third_party/musl/src/math/powl.c - cobalt - Git at Google

 /* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_powl.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.
  */
 /*                                                      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 "libm.h"

 #if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
 long double powl(long double x, long double y)
 {
 	return pow(x, y);
 }
 #elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384

 /* Table size */
 #define NXT 32

 /* 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 const long double P[] = {
  8.3319510773868690346226E-4L,
  4.9000050881978028599627E-1L,
  1.7500123722550302671919E0L,
  1.4000100839971580279335E0L,
 };
 static const 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 const 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 const 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 const 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 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 const long double huge = 0x1p10000L;
 /* XXX Prevent gcc from erroneously constant folding this. */
 static const volatile long double twom10000 = 0x1p-10000L;

 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;
 	volatile long double z=0;
 	long double w=0, W=0, Wa=0, Wb=0, ya=0, yb=0, u=0;

 	/* make sure no invalid exception is raised by nan comparision */
 	if (isnan(x)) {
 		if (!isnan(y) && y == 0.0)
 			return 1.0;
 		return x;
 	}
 	if (isnan(y)) {
 		if (x == 1.0)
 			return 1.0;
 		return y;
 	}
 	if (x == 1.0)
 		return 1.0; /* 1**y = 1, even if y is nan */
 	if (x == -1.0 && !isfinite(y))
 		return 1.0; /* -1**inf = 1 */
 	if (y == 0.0)
 		return 1.0; /* x**0 = 1, even if x is nan */
 	if (y == 1.0)
 		return x;
 	if (y >= LDBL_MAX) {
 		if (x > 1.0 || x < -1.0)
 			return INFINITY;
 		if (x != 0.0)
 			return 0.0;
 	}
 	if (y <= -LDBL_MAX) {
 		if (x > 1.0 || x < -1.0)
 			return 0.0;
 		if (x != 0.0 || y == -INFINITY)
 			return INFINITY;
 	}
 	if (x >= LDBL_MAX) {
 		if (y > 0.0)
 			return INFINITY;
 		return 0.0;
 	}

 	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(0.5 * ya);
 		yb = 0.5 * fabsl(w);
 		if( ya != yb )
 			yoddint = 1;
 	}

 	if (x <= -LDBL_MAX) {
 		if (y > 0.0) {
 			if (yoddint)
 				return -INFINITY;
 			return INFINITY;
 		}
 		if (y < 0.0) {
 			if (yoddint)
 				return -0.0;
 			return 0.0;
 		}
 	}
 	nflg = 0; /* (x<0)**(odd int) */
 	if (x <= 0.0) {
 		if (x == 0.0) {
 			if (y < 0.0) {
 				if (signbit(x) && yoddint)
 					/* (-0.0)**(-odd int) = -inf, divbyzero */
 					return -1.0/0.0;
 				/* (+-0.0)**(negative) = inf, divbyzero */
 				return 1.0/0.0;
 			}
 			if (signbit(x) && yoddint)
 				return -0.0;
 			return 0.0;
 		}
 		if (iyflg == 0)
 			return (x - x) / (x - x); /* (x<0)**(non-int) is NaN */
 		/* (x<0)**(integer) */
 		if (yoddint)
 			nflg = 1; /* negate result */
 		x = -x;
 	}
 	/* (+integer)**(integer)  */
 	if (iyflg && floorl(x) == x && fabsl(y) < 32768.0) {
 		w = powil(x, (int)y);
 		return nflg ? -w : w;
 	}

 	/* separate significand from exponent */
 	x = frexpl(x, &i);
 	e = i;

 	/* find significand in antilog table A[] */
 	i = 1;
 	if (x <= A[17])
 		i = 17;
 	if (x <= A[i+8])
 		i += 8;
 	if (x <= A[i+4])
 		i += 4;
 	if (x <= A[i+2])
 		i += 2;
 	if (x >= A[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 -= A[i];
 	x -= B[i/2];
 	x /= A[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 - 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 /= 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 = (Ga + Ha) * NXT;

 	/* 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 > 0.0) {
 		e += 1;
 		Hb -= 1.0/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 = A[e];
 	z = w * z;  /*  2**-e * ( 1 + (2**Hb-1) )  */
 	z = z + w;
 	z = scalbnl(z, i);  /* multiply by integer power of 2 */

 	if (nflg)
 		z = -z;
 	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 = x * NXT;
 	t = floorl(t);
 	t = t / NXT;
 	return t;
 }

 /*
  *      Positive real raised to integer power, long double precision
  *
  *
  * SYNOPSIS:
  *
  * long double x, y, powil();
  * int n;
  *
  * y = powil( x, n );
  *
  *
  * DESCRIPTION:
  *
  * Returns argument x>0 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, lx;

 	if (nn == 0)
 		return 1.0;

 	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 - 0.5 + 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+2.0) {
 		x = 1.0/x;
 		sign = -sign;
 	}

 	/* First bit of the power */
 	if (n & 1)
 		y = x;
 	else
 		y = 1.0;

 	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 (sign < 0)
 		y = 1.0/y;
 	return y;
 }
 #elif LDBL_MANT_DIG == 113 && LDBL_MAX_EXP == 16384
 // TODO: broken implementation to make things compile
 long double powl(long double x, long double y)
 {
 	return pow(x, y);
 }
 #endif
	/* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_powl.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.
	*/
	/* 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 "libm.h"

