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/* log2l.c
*
* Base 2 logarithm, long double precision
*
*
*
* SYNOPSIS:
*
* long double x, y, log2l();
*
* y = log2l( x );
*
*
*
* DESCRIPTION:
*
* Returns the base 2 logarithm of x.
*
* The argument is separated into its exponent and fractional
* parts. If the exponent is between -1 and +1, the (natural)
* 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 30000 9.8e-20 2.7e-20
* IEEE exp(+-10000) 70000 5.4e-20 2.3e-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 -INFINITYL
* log domain: x < 0; returns NANL
*/
/*
Cephes Math Library Release 2.8: May, 1998
Copyright 1984, 1991, 1998 by Stephen L. Moshier
*/
#include <math.h>
/* Coefficients for ln(1+x) = x - x**2/2 + x**3 P(x)/Q(x)
* 1/sqrt(2) <= x < sqrt(2)
* Theoretical peak relative error = 6.2e-22
*/
#ifdef UNK
static long double P[] = {
4.9962495940332550844739E-1L,
1.0767376367209449010438E1L,
7.7671073698359539859595E1L,
2.5620629828144409632571E2L,
4.2401812743503691187826E2L,
3.4258224542413922935104E2L,
1.0747524399916215149070E2L,
};
static long double Q[] = {
/* 1.0000000000000000000000E0,*/
2.3479774160285863271658E1L,
1.9444210022760132894510E2L,
7.7952888181207260646090E2L,
1.6911722418503949084863E3L,
2.0307734695595183428202E3L,
1.2695660352705325274404E3L,
3.2242573199748645407652E2L,
};
#endif
#ifdef IBMPC
static short P[] = {
0xfe72,0xce22,0xd7b9,0xffce,0x3ffd, XPD
0xb778,0x0e34,0x2c71,0xac47,0x4002, XPD
0xea8b,0xc751,0x96f8,0x9b57,0x4005, XPD
0xfeaf,0x6a02,0x67fb,0x801a,0x4007, XPD
0x6b5a,0xf252,0x51ff,0xd402,0x4007, XPD
0x39ce,0x9f76,0x8704,0xab4a,0x4007, XPD
0x1b39,0x740b,0x532e,0xd6f3,0x4005, XPD
};
static short Q[] = {
/*0x0000,0x0000,0x0000,0x8000,0x3fff,*/
0x2f3a,0xbf26,0x93d5,0xbbd6,0x4003, XPD
0x13c8,0x031a,0x2d7b,0xc271,0x4006, XPD
0x449d,0x1993,0xd933,0xc2e1,0x4008, XPD
0x5b65,0x574e,0x8301,0xd365,0x4009, XPD
0xa65d,0x3bd2,0xc043,0xfdd8,0x4009, XPD
0x3b21,0xffea,0x1cf5,0x9eb2,0x4009, XPD
0x545c,0xd708,0x7e62,0xa136,0x4007, XPD
};
#endif
#ifdef MIEEE
static long P[] = {
0x3ffd0000,0xffced7b9,0xce22fe72,
0x40020000,0xac472c71,0x0e34b778,
0x40050000,0x9b5796f8,0xc751ea8b,
0x40070000,0x801a67fb,0x6a02feaf,
0x40070000,0xd40251ff,0xf2526b5a,
0x40070000,0xab4a8704,0x9f7639ce,
0x40050000,0xd6f3532e,0x740b1b39,
};
static long Q[] = {
/*0x3fff0000,0x80000000,0x00000000,*/
0x40030000,0xbbd693d5,0xbf262f3a,
0x40060000,0xc2712d7b,0x031a13c8,
0x40080000,0xc2e1d933,0x1993449d,
0x40090000,0xd3658301,0x574e5b65,
0x40090000,0xfdd8c043,0x3bd2a65d,
0x40090000,0x9eb21cf5,0xffea3b21,
0x40070000,0xa1367e62,0xd708545c,
};
#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,
};
/* log2(e) - 1 */
#define LOG2EA 4.4269504088896340735992e-1L
#endif
#ifdef IBMPC
static short R[] = {
0x6ef4,0xf922,0x7763,0x817b,0x3ff6, XPD
0x15fd,0x1af9,0xde8f,0xb84b,0xbffe, XPD
0x8b96,0x4f8d,0xa53c,0xac6f,0x4002, XPD
0x8932,0xb4e3,0xe8ae,0x8ede,0xc004, XPD
};
static short S[] = {
/*0x0000,0x0000,0x0000,0x8000,0x3fff,*/
0x7ce4,0x1fc9,0xbdc5,0xd19b,0xc003, XPD
0x0af3,0x0d10,0x716f,0xc19e,0x4006, XPD
0x4d7d,0x0f55,0x5d06,0xd64e,0xc007, XPD
};
static short LG2EA[] = {0xc2ef,0x705f,0xeca5,0xe2a8,0x3ffd, XPD};
#define LOG2EA *(long double *)LG2EA
#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 LG2EA[] = {0x3ffd0000,0xe2a8eca5,0x705fc2ef};
#define LOG2EA *(long double *)LG2EA
#endif
#define SQRTH 0.70710678118654752440L
extern long double MINLOGL;
#ifdef ANSIPROT
extern long double frexpl ( long double, int * );
extern long double ldexpl ( long double, int );
extern long double polevll ( long double, void *, int );
extern long double p1evll ( long double, void *, int );
extern int isnanl ( long double );
#else
long double frexpl(), ldexpl(), polevll(), p1evll();
extern int isnanl ();
#endif
#ifdef INFINITIES
extern long double INFINITYL;
#endif
#ifdef NANS
extern long double NANL;
#endif
long double log2l(x)
long double x;
{
VOLATILE long double z;
long double y;
int e;
#ifdef NANS
if( isnanl(x) )
return(x);
#endif
#ifdef INFINITIES
if( x == INFINITYL )
return(x);
#endif
/* Test for domain */
if( x <= 0.0L )
{
if( x == 0.0L )
{
#ifdef INFINITIES
return( -INFINITYL );
#else
mtherr( "log2l", SING );
return( -16384.0L );
#endif
}
else
{
#ifdef NANS
return( NANL );
#else
mtherr( "log2l", DOMAIN );
return( -16384.0L );
#endif
}
}
/* 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;
y = x * ( z * polevll( z, R, 3 ) / p1evll( z, S, 3 ) );
goto done;
}
/* 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, 7 ) );
y = y - ldexpl( z, -1 ); /* -0.5x^2 + ... */
done:
/* Multiply log of fraction by log2(e)
* and base 2 exponent by 1
*
* ***CAUTION***
*
* This sequence of operations is critical and it may
* be horribly defeated by some compiler optimizers.
*/
z = y * LOG2EA;
z += x * LOG2EA;
z += y;
z += x;
z += e;
return( z );
}
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