/* md_pow.c
*
* Power function
*
*
*
* SYNOPSIS:
*
* double x, y, z, md_pow();
*
* z = md_pow( x, y );
*
*
*
* DESCRIPTION:
*
* Computes x raised to the yth power. Analytically,
*
* x**y = md_exp( y md_log(x) ).
*
* Following Cody and Waite, this program uses a lookup table
* of 2**-i/16 and pseudo extended precision arithmetic to
* obtain an extra three bits of accuracy in both the logarithm
* and the exponential.
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE -26,26 30000 4.2e-16 7.7e-17
* DEC -26,26 60000 4.8e-17 9.1e-18
* 1/26 < x < 26, with md_log(x) uniformly distributed.
* -26 < y < 26, y uniformly distributed.
* IEEE 0,8700 30000 1.5e-14 2.1e-15
* 0.99 < x < 1.01, 0 < y < 8700, uniformly distributed.
*
*
* ERROR MESSAGES:
*
* message condition value returned
* md_pow overflow x**y > MAXNUM INFINITY
* md_pow underflow x**y < 1/MAXNUM 0.0
* md_pow domain x<0 and y noninteger 0.0
*
*/
/*
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1995, 2000 by Stephen L. Moshier
*/
#include "mconf.h"
static char fname[] = {"md_pow"};
#define SQRTH 0.70710678118654752440
#ifdef UNK
static double P[] = {
4.97778295871696322025E-1,
3.73336776063286838734E0,
7.69994162726912503298E0,
4.66651806774358464979E0
};
static double Q[] = {
/* 1.00000000000000000000E0, */
9.33340916416696166113E0,
2.79999886606328401649E1,
3.35994905342304405431E1,
1.39995542032307539578E1
};
/* 2^(-i/16), IEEE precision */
static double A[] = {
1.00000000000000000000E0,
9.57603280698573700036E-1,
9.17004043204671215328E-1,
8.78126080186649726755E-1,
8.40896415253714502036E-1,
8.05245165974627141736E-1,
7.71105412703970372057E-1,
7.38413072969749673113E-1,
7.07106781186547572737E-1,
6.77127773468446325644E-1,
6.48419777325504820276E-1,
6.20928906036742001007E-1,
5.94603557501360513449E-1,
5.69394317378345782288E-1,
5.45253866332628844837E-1,
5.22136891213706877402E-1,
5.00000000000000000000E-1
};
static double B[] = {
0.00000000000000000000E0,
1.64155361212281360176E-17,
4.09950501029074826006E-17,
3.97491740484881042808E-17,
-4.83364665672645672553E-17,
1.26912513974441574796E-17,
1.99100761573282305549E-17,
-1.52339103990623557348E-17,
0.00000000000000000000E0
};
static double R[] = {
1.49664108433729301083E-5,
1.54010762792771901396E-4,
1.33335476964097721140E-3,
9.61812908476554225149E-3,
5.55041086645832347466E-2,
2.40226506959099779976E-1,
6.93147180559945308821E-1
};
#define douba(k) A[k]
#define doubb(k) B[k]
#define MEXP 16383.0
#ifdef DENORMAL
#define MNEXP -17183.0
#else
#define MNEXP -16383.0
#endif
#endif
#ifdef DEC
static unsigned short P[] = {
0037776,0156313,0175332,0163602,
0040556,0167577,0052366,0174245,
0040766,0062753,0175707,0055564,
0040625,0052035,0131344,0155636,
};
static unsigned short Q[] = {
/*0040200,0000000,0000000,0000000,*/
0041025,0052644,0154404,0105155,
0041337,0177772,0007016,0047646,
0041406,0062740,0154273,0020020,
