Category : C Source Code
Archive   : CEPHES22.ZIP
Filename : EXPL.C

/* expl.c
*
* Exponential function, long double precision
*
*
*
* SYNOPSIS:
*
* long double x, y, expl();
*
* y = expl( x );
*
*
*
* DESCRIPTION:
*
* Returns e (2.71828...) raised to the x power.
*
* Range reduction is accomplished by separating the argument
* into an integer k and fraction f such that
*
* x k f
* e = 2 e.
*
* A Pade' form of degree 2/3 is used to approximate exp(f) - 1
* in the basic range [-0.5 ln 2, 0.5 ln 2].
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE +-10000 50000 1.12e-19 2.81e-20
*
*
* Error amplification in the exponential function can be
* a serious matter. The error propagation involves
* exp( X(1+delta) ) = exp(X) ( 1 + X*delta + ... ),
* which shows that a 1 lsb error in representing X produces
* a relative error of X times 1 lsb in the function.
* While the routine gives an accurate result for arguments
* that are exactly represented by a long double precision
* computer number, the result contains amplified roundoff
* error for large arguments not exactly represented.
*
*
* ERROR MESSAGES:
*
* message condition value returned
* exp underflow x < MINLOG 0.0
* exp overflow x > MAXLOG MAXNUM
*
*/

/*
Cephes Math Library Release 2.2: December, 1990
Copyright 1984, 1990 by Stephen L. Moshier
Direct inquiries to 30 Frost Street, Cambridge, MA 02140
*/


/* Exponential function */

#include "mconf.h"
static char fname[] = {"expl"};

#ifdef UNK
static long double P[3] = {
1.2617719307481059087798E-4L,
3.0299440770744196129956E-2L,
9.9999999999999999991025E-1L,
};
static long double Q[4] = {
3.0019850513866445504159E-6L,
2.5244834034968410419224E-3L,
2.2726554820815502876593E-1L,
2.0000000000000000000897E0L,
};
static long double C1 = 6.9314575195312500000000E-1;
static long double C2 = 1.4286068203094172321215E-6;
#endif

#ifdef DEC
not supported in long double precision
#endif

#ifdef IBMPC
static short P[15] = {
0x424e,0x225f,0x6eaf,0x844e,0x3ff2,
0xf39e,0x5163,0x8866,0xf836,0x3ff9,
0xfffe,0xffff,0xffff,0xffff,0x3ffe,
};
static short Q[20] = {
0xff1e,0xb2fc,0xb5e1,0xc975,0x3fec,
0xff3e,0x45b5,0xcda8,0xa571,0x3ff6,
0x9ee1,0x3f03,0x4cc4,0xe8b8,0x3ffc,
0x0000,0x0000,0x0000,0x8000,0x4000,
};
static short sc1[] = {0x0000,0x0000,0x0000,0xb172,0x3ffe};
#define C1 (*(long double *)sc1)
static short sc2[] = {0x4f1e,0xcd5e,0x8e7b,0xbfbe,0x3feb};
#define C2 (*(long double *)sc2)
#endif

#ifdef MIEEE
static long A[9] = {
0x3ff20000,0x844e6eaf,0x225f424e,
0x3ff90000,0xf8368866,0x5163f39e,
0x3ffe0000,0xffffffff,0xfffffffe,
};
static long B[12] = {
0x3fec0000,0xc975b5e1,0xb2fcff1e,
0x3ff60000,0xa571cda8,0x45b5ff3e,
0x3ffc0000,0xe8b84cc4,0x3f039ee1,
0x40000000,0x80000000,0x00000000,
};
static long sc1[] = {0x3ffe0000,0xb1720000,0x00000000};
#define C1 (*(long double *)sc1)
static long sc2[] = {0x3feb0000,0xbfbe8e7b,0xcd5e4f1e};
#define C2 (*(long double *)sc2)
#endif

extern long double LOG2EL, MAXLOGL, MINLOGL, MAXNUML;

long double expl(x)
long double x;
{
long double px, xx;
int n;
long double polevll(), floorl(), ldexpl();

if( x > MAXLOGL)
{
mtherr( fname, OVERFLOW );
return( MAXNUML );
}

if( x < MINLOGL )
{
mtherr( fname, UNDERFLOW );
return(0.0L);
}

/* Express e**x = e**g 2**n
* = e**g e**( n loge(2) )
* = e**( g + n loge(2) )
*/
px = floorl( LOG2EL * x + 0.5L ); /* floor() truncates toward -infinity. */
n = px;
x -= px * C1;
x -= px * C2;


/* rational approximation for exponential
* of the fractional part:
* e**x = 1 + 2x P(x**2)/( Q(x**2) - P(x**2) )
*/
xx = x * x;
px = x * polevll( xx, P, 2 );
x = px/( polevll( xx, Q, 3 ) - px );
x = 1.0L + ldexpl( x, 1 );

x = ldexpl( x, n );
return(x);
}