663 lines
13 KiB
C
663 lines
13 KiB
C
/*-
|
|
* Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
|
|
*
|
|
* Permission to use, copy, modify, and distribute this software for any
|
|
* purpose with or without fee is hereby granted, provided that the above
|
|
* copyright notice and this permission notice appear in all copies.
|
|
*
|
|
* THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
|
|
* WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
|
|
* MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
|
|
* ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
|
|
* WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
|
|
* ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
|
|
* OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
|
|
*/
|
|
|
|
#include <sys/cdefs.h>
|
|
__FBSDID("$FreeBSD$");
|
|
|
|
#include <math.h>
|
|
|
|
#include "../ld/math_private.h"
|
|
|
|
/*
|
|
* Polynomial evaluator:
|
|
* P[0] x^n + P[1] x^(n-1) + ... + P[n]
|
|
*/
|
|
static inline long double
|
|
__polevll(long double x, long double *PP, int n)
|
|
{
|
|
long double y;
|
|
long double *P;
|
|
|
|
P = PP;
|
|
y = *P++;
|
|
do {
|
|
y = y * x + *P++;
|
|
} while (--n);
|
|
|
|
return (y);
|
|
}
|
|
|
|
/*
|
|
* Polynomial evaluator:
|
|
* x^n + P[0] x^(n-1) + P[1] x^(n-2) + ... + P[n]
|
|
*/
|
|
static inline long double
|
|
__p1evll(long double x, long double *PP, int n)
|
|
{
|
|
long double y;
|
|
long double *P;
|
|
|
|
P = PP;
|
|
n -= 1;
|
|
y = x + *P++;
|
|
do {
|
|
y = y * x + *P++;
|
|
} while (--n);
|
|
|
|
return (y);
|
|
}
|
|
|
|
/* 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
|
|
*
|
|
*/
|
|
|
|
#include <sys/cdefs.h>
|
|
__FBSDID("$FreeBSD$");
|
|
|
|
#include <float.h>
|
|
#include <math.h>
|
|
|
|
#include "../ld/math_private.h"
|
|
|
|
/* Table size */
|
|
#define NXT 32
|
|
/* log2(Table size) */
|
|
#define LNXT 5
|
|
|
|
/* 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: */
|
|
#define MNEXP (-NXT*(16384.0L+64.0L))
|
|
/* log2(e) - 1 */
|
|
#define LOG2EA 0.44269504088896340735992L
|
|
|
|
#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
|
|
|
|
static const long double MAXLOGL = 1.1356523406294143949492E4L;
|
|
static const long double MINLOGL = -1.13994985314888605586758E4L;
|
|
static const long double LOGE2L = 6.9314718055994530941723E-1L;
|
|
static volatile long double z;
|
|
static long double w, W, Wa, Wb, ya, yb, u;
|
|
static const long double huge = 0x1p10000L;
|
|
#if 0 /* XXX Prevent gcc from erroneously constant folding this. */
|
|
static const long double twom10000 = 0x1p-10000L;
|
|
#else
|
|
static volatile long double twom10000 = 0x1p-10000L;
|
|
#endif
|
|
|
|
static long double reducl( long double );
|
|
static long double powil ( long double, int );
|
|
|
|
long double
|
|
powl(long double x, long double 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 );
|
|
|
|
if( x == 1.0L )
|
|
return( 1.0L );
|
|
|
|
if( isnan(x) )
|
|
return ( nan_mix(x, y) );
|
|
if( isnan(y) )
|
|
return ( nan_mix(x, y) );
|
|
|
|
if( y == 1.0L )
|
|
return( x );
|
|
|
|
if( !isfinite(y) && x == -1.0L )
|
|
return( 1.0L );
|
|
|
|
if( y >= LDBL_MAX )
|
|
{
|
|
if( x > 1.0L )
|
|
return( INFINITY );
|
|
if( x > 0.0L && x < 1.0L )
|
|
return( 0.0L );
|
|
if( x < -1.0L )
|
|
return( INFINITY );
|
|
if( x > -1.0L && x < 0.0L )
|
|
return( 0.0L );
|
|
}
|
|
if( y <= -LDBL_MAX )
|
|
{
|
|
if( x > 1.0L )
|
|
return( 0.0L );
|
|
if( x > 0.0L && x < 1.0L )
|
|
return( INFINITY );
|
|
if( x < -1.0L )
|
|
return( 0.0L );
|
|
if( x > -1.0L && x < 0.0L )
|
|
return( INFINITY );
|
|
}
|
|
if( x >= LDBL_MAX )
|
|
{
|
|
if( y > 0.0L )
|
|
return( INFINITY );
|
|
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 <= -LDBL_MAX )
|
|
{
|
|
if( y > 0.0L )
|
|
{
|
|
if( yoddint )
|
|
return( -INFINITY );
|
|
return( INFINITY );
|
|
}
|
|
if( y < 0.0L )
|
|
{
|
|
if( yoddint )
|
|
return( -0.0L );
|
|
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 )
|
|
{
|
|
if( signbit(x) && yoddint )
|
|
return( -INFINITY );
|
|
return( INFINITY );
|
|
}
|
|
if( y > 0.0 )
|
|
{
|
|
if( signbit(x) && yoddint )
|
|
return( -0.0L );
|
|
return( 0.0 );
|
|
}
|
|
if( y == 0.0L )
|
|
return( 1.0L ); /* 0**0 */
|
|
else
|
|
return( 0.0L ); /* 0**y */
|
|
}
|
|
else
|
|
{
|
|
if( iyflg == 0 )
|
|
return (x - x) / (x - x); /* (x<0)**(non-int) is NaN */
|
|
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 )
|
|
return (huge * huge); /* overflow */
|
|
|
|
if( w < MNEXP )
|
|
return (twom10000 * twom10000); /* underflow */
|
|
|
|
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 );
|
|
}
|
|
|
|
|
|
/* Find a multiple of 1/NXT that is within 1/NXT of x. */
|
|
static inline long double
|
|
reducl(long double x)
|
|
{
|
|
long double t;
|
|
|
|
t = ldexpl( x, LNXT );
|
|
t = floorl( t );
|
|
t = ldexpl( t, -LNXT );
|
|
return(t);
|
|
}
|
|
|
|
/* powil.c
|
|
*
|
|
* Real raised to integer power, long double precision
|
|
*
|
|
*
|
|
*
|
|
* SYNOPSIS:
|
|
*
|
|
* long double x, y, powil();
|
|
* int n;
|
|
*
|
|
* y = powil( x, n );
|
|
*
|
|
*
|
|
*
|
|
* DESCRIPTION:
|
|
*
|
|
* Returns argument x raised to the nth power.
