/*
 * ====================================================
 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
 *
 * Developed at SunSoft, a Sun Microsystems, Inc. business.
 * Permission to use, copy, modify, and distribute this
 * software is freely granted, provided that this notice 
 * is preserved.
 * ====================================================
 */

/* Sometimes it's necessary to define __LITTLE_ENDIAN explicitly
   but these catch some common cases. */

#if defined(_M_IX86) || defined(__i386__) || defined(_M_X64) || defined(__x86_64__) || defined(_M_ARM64) || defined(__aarch64__) || defined(_M_ARM) || defined(__arm__) || defined(__wasm__) || defined(__EMSCRIPTEN__)
	#define __LITTLE_ENDIAN
#endif

typedef union {
	double d;
	int i[2];
} fdlibm_bits;

#ifdef __LITTLE_ENDIAN
	#define __HI(x) (((fdlibm_bits*)&(x))->i[1])
	#define __LO(x) (((fdlibm_bits*)&(x))->i[0])
#else
	#define __HI(x) (((fdlibm_bits*)&(x))->i[0])
	#define __LO(x) (((fdlibm_bits*)&(x))->i[1])
#endif

/* 
 * set X_TLOSS = pi*2**52, which is possibly defined in <values.h>
 * (one may replace the following line by "#include <values.h>")
 */

double fdlibm_copysign(double x, double y)
{
	__HI(x) = (__HI(x)&0x7fffffff)|(__HI(y)&0x80000000);
        return x;
}

double fdlibm_fabs(double x)
{
	__HI(x) &= 0x7fffffff;
	return x;
}

double fdlibm_scalbn(double x, int n)
{
	static const double
		two54   =  1.80143985094819840000e+16, /* 0x43500000, 0x00000000 */
		twom54  =  5.55111512312578270212e-17, /* 0x3C900000, 0x00000000 */
		huge   = 1.0e+300,
		tiny   = 1.0e-300;

	int  k,hx,lx;
	hx = __HI(x);
	lx = __LO(x);
        k = (hx&0x7ff00000)>>20;		/* extract exponent */
        if (k==0) {				/* 0 or subnormal x */
            if ((lx|(hx&0x7fffffff))==0) return x; /* +-0 */
	    x *= two54; 
	    hx = __HI(x);
	    k = ((hx&0x7ff00000)>>20) - 54; 
            if (n< -50000) return tiny*x; 	/*underflow*/
	    }
        if (k==0x7ff) return x+x;		/* NaN or Inf */
        k = k+n; 
        if (k >  0x7fe) return huge*fdlibm_copysign(huge,x); /* overflow  */
        if (k > 0) 				/* normal result */
	    {__HI(x) = (hx&0x800fffff)|(k<<20); return x;}
        if (k <= -54) {
            if (n > 50000) 	/* in case integer overflow in n+k */
			return huge*fdlibm_copysign(huge,x);	/*overflow*/
			else return tiny*fdlibm_copysign(tiny,x); 	/*underflow*/
		}
        k += 54;				/* subnormal result */
        __HI(x) = (hx&0x800fffff)|(k<<20);
        return x*twom54;
}

double fdlibm_floor(double x)
{
	static const double huge = 1.0e300;

	int i0,i1,j0;
	unsigned i,j;
	i0 =  __HI(x);
	i1 =  __LO(x);
	j0 = ((i0>>20)&0x7ff)-0x3ff;
	if(j0<20) {
	    if(j0<0) { 	/* raise inexact if x != 0 */
		if(huge+x>0.0) {/* return 0*sign(x) if |x|<1 */
		    if(i0>=0) {i0=i1=0;} 
		    else if(((i0&0x7fffffff)|i1)!=0)
			{ i0=0xbff00000;i1=0;}
		}
	    } else {
		i = (0x000fffff)>>j0;
		if(((i0&i)|i1)==0) return x; /* x is integral */
		if(huge+x>0.0) {	/* raise inexact flag */
		    if(i0<0) i0 += (0x00100000)>>j0;
		    i0 &= (~i); i1=0;
		}
	    }
	} else if (j0>51) {
	    if(j0==0x400) return x+x;	/* inf or NaN */
	    else return x;		/* x is integral */
	} else {
	    i = ((unsigned)(0xffffffff))>>(j0-20);
	    if((i1&i)==0) return x;	/* x is integral */
	    if(huge+x>0.0) { 		/* raise inexact flag */
		if(i0<0) {
		    if(j0==20) i0+=1; 
		    else {
			j = i1+(1<<(52-j0));
			if(j<(unsigned)i1) i0 +=1 ; 	/* got a carry */
			i1=j;
		    }
		}
		i1 &= (~i);
	    }
	}
	__HI(x) = i0;
	__LO(x) = i1;
	return x;
}

double fdlibm_frexp(double x, int *eptr)
{
	static const double
		two54 =  1.80143985094819840000e+16; /* 0x43500000, 0x00000000 */

	int  hx, ix, lx;
	hx = __HI(x);
	ix = 0x7fffffff&hx;
	lx = __LO(x);
	*eptr = 0;
	if(ix>=0x7ff00000||((ix|lx)==0)) return x;	/* 0,inf,nan */
	if (ix<0x00100000) {		/* subnormal */
	    x *= two54;
	    hx = __HI(x);
	    ix = hx&0x7fffffff;
	    *eptr = -54;
	}
	*eptr += (ix>>20)-1022;
	hx = (hx&0x800fffff)|0x3fe00000;
	__HI(x) = hx;
	return x;
}

double fdlibm_atan(double x)
{

	static const double atanhi[] = {
	  4.63647609000806093515e-01, /* atan(0.5)hi 0x3FDDAC67, 0x0561BB4F */
	  7.85398163397448278999e-01, /* atan(1.0)hi 0x3FE921FB, 0x54442D18 */
	  9.82793723247329054082e-01, /* atan(1.5)hi 0x3FEF730B, 0xD281F69B */
	  1.57079632679489655800e+00, /* atan(inf)hi 0x3FF921FB, 0x54442D18 */
	};

	static const double atanlo[] = {
	  2.26987774529616870924e-17, /* atan(0.5)lo 0x3C7A2B7F, 0x222F65E2 */
	  3.06161699786838301793e-17, /* atan(1.0)lo 0x3C81A626, 0x33145C07 */
	  1.39033110312309984516e-17, /* atan(1.5)lo 0x3C700788, 0x7AF0CBBD */
	  6.12323399573676603587e-17, /* atan(inf)lo 0x3C91A626, 0x33145C07 */
	};

