/* @(#)k_rem_pio2.c 1.3 95/01/18 */

/*

 * ====================================================

 * 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.

 * ====================================================

 */

/*

 * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)

 * double x[],y[]; int e0,nx,prec; int ipio2[];

 * 

 * __kernel_rem_pio2 return the last three digits of N with 

 *		y = x - N*pi/2

 * so that |y| < pi/2.

 *

 * The method is to compute the integer (mod 8) and fraction parts of 

 * (2/pi)*x without doing the full multiplication. In general we

 * skip the part of the product that are known to be a huge integer (

 * more accurately, = 0 mod 8 ). Thus the number of operations are

 * independent of the exponent of the input.

 *

 * (2/pi) is represented by an array of 24-bit integers in ipio2[].

 *

 * Input parameters:

 * 	x[]	The input value (must be positive) is broken into nx 

 *		pieces of 24-bit integers in double precision format.

 *		x[i] will be the i-th 24 bit of x. The scaled exponent 

 *		of x[0] is given in input parameter e0 (i.e., x[0]*2^e0 

 *		match x's up to 24 bits.

 *

 *		Example of breaking a double positive z into x[0]+x[1]+x[2]:

 *			e0 = ilogb(z)-23

 *			z  = scalbn(z,-e0)

 *		for i = 0,1,2

 *			x[i] = floor(z)

 *			z    = (z-x[i])*2**24

 *

 *

 *	y[]	ouput result in an array of double precision numbers.

 *		The dimension of y[] is:

 *			24-bit  precision	1

 *			53-bit  precision	2

 *			64-bit  precision	2

 *			113-bit precision	3

 *		The actual value is the sum of them. Thus for 113-bit

 *		precison, one may have to do something like:

 *

 *		long double t,w,r_head, r_tail;

 *		t = (long double)y[2] + (long double)y[1];

 *		w = (long double)y[0];

 *		r_head = t+w;

 *		r_tail = w - (r_head - t);

 *

 *	e0	The exponent of x[0]

 *

 *	nx	dimension of x[]

 *

 *  	prec	an integer indicating the precision:

 *			0	24  bits (single)

 *			1	53  bits (double)

 *			2	64  bits (extended)

 *			3	113 bits (quad)

 *

 *	ipio2[]

 *		integer array, contains the (24*i)-th to (24*i+23)-th 

 *		bit of 2/pi after binary point. The corresponding 

 *		floating value is

 *

 *			ipio2[i] * 2^(-24(i+1)).

 *

 * External function:

 *	double scalbn(), floor();

 *

 *

 * Here is the description of some local variables:

 *

 * 	jk	jk+1 is the initial number of terms of ipio2[] needed

 *		in the computation. The recommended value is 2,3,4,

 *		6 for single, double, extended,and quad.

 *

 * 	jz	local integer variable indicating the number of 

 *		terms of ipio2[] used. 

 *

 *	jx	nx - 1

 *

 *	jv	index for pointing to the suitable ipio2[] for the

 *		computation. In general, we want

 *			( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8

 *		is an integer. Thus

 *			e0-3-24*jv >= 0 or (e0-3)/24 >= jv

 *		Hence jv = max(0,(e0-3)/24).

 *

 *	jp	jp+1 is the number of terms in PIo2[] needed, jp = jk.

 *

 * 	q[]	double array with integral value, representing the

 *		24-bits chunk of the product of x and 2/pi.

 *

 *	q0	the corresponding exponent of q[0]. Note that the

 *		exponent for q[i] would be q0-24*i.

 *

 *	PIo2[]	double precision array, obtained by cutting pi/2

 *		into 24 bits chunks. 

 *

 *	f[]	ipio2[] in floating point 

 *

 *	iq[]	integer array by breaking up q[] in 24-bits chunk.

 *

 *	fq[]	final product of x*(2/pi) in fq[0],..,fq[jk]

 *

 *	ih	integer. If >0 it indicates q[] is >= 0.5, hence

 *		it also indicates the *sign* of the result.

 *

 */

/*

 * Constants:

 * The hexadecimal values are the intended ones for the following 

 * constants. The decimal values may be used, provided that the 

 * compiler will convert from decimal to binary accurately enough 

 * to produce the hexadecimal values shown.

 */

#include "fdlibm.h"

#ifdef __STDC__

static const int init_jk[] = {2,3,4,6}; /* initial value for jk */

#else

static int init_jk[] = {2,3,4,6}; 

#endif

#ifdef __STDC__

static const double PIo2[] = {

#else

static double PIo2[] = {

#endif

  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 */

};

#ifdef __STDC__

static const double			

#else

static double			

#endif

zero   = 0.0,

one    = 1.0,

two24   =  1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */

twon24  =  5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */

#ifdef __STDC__

	int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, const int *ipio2) 

#else

	int __kernel_rem_pio2(x,y,e0,nx,prec,ipio2) 	

	double x[], y[]; int e0,nx,prec; int ipio2[];

#endif

{

	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  = scalbn(z,q0);		/* actual value of z */

	z -= 8.0*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 -= 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 = 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 = 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;

}