0,0 → 1,804 |
/* powl.c |
* |
* Power function, long double precision |
* |
* |
* |
* SYNOPSIS: |
* |
* long double x, y, z, powl(); |
* |
* z = powl( x, y ); |
* |
* |
* |
* DESCRIPTION: |
* |
* Computes x raised to the yth power. Analytically, |
* |
* x**y = exp( y log(x) ). |
* |
* Following Cody and Waite, this program uses a lookup table |
* of 2**-i/32 and pseudo extended precision arithmetic to |
* obtain several extra bits of accuracy in both the logarithm |
* and the exponential. |
* |
* |
* |
* ACCURACY: |
* |
* The relative error of pow(x,y) can be estimated |
* by y dl ln(2), where dl is the absolute error of |
* the internally computed base 2 logarithm. At the ends |
* of the approximation interval the logarithm equal 1/32 |
* and its relative error is about 1 lsb = 1.1e-19. Hence |
* the predicted relative error in the result is 2.3e-21 y . |
* |
* Relative error: |
* arithmetic domain # trials peak rms |
* |
* IEEE +-1000 40000 2.8e-18 3.7e-19 |
* .001 < x < 1000, with log(x) uniformly distributed. |
* -1000 < y < 1000, y uniformly distributed. |
* |
* IEEE 0,8700 60000 6.5e-18 1.0e-18 |
* 0.99 < x < 1.01, 0 < y < 8700, uniformly distributed. |
* |
* |
* ERROR MESSAGES: |
* |
* message condition value returned |
* pow overflow x**y > MAXNUM INFINITY |
* pow underflow x**y < 1/MAXNUM 0.0 |
* pow domain x<0 and y noninteger 0.0 |
* |
*/ |
|
/* |
Cephes Math Library Release 2.7: May, 1998 |
Copyright 1984, 1991, 1998 by Stephen L. Moshier |
*/ |
|
/* |
Modified for mingw |
2002-07-22 Danny Smith <dannysmith@users.sourceforge.net> |
*/ |
|
#ifdef __MINGW32__ |
#include "cephes_mconf.h" |
#else |
#include "mconf.h" |
|
static char fname[] = {"powl"}; |
#endif |
|
#ifndef _SET_ERRNO |
#define _SET_ERRNO(x) |
#endif |
|
|
/* Table size */ |
#define NXT 32 |
/* log2(Table size) */ |
#define LNXT 5 |
|
#ifdef UNK |
/* log(1+x) = x - .5x^2 + x^3 * P(z)/Q(z) |
* on the domain 2^(-1/32) - 1 <= x <= 2^(1/32) - 1 |
*/ |
static long double P[] = { |
8.3319510773868690346226E-4L, |
4.9000050881978028599627E-1L, |
1.7500123722550302671919E0L, |
1.4000100839971580279335E0L, |
}; |
static long double Q[] = { |
/* 1.0000000000000000000000E0L,*/ |
5.2500282295834889175431E0L, |
8.4000598057587009834666E0L, |
4.2000302519914740834728E0L, |
}; |
/* A[i] = 2^(-i/32), rounded to IEEE long double precision. |
* If i is even, A[i] + B[i/2] gives additional accuracy. |
*/ |
static long double A[33] = { |
1.0000000000000000000000E0L, |
9.7857206208770013448287E-1L, |
9.5760328069857364691013E-1L, |
9.3708381705514995065011E-1L, |
9.1700404320467123175367E-1L, |
8.9735453750155359320742E-1L, |
8.7812608018664974155474E-1L, |
8.5930964906123895780165E-1L, |