	static const double aT[] = {
	  3.33333333333329318027e-01, /* 0x3FD55555, 0x5555550D */
	 -1.99999999998764832476e-01, /* 0xBFC99999, 0x9998EBC4 */
	  1.42857142725034663711e-01, /* 0x3FC24924, 0x920083FF */
	 -1.11111104054623557880e-01, /* 0xBFBC71C6, 0xFE231671 */
	  9.09088713343650656196e-02, /* 0x3FB745CD, 0xC54C206E */
	 -7.69187620504482999495e-02, /* 0xBFB3B0F2, 0xAF749A6D */
	  6.66107313738753120669e-02, /* 0x3FB10D66, 0xA0D03D51 */
	 -5.83357013379057348645e-02, /* 0xBFADDE2D, 0x52DEFD9A */
	  4.97687799461593236017e-02, /* 0x3FA97B4B, 0x24760DEB */
	 -3.65315727442169155270e-02, /* 0xBFA2B444, 0x2C6A6C2F */
	  1.62858201153657823623e-02, /* 0x3F90AD3A, 0xE322DA11 */
	};

	static const double 
		one   = 1.0,
		huge   = 1.0e300;

	double w,s1,s2,z;
	int ix,hx,id;

	hx = __HI(x);
	ix = hx&0x7fffffff;
	if(ix>=0x44100000) {	/* if |x| >= 2^66 */
	    if(ix>0x7ff00000||
		(ix==0x7ff00000&&(__LO(x)!=0)))
		return x+x;		/* NaN */
	    if(hx>0) return  atanhi[3]+atanlo[3];
	    else     return -atanhi[3]-atanlo[3];
	} if (ix < 0x3fdc0000) {	/* |x| < 0.4375 */
	    if (ix < 0x3e200000) {	/* |x| < 2^-29 */
		if(huge+x>one) return x;	/* raise inexact */
	    }
	    id = -1;
	} else {
	x = fdlibm_fabs(x);
	if (ix < 0x3ff30000) {		/* |x| < 1.1875 */
	    if (ix < 0x3fe60000) {	/* 7/16 <=|x|<11/16 */
		id = 0; x = (2.0*x-one)/(2.0+x); 
	    } else {			/* 11/16<=|x|< 19/16 */
		id = 1; x  = (x-one)/(x+one); 
	    }
	} else {
	    if (ix < 0x40038000) {	/* |x| < 2.4375 */
		id = 2; x  = (x-1.5)/(one+1.5*x);
	    } else {			/* 2.4375 <= |x| < 2^66 */
		id = 3; x  = -1.0/x;
	    }
	}}
    /* end of argument reduction */
	z = x*x;
	w = z*z;
    /* break sum from i=0 to 10 aT[i]z**(i+1) into odd and even poly */
	s1 = z*(aT[0]+w*(aT[2]+w*(aT[4]+w*(aT[6]+w*(aT[8]+w*aT[10])))));
	s2 = w*(aT[1]+w*(aT[3]+w*(aT[5]+w*(aT[7]+w*aT[9]))));
	if (id<0) return x - x*(s1+s2);
	else {
	    z = atanhi[id] - ((x*(s1+s2) - atanlo[id]) - x);
	    return (hx<0)? -z:z;
	}
}

static double __ieee754_sqrt(double x)
{
	static	const double	one	= 1.0, tiny=1.0e-300;

	double z;
	int 	sign = (int)0x80000000; 
	unsigned r,t1,s1,ix1,q1;
	int ix0,s0,q,m,t,i;

	ix0 = __HI(x);			/* high word of x */
	ix1 = __LO(x);		/* low word of x */

    /* take care of Inf and NaN */
	if((ix0&0x7ff00000)==0x7ff00000) {			
	    return x*x+x;		/* sqrt(NaN)=NaN, sqrt(+inf)=+inf
					   sqrt(-inf)=sNaN */
	} 
    /* take care of zero */
	if(ix0<=0) {
	    if(((ix0&(~sign))|ix1)==0) return x;/* sqrt(+-0) = +-0 */
	    else if(ix0<0)
		return (x-x)/(x-x);		/* sqrt(-ve) = sNaN */
	}
    /* normalize x */
	m = (ix0>>20);
	if(m==0) {				/* subnormal x */
	    while(ix0==0) {
		m -= 21;
		ix0 |= (ix1>>11); ix1 <<= 21;
	    }
	    for(i=0;(ix0&0x00100000)==0;i++) ix0<<=1;
	    m -= i-1;
	    ix0 |= (ix1>>(32-i));
	    ix1 <<= i;
	}
	m -= 1023;	/* unbias exponent */
	ix0 = (ix0&0x000fffff)|0x00100000;
	if(m&1){	/* odd m, double x to make it even */
	    ix0 += ix0 + ((ix1&sign)>>31);
	    ix1 += ix1;
	}
	m >>= 1;	/* m = [m/2] */

    /* generate sqrt(x) bit by bit */
	ix0 += ix0 + ((ix1&sign)>>31);
	ix1 += ix1;
	q = q1 = s0 = s1 = 0;	/* [q,q1] = sqrt(x) */
	r = 0x00200000;		/* r = moving bit from right to left */

	while(r!=0) {
	    t = s0+r; 
	    if(t<=ix0) { 
		s0   = t+r; 
		ix0 -= t; 
		q   += r; 
	    } 
	    ix0 += ix0 + ((ix1&sign)>>31);
	    ix1 += ix1;
	    r>>=1;
	}

	r = sign;
	while(r!=0) {
	    t1 = s1+r; 
	    t  = s0;
	    if((t<ix0)||((t==ix0)&&(t1<=ix1))) { 
		s1  = t1+r;
		if(((t1&sign)==sign)&&(s1&sign)==0) s0 += 1;
		ix0 -= t;
		if (ix1 < t1) ix0 -= 1;
		ix1 -= t1;
		q1  += r;
	    }
	    ix0 += ix0 + ((ix1&sign)>>31);
	    ix1 += ix1;
	    r>>=1;
	}

