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/*-
* 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.
*/
/* 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 "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( x );
if( isnan(y) )
return( 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 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);
}