/*							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      MAXNUM * pow underflow   x**y < 1/MAXNUM       0.0 * pow domain      x<0 and y noninteger  0.0 * *//*Cephes Math Library Release 2.2:  January, 1991Copyright 1984, 1991 by Stephen L. MoshierDirect inquiries to 30 Frost Street, Cambridge, MA 02140*/#include "mconf.h"static char fname[] = {"powl"};/* 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: */#define MNEXP (-NXT*(16384.0L-64.0L))/* log2(e) - 1 */#define LOG2EA 0.44269504088896340735992L#endif#ifdef IBMPCstatic short P[] = {0xb804,0xa8b7,0xc6f4,0xda6a,0x3ff4, XPD0x7de9,0xcf02,0x58c0,0xfae1,0x3ffd, XPD0x405a,0x3722,0x67c9,0xe000,0x3fff, XPD0xcd99,0x6b43,0x87ca,0xb333,0x3fff, XPD};static short Q[] = {/* 0x0000,0x0000,0x0000,0x8000,0x3fff, */0x6307,0xa469,0x3b33,0xa800,0x4001, XPD0xfec2,0x62d7,0xa51c,0x8666,0x4002, XPD0xda32,0xd072,0xa5d7,0x8666,0x4001, XPD};static short A[] = {0x0000,0x0000,0x0000,0x8000,0x3fff, XPD0x033a,0x722a,0xb2db,0xfa83,0x3ffe, XPD0xcc2c,0x2486,0x7d15,0xf525,0x3ffe, XPD0xf5cb,0xdcda,0xb99b,0xefe4,0x3ffe, XPD0x392f,0xdd24,0xc6e7,0xeac0,0x3ffe, XPD0x48a8,0x7c83,0x06e7,0xe5b9,0x3ffe, XPD0xe111,0x2a94,0xdeec,0xe0cc,0x3ffe, XPD0x3755,0xdaf2,0xb797,0xdbfb,0x3ffe, XPD0x6af4,0xd69d,0xfcca,0xd744,0x3ffe, XPD0xe45a,0xf12a,0x1d91,0xd2a8,0x3ffe, XPD0x80e4,0x1f84,0x8c15,0xce24,0x3ffe, XPD0x27a3,0x6e2f,0xbd86,0xc9b9,0x3ffe, XPD0xdadd,0x5506,0x2a11,0xc567,0x3ffe, XPD0x9456,0x6670,0x4cca,0xc12c,0x3ffe, XPD0x36bf,0x580c,0xa39f,0xbd08,0x3ffe, XPD0x9ee9,0x62fb,0xaf47,0xb8fb,0x3ffe, XPD0x6484,0xf9de,0xf333,0xb504,0x3ffe, XPD0x2590,0xd2ac,0xf581,0xb123,0x3ffe, XPD0x4ac6,0x42a1,0x3eea,0xad58,0x3ffe, XPD0x0ef8,0xea7c,0x5ab4,0xa9a1,0x3ffe, XPD0x38ea,0xb151,0xd6a9,0xa5fe,0x3ffe, XPD0x6819,0x0c49,0x4303,0xa270,0x3ffe, XPD0x11ae,0x91a1,0x3260,0x9ef5,0x3ffe, XPD0x5539,0xd54e,0x39b9,0x9b8d,0x3ffe, XPD0xa96f,0x8db8,0xf051,0x9837,0x3ffe, XPD0x0961,0xfef7,0xefa8,0x94f4,0x3ffe, XPD0xc336,0xab11,0xd373,0x91c3,0x3ffe, XPD0x53c0,0x45cd,0x398b,0x8ea4,0x3ffe, XPD0xd6e7,0xea8b,0xc1e3,0x8b95,0x3ffe, XPD0x8527,0x92da,0x0e80,0x8898,0x3ffe, XPD0x7b15,0xcc48,0xc367,0x85aa,0x3ffe, XPD0xa1d7,0xac2b,0x8698,0x82cd,0x3ffe, XPD0x0000,0x0000,0x0000,0x8000,0x3ffe, XPD};static short