8.4089641525371454301892E-1L, |
8.2287773907698242225554E-1L, |
8.0524516597462715409607E-1L, |
7.8799042255394324325455E-1L, |
7.7110541270397041179298E-1L, |
7.5458221379671136985669E-1L, |
7.3841307296974965571198E-1L, |
7.2259040348852331001267E-1L, |
7.0710678118654752438189E-1L, |
6.9195494098191597746178E-1L, |
6.7712777346844636413344E-1L, |
6.6261832157987064729696E-1L, |
6.4841977732550483296079E-1L, |
6.3452547859586661129850E-1L, |
6.2092890603674202431705E-1L, |
6.0762367999023443907803E-1L, |
5.9460355750136053334378E-1L, |
5.8186242938878875689693E-1L, |
5.6939431737834582684856E-1L, |
5.5719337129794626814472E-1L, |
5.4525386633262882960438E-1L, |
5.3357020033841180906486E-1L, |
5.2213689121370692017331E-1L, |
5.1094857432705833910408E-1L, |
5.0000000000000000000000E-1L, |
}; |
static long double B[17] = { |
0.0000000000000000000000E0L, |
2.6176170809902549338711E-20L, |
-1.0126791927256478897086E-20L, |
1.3438228172316276937655E-21L, |
1.2207982955417546912101E-20L, |
-6.3084814358060867200133E-21L, |
1.3164426894366316434230E-20L, |
-1.8527916071632873716786E-20L, |
1.8950325588932570796551E-20L, |
1.5564775779538780478155E-20L, |
6.0859793637556860974380E-21L, |
-2.0208749253662532228949E-20L, |
1.4966292219224761844552E-20L, |
3.3540909728056476875639E-21L, |
-8.6987564101742849540743E-22L, |
-1.2327176863327626135542E-20L, |
0.0000000000000000000000E0L, |
}; |
|
/* 2^x = 1 + x P(x), |
* on the interval -1/32 <= x <= 0 |
*/ |
static long double R[] = { |
1.5089970579127659901157E-5L, |
1.5402715328927013076125E-4L, |
1.3333556028915671091390E-3L, |
9.6181291046036762031786E-3L, |
5.5504108664798463044015E-2L, |
2.4022650695910062854352E-1L, |
6.9314718055994530931447E-1L, |
}; |
|
#define douba(k) A[k] |
#define doubb(k) B[k] |
#define MEXP (NXT*16384.0L) |
/* The following if denormal numbers are supported, else -MEXP: */ |
#ifdef DENORMAL |
#define MNEXP (-NXT*(16384.0L+64.0L)) |
#else |
#define MNEXP (-NXT*16384.0L) |
#endif |
/* log2(e) - 1 */ |
#define LOG2EA 0.44269504088896340735992L |
#endif |
|
|
#ifdef IBMPC |
static const unsigned short P[] = { |
0xb804,0xa8b7,0xc6f4,0xda6a,0x3ff4, XPD |
0x7de9,0xcf02,0x58c0,0xfae1,0x3ffd, XPD |
0x405a,0x3722,0x67c9,0xe000,0x3fff, XPD |
0xcd99,0x6b43,0x87ca,0xb333,0x3fff, XPD |
}; |
static const unsigned short Q[] = { |
/* 0x0000,0x0000,0x0000,0x8000,0x3fff, */ |
0x6307,0xa469,0x3b33,0xa800,0x4001, XPD |
0xfec2,0x62d7,0xa51c,0x8666,0x4002, XPD |
0xda32,0xd072,0xa5d7,0x8666,0x4001, XPD |
}; |
static const unsigned short A[] = { |
0x0000,0x0000,0x0000,0x8000,0x3fff, XPD |
0x033a,0x722a,0xb2db,0xfa83,0x3ffe, XPD |
0xcc2c,0x2486,0x7d15,0xf525,0x3ffe, XPD |
0xf5cb,0xdcda,0xb99b,0xefe4,0x3ffe, XPD |