    /* use floating add to find out rounding direction */
	if((ix0|ix1)!=0) {
	    z = one-tiny; /* trigger inexact flag */
	    if (z>=one) {
	        z = one+tiny;
	        if (q1==(unsigned)0xffffffff) { q1=0; q += 1;}
		else if (z>one) {
		    if (q1==(unsigned)0xfffffffe) q+=1;
		    q1+=2; 
		} else
	            q1 += (q1&1);
	    }
	}
	ix0 = (q>>1)+0x3fe00000;
	ix1 =  q1>>1;
	if ((q&1)==1) ix1 |= sign;
	ix0 += (m <<20);
	__HI(z) = ix0;
	__LO(z) = ix1;
	return z;
}

double fdlibm_sqrt(double x)		/* wrapper sqrt */
{
	return __ieee754_sqrt(x);
}

static double __ieee754_pow(double x, double y)
{
	static const double 
	bp[] = {1.0, 1.5,},
	dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
	dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */
	zero    =  0.0,
	one	=  1.0,
	two	=  2.0,
	two53	=  9007199254740992.0,	/* 0x43400000, 0x00000000 */
	huge	=  1.0e300,
	tiny    =  1.0e-300,
		/* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
	L1  =  5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */
	L2  =  4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */
	L3  =  3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */
	L4  =  2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */
	L5  =  2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */
	L6  =  2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */
	P1   =  1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
	P2   = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
	P3   =  6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
	P4   = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
	P5   =  4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */
	lg2  =  6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
	lg2_h  =  6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */
	lg2_l  = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */
	ovt =  8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */
	cp    =  9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
	cp_h  =  9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */
	cp_l  = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/
	ivln2    =  1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */
	ivln2_h  =  1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/
	ivln2_l  =  1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/

	double z,ax,z_h,z_l,p_h,p_l;
	double y1,t1,t2,r,s,t,u,v,w;
	int i0,i1,i,j,k,yisint,n;
	int hx,hy,ix,iy;
	unsigned lx,ly;

	i0 = ((*(int*)&one)>>29)^1; i1=1-i0;
	hx = __HI(x); lx = __LO(x);
	hy = __HI(y); ly = __LO(y);
	ix = hx&0x7fffffff;  iy = hy&0x7fffffff;

    /* y==zero: x**0 = 1 */
	if((iy|ly)==0) return one; 	

    /* +-NaN return x+y */
	if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) ||
	   iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0))) 
		return x+y;	

    /* determine if y is an odd int when x < 0
     * yisint = 0	... y is not an integer
     * yisint = 1	... y is an odd int
     * yisint = 2	... y is an even int
     */
	yisint  = 0;
	if(hx<0) {	
	    if(iy>=0x43400000) yisint = 2; /* even integer y */
	    else if(iy>=0x3ff00000) {
		k = (iy>>20)-0x3ff;	   /* exponent */
		if(k>20) {
		    j = ly>>(52-k);
		    if((j<<(52-k))==ly) yisint = 2-(j&1);
		} else if(ly==0) {
		    j = iy>>(20-k);
		    if((j<<(20-k))==iy) yisint = 2-(j&1);
		}
	    }		
	} 

    /* special value of y */
	if(ly==0) { 	
	    if (iy==0x7ff00000) {	/* y is +-inf */
	        if(((ix-0x3ff00000)|lx)==0)
		    return  y - y;	/* inf**+-1 is NaN */
	        else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */
		    return (hy>=0)? y: zero;
	        else			/* (|x|<1)**-,+inf = inf,0 */
		    return (hy<0)?-y: zero;
	    } 
	    if(iy==0x3ff00000) {	/* y is  +-1 */
		if(hy<0) return one/x; else return x;
	    }
	    if(hy==0x40000000) return x*x; /* y is  2 */
	    if(hy==0x3fe00000) {	/* y is  0.5 */
		if(hx>=0)	/* x >= +0 */
		return fdlibm_sqrt(x);	
	    }
	}

	ax   = fdlibm_fabs(x);
    /* special value of x */
	if(lx==0) {
	    if(ix==0x7ff00000||ix==0||ix==0x3ff00000){
		z = ax;			/*x is +-0,+-inf,+-1*/
		if(hy<0) z = one/z;	/* z = (1/|x|) */
		if(hx<0) {
		    if(((ix-0x3ff00000)|yisint)==0) {
			z = (z-z)/(z-z); /* (-1)**non-int is NaN */
		    } else if(yisint==1) 
			z = -z;		/* (x<0)**odd = -(|x|**odd) */
		}
		return z;
	    }
	}
    
	n = (hx>>31)+1;

    /* (x<0)**(non-int) is NaN */
	if((n|yisint)==0) return (x-x)/(x-x);

	s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
	if((n|(yisint-1))==0) s = -one;/* (-ve)**(odd int) */

    /* |y| is huge */
	if(iy>0x41e00000) { /* if |y| > 2**31 */
	    if(iy>0x43f00000){	/* if |y| > 2**64, must o/uflow */
		if(ix<=0x3fefffff) return (hy<0)? huge*huge:tiny*tiny;
		if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny;
	    }
	/* over/underflow if x is not close to one */
	    if(ix<0x3fefffff) return (hy<0)? s*huge*huge:s*tiny*tiny;
	    if(ix>0x3ff00000) return (hy>0)? s*huge*huge:s*tiny*tiny;
	/* now |1-x| is tiny <= 2**-20, suffice to compute 
	   log(x) by x-x^2/2+x^3/3-x^4/4 */
	    t = ax-one;		/* t has 20 trailing zeros */
	    w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25));
	    u = ivln2_h*t;	/* ivln2_h has 21 sig. bits */
	    v = t*ivln2_l-w*ivln2;
	    t1 = u+v;
	    __LO(t1) = 0;
	    t2 = v-(t1-u);
	} else {
	    double ss,s2,s_h,s_l,t_h,t_l;
	    n = 0;
	/* take care subnormal number */
	    if(ix<0x00100000)
		{ax *= two53; n -= 53; ix = __HI(ax); }
	    n  += ((ix)>>20)-0x3ff;
	    j  = ix&0x000fffff;
	/* determine interval */
	    ix = j|0x3ff00000;		/* normalize ix */
	    if(j<=0x3988E) k=0;		/* |x|<sqrt(3/2) */
	    else if(j<0xBB67A) k=1;	/* |x|<sqrt(3)   */
	    else {k=0;n+=1;ix -= 0x00100000;}
	    __HI(ax) = ix;