0041137,0177054,0106127,0044555,
};
static unsigned short A[] = {
0040200,0000000,0000000,0000000,
0040165,0022575,0012444,0103314,
0040152,0140306,0163735,0022071,
0040140,0146336,0166052,0112341,
0040127,0042374,0145326,0116553,
0040116,0022214,0012437,0102201,
0040105,0063452,0010525,0003333,
0040075,0004243,0117530,0006067,
0040065,0002363,0031771,0157145,
0040055,0054076,0165102,0120513,
0040045,0177326,0124661,0050471,
0040036,0172462,0060221,0120422,
0040030,0033760,0050615,0134251,
0040021,0141723,0071653,0010703,
0040013,0112701,0161752,0105727,
0040005,0125303,0063714,0044173,
0040000,0000000,0000000,0000000
};
static unsigned short B[] = {
0000000,0000000,0000000,0000000,
0021473,0040265,0153315,0140671,
0121074,0062627,0042146,0176454,
0121413,0003524,0136332,0066212,
0121767,0046404,0166231,0012553,
0121257,0015024,0002357,0043574,
0021736,0106532,0043060,0056206,
0121310,0020334,0165705,0035326,
0000000,0000000,0000000,0000000
};
static unsigned short R[] = {
0034173,0014076,0137624,0115771,
0035041,0076763,0003744,0111311,
0035656,0141766,0041127,0074351,
0036435,0112533,0073611,0116664,
0037143,0054106,0134040,0152223,
0037565,0176757,0176026,0025551,
0040061,0071027,0173721,0147572
};
/*
static double R[] = {
0.14928852680595608186e-4,
0.15400290440989764601e-3,
0.13333541313585784703e-2,
0.96181290595172416964e-2,
0.55504108664085595326e-1,
0.24022650695909537056e0,
0.69314718055994529629e0
};
*/
#define douba(k) (*(double *)&A[(k)<<2])
#define doubb(k) (*(double *)&B[(k)<<2])
#define MEXP 2031.0
#define MNEXP -2031.0
#endif
#ifdef IBMPC
static unsigned short P[] = {
0x5cf0,0x7f5b,0xdb99,0x3fdf,
0xdf15,0xea9e,0xddef,0x400d,
0xeb6f,0x7f78,0xccbd,0x401e,
0x9b74,0xb65c,0xaa83,0x4012,
};
static unsigned short Q[] = {
/*0x0000,0x0000,0x0000,0x3ff0,*/
0x914e,0x9b20,0xaab4,0x4022,
0xc9f5,0x41c1,0xffff,0x403b,
0x6402,0x1b17,0xccbc,0x4040,
0xe92e,0x918a,0xffc5,0x402b,
};
static unsigned short A[] = {
0x0000,0x0000,0x0000,0x3ff0,
0x90da,0xa2a4,0xa4af,0x3fee,
0xa487,0xdcfb,0x5818,0x3fed,
0x529c,0xdd85,0x199b,0x3fec,
0xd3ad,0x995a,0xe89f,0x3fea,
0xf090,0x82a3,0xc491,0x3fe9,
0xa0db,0x422a,0xace5,0x3fe8,
0x0187,0x73eb,0xa114,0x3fe7,
0x3bcd,0x667f,0xa09e,0x3fe6,
0x5429,0xdd48,0xab07,0x3fe5,
0x2a27,0xd536,0xbfda,0x3fe4,
0x3422,0x4c12,0xdea6,0x3fe3,
0xb715,0x0a31,0x06fe,0x3fe3,
0x6238,0x6e75,0x387a,0x3fe2,
0x517b,0x3c7d,0x72b8,0x3fe1,
0x890f,0x6cf9,0xb558,0x3fe0,
0x0000,0x0000,0x0000,0x3fe0
};
static unsigned short B[] = {
0x0000,0x0000,0x0000,0x0000,
0x3707,0xd75b,0xed02,0x3c72,
0xcc81,0x345d,0xa1cd,0x3c87,
0x4b27,0x5686,0xe9f1,0x3c86,
0x6456,0x13b2,0xdd34,0xbc8b,
0x42e2,0xafec,0x4397,0x3c6d,
0x82e4,0xd231,0xf46a,0x3c76,
0x8a76,0xb9d7,0x9041,0xbc71,
0x0000,0x0000,0x0000,0x0000
};
static unsigned short R[] = {
0x937f,0xd7f2,0x6307,0x3eef,
0x9259,0x60fc,0x2fbe,0x3f24,
0xef1d,0xc84a,0xd87e,0x3f55,
