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raze/source/common/thirdparty/math/pow.c
Christoph Oelckers f671eb622f - migrated to GZDoom's utility code.
2020-04-12 08:30:39 +02:00

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/* pow.c
*
* Power function
*
*
*
* SYNOPSIS:
*
* double x, y, z, pow();
*
* z = pow( 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/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 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
* 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 2.8: June, 2000
Copyright 1984, 1995, 2000 by Stephen L. Moshier
Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are met:
1. Redistributions of source code must retain the above copyright notice,
this list of conditions and the following disclaimer.
2. Redistributions in binary form must reproduce the above copyright
notice, this list of conditions and the following disclaimer in the
documentation and/or other materials provided with the distribution.
3. Neither the name of the <ORGANIZATION> nor the names of its
contributors may be used to endorse or promote products derived from
this software without specific prior written permission.
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
POSSIBILITY OF SUCH DAMAGE.
*/
#include "mconf.h"
static char fname[] = {"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
/* 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 floor ( double );
extern double fabs ( double );
extern double frexp ( double, int * );
extern double ldexp ( double, int );
extern double polevl ( double, void *, int );
extern double p1evl ( double, void *, int );
extern double c_powi ( double, int );
extern int signbit ( double );
extern int isnan ( double );
extern int isfinite ( double );
static double reduc ( double );
#else
double floor(), fabs(), frexp(), ldexp();
double polevl(), p1evl(), c_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 c_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( "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 = floor(y);
if( w == y )
iyflg = 1;
/* Test for odd integer y. */
yoddint = 0;
if( iyflg )
{
ya = fabs(y);
ya = floor(0.5 * ya);
yb = 0.5 * 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 = (int)w;
w = floor(x);
if( (w == x) && (fabs(y) < 32768.0) )
{
w = c_powi( x, (int) y );
return( w );
}
}
if( nflg )
x = fabs(x);
/* For results close to 1, use a series expansion. */
w = x - 1.0;
aw = fabs(w);
ay = fabs(y);
wy = w * y;
ya = 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 * log(x);
if (aw > 1.0e-3 && 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 = 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 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 * polevl( x, P, 3 ) / p1evl( x, Q, 4 ) );
w = w - ldexp( z, -1 ); /* w - 0.5 * z */
/* Convert to base 2 logarithm:
* multiply by log2(e)
*/
w = w + LOG2EA * w;
/* Note x was not yet added in
* to above rational approximation,
* so do it now, while multiplying
* by log2(e).
*/
z = w + LOG2EA * x;
z = z + x;
/* Compute exponent term of the base 2 logarithm. */
w = -i;
w = ldexp( w, -4 ); /* divide by 16 */
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 1/16
*/
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 = 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 = (int)w;
Hb = H - Ha;
if( Hb > 0.0 )
{
e += 1;
Hb -= 0.0625;
}
/* Now the product y * 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 = 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 = ldexp( x, 4 );
t = floor( t );
t = ldexp( t, -4 );
return(t);
}
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