Output of file : LOGL.C contained in archive : CEPHES22.ZIP
/* logl.c
*
* Natural logarithm, long double precision
*
*
*
* SYNOPSIS:
*
* long double x, y, logl();
*
* y = logl( x );
*
*
*
* DESCRIPTION:
*
* Returns the base e (2.718...) logarithm of x.
*
* The argument is separated into its exponent and fractional
* parts. If the exponent is between -1 and +1, the logarithm
* of the fraction is approximated by
*
* log(1+x) = x - 0.5 x**2 + x**3 P(x)/Q(x).
*
* Otherwise, setting z = 2(x-1)/x+1),
*
* log(x) = z + z**3 P(z)/Q(z).
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE 0.5, 2.0 150000 8.71e-20 2.75e-20
* IEEE exp(+-10000) 100000 5.39e-20 2.34e-20
*
* In the tests over the interval exp(+-10000), the logarithms
* of the random arguments were uniformly distributed over
* [-10000, +10000].
*
* ERROR MESSAGES:
*
* log singularity: x = 0; returns MINLOG
* log domain: x < 0; returns MINLOG
*/

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

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

/* Coefficients for log(1+x) = x - x**2/2 + x**3 P(x)/Q(x)
* 1/sqrt(2) <= x < sqrt(2)
* Theoretical peak relative error = 2.32e-20
*/
#ifdef UNK
static long double P[] = {
4.5270000862445199635215E-5,
4.9854102823193375972212E-1,
6.5787325942061044846969E0,
2.9911919328553073277375E1,
6.0949667980987787057556E1,
5.7112963590585538103336E1,
2.0039553499201281259648E1,
};
static long double Q[] = {
/* 1.0000000000000000000000E0,*/
1.5062909083469192043167E1,
8.3047565967967209469434E1,
2.2176239823732856465394E2,
3.0909872225312059774938E2,
2.1642788614495947685003E2,
6.0118660497603843919306E1,
};
#endif

#ifdef IBMPC
static short P[] = {
0x51b9,0x9cae,0x4b15,0xbde0,0x3ff0,
0x19cf,0xf0d4,0xc507,0xff40,0x3ffd,
0x9942,0xa7d2,0xfa37,0xd284,0x4001,
0x8445,0x619a,0x75c3,0xf3cc,0x4004,
0x81ab,0x3cd0,0xacba,0xe473,0x4004,
0x4cbf,0xcc18,0x016c,0xa051,0x4003,
};
static short Q[] = {
/*0x0000,0x0000,0x0000,0x8000,0x3fff,*/
0xb8b7,0x81f1,0xacf4,0xf101,0x4002,
0xbc31,0x09a4,0x5a91,0xa618,0x4005,
0xaeec,0xe7da,0x2c87,0xddc3,0x4006,
0x2bde,0x4845,0xa2ee,0x9a8c,0x4007,
0x3120,0x4703,0x89f2,0xd86d,0x4006,
0x7347,0x3224,0x8223,0xf079,0x4004,
};
#endif

#ifdef MIEEE
static long P[] = {
0x3ff00000,0xbde04b15,0x9cae51b9,
0x3ffd0000,0xff40c507,0xf0d419cf,
0x40010000,0xd284fa37,0xa7d29942,
0x40040000,0xf3cc75c3,0x619a8445,
0x40040000,0xe473acba,0x3cd081ab,
0x40030000,0xa051016c,0xcc184cbf,
};
static long Q[] = {
/*0x3fff0000,0x80000000,0x00000000,*/
0x40020000,0xf101acf4,0x81f1b8b7,
0x40050000,0xa6185a91,0x09a4bc31,
0x40060000,0xddc32c87,0xe7daaeec,
0x40070000,0x9a8ca2ee,0x48452bde,
0x40060000,0xd86d89f2,0x47033120,
0x40040000,0xf0798223,0x32247347,
};
#endif

/* Coefficients for log(x) = z + z^3 P(z^2)/Q(z^2),
* where z = 2(x-1)/(x+1)
* 1/sqrt(2) <= x < sqrt(2)
* Theoretical peak relative error = 6.16e-22
*/

