that the
* compiler will convert from decimal to binary accurately enough
* to produce the hexadecimal values shown.
*/

#include <float.h>

#include "math_private.h"

static const double
one = 1.0,
halF[2] = {0.5,-0.5,},
o_threshold=  7.09782712893383973096e+02,  /* 0x40862E42, 0xFEFA39EF */
u_threshold= -7.45133219101941108420e+02,  /* 0xc0874910, 0xD52D3051 */
ln2HI[2]   ={ 6.93147180369123816490e-01,  /* 0x3fe62e42, 0xfee00000 */
      -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */
ln2LO[2]   ={ 1.90821492927058770002e-10,  /* 0x3dea39ef, 0x35793c76 */
      -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */
invln2 =  1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
P1   =  1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
P2   = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
P3   =  6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
P4   = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
P5   =  4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */

static const double E = 2.7182818284590452354; /* e */

static volatile double
huge = 1.0e+300,
twom1000= 9.33263618503218878990e-302;     /* 2**-1000=0x01700000,0*/

double
__ieee754_exp(double x) /* default IEEE double exp */
{
double y,hi=0.0,lo=0.0,c,t,twopk;
int32_t k=0,xsb;
u_int32_t hx;

GET_HIGH_WORD(hx,x);
xsb = (hx>>31)&1;  /* sign bit of x */
hx &= 0x7fffffff;  /* high word of |x| */

    /* filter out non-finite argument */
if(hx >= 0x40862E42) {   /* if |x|>=709.78... */
            if(hx>=0x7ff00000) {
         u_int32_t lx;
  GET_LOW_WORD(lx,x);
  if(((hx&0xfffff)|lx)!=0)
       return x+x;   /* NaN */
  else return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */
     }
     if(x > o_threshold) return huge*huge; /* overflow */
     if(x < u_threshold) return twom1000*twom1000; /* underflow */
}

    /* argument reduction */
if(hx > 0x3fd62e42) {  /* if  |x| > 0.5 ln2 */
     if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
  if (x == 1.0) return E;
  hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb;
     } else {
  k  = (int)(invln2*x+halF[xsb]);
  t  = k;
  hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */
  lo = t*ln2LO[0];
     }
     STRICT_ASSIGN(double, x, hi - lo);
}
else if(hx < 0x3e300000)  { /* when |x|<2**-28 */
     if(huge+x>one) return one+x;/* trigger inexact */
}
else k = 0;

    /* x is now in primary range */
t  = x*x;
if(k >= -1021)
     INSERT_WORDS(twopk,((u_int32_t)(0x3ff+k))<<20, 0);
else
     INSERT_WORDS(twopk,((u_int32_t)(0x3ff+(k+1000)))<<20, 0);
c  = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
if(k==0)  return one-((x*c)/(c-2.0)-x);
else   y = one-((lo-(x*c)/(2.0-c))-hi);
if(k >= -1021) {
     if (k==1024) {
         double const_0x1p1023 = pow(2, 1023);
         return y*2.0*const_0x1p1023;
     }
     return y*twopk;
} else {
     return y*twopk*twom1000;
}
}

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