0x33b7,0x6ef1,0xb2ab,0x3f83,
0x1a92,0xd704,0x6b08,0x3fac,
0xc56d,0xff82,0xbfbd,0x3fce,
0x39ef,0xfefa,0x2e42,0x3fe6
};
#define douba(k) (*(double *)&A[(k)<<2])
#define doubb(k) (*(double *)&B[(k)<<2])
#define MEXP 16383.0
#ifdef DENORMAL
#define MNEXP -17183.0
#else
#define MNEXP -16383.0
#endif
#endif
#ifdef MIEEE
static unsigned short P[] = {
0x3fdf,0xdb99,0x7f5b,0x5cf0,
0x400d,0xddef,0xea9e,0xdf15,
0x401e,0xccbd,0x7f78,0xeb6f,
0x4012,0xaa83,0xb65c,0x9b74
};
static unsigned short Q[] = {
0x4022,0xaab4,0x9b20,0x914e,
0x403b,0xffff,0x41c1,0xc9f5,
0x4040,0xccbc,0x1b17,0x6402,
0x402b,0xffc5,0x918a,0xe92e
};
static unsigned short A[] = {
0x3ff0,0x0000,0x0000,0x0000,
0x3fee,0xa4af,0xa2a4,0x90da,
0x3fed,0x5818,0xdcfb,0xa487,
0x3fec,0x199b,0xdd85,0x529c,
0x3fea,0xe89f,0x995a,0xd3ad,
0x3fe9,0xc491,0x82a3,0xf090,
0x3fe8,0xace5,0x422a,0xa0db,
0x3fe7,0xa114,0x73eb,0x0187,
0x3fe6,0xa09e,0x667f,0x3bcd,
0x3fe5,0xab07,0xdd48,0x5429,
0x3fe4,0xbfda,0xd536,0x2a27,
0x3fe3,0xdea6,0x4c12,0x3422,
0x3fe3,0x06fe,0x0a31,0xb715,
0x3fe2,0x387a,0x6e75,0x6238,
0x3fe1,0x72b8,0x3c7d,0x517b,
0x3fe0,0xb558,0x6cf9,0x890f,
0x3fe0,0x0000,0x0000,0x0000
};
static unsigned short B[] = {
0x0000,0x0000,0x0000,0x0000,
0x3c72,0xed02,0xd75b,0x3707,
0x3c87,0xa1cd,0x345d,0xcc81,
0x3c86,0xe9f1,0x5686,0x4b27,
0xbc8b,0xdd34,0x13b2,0x6456,
0x3c6d,0x4397,0xafec,0x42e2,
0x3c76,0xf46a,0xd231,0x82e4,
0xbc71,0x9041,0xb9d7,0x8a76,
0x0000,0x0000,0x0000,0x0000
};
static unsigned short R[] = {
0x3eef,0x6307,0xd7f2,0x937f,
0x3f24,0x2fbe,0x60fc,0x9259,
0x3f55,0xd87e,0xc84a,0xef1d,
0x3f83,0xb2ab,0x6ef1,0x33b7,
0x3fac,0x6b08,0xd704,0x1a92,
0x3fce,0xbfbd,0xff82,0xc56d,
0x3fe6,0x2e42,0xfefa,0x39ef
};
#define douba(k) (*(double *)&A[(k)<<2])
#define doubb(k) (*(double *)&B[(k)<<2])
#define MEXP 16383.0
#ifdef DENORMAL
#define MNEXP -17183.0
#else
#define MNEXP -16383.0
#endif
#endif
/* md_log2(e) - 1 */
#define LOG2EA 0.44269504088896340736
#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
#ifdef ANSIPROT
extern double md_floor ( double );
extern double md_fabs ( double );
extern double md_frexp ( double, int * );
extern double md_ldexp ( double, int );
extern double polevl ( double, void *, int );
extern double p1evl ( double, void *, int );
extern double md_powi ( double, int );
extern int signbit ( double );
extern int isnan ( double );
extern int isfinite ( double );
static double reduc ( double );
#else
double md_floor(), md_fabs(), md_frexp(), md_ldexp();
double polevl(), p1evl(), md_powi();
int signbit(), isnan(), isfinite();
static double reduc();
#endif
extern double MAXNUM;
#ifdef INFINITIES
extern double INFINITY;
#endif
#ifdef NANS
extern double NAN;
#endif
#ifdef MINUSZERO
extern double NEGZERO;
#endif
double md_pow( x, y )
double x, y;
{
double w, z, W, Wa, Wb, ya, yb, u;
/* double F, Fa, Fb, G, Ga, Gb, H, Ha, Hb */
double aw, ay, wy;
int e, i, nflg, iyflg, yoddint;
if( y == 0.0 )