	/* compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
	    u = ax-bp[k];		/* bp[0]=1.0, bp[1]=1.5 */
	    v = one/(ax+bp[k]);
	    ss = u*v;
	    s_h = ss;
	    __LO(s_h) = 0;
	/* t_h=ax+bp[k] High */
	    t_h = zero;
	    __HI(t_h)=((ix>>1)|0x20000000)+0x00080000+(k<<18); 
	    t_l = ax - (t_h-bp[k]);
	    s_l = v*((u-s_h*t_h)-s_h*t_l);
	/* compute log(ax) */
	    s2 = ss*ss;
	    r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
	    r += s_l*(s_h+ss);
	    s2  = s_h*s_h;
	    t_h = 3.0+s2+r;
	    __LO(t_h) = 0;
	    t_l = r-((t_h-3.0)-s2);
	/* u+v = ss*(1+...) */
	    u = s_h*t_h;
	    v = s_l*t_h+t_l*ss;
	/* 2/(3log2)*(ss+...) */
	    p_h = u+v;
	    __LO(p_h) = 0;
	    p_l = v-(p_h-u);
	    z_h = cp_h*p_h;		/* cp_h+cp_l = 2/(3*log2) */
	    z_l = cp_l*p_h+p_l*cp+dp_l[k];
	/* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */
	    t = (double)n;
	    t1 = (((z_h+z_l)+dp_h[k])+t);
	    __LO(t1) = 0;
	    t2 = z_l-(((t1-t)-dp_h[k])-z_h);
	}

    /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
	y1  = y;
	__LO(y1) = 0;
	p_l = (y-y1)*t1+y*t2;
	p_h = y1*t1;
	z = p_l+p_h;
	j = __HI(z);
	i = __LO(z);
	if (j>=0x40900000) {				/* z >= 1024 */
	    if(((j-0x40900000)|i)!=0)			/* if z > 1024 */
		return s*huge*huge;			/* overflow */
	    else {
		if(p_l+ovt>z-p_h) return s*huge*huge;	/* overflow */
	    }
	} else if((j&0x7fffffff)>=0x4090cc00 ) {	/* z <= -1075 */
	    if(((j-0xc090cc00)|i)!=0) 		/* z < -1075 */
		return s*tiny*tiny;		/* underflow */
	    else {
		if(p_l<=z-p_h) return s*tiny*tiny;	/* underflow */
	    }
	}
    /*
     * compute 2**(p_h+p_l)
     */
	i = j&0x7fffffff;
	k = (i>>20)-0x3ff;
	n = 0;
	if(i>0x3fe00000) {		/* if |z| > 0.5, set n = [z+0.5] */
	    n = j+(0x00100000>>(k+1));
	    k = ((n&0x7fffffff)>>20)-0x3ff;	/* new k for n */
	    t = zero;
	    __HI(t) = (n&~(0x000fffff>>k));
	    n = ((n&0x000fffff)|0x00100000)>>(20-k);
	    if(j<0) n = -n;
	    p_h -= t;
	} 
	t = p_l+p_h;
	__LO(t) = 0;
	u = t*lg2_h;
	v = (p_l-(t-p_h))*lg2+t*lg2_l;
	z = u+v;
	w = v-(z-u);
	t  = z*z;
	t1  = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
	r  = (z*t1)/(t1-two)-(w+z*w);
	z  = one-(r-z);
	j  = __HI(z);
	j += (n<<20);
	if((j>>20)<=0) z = fdlibm_scalbn(z,n);	/* subnormal output */
	else __HI(z) += (n<<20);
	return s*z;
}

static double __ieee754_atan2(double y, double x)
{  
	static const double 
		tiny  = 1.0e-300,
		zero  = 0.0,
		pi_o_4  = 7.8539816339744827900E-01, /* 0x3FE921FB, 0x54442D18 */
		pi_o_2  = 1.5707963267948965580E+00, /* 0x3FF921FB, 0x54442D18 */
		pi      = 3.1415926535897931160E+00, /* 0x400921FB, 0x54442D18 */
		pi_lo   = 1.2246467991473531772E-16; /* 0x3CA1A626, 0x33145C07 */

	double z;
	int k,m,hx,hy,ix,iy;
	unsigned lx,ly;

	hx = __HI(x); ix = hx&0x7fffffff;
	lx = __LO(x);
	hy = __HI(y); iy = hy&0x7fffffff;
	ly = __LO(y);
	if(((ix|((lx|(unsigned)-(int)lx)>>31))>0x7ff00000)||
	   ((iy|((ly|(unsigned)-(int)ly)>>31))>0x7ff00000))	/* x or y is NaN */
	   return x+y;
	if((hx-0x3ff00000|lx)==0) return fdlibm_atan(y);   /* x=1.0 */
	m = ((hy>>31)&1)|((hx>>30)&2);	/* 2*sign(x)+sign(y) */

    /* when y = 0 */
	if((iy|ly)==0) {
	    switch(m) {
		case 0: 
		case 1: return y; 	/* atan(+-0,+anything)=+-0 */
		case 2: return  pi+tiny;/* atan(+0,-anything) = pi */
		case 3: return -pi-tiny;/* atan(-0,-anything) =-pi */
	    }
	}
    /* when x = 0 */
	if((ix|lx)==0) return (hy<0)?  -pi_o_2-tiny: pi_o_2+tiny;
	    
    /* when x is INF */
	if(ix==0x7ff00000) {
	    if(iy==0x7ff00000) {
		switch(m) {
		    case 0: return  pi_o_4+tiny;/* atan(+INF,+INF) */
		    case 1: return -pi_o_4-tiny;/* atan(-INF,+INF) */
		    case 2: return  3.0*pi_o_4+tiny;/*atan(+INF,-INF)*/
		    case 3: return -3.0*pi_o_4-tiny;/*atan(-INF,-INF)*/
		}
	    } else {
		switch(m) {
		    case 0: return  zero  ;	/* atan(+...,+INF) */
		    case 1: return -zero  ;	/* atan(-...,+INF) */
		    case 2: return  pi+tiny  ;	/* atan(+...,-INF) */
		    case 3: return -pi-tiny  ;	/* atan(-...,-INF) */
		}
	    }
	}
    /* when y is INF */
	if(iy==0x7ff00000) return (hy<0)? -pi_o_2-tiny: pi_o_2+tiny;

