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/* expm1l.c
*
* Exponential function, minus 1
* Long double precision
*
*
* SYNOPSIS:
*
* long double x, y, expm1l();
*
* y = expm1l( x );
*
*
*
* DESCRIPTION:
*
* Returns e (2.71828...) raised to the x power, minus 1.
*
* Range reduction is accomplished by separating the argument
* into an integer k and fraction f such that
*
* x k f
* e = 2 e.
*
* An expansion x + .5 x^2 + x^3 R(x) approximates exp(f) - 1
* in the basic range [-0.5 ln 2, 0.5 ln 2].
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE -45,+MAXLOG 200,000 1.2e-19 2.5e-20
*
* ERROR MESSAGES:
*
* message condition value returned
* expm1l overflow x > MAXLOG MAXNUM
*
*/
/*
Cephes Math Library Release 2.9: April, 2001
Copyright 2001 by Stephen L. Moshier
*/
#include "mconf.h"
#ifndef ANSIPROT
long double ldexpl(), floorl();
#else
extern long double ldexpl(long double, int);
extern long double floorl(long double);
#endif
extern long double MAXLOGL, MAXNUML;
char *fname = "expm1l";
/* exp(x) - 1 = x + 0.5 x^2 + x^3 P(x)/Q(x)
-.5 ln 2 < x < .5 ln 2
Theoretical peak relative error = 3.4e-22 */
static long double
P0 = -1.586135578666346600772998894928250240826E4L,
P1 = 2.642771505685952966904660652518429479531E3L,
P2 = -3.423199068835684263987132888286791620673E2L,
P3 = 1.800826371455042224581246202420972737840E1L,
P4 = -5.238523121205561042771939008061958820811E-1L,
Q0 = -9.516813471998079611319047060563358064497E4L,
Q1 = 3.964866271411091674556850458227710004570E4L,
Q2 = -7.207678383830091850230366618190187434796E3L,
Q3 = 7.206038318724600171970199625081491823079E2L,
Q4 = -4.002027679107076077238836622982900945173E1L,
/* Q5 = 1.000000000000000000000000000000000000000E0 */
/* C1 + C2 = ln 2 */
C1 = 6.93145751953125E-1L,
C2 = 1.428606820309417232121458176568075500134E-6L,
/* ln 2^-65 */
minarg = -4.5054566736396445112120088E1L;
#ifdef INFINITIES
extern long double INFINITYL;
#endif
long double expm1l(x)
long double x;
{
long double px, qx, xx;
int k;
/* Overflow. */
if (x > MAXLOGL)
{
mtherr (fname, OVERFLOW);
#ifdef INFINITIES
return (INFINITYL);
#else
return MAXNUML;
#endif
}
#ifdef MINUSZERO
if (x == 0.0)
return x;
#endif
/* Minimum value. */
if (x < minarg)
return -1.0L;
xx = C1 + C2;
/* Express x = ln 2 (k + remainder), remainder not exceeding 1/2. */
px = floorl (0.5 + x / xx);
k = px;
/* remainder times ln 2 */
x -= px * C1;
x -= px * C2;
/* Approximate exp(remainder ln 2). */
px = (((( P4 * x
+ P3) * x
+ P2) * x
+ P1) * x
+ P0) * x;
qx = (((( x
+ Q4) * x
+ Q3) * x
+ Q2) * x
+ Q1) * x
+ Q0;
xx = x * x;
qx = x + (0.5 * xx + xx * px / qx);
/* exp(x) = exp(k ln 2) exp(remainder ln 2) = 2^k exp(remainder ln 2).
We have qx = exp(remainder ln 2) - 1, so
exp(x) - 1 = 2^k (qx + 1) - 1 = 2^k qx + 2^k - 1. */
px = ldexpl(1.0L, k);
x = px * qx + (px - 1.0);
return x;
}
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