0x392f,0xdd24,0xc6e7,0xeac0,0x3ffe, XPD |
0x48a8,0x7c83,0x06e7,0xe5b9,0x3ffe, XPD |
0xe111,0x2a94,0xdeec,0xe0cc,0x3ffe, XPD |
0x3755,0xdaf2,0xb797,0xdbfb,0x3ffe, XPD |
0x6af4,0xd69d,0xfcca,0xd744,0x3ffe, XPD |
0xe45a,0xf12a,0x1d91,0xd2a8,0x3ffe, XPD |
0x80e4,0x1f84,0x8c15,0xce24,0x3ffe, XPD |
0x27a3,0x6e2f,0xbd86,0xc9b9,0x3ffe, XPD |
0xdadd,0x5506,0x2a11,0xc567,0x3ffe, XPD |
0x9456,0x6670,0x4cca,0xc12c,0x3ffe, XPD |
0x36bf,0x580c,0xa39f,0xbd08,0x3ffe, XPD |
0x9ee9,0x62fb,0xaf47,0xb8fb,0x3ffe, XPD |
0x6484,0xf9de,0xf333,0xb504,0x3ffe, XPD |
0x2590,0xd2ac,0xf581,0xb123,0x3ffe, XPD |
0x4ac6,0x42a1,0x3eea,0xad58,0x3ffe, XPD |
0x0ef8,0xea7c,0x5ab4,0xa9a1,0x3ffe, XPD |
0x38ea,0xb151,0xd6a9,0xa5fe,0x3ffe, XPD |
0x6819,0x0c49,0x4303,0xa270,0x3ffe, XPD |
0x11ae,0x91a1,0x3260,0x9ef5,0x3ffe, XPD |
0x5539,0xd54e,0x39b9,0x9b8d,0x3ffe, XPD |
0xa96f,0x8db8,0xf051,0x9837,0x3ffe, XPD |
0x0961,0xfef7,0xefa8,0x94f4,0x3ffe, XPD |
0xc336,0xab11,0xd373,0x91c3,0x3ffe, XPD |
0x53c0,0x45cd,0x398b,0x8ea4,0x3ffe, XPD |
0xd6e7,0xea8b,0xc1e3,0x8b95,0x3ffe, XPD |
0x8527,0x92da,0x0e80,0x8898,0x3ffe, XPD |
0x7b15,0xcc48,0xc367,0x85aa,0x3ffe, XPD |
0xa1d7,0xac2b,0x8698,0x82cd,0x3ffe, XPD |
0x0000,0x0000,0x0000,0x8000,0x3ffe, XPD |
}; |
static const unsigned short B[] = { |
0x0000,0x0000,0x0000,0x0000,0x0000, XPD |
0x1f87,0xdb30,0x18f5,0xf73a,0x3fbd, XPD |
0xac15,0x3e46,0x2932,0xbf4a,0xbfbc, XPD |
0x7944,0xba66,0xa091,0xcb12,0x3fb9, XPD |
0xff78,0x40b4,0x2ee6,0xe69a,0x3fbc, XPD |
0xc895,0x5069,0xe383,0xee53,0xbfbb, XPD |
0x7cde,0x9376,0x4325,0xf8ab,0x3fbc, XPD |
0xa10c,0x25e0,0xc093,0xaefd,0xbfbd, XPD |
0x7d3e,0xea95,0x1366,0xb2fb,0x3fbd, XPD |
0x5d89,0xeb34,0x5191,0x9301,0x3fbd, XPD |
0x80d9,0xb883,0xfb10,0xe5eb,0x3fbb, XPD |
0x045d,0x288c,0xc1ec,0xbedd,0xbfbd, XPD |
0xeded,0x5c85,0x4630,0x8d5a,0x3fbd, XPD |
0x9d82,0xe5ac,0x8e0a,0xfd6d,0x3fba, XPD |
0x6dfd,0xeb58,0xaf14,0x8373,0xbfb9, XPD |
0xf938,0x7aac,0x91cf,0xe8da,0xbfbc, XPD |
0x0000,0x0000,0x0000,0x0000,0x0000, XPD |
}; |
static const unsigned short R[] = { |
0xa69b,0x530e,0xee1d,0xfd2a,0x3fee, XPD |
0xc746,0x8e7e,0x5960,0xa182,0x3ff2, XPD |
0x63b6,0xadda,0xfd6a,0xaec3,0x3ff5, XPD |
0xc104,0xfd99,0x5b7c,0x9d95,0x3ff8, XPD |
0xe05e,0x249d,0x46b8,0xe358,0x3ffa, XPD |
0x5d1d,0x162c,0xeffc,0xf5fd,0x3ffc, XPD |
0x79aa,0xd1cf,0x17f7,0xb172,0x3ffe, XPD |
}; |
|
/* 10 byte sizes versus 12 byte */ |
#define douba(k) (*(long double *)(&A[(sizeof( long double )/2)*(k)])) |
#define doubb(k) (*(long double *)(&B[(sizeof( long double )/2)*(k)])) |
#define MEXP (NXT*16384.0L) |
#ifdef DENORMAL |
#define MNEXP (-NXT*(16384.0L+64.0L)) |
#else |
#define MNEXP (-NXT*16384.0L) |
#endif |
static const |
union |
{ |
unsigned short L[6]; |
long double ld; |