    /* compute y/x */
	k = (iy-ix)>>20;
	if(k > 60) z=pi_o_2+0.5*pi_lo; 	/* |y/x| >  2**60 */
	else if(hx<0&&k<-60) z=0.0; 	/* |y|/x < -2**60 */
	else z=fdlibm_atan(fdlibm_fabs(y/x));		/* safe to do y/x */
	switch (m) {
	    case 0: return       z  ;	/* atan(+,+) */
	    case 1: __HI(z) ^= 0x80000000;
		    return       z  ;	/* atan(-,+) */
	    case 2: return  pi-(z-pi_lo);/* atan(+,-) */
	    default: /* case 3 */
	    	    return  (z-pi_lo)-pi;/* atan(-,-) */
	}
}

static double __ieee754_asin(double x)
{
	static const double 
		one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
		huge =  1.000e+300,
		pio2_hi =  1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */
		pio2_lo =  6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */
		pio4_hi =  7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
			/* coefficient for R(x^2) */
		pS0 =  1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */
		pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */
		pS2 =  2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */
		pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */
		pS4 =  7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */
		pS5 =  3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */
		qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */
		qS2 =  2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */
		qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */
		qS4 =  7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */

	double t,w,p,q,c,r,s;
	int hx,ix;
	hx = __HI(x);
	ix = hx&0x7fffffff;
	if(ix>= 0x3ff00000) {		/* |x|>= 1 */
	    if(((ix-0x3ff00000)|__LO(x))==0)
		    /* asin(1)=+-pi/2 with inexact */
		return x*pio2_hi+x*pio2_lo;	
	    return (x-x)/(x-x);		/* asin(|x|>1) is NaN */   
	} else if (ix<0x3fe00000) {	/* |x|<0.5 */
	    if(ix<0x3e400000) {		/* if |x| < 2**-27 */
		if(huge+x>one) return x;/* return x with inexact if x!=0*/
	    } else 
		t = x*x;
		p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
		q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
		w = p/q;
		return x+x*w;
	}
	/* 1> |x|>= 0.5 */
	w = one-fdlibm_fabs(x);
	t = w*0.5;
	p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
	q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
	s = fdlibm_sqrt(t);
	if(ix>=0x3FEF3333) { 	/* if |x| > 0.975 */
	    w = p/q;
	    t = pio2_hi-(2.0*(s+s*w)-pio2_lo);
	} else {
	    w  = s;
	    __LO(w) = 0;
	    c  = (t-w*w)/(s+w);
	    r  = p/q;
	    p  = 2.0*s*r-(pio2_lo-2.0*c);
	    q  = pio4_hi-2.0*w;
	    t  = pio4_hi-(p-q);
	}    
	if(hx>0) return t; else return -t;    
}

static double __ieee754_acos(double x)
{
	static const double 
		one=  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
		pi =  3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */
		pio2_hi =  1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */
		pio2_lo =  6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */
		pS0 =  1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */
		pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */
		pS2 =  2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */
		pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */
		pS4 =  7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */
		pS5 =  3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */
		qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */
		qS2 =  2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */
		qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */
		qS4 =  7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */

	double z,p,q,r,w,s,c,df;
	int hx,ix;
	hx = __HI(x);
	ix = hx&0x7fffffff;
	if(ix>=0x3ff00000) {	/* |x| >= 1 */
	    if(((ix-0x3ff00000)|__LO(x))==0) {	/* |x|==1 */
		if(hx>0) return 0.0;		/* acos(1) = 0  */
		else return pi+2.0*pio2_lo;	/* acos(-1)= pi */
	    }
	    return (x-x)/(x-x);		/* acos(|x|>1) is NaN */
	}
	if(ix<0x3fe00000) {	/* |x| < 0.5 */
	    if(ix<=0x3c600000) return pio2_hi+pio2_lo;/*if|x|<2**-57*/
	    z = x*x;
	    p = z*(pS0+z*(pS1+z*(pS2+z*(pS3+z*(pS4+z*pS5)))));
	    q = one+z*(qS1+z*(qS2+z*(qS3+z*qS4)));
	    r = p/q;
	    return pio2_hi - (x - (pio2_lo-x*r));
	} else  if (hx<0) {		/* x < -0.5 */
	    z = (one+x)*0.5;
	    p = z*(pS0+z*(pS1+z*(pS2+z*(pS3+z*(pS4+z*pS5)))));
	    q = one+z*(qS1+z*(qS2+z*(qS3+z*qS4)));
	    s = fdlibm_sqrt(z);
	    r = p/q;
	    w = r*s-pio2_lo;
	    return pi - 2.0*(s+w);
	} else {			/* x > 0.5 */
	    z = (one-x)*0.5;
	    s = fdlibm_sqrt(z);
	    df = s;
	    __LO(df) = 0;
	    c  = (z-df*df)/(s+df);
	    p = z*(pS0+z*(pS1+z*(pS2+z*(pS3+z*(pS4+z*pS5)))));
	    q = one+z*(qS1+z*(qS2+z*(qS3+z*qS4)));
	    r = p/q;
	    w = r*s+c;
	    return 2.0*(df+w);
	}
}

/*
 * Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi 
 */
static const int two_over_pi[] = {
0xA2F983, 0x6E4E44, 0x1529FC, 0x2757D1, 0xF534DD, 0xC0DB62, 
0x95993C, 0x439041, 0xFE5163, 0xABDEBB, 0xC561B7, 0x246E3A, 
0x424DD2, 0xE00649, 0x2EEA09, 0xD1921C, 0xFE1DEB, 0x1CB129, 
0xA73EE8, 0x8235F5, 0x2EBB44, 0x84E99C, 0x7026B4, 0x5F7E41, 
0x3991D6, 0x398353, 0x39F49C, 0x845F8B, 0xBDF928, 0x3B1FF8, 
0x97FFDE, 0x05980F, 0xEF2F11, 0x8B5A0A, 0x6D1F6D, 0x367ECF, 
0x27CB09, 0xB74F46, 0x3F669E, 0x5FEA2D, 0x7527BA, 0xC7EBE5, 
0xF17B3D, 0x0739F7, 0x8A5292, 0xEA6BFB, 0x5FB11F, 0x8D5D08, 
0x560330, 0x46FC7B, 0x6BABF0, 0xCFBC20, 0x9AF436, 0x1DA9E3, 
0x91615E, 0xE61B08, 0x659985, 0x5F14A0, 0x68408D, 0xFFD880, 
0x4D7327, 0x310606, 0x1556CA, 0x73A8C9, 0x60E27B, 0xC08C6B, 
};

static const int npio2_hw[] = {
0x3FF921FB, 0x400921FB, 0x4012D97C, 0x401921FB, 0x401F6A7A, 0x4022D97C,
0x4025FDBB, 0x402921FB, 0x402C463A, 0x402F6A7A, 0x4031475C, 0x4032D97C,
0x40346B9C, 0x4035FDBB, 0x40378FDB, 0x403921FB, 0x403AB41B, 0x403C463A,
0x403DD85A, 0x403F6A7A, 0x40407E4C, 0x4041475C, 0x4042106C, 0x4042D97C,
0x4043A28C, 0x40446B9C, 0x404534AC, 0x4045FDBB, 0x4046C6CB, 0x40478FDB,
0x404858EB, 0x404921FB,
};

static int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, const int *ipio2) 
{
	static const int init_jk[] = {2,3,4,6}; /* initial value for jk */

	static const double PIo2[] = {
	  1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */
	  7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */
	  5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */
	  3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */
	  1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */
	  1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */
	  2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */
	  2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */
	};

	static const double			
	zero   = 0.0,
	one    = 1.0,
	two24   =  1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
	twon24  =  5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */

	int jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih;
	double z,fw,f[20],fq[20],q[20];