return( 1.0 );
#ifdef NANS
if( isnan(x) )
return( x );
if( isnan(y) )
return( y );
#endif
if( y == 1.0 )
return( x );
#ifdef INFINITIES
if( !isfinite(y) && (x == 1.0 || x == -1.0) )
{
mtherr( "md_pow", DOMAIN );
#ifdef NANS
return( NAN );
#else
return( INFINITY );
#endif
}
#endif
if( x == 1.0 )
return( 1.0 );
if( y >= MAXNUM )
{
#ifdef INFINITIES
if( x > 1.0 )
return( INFINITY );
#else
if( x > 1.0 )
return( MAXNUM );
#endif
if( x > 0.0 && x < 1.0 )
return( 0.0);
if( x < -1.0 )
{
#ifdef INFINITIES
return( INFINITY );
#else
return( MAXNUM );
#endif
}
if( x > -1.0 && x < 0.0 )
return( 0.0 );
}
if( y <= -MAXNUM )
{
if( x > 1.0 )
return( 0.0 );
#ifdef INFINITIES
if( x > 0.0 && x < 1.0 )
return( INFINITY );
#else
if( x > 0.0 && x < 1.0 )
return( MAXNUM );
#endif
if( x < -1.0 )
return( 0.0 );
#ifdef INFINITIES
if( x > -1.0 && x < 0.0 )
return( INFINITY );
#else
if( x > -1.0 && x < 0.0 )
return( MAXNUM );
#endif
}
if( x >= MAXNUM )
{
#if INFINITIES
if( y > 0.0 )
return( INFINITY );
#else
if( y > 0.0 )
return( MAXNUM );
#endif
return(0.0);
}
/* Set iyflg to 1 if y is an integer. */
iyflg = 0;
w = md_floor(y);
if( w == y )
iyflg = 1;
/* Test for odd integer y. */
yoddint = 0;
if( iyflg )
{
ya = md_fabs(y);
ya = md_floor(0.5 * ya);
yb = 0.5 * md_fabs(w);
if( ya != yb )
yoddint = 1;
}
if( x <= -MAXNUM )
{
if( y > 0.0 )
{
#ifdef INFINITIES
if( yoddint )
return( -INFINITY );
return( INFINITY );
#else
if( yoddint )
return( -MAXNUM );
return( MAXNUM );
#endif
}
if( y < 0.0 )
{
#ifdef MINUSZERO
if( yoddint )
return( NEGZERO );
#endif
return( 0.0 );
}
}
nflg = 0; /* flag = 1 if x<0 raised to integer power */
if( x <= 0.0 )
{
if( x == 0.0 )
{
if( y < 0.0 )
{
#ifdef MINUSZERO
if( signbit(x) && yoddint )
return( -INFINITY );
#endif
#ifdef INFINITIES
return( INFINITY );
#else
return( MAXNUM );
#endif
}
if( y > 0.0 )
{
#ifdef MINUSZERO
if( signbit(x) && yoddint )
return( NEGZERO );
#endif
return( 0.0 );
}
return( 1.0 );
}
else
{
if( iyflg == 0 )
{ /* noninteger power of negative number */
mtherr( fname, DOMAIN );
#ifdef NANS
return(NAN);
#else
return(0.0L);
#endif
}
nflg = 1;
}
}
/* Integer power of an integer. */
if( iyflg )
{
i = w;
w = md_floor(x);
if( (w == x) && (md_fabs(y) < 32768.0) )
{
w = md_powi( x, (int) y );
return( w );
}
}
if( nflg )
x = md_fabs(x);
/* For results close to 1, use a series expansion. */
w = x - 1.0;
aw = md_fabs(w);
ay = md_fabs(y);
wy = w * y;
ya = md_fabs(wy);
if((aw <= 1.0e-3 && ay <= 1.0)
|| (ya <= 1.0e-3 && ay >= 1.0))
{
z = (((((w*(y-5.)/720. + 1./120.)*w*(y-4.) + 1./24.)*w*(y-3.)
+ 1./6.)*w*(y-2.) + 0.5)*w*(y-1.) )*wy + wy + 1.;
goto done;
}
/* These are probably too much trouble. */
#if 0
w = y * md_log(x);
if (aw > 1.0e-3 && md_fabs(w) < 1.0e-3)
{
z = ((((((
w/7. + 1.)*w/6. + 1.)*w/5. + 1.)*w/4. + 1.)*w/3. + 1.)*w/2. + 1.)*w + 1.;
goto done;
}
if(ya <= 1.0e-3 && aw <= 1.0e-4)
{
z = (((((
wy*1./720.