} log2ea = {{0xc2ef,0x705f,0xeca5,0xe2a8,0x3ffd, XPD}}; |
|
#define LOG2EA (log2ea.ld) |
/* |
#define LOG2EA 0.44269504088896340735992L |
*/ |
#endif |
|
#ifdef MIEEE |
static long P[] = { |
0x3ff40000,0xda6ac6f4,0xa8b7b804, |
0x3ffd0000,0xfae158c0,0xcf027de9, |
0x3fff0000,0xe00067c9,0x3722405a, |
0x3fff0000,0xb33387ca,0x6b43cd99, |
}; |
static long Q[] = { |
/* 0x3fff0000,0x80000000,0x00000000, */ |
0x40010000,0xa8003b33,0xa4696307, |
0x40020000,0x8666a51c,0x62d7fec2, |
0x40010000,0x8666a5d7,0xd072da32, |
}; |
static long A[] = { |
0x3fff0000,0x80000000,0x00000000, |
0x3ffe0000,0xfa83b2db,0x722a033a, |
0x3ffe0000,0xf5257d15,0x2486cc2c, |
0x3ffe0000,0xefe4b99b,0xdcdaf5cb, |
0x3ffe0000,0xeac0c6e7,0xdd24392f, |
0x3ffe0000,0xe5b906e7,0x7c8348a8, |
0x3ffe0000,0xe0ccdeec,0x2a94e111, |
0x3ffe0000,0xdbfbb797,0xdaf23755, |
0x3ffe0000,0xd744fcca,0xd69d6af4, |
0x3ffe0000,0xd2a81d91,0xf12ae45a, |
0x3ffe0000,0xce248c15,0x1f8480e4, |
0x3ffe0000,0xc9b9bd86,0x6e2f27a3, |
0x3ffe0000,0xc5672a11,0x5506dadd, |
0x3ffe0000,0xc12c4cca,0x66709456, |
0x3ffe0000,0xbd08a39f,0x580c36bf, |
0x3ffe0000,0xb8fbaf47,0x62fb9ee9, |
0x3ffe0000,0xb504f333,0xf9de6484, |
0x3ffe0000,0xb123f581,0xd2ac2590, |
0x3ffe0000,0xad583eea,0x42a14ac6, |
0x3ffe0000,0xa9a15ab4,0xea7c0ef8, |
0x3ffe0000,0xa5fed6a9,0xb15138ea, |
0x3ffe0000,0xa2704303,0x0c496819, |
0x3ffe0000,0x9ef53260,0x91a111ae, |
0x3ffe0000,0x9b8d39b9,0xd54e5539, |
0x3ffe0000,0x9837f051,0x8db8a96f, |
0x3ffe0000,0x94f4efa8,0xfef70961, |
0x3ffe0000,0x91c3d373,0xab11c336, |
0x3ffe0000,0x8ea4398b,0x45cd53c0, |
0x3ffe0000,0x8b95c1e3,0xea8bd6e7, |
0x3ffe0000,0x88980e80,0x92da8527, |
0x3ffe0000,0x85aac367,0xcc487b15, |
0x3ffe0000,0x82cd8698,0xac2ba1d7, |
0x3ffe0000,0x80000000,0x00000000, |
}; |
static long B[51] = { |
0x00000000,0x00000000,0x00000000, |
0x3fbd0000,0xf73a18f5,0xdb301f87, |
0xbfbc0000,0xbf4a2932,0x3e46ac15, |
0x3fb90000,0xcb12a091,0xba667944, |
0x3fbc0000,0xe69a2ee6,0x40b4ff78, |
0xbfbb0000,0xee53e383,0x5069c895, |
0x3fbc0000,0xf8ab4325,0x93767cde, |
0xbfbd0000,0xaefdc093,0x25e0a10c, |
0x3fbd0000,0xb2fb1366,0xea957d3e, |
0x3fbd0000,0x93015191,0xeb345d89, |
0x3fbb0000,0xe5ebfb10,0xb88380d9, |
0xbfbd0000,0xbeddc1ec,0x288c045d, |
0x3fbd0000,0x8d5a4630,0x5c85eded, |
0x3fba0000,0xfd6d8e0a,0xe5ac9d82, |
0xbfb90000,0x8373af14,0xeb586dfd, |
0xbfbc0000,0xe8da91cf,0x7aacf938, |
0x00000000,0x00000000,0x00000000, |
}; |
static long R[] = { |
0x3fee0000,0xfd2aee1d,0x530ea69b, |
0x3ff20000,0xa1825960,0x8e7ec746, |
0x3ff50000,0xaec3fd6a,0xadda63b6, |
0x3ff80000,0x9d955b7c,0xfd99c104, |
0x3ffa0000,0xe35846b8,0x249de05e, |
0x3ffc0000,0xf5fdeffc,0x162c5d1d, |
0x3ffe0000,0xb17217f7,0xd1cf79aa, |
}; |
|
#define douba(k) (*(long double *)&A[3*(k)]) |