    /* initialize jk*/
	jk = init_jk[prec];
	jp = jk;

    /* determine jx,jv,q0, note that 3>q0 */
	jx =  nx-1;
	jv = (e0-3)/24; if(jv<0) jv=0;
	q0 =  e0-24*(jv+1);

    /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
	j = jv-jx; m = jx+jk;
	for(i=0;i<=m;i++,j++) f[i] = (j<0)? zero : (double) ipio2[j];

    /* compute q[0],q[1],...q[jk] */
	for (i=0;i<=jk;i++) {
	    for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; q[i] = fw;
	}

	jz = jk;
recompute:
    /* distill q[] into iq[] reversingly */
	for(i=0,j=jz,z=q[jz];j>0;i++,j--) {
	    fw    =  (double)((int)(twon24* z));
	    iq[i] =  (int)(z-two24*fw);
	    z     =  q[j-1]+fw;
	}

    /* compute n */
	z  = fdlibm_scalbn(z,q0);		/* actual value of z */
	z -= 8.0*fdlibm_floor(z*0.125);		/* trim off integer >= 8 */
	n  = (int) z;
	z -= (double)n;
	ih = 0;
	if(q0>0) {	/* need iq[jz-1] to determine n */
	    i  = (iq[jz-1]>>(24-q0)); n += i;
	    iq[jz-1] -= i<<(24-q0);
	    ih = iq[jz-1]>>(23-q0);
	} 
	else if(q0==0) ih = iq[jz-1]>>23;
	else if(z>=0.5) ih=2;

	if(ih>0) {	/* q > 0.5 */
	    n += 1; carry = 0;
	    for(i=0;i<jz ;i++) {	/* compute 1-q */
		j = iq[i];
		if(carry==0) {
		    if(j!=0) {
			carry = 1; iq[i] = 0x1000000- j;
		    }
		} else  iq[i] = 0xffffff - j;
	    }
	    if(q0>0) {		/* rare case: chance is 1 in 12 */
	        switch(q0) {
	        case 1:
	    	   iq[jz-1] &= 0x7fffff; break;
	    	case 2:
	    	   iq[jz-1] &= 0x3fffff; break;
	        }
	    }
	    if(ih==2) {
		z = one - z;
		if(carry!=0) z -= fdlibm_scalbn(one,q0);
	    }
	}

    /* check if recomputation is needed */
	if(z==zero) {
	    j = 0;
	    for (i=jz-1;i>=jk;i--) j |= iq[i];
	    if(j==0) { /* need recomputation */
		for(k=1;iq[jk-k]==0;k++);   /* k = no. of terms needed */

		for(i=jz+1;i<=jz+k;i++) {   /* add q[jz+1] to q[jz+k] */
		    f[jx+i] = (double) ipio2[jv+i];
		    for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j];
		    q[i] = fw;
		}
		jz += k;
		goto recompute;
	    }
	}

    /* chop off zero terms */
	if(z==0.0) {
	    jz -= 1; q0 -= 24;
	    while(iq[jz]==0) { jz--; q0-=24;}
	} else { /* break z into 24-bit if necessary */
	    z = fdlibm_scalbn(z,-q0);
	    if(z>=two24) { 
		fw = (double)((int)(twon24*z));
		iq[jz] = (int)(z-two24*fw);
		jz += 1; q0 += 24;
		iq[jz] = (int) fw;
	    } else iq[jz] = (int) z ;
	}

    /* convert integer "bit" chunk to floating-point value */
	fw = fdlibm_scalbn(one,q0);
	for(i=jz;i>=0;i--) {
	    q[i] = fw*(double)iq[i]; fw*=twon24;
	}

    /* compute PIo2[0,...,jp]*q[jz,...,0] */
	for(i=jz;i>=0;i--) {
	    for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k];
	    fq[jz-i] = fw;
	}

    /* compress fq[] into y[] */
	switch(prec) {
	    case 0:
		fw = 0.0;
		for (i=jz;i>=0;i--) fw += fq[i];
		y[0] = (ih==0)? fw: -fw; 
		break;
	    case 1:
	    case 2:
		fw = 0.0;
		for (i=jz;i>=0;i--) fw += fq[i]; 
		y[0] = (ih==0)? fw: -fw; 
		fw = fq[0]-fw;
		for (i=1;i<=jz;i++) fw += fq[i];
		y[1] = (ih==0)? fw: -fw; 
		break;
	    case 3:	/* painful */
		for (i=jz;i>0;i--) {
		    fw      = fq[i-1]+fq[i]; 
		    fq[i]  += fq[i-1]-fw;
		    fq[i-1] = fw;
		}
		for (i=jz;i>1;i--) {
		    fw      = fq[i-1]+fq[i]; 
		    fq[i]  += fq[i-1]-fw;
		    fq[i-1] = fw;
		}
		for (fw=0.0,i=jz;i>=2;i--) fw += fq[i]; 
		if(ih==0) {
		    y[0] =  fq[0]; y[1] =  fq[1]; y[2] =  fw;
		} else {
		    y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw;
		}
	}
	return n&7;
}