B[] = {0x0000,0x0000,0x0000,0x0000,0x0000, XPD0x1f87,0xdb30,0x18f5,0xf73a,0x3fbd, XPD0xac15,0x3e46,0x2932,0xbf4a,0xbfbc, XPD0x7944,0xba66,0xa091,0xcb12,0x3fb9, XPD0xff78,0x40b4,0x2ee6,0xe69a,0x3fbc, XPD0xc895,0x5069,0xe383,0xee53,0xbfbb, XPD0x7cde,0x9376,0x4325,0xf8ab,0x3fbc, XPD0xa10c,0x25e0,0xc093,0xaefd,0xbfbd, XPD0x7d3e,0xea95,0x1366,0xb2fb,0x3fbd, XPD0x5d89,0xeb34,0x5191,0x9301,0x3fbd, XPD0x80d9,0xb883,0xfb10,0xe5eb,0x3fbb, XPD0x045d,0x288c,0xc1ec,0xbedd,0xbfbd, XPD0xeded,0x5c85,0x4630,0x8d5a,0x3fbd, XPD0x9d82,0xe5ac,0x8e0a,0xfd6d,0x3fba, XPD0x6dfd,0xeb58,0xaf14,0x8373,0xbfb9, XPD0xf938,0x7aac,0x91cf,0xe8da,0xbfbc, XPD0x0000,0x0000,0x0000,0x0000,0x0000, XPD};static short R[] = {0xa69b,0x530e,0xee1d,0xfd2a,0x3fee, XPD0xc746,0x8e7e,0x5960,0xa182,0x3ff2, XPD0x63b6,0xadda,0xfd6a,0xaec3,0x3ff5, XPD0xc104,0xfd99,0x5b7c,0x9d95,0x3ff8, XPD0xe05e,0x249d,0x46b8,0xe358,0x3ffa, XPD0x5d1d,0x162c,0xeffc,0xf5fd,0x3ffc, XPD0x79aa,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)#define MNEXP (-NXT*16384.0L)static short L[] = {0xc2ef,0x705f,0xeca5,0xe2a8,0x3ffd, XPD};#define LOG2EA (*(long double *)(&L[0]))#endif#ifdef MIEEEstatic 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)#define MNEXP (-NXT*16382.0L)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 Wbextern long double MAXNUML;static VOLATILE long double z;static long double w, W, Wa, Wb, ya, yb, u;long double floorl(), fabsl(), frexpl(), ldexpl();long double polevll(), p1evll(), powil();static long double reducl();long double powl( x, y )long double x, y;{/* double F, Fa, Fb, G, Ga, Gb, H, Ha, Hb */int i, nflg;long e;nflg = 0;	/* flag = 1 if x<0 raised to integer power */w = floorl(y);if( (w == y) && (fabsl(w) < 32768.0L) )	{	i = w;	w = powil( x, i );	return( w );	}if( x <= 0.0L )	{	if( x == 0.0L )		{		if( y == 0.0L )			return( 1.0L );  /*   0**0   */		else  			return( 0.0L );  /*   0**y   */		}	else		{		if( w != y )			{ /* noninteger power of negative number */			mtherr( fname, DOMAIN );			return(0.0L);			}		nflg = 1;		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 = (x *  z * polevll( x, P, 3 )) / p1evll( x, Q, 4 );*/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 )	{/*	printf( "w = %.4Le ", w ); */	mtherr( fname, OVERFLOW );	return( MAXNUML );	}if( w < MNEXP )	{/*	printf( "w = %.4Le ", w ); */	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 );}/* Find a multiple of 1/NXT that is within 1/NXT of x. */static long double reducl(x)long double x;{long double t;t = ldexpl( x, LNXT );t = floorl( t );t = ldexpl( t, -LNXT );return(t);}