|
|
* The routine efficiently decomposes n as a sum of powers of
|
|
* two. The desired power is a product of two-to-the-kth
|
|
* powers of x. Thus to compute the 32767 power of x requires
|
|
* 28 multiplications instead of 32767 multiplications.
|
|
*
|
|
*
|
|
*
|
|
* ACCURACY:
|
|
*
|
|
*
|
|
* Relative error:
|
|
* arithmetic x domain n domain # trials peak rms
|
|
* IEEE .001,1000 -1022,1023 50000 4.3e-17 7.8e-18
|
|
* IEEE 1,2 -1022,1023 20000 3.9e-17 7.6e-18
|
|
* IEEE .99,1.01 0,8700 10000 3.6e-16 7.2e-17
|
|
*
|
|
* Returns MAXNUM on overflow, zero on underflow.
|
|
*
|
|
*/
|
|
|
|
static long double
|
|
powil(long double x, int nn)
|
|
{
|
|
long double ww, y;
|
|
long double s;
|
|
int n, e, sign, asign, lx;
|
|
|
|
if( x == 0.0L )
|
|
{
|
|
if( nn == 0 )
|
|
return( 1.0L );
|
|
else if( nn < 0 )
|
|
return( LDBL_MAX );
|
|
else
|
|
return( 0.0L );
|
|
}
|
|
|
|
if( nn == 0 )
|
|
return( 1.0L );
|
|
|
|
|
|
if( x < 0.0L )
|
|
{
|
|
asign = -1;
|
|
x = -x;
|
|
}
|
|
else
|
|
asign = 0;
|
|
|
|
|
|
if( nn < 0 )
|
|
{
|
|
sign = -1;
|
|
n = -nn;
|
|
}
|
|
else
|
|
{
|
|
sign = 1;
|
|
n = nn;
|
|
}
|
|
|
|
/* Overflow detection */
|
|
|
|
/* Calculate approximate logarithm of answer */
|
|
s = x;
|
|
s = frexpl( s, &lx );
|
|
e = (lx - 1)*n;
|
|
if( (e == 0) || (e > 64) || (e < -64) )
|
|
{
|
|
s = (s - 7.0710678118654752e-1L) / (s + 7.0710678118654752e-1L);
|
|
s = (2.9142135623730950L * s - 0.5L + lx) * nn * LOGE2L;
|
|
}
|
|
else
|
|
{
|
|
s = LOGE2L * e;
|
|
}
|
|
|
|
if( s > MAXLOGL )
|
|
return (huge * huge); /* overflow */
|
|
|
|
if( s < MINLOGL )
|
|
return (twom10000 * twom10000); /* underflow */
|
|
/* Handle tiny denormal answer, but with less accuracy
|
|
* since roundoff error in 1.0/x will be amplified.
|
|
* The precise demarcation should be the gradual underflow threshold.
|
|
*/
|
|
if( s < (-MAXLOGL+2.0L) )
|
|
{
|
|
x = 1.0L/x;
|
|
sign = -sign;
|
|
}
|
|
|
|
/* First bit of the power */
|
|
if( n & 1 )
|
|
y = x;
|
|
|
|
else
|
|
{
|
|
y = 1.0L;
|
|
asign = 0;
|
|
}
|
|
|
|
ww = x;
|
|
n >>= 1;
|
|
while( n )
|
|
{
|
|
ww = ww * ww; /* arg to the 2-to-the-kth power */
|
|
if( n & 1 ) /* if that bit is set, then include in product */
|
|
y *= ww;
|
|
n >>= 1;
|
|
}
|
|
|
|
if( asign )
|
|
y = -y; /* odd power of negative number */
|
|
if( sign < 0 )
|
|
y = 1.0L/y;
|
|
return(y);
|
|
}
|