/*
 * invpio2:  53 bits of 2/pi
 * pio2_1:   first  33 bit of pi/2
 * pio2_1t:  pi/2 - pio2_1
 * pio2_2:   second 33 bit of pi/2
 * pio2_2t:  pi/2 - (pio2_1+pio2_2)
 * pio2_3:   third  33 bit of pi/2
 * pio2_3t:  pi/2 - (pio2_1+pio2_2+pio2_3)
 */

int __ieee754_rem_pio2(double x, double *y)
{
	static const double 
	zero =  0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
	half =  5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
	two24 =  1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
	invpio2 =  6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
	pio2_1  =  1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */
	pio2_1t =  6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */
	pio2_2  =  6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */
	pio2_2t =  2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */
	pio2_3  =  2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */
	pio2_3t =  8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */

	double z,w,t,r,fn;
	double tx[3];
	int e0,i,j,nx,n,ix,hx;

	hx = __HI(x);		/* high word of x */
	ix = hx&0x7fffffff;
	if(ix<=0x3fe921fb)   /* |x| ~<= pi/4 , no need for reduction */
	    {y[0] = x; y[1] = 0; return 0;}
	if(ix<0x4002d97c) {  /* |x| < 3pi/4, special case with n=+-1 */
	    if(hx>0) { 
		z = x - pio2_1;
		if(ix!=0x3ff921fb) { 	/* 33+53 bit pi is good enough */
		    y[0] = z - pio2_1t;
		    y[1] = (z-y[0])-pio2_1t;
		} else {		/* near pi/2, use 33+33+53 bit pi */
		    z -= pio2_2;
		    y[0] = z - pio2_2t;
		    y[1] = (z-y[0])-pio2_2t;
		}
		return 1;
	    } else {	/* negative x */
		z = x + pio2_1;
		if(ix!=0x3ff921fb) { 	/* 33+53 bit pi is good enough */
		    y[0] = z + pio2_1t;
		    y[1] = (z-y[0])+pio2_1t;
		} else {		/* near pi/2, use 33+33+53 bit pi */
		    z += pio2_2;
		    y[0] = z + pio2_2t;
		    y[1] = (z-y[0])+pio2_2t;
		}
		return -1;
	    }
	}
	if(ix<=0x413921fb) { /* |x| ~<= 2^19*(pi/2), medium size */
	    t  = fdlibm_fabs(x);
	    n  = (int) (t*invpio2+half);
	    fn = (double)n;
	    r  = t-fn*pio2_1;
	    w  = fn*pio2_1t;	/* 1st round good to 85 bit */
	    if(n<32&&ix!=npio2_hw[n-1]) {	
		y[0] = r-w;	/* quick check no cancellation */
	    } else {
	        j  = ix>>20;
	        y[0] = r-w; 
	        i = j-(((__HI(y[0]))>>20)&0x7ff);
	        if(i>16) {  /* 2nd iteration needed, good to 118 */
		    t  = r;
		    w  = fn*pio2_2;	
		    r  = t-w;
		    w  = fn*pio2_2t-((t-r)-w);	
		    y[0] = r-w;
		    i = j-(((__HI(y[0]))>>20)&0x7ff);
		    if(i>49)  {	/* 3rd iteration need, 151 bits acc */
		    	t  = r;	/* will cover all possible cases */
		    	w  = fn*pio2_3;	
		    	r  = t-w;
		    	w  = fn*pio2_3t-((t-r)-w);	
		    	y[0] = r-w;
		    }
		}
	    }
	    y[1] = (r-y[0])-w;
	    if(hx<0) 	{y[0] = -y[0]; y[1] = -y[1]; return -n;}
	    else	 return n;
	}
    /* 
     * all other (large) arguments
     */
	if(ix>=0x7ff00000) {		/* x is inf or NaN */
	    y[0]=y[1]=x-x; return 0;
	}
    /* set z = scalbn(|x|,ilogb(x)-23) */
	__LO(z) = __LO(x);
	e0 	= (ix>>20)-1046;	/* e0 = ilogb(z)-23; */
	__HI(z) = ix - (e0<<20);
	for(i=0;i<2;i++) {
		tx[i] = (double)((int)(z));
		z     = (z-tx[i])*two24;
	}
	tx[2] = z;
	nx = 3;
	while(tx[nx-1]==zero) nx--;	/* skip zero term */
	n  =  __kernel_rem_pio2(tx,y,e0,nx,2,two_over_pi);
	if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;}
	return n;
}

double __kernel_sin(double x, double y, int iy)
{

	static const double 
	half =  5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
	S1  = -1.66666666666666324348e-01, /* 0xBFC55555, 0x55555549 */
	S2  =  8.33333333332248946124e-03, /* 0x3F811111, 0x1110F8A6 */
	S3  = -1.98412698298579493134e-04, /* 0xBF2A01A0, 0x19C161D5 */
	S4  =  2.75573137070700676789e-06, /* 0x3EC71DE3, 0x57B1FE7D */
	S5  = -2.50507602534068634195e-08, /* 0xBE5AE5E6, 0x8A2B9CEB */
	S6  =  1.58969099521155010221e-10; /* 0x3DE5D93A, 0x5ACFD57C */

	double z,r,v;
	int ix;
	ix = __HI(x)&0x7fffffff;	/* high word of x */
	if(ix<0x3e400000)			/* |x| < 2**-27 */
	   {if((int)x==0) return x;}		/* generate inexact */
	z	=  x*x;
	v	=  z*x;
	r	=  S2+z*(S3+z*(S4+z*(S5+z*S6)));
	if(iy==0) return x+v*(S1+z*r);
	else      return x-((z*(half*y-v*r)-y)-v*S1);
}

double __kernel_cos(double x, double y)
{
	static const double 
	one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
	C1  =  4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */
	C2  = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */
	C3  =  2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */
	C4  = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */
	C5  =  2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */
	C6  = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */

	double a,hz,z,r,qx;
	int ix;
	ix = __HI(x)&0x7fffffff;	/* ix = |x|'s high word*/
	if(ix<0x3e400000) {			/* if x < 2**27 */
	    if(((int)x)==0) return one;		/* generate inexact */
	}
	z  = x*x;
	r  = z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*C6)))));
	if(ix < 0x3FD33333) 			/* if |x| < 0.3 */ 
	    return one - (0.5*z - (z*r - x*y));
	else {
	    if(ix > 0x3fe90000) {		/* x > 0.78125 */
		qx = 0.28125;
	    } else {
	        __HI(qx) = ix-0x00200000;	/* x/4 */
	        __LO(qx) = 0;
	    }
	    hz = 0.5*z-qx;
	    a  = one-qx;
	    return a - (hz - (z*r-x*y));
	}
}

double
__kernel_tan(double x, double y, int iy) {
	double z, r, v, w, s;
	int ix, hx;