	#if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
	long double powl(long double x, long double y)
	{
	return pow(x, y);
	}
	#elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384

	/* Table size */
	#define NXT 32

	/* 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 const long double P[] = {
	8.3319510773868690346226E-4L,
	4.9000050881978028599627E-1L,
	1.7500123722550302671919E0L,
	1.4000100839971580279335E0L,
	};
	static const 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 const 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 const 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 const 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 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 const long double huge = 0x1p10000L;
	/* XXX Prevent gcc from erroneously constant folding this. */
	static const volatile long double twom10000 = 0x1p-10000L;

	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;
	volatile long double z=0;
	long double w=0, W=0, Wa=0, Wb=0, ya=0, yb=0, u=0;

	/* make sure no invalid exception is raised by nan comparision */
	if (isnan(x)) {
	if (!isnan(y) && y == 0.0)
	return 1.0;
	return x;
	}
	if (isnan(y)) {
	if (x == 1.0)
	return 1.0;
	return y;
	}
	if (x == 1.0)
	return 1.0; /* 1*y = 1, even if y is nan /
	if (x == -1.0 && !isfinite(y))
	return 1.0; /* -1*inf = 1 /
	if (y == 0.0)
	return 1.0; /* x*0 = 1, even if x is nan /
	if (y == 1.0)
	return x;
	if (y >= LDBL_MAX) {
	if (x > 1.0 \|\| x < -1.0)
	return INFINITY;
	if (x != 0.0)
	return 0.0;
	}
	if (y <= -LDBL_MAX) {
	if (x > 1.0 \|\| x < -1.0)
	return 0.0;
	if (x != 0.0 \|\| y == -INFINITY)
	return INFINITY;
	}
	if (x >= LDBL_MAX) {
	if (y > 0.0)
	return INFINITY;
	return 0.0;
	}

	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(0.5 * ya);
	yb = 0.5 * fabsl(w);
	if( ya != yb )
	yoddint = 1;
	}

	if (x <= -LDBL_MAX) {
	if (y > 0.0) {
	if (yoddint)
	return -INFINITY;
	return INFINITY;
	}
	if (y < 0.0) {
	if (yoddint)
	return -0.0;
	return 0.0;
	}
	}
	nflg = 0; /* (x<0)*(odd int) /
	if (x <= 0.0) {
	if (x == 0.0) {
	if (y < 0.0) {
	if (signbit(x) && yoddint)
	/* (-0.0)*(-odd int) = -inf, divbyzero /
	return -1.0/0.0;
	/* (+-0.0)*(negative) = inf, divbyzero /
	return 1.0/0.0;
	}
	if (signbit(x) && yoddint)
	return -0.0;
	return 0.0;
	}
	if (iyflg == 0)
	return (x - x) / (x - x); /* (x<0)*(non-int) is NaN /
	/* (x<0)*(integer) /
	if (yoddint)
	nflg = 1; /* negate result */
	x = -x;
	}
	/* (+integer)*(integer) /
	if (iyflg && floorl(x) == x && fabsl(y) < 32768.0) {
	w = powil(x, (int)y);
	return nflg ? -w : w;
	}

	/* separate significand from exponent */
	x = frexpl(x, &i);
	e = i;

	/* find significand in antilog table A[] */
	i = 1;
	if (x <= A[17])
	i = 17;
	if (x <= A[i+8])
	i += 8;
	if (x <= A[i+4])
	i += 4;
	if (x <= A[i+2])
	i += 2;
	if (x >= A[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 -= A[i];
	x -= B[i/2];
	x /= A[i];

	/* rational approximation for log(1+v):
	*
	* log(1+v) = v - v2/2 + v3 P(v) / Q(v)
	*/
	z = x*x;
	w = x * (z * __polevll(x, P, 3) / __p1evll(x, Q, 3));
	w = 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 /= 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)
	* = wya + wyb + 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 = (Ga + Ha) * NXT;

	/* 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 > 0.0) {
	e += 1;
	Hb -= 1.0/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 = A[e];
	z = w * z; /* 2*-e ( 1 + (2*Hb-1) ) /
	z = z + w;
	z = scalbnl(z, i); /* multiply by integer power of 2 */

	if (nflg)
	z = -z;
	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 = x * NXT;
	t = floorl(t);
	t = t / NXT;
	return t;
	}

	/*
	* Positive real raised to integer power, long double precision
	*
	*
	* SYNOPSIS:
	*
	* long double x, y, powil();
	* int n;
	*
	* y = powil( x, n );
	*
	*
	* DESCRIPTION:
	*
	* Returns argument x>0 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, lx;

	if (nn == 0)
	return 1.0;

	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 - 0.5 + 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+2.0) {
	x = 1.0/x;
	sign = -sign;
	}

	/* First bit of the power */
	if (n & 1)
	y = x;
	else
	y = 1.0;

	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 (sign < 0)
	y = 1.0/y;
	return y;
	}
	#elif LDBL_MANT_DIG == 113 && LDBL_MAX_EXP == 16384
	// TODO: broken implementation to make things compile
	long double powl(long double x, long double y)
	{
	return pow(x, y);
	}
	#endif