#ifdef UNK
static long double R[4] = {
1.9757429581415468984296E-3L,
-7.1990767473014147232598E-1L,
1.0777257190312272158094E1L,
-3.5717684488096787370998E1L,
};
static long double S[4] = {
/* 1.00000000000000000000E0L,*/
-2.6201045551331104417768E1L,
1.9361891836232102174846E2L,
-4.2861221385716144629696E2L,
};
static long double C1 = 6.9314575195312500000000E-1;
static long double C2 = 1.4286068203094172321215E-6;
#endif
#ifdef IBMPC
static short R[20] = {
0x6ef4,0xf922,0x7763,0x817b,0x3ff6,
0x15fd,0x1af9,0xde8f,0xb84b,0xbffe,
0x8b96,0x4f8d,0xa53c,0xac6f,0x4002,
0x8932,0xb4e3,0xe8ae,0x8ede,0xc004,
};
static short S[15] = {
/*0x0000,0x0000,0x0000,0x8000,0x3fff,*/
0x7ce4,0x1fc9,0xbdc5,0xd19b,0xc003,
0x0af3,0x0d10,0x716f,0xc19e,0x4006,
0x4d7d,0x0f55,0x5d06,0xd64e,0xc007,
};
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 R[12] = {
0x3ff60000,0x817b7763,0xf9226ef4,
0xbffe0000,0xb84bde8f,0x1af915fd,
0x40020000,0xac6fa53c,0x4f8d8b96,
0xc0040000,0x8edee8ae,0xb4e38932,
};
static long S[9] = {
/*0x3fff0000,0x80000000,0x00000000,*/
0xc0030000,0xd19bbdc5,0x1fc97ce4,
0x40060000,0xc19e716f,0x0d100af3,
0xc0070000,0xd64e5d06,0x0f554d7d,
};
static long sc1[] = {0x3ffe0000,0xb1720000,0x00000000};
#define C1 (*(long double *)sc1)
static long sc2[] = {0x3feb0000,0xbfbe8e7b,0xcd5e4f1e};
#define C2 (*(long double *)sc2)
#endif

#define SQRTH 0.70710678118654752440
extern long double MINLOGL;
long double frexpl(), ldexpl(), polevll(), p1evll();

long double logl(x)
long double x;
{
long double y, z;
int e;

/* Test for domain */
if( x <= 0.0L )
{
if( x == 0.0L )
mtherr( fname, SING );
else
mtherr( fname, DOMAIN );
return( MINLOGL );
}

/* separate mantissa from exponent */

/* Note, frexp is used so that denormal numbers
* will be handled properly.
*/
x = frexpl( x, &e );

/* logarithm using log(x) = z + z**3 P(z)/Q(z),
* where z = 2(x-1)/x+1)
*/
if( (e > 2) || (e < -2) )
{
if( x < SQRTH )
{ /* 2( 2x-1 )/( 2x+1 ) */
e -= 1;
z = x - 0.5L;
y = 0.5L * z + 0.5L;
}
else
{ /* 2 (x-1)/(x+1) */
z = x - 0.5L;
z -= 0.5L;
y = 0.5L * x + 0.5L;
}
x = z / y;
z = x*x;
z = x * ( z * polevll( z, R, 3 ) / p1evll( z, S, 3 ) );
z = z + e * C2;
z = z + x;
z = z + e * C1;
return( z );
}

/* logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) */

if( x < SQRTH )
{
e -= 1;
x = ldexpl( x, 1 ) - 1.0L; /* 2x - 1 */
}
else
{
x = x - 1.0L;
}
z = x*x;
y = x * ( z * polevll( x, P, 6 ) / p1evll( x, Q, 6 ) );
y = y + e * C2;
z = y - ldexpl( z, -1 ); /* y - 0.5 * z */
/* Note, the sum of above terms does not exceed x/4,
* so it contributes at most about 1/4 lsb to the error.
*/
z = z + x;
z = z + e * C1; /* This sum has an error of 1/2 lsb. */
return( z );
}

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