	static const double xxx[] = {
			 3.33333333333334091986e-01,	/* 3FD55555, 55555563 */
			 1.33333333333201242699e-01,	/* 3FC11111, 1110FE7A */
			 5.39682539762260521377e-02,	/* 3FABA1BA, 1BB341FE */
			 2.18694882948595424599e-02,	/* 3F9664F4, 8406D637 */
			 8.86323982359930005737e-03,	/* 3F8226E3, E96E8493 */
			 3.59207910759131235356e-03,	/* 3F6D6D22, C9560328 */
			 1.45620945432529025516e-03,	/* 3F57DBC8, FEE08315 */
			 5.88041240820264096874e-04,	/* 3F4344D8, F2F26501 */
			 2.46463134818469906812e-04,	/* 3F3026F7, 1A8D1068 */
			 7.81794442939557092300e-05,	/* 3F147E88, A03792A6 */
			 7.14072491382608190305e-05,	/* 3F12B80F, 32F0A7E9 */
			-1.85586374855275456654e-05,	/* BEF375CB, DB605373 */
			 2.59073051863633712884e-05,	/* 3EFB2A70, 74BF7AD4 */
	/* one */	 1.00000000000000000000e+00,	/* 3FF00000, 00000000 */
	/* pio4 */	 7.85398163397448278999e-01,	/* 3FE921FB, 54442D18 */
	/* pio4lo */	 3.06161699786838301793e-17	/* 3C81A626, 33145C07 */
	};
	#define	one	xxx[13]
	#define	pio4	xxx[14]
	#define	pio4lo	xxx[15]
	#define	T	xxx
	/* INDENT ON */

	hx = __HI(x);		/* high word of x */
	ix = hx & 0x7fffffff;			/* high word of |x| */
	if (ix < 0x3e300000) {			/* x < 2**-28 */
		if ((int) x == 0) {		/* generate inexact */
			if (((ix | __LO(x)) | (iy + 1)) == 0)
				return one / fdlibm_fabs(x);
			else {
				if (iy == 1)
					return x;
				else {	/* compute -1 / (x+y) carefully */
					double a, t;

					z = w = x + y;
					__LO(z) = 0;
					v = y - (z - x);
					t = a = -one / w;
					__LO(t) = 0;
					s = one + t * z;
					return t + a * (s + t * v);
				}
			}
		}
	}
	if (ix >= 0x3FE59428) {	/* |x| >= 0.6744 */
		if (hx < 0) {
			x = -x;
			y = -y;
		}
		z = pio4 - x;
		w = pio4lo - y;
		x = z + w;
		y = 0.0;
	}
	z = x * x;
	w = z * z;
	/*
	 * Break x^5*(T[1]+x^2*T[2]+...) into
	 * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
	 * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
	 */
	r = T[1] + w * (T[3] + w * (T[5] + w * (T[7] + w * (T[9] +
		w * T[11]))));
	v = z * (T[2] + w * (T[4] + w * (T[6] + w * (T[8] + w * (T[10] +
		w * T[12])))));
	s = z * x;
	r = y + z * (s * (r + v) + y);
	r += T[0] * s;
	w = x + r;
	if (ix >= 0x3FE59428) {
		v = (double) iy;
		return (double) (1 - ((hx >> 30) & 2)) *
			(v - 2.0 * (x - (w * w / (w + v) - r)));
	}
	if (iy == 1)
		return w;
	else {
		/*
		 * if allow error up to 2 ulp, simply return
		 * -1.0 / (x+r) here
		 */
		/* compute -1.0 / (x+r) accurately */
		double a, t;
		z = w;
		__LO(z) = 0;
		v = r - (z - x);	/* z+v = r+x */
		t = a = -1.0 / w;	/* a = -1.0/w */
		__LO(t) = 0;
		s = 1.0 + t * z;
		return t + a * (s + t * v);
	}

	#undef one
	#undef pio4
	#undef pio4lo
	#undef T
}

double fdlibm_sin(double x)
{
	double y[2],z=0.0;
	int n, ix;

    /* High word of x. */
	ix = __HI(x);

    /* |x| ~< pi/4 */
	ix &= 0x7fffffff;
	if(ix <= 0x3fe921fb) return __kernel_sin(x,z,0);

    /* sin(Inf or NaN) is NaN */
	else if (ix>=0x7ff00000) return x-x;

    /* argument reduction needed */
	else {
	    n = __ieee754_rem_pio2(x,y);
	    switch(n&3) {
		case 0: return  __kernel_sin(y[0],y[1],1);
		case 1: return  __kernel_cos(y[0],y[1]);
		case 2: return -__kernel_sin(y[0],y[1],1);
		default:
			return -__kernel_cos(y[0],y[1]);
	    }
	}
}

double fdlibm_cos(double x)
{
	double y[2],z=0.0;
	int n, ix;

    /* High word of x. */
	ix = __HI(x);

    /* |x| ~< pi/4 */
	ix &= 0x7fffffff;
	if(ix <= 0x3fe921fb) return __kernel_cos(x,z);

    /* cos(Inf or NaN) is NaN */
	else if (ix>=0x7ff00000) return x-x;

    /* argument reduction needed */
	else {
	    n = __ieee754_rem_pio2(x,y);
	    switch(n&3) {
		case 0: return  __kernel_cos(y[0],y[1]);
		case 1: return -__kernel_sin(y[0],y[1],1);
		case 2: return -__kernel_cos(y[0],y[1]);
		default:
		        return  __kernel_sin(y[0],y[1],1);
	    }
	}
}

double fdlibm_tan(double x)
{
	double y[2],z=0.0;
	int n, ix;

    /* High word of x. */
	ix = __HI(x);

    /* |x| ~< pi/4 */
	ix &= 0x7fffffff;
	if(ix <= 0x3fe921fb) return __kernel_tan(x,z,1);

    /* tan(Inf or NaN) is NaN */
	else if (ix>=0x7ff00000) return x-x;		/* NaN */

    /* argument reduction needed */
	else {
	    n = __ieee754_rem_pio2(x,y);
	    return __kernel_tan(y[0],y[1],1-((n&1)<<1)); /*   1 -- n even
							-1 -- n odd */
	}
}

double fdlibm_asin(double x)		/* wrapper asin */
{
	return __ieee754_asin(x);
}

double fdlibm_acos(double x)		/* wrapper acos */
{
	return __ieee754_acos(x);
}

double fdlibm_atan2(double y, double x)	/* wrapper atan2 */
{
	return __ieee754_atan2(y,x);
}

double fdlibm_pow(double x, double y)	/* wrapper pow */
{
	return  __ieee754_pow(x,y);
}

double fdlibm_fmin(double a, double b)
{
	return a < b ? a : b;
}

double fdlibm_fmax(double a, double b)
{
	return a < b ? b : a;
}