+ (-w*1./48. + 1./120.) )*wy
+ ((w*17./144. - 1./12.)*w + 1./24.) )*wy
+ (((-w*5./16. + 7./24.)*w - 1./4.)*w + 1./6.) )*wy
+ ((((w*137./360. - 5./12.)*w + 11./24.)*w - 1./2.)*w + 1./2.) )*wy
+ (((((-w*1./6. + 1./5.)*w - 1./4)*w + 1./3.)*w -1./2.)*w ) )*wy
+ wy + 1.0;
goto done;
}
#endif
/* separate significand from exponent */
x = md_frexp( x, &e );
#if 0
/* For debugging, check for gross overflow. */
if( (e * y) > (MEXP + 1024) )
goto overflow;
#endif
/* Find significand of x in antilog table A[]. */
i = 1;
if( x <= douba(9) )
i = 9;
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 md_log(x/A[i]):
*
* md_log(x) = md_log( a x/a ) = md_log(a) + md_log(x/a)
*
* md_log(x/a) = md_log(1+v), v = x/a - 1 = (x-a)/a
*/
x -= douba(i);
x -= doubb(i/2);
x /= douba(i);
/* rational approximation for md_log(1+v):
*
* md_log(1+v) = v - v**2/2 + v**3 P(v) / Q(v)
*/
z = x*x;
w = x * ( z * polevl( x, P, 3 ) / p1evl( x, Q, 4 ) );
w = w - md_ldexp( z, -1 ); /* w - 0.5 * z */
/* Convert to base 2 logarithm:
* multiply by md_log2(e)
*/
w = w + LOG2EA * w;
/* Note x was not yet added in
* to above rational approximation,
* so do it now, while multiplying
* by md_log2(e).
*/
z = w + LOG2EA * x;
z = z + x;
/* Compute exponent term of the base 2 logarithm. */
w = -i;
w = md_ldexp( w, -4 ); /* divide by 16 */
w += e;
/* Now base 2 md_log of x is w + z. */
/* Multiply base 2 md_log by y, in extended precision. */
/* separate y into large part ya
* and small part yb less than 1/16
*/
ya = reduc(y);
yb = y - ya;
F = z * y + w * yb;
Fa = reduc(F);
Fb = F - Fa;
G = Fa + w * ya;
Ga = reduc(G);
Gb = G - Ga;
H = Fb + Gb;
Ha = reduc(H);
w = md_ldexp( Ga+Ha, 4 );
/* Test the power of 2 for overflow */
if( w > MEXP )
{
#ifndef INFINITIES
mtherr( fname, OVERFLOW );
#endif
#ifdef INFINITIES
if( nflg && yoddint )
return( -INFINITY );
return( INFINITY );
#else
if( nflg && yoddint )
return( -MAXNUM );
return( MAXNUM );
#endif
}
if( w < (MNEXP - 1) )
{
#ifndef DENORMAL
mtherr( fname, UNDERFLOW );
#endif
#ifdef MINUSZERO
if( nflg && yoddint )
return( NEGZERO );
#endif
return( 0.0 );
}
e = w;
Hb = H - Ha;
if( Hb > 0.0 )
{
e += 1;
Hb -= 0.0625;
}
/* Now the product y * md_log2(x) = Hb + e/16.0.
*
* Compute base 2 exponential of Hb,
* where -0.0625 <= Hb <= 0.
*/
z = Hb * polevl( Hb, R, 6 ); /* z = 2**Hb - 1 */
/* Express e/16 as an integer plus a negative number of 16ths.
* Find lookup table entry for the fractional power of 2.
*/
if( e < 0 )
i = 0;
else
i = 1;
i = e/16 + i;
e = 16*i - e;
w = douba( e );
z = w + w * z; /* 2**-e * ( 1 + (2**Hb-1) ) */
z = md_ldexp( z, i ); /* multiply by integer power of 2 */
done:
/* Negate if odd integer power of negative number */
if( nflg && yoddint )
{
#ifdef MINUSZERO
if( z == 0.0 )
z = NEGZERO;
else
#endif
z = -z;
}
return( z );
}
/* Find a multiple of 1/16 that is within 1/16 of x. */
static double reduc(x)
double x;
{
double t;
t = md_ldexp( x, 4 );
t = md_floor( t );
t = md_ldexp( t, -4 );
return(t);
}