#define doubb(k) (*(long double *)&B[3*(k)]) |
#define MEXP (NXT*16384.0L) |
#ifdef DENORMAL |
#define MNEXP (-NXT*(16384.0L+64.0L)) |
#else |
#define MNEXP (-NXT*16382.0L) |
#endif |
static long L[3] = {0x3ffd0000,0xe2a8eca5,0x705fc2ef}; |
#define LOG2EA (*(long double *)(&L[0])) |
#endif |
|
|
#define F W |
#define Fa Wa |
#define Fb Wb |
#define G W |
#define Ga Wa |
#define Gb u |
#define H W |
#define Ha Wb |
#define Hb Wb |
|
#ifndef __MINGW32__ |
extern long double MAXNUML; |
#endif |
|
static VOLATILE long double z; |
static long double w, W, Wa, Wb, ya, yb, u; |
|
#ifdef __MINGW32__ |
static __inline__ long double reducl( long double ); |
extern long double __powil ( long double, int ); |
extern long double powl ( long double x, long double y); |
#else |
#ifdef ANSIPROT |
extern long double floorl ( long double ); |
extern long double fabsl ( long double ); |
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 long double __powil ( long double, int ); |
extern int isnanl ( long double ); |
extern int isfinitel ( long double ); |
static long double reducl( long double ); |
extern int signbitl ( long double ); |
#else |
long double floorl(), fabsl(), frexpl(), ldexpl(); |
long double polevll(), p1evll(), __powil(); |
static long double reducl(); |
int isnanl(), isfinitel(), signbitl(); |
#endif /* __MINGW32__ */ |
|
#ifdef INFINITIES |
extern long double INFINITYL; |
#else |
#define INFINITYL MAXNUML |
#endif |
|
#ifdef NANS |
extern long double NANL; |
#endif |
#ifdef MINUSZERO |
extern long double NEGZEROL; |
#endif |
|
#endif /* __MINGW32__ */ |
|
#ifdef __MINGW32__ |
|
/* No error checking. We handle Infs and zeros ourselves. */ |
static __inline__ long double |
__fast_ldexpl (long double x, int expn) |
{ |
long double res; |
__asm__ ("fscale" |
: "=t" (res) |
: "0" (x), "u" ((long double) expn)); |
return res; |
} |
|
#define ldexpl __fast_ldexpl |
|
#endif |
|
|
long double powl( x, y ) |
long double x, y; |
{ |
/* double F, Fa, Fb, G, Ga, Gb, H, Ha, Hb */ |
int i, nflg, iyflg, yoddint; |
long e; |
|
if( y == 0.0L ) |
return( 1.0L ); |
|
#ifdef NANS |
if( isnanl(x) ) |
{ |
_SET_ERRNO (EDOM); |
return( x ); |
} |
if( isnanl(y) ) |
{ |
_SET_ERRNO (EDOM); |
return( y ); |
} |
#endif |
|
if( y == 1.0L ) |
return( x ); |
|
if( isinfl(y) && (x == -1.0L || x == 1.0L) ) |
return( y ); |
|
if( x == 1.0L ) |
return( 1.0L ); |
|
if( y >= MAXNUML ) |
{ |
_SET_ERRNO (ERANGE); |
#ifdef INFINITIES |
if( x > 1.0L ) |
return( INFINITYL ); |
#else |
if( x > 1.0L ) |
return( MAXNUML ); |
#endif |
if( x > 0.0L && x < 1.0L ) |
return( 0.0L ); |
#ifdef INFINITIES |
if( x < -1.0L ) |
return( INFINITYL ); |
#else |
if( x < -1.0L ) |
return( MAXNUML ); |
