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/* md_exp.c
*
* Exponential function
*
*
*
* SYNOPSIS:
*
* double x, y, md_exp();
*
* y = md_exp( 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 1 + 2x P(x**2)/( Q(x**2) - P(x**2) )
* of degree 2/3 is used to approximate md_exp(f) in the basic
* interval [-0.5, 0.5].
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC +- 88 50000 2.8e-17 7.0e-18
* IEEE +- 708 40000 2.0e-16 5.6e-17
*
*
* Error amplification in the exponential function can be
* a serious matter. The error propagation involves
* md_exp( X(1+delta) ) = md_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 double precision
* computer number, the result contains amplified roundoff
* error for large arguments not exactly represented.
*
*
* ERROR MESSAGES:
*
* message condition value returned
* md_exp underflow x < MINLOG 0.0
* md_exp overflow x > MAXLOG INFINITY
*
*/
/*
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1995, 2000 by Stephen L. Moshier
*/
/* Exponential function */
#include "mconf.h"
#ifdef UNK
static double P[] = {
1.26177193074810590878E-4,
3.02994407707441961300E-2,
9.99999999999999999910E-1,
};
static double Q[] = {
3.00198505138664455042E-6,
2.52448340349684104192E-3,
2.27265548208155028766E-1,
2.00000000000000000009E0,
};
static double C1 = 6.93145751953125E-1;
static double C2 = 1.42860682030941723212E-6;
#endif
#ifdef DEC
static unsigned short P[] = {
0035004,0047156,0127442,0057502,
0036770,0033210,0063121,0061764,
0040200,0000000,0000000,0000000,
};
static unsigned short Q[] = {
0033511,0072665,0160662,0176377,
0036045,0070715,0124105,0132777,
0037550,0134114,0142077,0001637,
0040400,0000000,0000000,0000000,
};
static unsigned short sc1[] = {0040061,0071000,0000000,0000000};
#define C1 (*(double *)sc1)
static unsigned short sc2[] = {0033277,0137216,0075715,0057117};
#define C2 (*(double *)sc2)
#endif
#ifdef IBMPC
static unsigned short P[] = {
0x4be8,0xd5e4,0x89cd,0x3f20,
0x2c7e,0x0cca,0x06d1,0x3f9f,
0x0000,0x0000,0x0000,0x3ff0,
};
static unsigned short Q[] = {
0x5fa0,0xbc36,0x2eb6,0x3ec9,
0xb6c0,0xb508,0xae39,0x3f64,
0xe074,0x9887,0x1709,0x3fcd,
0x0000,0x0000,0x0000,0x4000,
};
static unsigned short sc1[] = {0x0000,0x0000,0x2e40,0x3fe6};
#define C1 (*(double *)sc1)
static unsigned short sc2[] = {0xabca,0xcf79,0xf7d1,0x3eb7};
#define C2 (*(double *)sc2)
#endif
#ifdef MIEEE
static unsigned short P[] = {
0x3f20,0x89cd,0xd5e4,0x4be8,
0x3f9f,0x06d1,0x0cca,0x2c7e,
0x3ff0,0x0000,0x0000,0x0000,
};
static unsigned short Q[] = {
0x3ec9,0x2eb6,0xbc36,0x5fa0,
0x3f64,0xae39,0xb508,0xb6c0,
0x3fcd,0x1709,0x9887,0xe074,
0x4000,0x0000,0x0000,0x0000,
};
static unsigned short sc1[] = {0x3fe6,0x2e40,0x0000,0x0000};
#define C1 (*(double *)sc1)
static unsigned short sc2[] = {0x3eb7,0xf7d1,0xcf79,0xabca};
#define C2 (*(double *)sc2)
#endif
#ifdef ANSIPROT
extern double polevl ( double, void *, int );
extern double p1evl ( double, void *, int );
extern double md_floor ( double );
extern double md_ldexp ( double, int );
extern int isnan ( double );
extern int isfinite ( double );
#else
double polevl(), p1evl(), md_floor(), md_ldexp();
int isnan(), isfinite();
#endif
extern double LOGE2, LOG2E, MAXLOG, MINLOG, MAXNUM;
#ifdef INFINITIES
extern double INFINITY;
#endif
double md_exp(x)
double x;
{
double px, xx;
int n;
#ifdef NANS
if( isnan(x) )
return(x);
#endif
if( x > MAXLOG)
{
#ifdef INFINITIES
return( INFINITY );
#else
mtherr( "md_exp", OVERFLOW );
return( MAXNUM );
#endif
}
if( x < MINLOG )
{
#ifndef INFINITIES
mtherr( "md_exp", UNDERFLOW );
#endif
return(0.0);
}
/* Express e**x = e**g 2**n
* = e**g e**( n loge(2) )
* = e**( g + n loge(2) )
*/
px = md_floor( LOG2E * x + 0.5 ); /* md_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 * polevl( xx, P, 2 );
x = px/( polevl( xx, Q, 3 ) - px );
x = 1.0 + 2.0 * x;
/* multiply by power of 2 */
x = md_ldexp( x, n );
return(x);
}
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