#endif |
if( x > -1.0L && x < 0.0L ) |
return( 0.0L ); |
} |
if( y <= -MAXNUML ) |
{ |
_SET_ERRNO (ERANGE); |
if( x > 1.0L ) |
return( 0.0L ); |
#ifdef INFINITIES |
if( x > 0.0L && x < 1.0L ) |
return( INFINITYL ); |
#else |
if( x > 0.0L && x < 1.0L ) |
return( MAXNUML ); |
#endif |
if( x < -1.0L ) |
return( 0.0L ); |
#ifdef INFINITIES |
if( x > -1.0L && x < 0.0L ) |
return( INFINITYL ); |
#else |
if( x > -1.0L && x < 0.0L ) |
return( MAXNUML ); |
#endif |
} |
if( x >= MAXNUML ) |
{ |
#if INFINITIES |
if( y > 0.0L ) |
return( INFINITYL ); |
#else |
if( y > 0.0L ) |
return( MAXNUML ); |
#endif |
return( 0.0L ); |
} |
|
w = floorl(y); |
/* Set iyflg to 1 if y is an integer. */ |
iyflg = 0; |
if( w == y ) |
iyflg = 1; |
|
/* Test for odd integer y. */ |
yoddint = 0; |
if( iyflg ) |
{ |
ya = fabsl(y); |
ya = floorl(0.5L * ya); |
yb = 0.5L * fabsl(w); |
if( ya != yb ) |
yoddint = 1; |
} |
|
if( x <= -MAXNUML ) |
{ |
if( y > 0.0L ) |
{ |
#ifdef INFINITIES |
if( yoddint ) |
return( -INFINITYL ); |
return( INFINITYL ); |
#else |
if( yoddint ) |
return( -MAXNUML ); |
return( MAXNUML ); |
#endif |
} |
if( y < 0.0L ) |
{ |
#ifdef MINUSZERO |
if( yoddint ) |
return( NEGZEROL ); |
#endif |
return( 0.0 ); |
} |
} |
|
|
nflg = 0; /* flag = 1 if x<0 raised to integer power */ |
if( x <= 0.0L ) |
{ |
if( x == 0.0L ) |
{ |
if( y < 0.0 ) |
{ |
#ifdef MINUSZERO |
if( signbitl(x) && yoddint ) |
return( -INFINITYL ); |
#endif |
#ifdef INFINITIES |
return( INFINITYL ); |
#else |
return( MAXNUML ); |
#endif |
} |
if( y > 0.0 ) |
{ |
#ifdef MINUSZERO |
if( signbitl(x) && yoddint ) |
return( NEGZEROL ); |
#endif |
return( 0.0 ); |
} |
if( y == 0.0L ) |
return( 1.0L ); /* 0**0 */ |
else |
return( 0.0L ); /* 0**y */ |
} |
else |
{ |
if( iyflg == 0 ) |
{ /* noninteger power of negative number */ |
mtherr( fname, DOMAIN ); |
_SET_ERRNO (EDOM); |
#ifdef NANS |
return(NANL); |
#else |
return(0.0L); |
#endif |
} |
nflg = 1; |
} |
} |
|
/* Integer power of an integer. */ |
|
if( iyflg ) |
{ |
i = w; |
w = floorl(x); |
if( (w == x) && (fabsl(y) < 32768.0) ) |
{ |
w = __powil( x, (int) y ); |
return( w ); |
} |
} |
|
|
if( nflg ) |
x = fabsl(x); |
|
/* separate significand from exponent */ |
x = frexpl( x, &i ); |
e = i; |
|
/* find significand in antilog table A[] */ |
i = 1; |
if( x <= douba(17) ) |
i = 17; |
if( x <= douba(i+8) ) |
i += 8; |
if( x <= douba(i+4) ) |
i += 4; |
if( x <= douba(i+2) ) |
i += 2; |
if( x >= douba(1) ) |
i = -1; |
i += 1; |
|
|
/* Find (x - A[i])/A[i] |
* in order to compute log(x/A[i]): |
* |
* log(x) = log( a x/a ) = log(a) + log(x/a) |
* |
* log(x/a) = log(1+v), v = x/a - 1 = (x-a)/a |
*/ |
x -= douba(i); |
x -= doubb(i/2); |
x /= douba(i); |
|
|
/* rational approximation for log(1+v): |
* |
* log(1+v) = v - v**2/2 + v**3 P(v) / Q(v) |
*/ |
z = x*x; |
w = x * ( z * polevll( x, P, 3 ) / p1evll( x, Q, 3 ) ); |
w = w - ldexpl( z, -1 ); /* w - 0.5 * z */ |
|
/* Convert to base 2 logarithm: |
* multiply by log2(e) = 1 + LOG2EA |
*/ |
z = LOG2EA * w; |
z += w; |
z += LOG2EA * x; |
z += x; |
|
/* Compute exponent term of the base 2 logarithm. */ |
w = -i; |
w = ldexpl( w, -LNXT ); /* divide by NXT */ |
w += e; |
/* Now base 2 log of x is w + z. */ |
|
/* Multiply base 2 log by y, in extended precision. */ |
|
/* separate y into large part ya |
* and small part yb less than 1/NXT |
*/ |
ya = reducl(y); |
yb = y - ya; |
|
/* (w+z)(ya+yb) |
* = w*ya + w*yb + z*y |
*/ |
F = z * y + w * yb; |
Fa = reducl(F); |
Fb = F - Fa; |
|
G = Fa + w * ya; |
Ga = reducl(G); |
Gb = G - Ga; |
|
H = Fb + Gb; |
Ha = reducl(H); |
w = ldexpl( Ga + Ha, LNXT ); |
|
/* Test the power of 2 for overflow */ |
if( w > MEXP ) |
{ |
_SET_ERRNO (ERANGE); |
mtherr( fname, OVERFLOW ); |
return( MAXNUML ); |
} |
|
if( w < MNEXP ) |
{ |
_SET_ERRNO (ERANGE); |
mtherr( fname, UNDERFLOW ); |
return( 0.0L ); |
} |
|
e = w; |
Hb = H - Ha; |
|
if( Hb > 0.0L ) |
{ |
e += 1; |
Hb -= (1.0L/NXT); /*0.0625L;*/ |
} |
|
/* Now the product y * log2(x) = Hb + e/NXT. |
* |
* Compute base 2 exponential of Hb, |
* where -0.0625 <= Hb <= 0. |
*/ |
z = Hb * polevll( Hb, R, 6 ); /* z = 2**Hb - 1 */ |
|
/* Express e/NXT as an integer plus a negative number of (1/NXT)ths. |
* Find lookup table entry for the fractional power of 2. |
*/ |
if( e < 0 ) |
i = 0; |
else |
i = 1; |
i = e/NXT + i; |
e = NXT*i - e; |
w = douba( e ); |
z = w * z; /* 2**-e * ( 1 + (2**Hb-1) ) */ |
z = z + w; |
z = ldexpl( z, i ); /* multiply by integer power of 2 */ |
|
if( nflg ) |
{ |
/* For negative x, |
* find out if the integer exponent |
* is odd or even. |
*/ |
w = ldexpl( y, -1 ); |
w = floorl(w); |
w = ldexpl( w, 1 ); |
if( w != y ) |
z = -z; /* odd exponent */ |
} |
|
return( z ); |
} |
|
static __inline__ long double |
__convert_inf_to_maxnum(long double x) |
{ |
if (isinf(x)) |
return (x > 0.0L ? MAXNUML : -MAXNUML); |
else |
return x; |
} |
|
|
/* Find a multiple of 1/NXT that is within 1/NXT of x. */ |
static __inline__ long double reducl(x) |
long double x; |
{ |
long double t; |
|
/* If the call to ldexpl overflows, set it to MAXNUML. |
This avoids Inf - Inf = Nan result when calculating the 'small' |
part of a reduction. Instead, the small part becomes Inf, |
causing under/overflow when adding it to the 'large' part. |
There must be a cleaner way of doing this. */ |
t = __convert_inf_to_maxnum (ldexpl( x, LNXT )); |
t = floorl( t ); |
t = ldexpl( t, -LNXT ); |
return(t); |
} |