/* @(#)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
 *		precision, 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.
 *
 */


#ifndef __FDLIBM_H__
#include "fdlibm.h"
#endif


int32_t __kernel_rem_pio2(double *x, double *y, int32_t e0, int32_t nx, int prec)
{
	int32_t jz, jx, jv, jp, jk, carry, n, iq[20], i, j, k, m, q0, ih;
	double z, fw, f[20], fq[20], q[20];

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

	/*
	 * 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.
	 */
	
	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 */
	};

	/*
	 * Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi
	 */
	static const int32_t two_over_pi[] = {
		IC(0xA2F983), IC(0x6E4E44), IC(0x1529FC), IC(0x2757D1), IC(0xF534DD), IC(0xC0DB62),
		IC(0x95993C), IC(0x439041), IC(0xFE5163), IC(0xABDEBB), IC(0xC561B7), IC(0x246E3A),
		IC(0x424DD2), IC(0xE00649), IC(0x2EEA09), IC(0xD1921C), IC(0xFE1DEB), IC(0x1CB129),
		IC(0xA73EE8), IC(0x8235F5), IC(0x2EBB44), IC(0x84E99C), IC(0x7026B4), IC(0x5F7E41),
		IC(0x3991D6), IC(0x398353), IC(0x39F49C), IC(0x845F8B), IC(0xBDF928), IC(0x3B1FF8),
		IC(0x97FFDE), IC(0x05980F), IC(0xEF2F11), IC(0x8B5A0A), IC(0x6D1F6D), IC(0x367ECF),
		IC(0x27CB09), IC(0xB74F46), IC(0x3F669E), IC(0x5FEA2D), IC(0x7527BA), IC(0xC7EBE5),
		IC(0xF17B3D), IC(0x0739F7), IC(0x8A5292), IC(0xEA6BFB), IC(0x5FB11F), IC(0x8D5D08),
		IC(0x560330), IC(0x46FC7B), IC(0x6BABF0), IC(0xCFBC20), IC(0x9AF436), IC(0x1DA9E3),
		IC(0x91615E), IC(0xE61B08), IC(0x659985), IC(0x5F14A0), IC(0x68408D), IC(0xFFD880),
		IC(0x4D7327), IC(0x310606), IC(0x1556CA), IC(0x73A8C9), IC(0x60E27B), IC(0xC08C6B)
	};

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

	/* 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] = two_over_pi[jv+jk] */
	j = jv - jx;
	m = jx + jk;
	for (i = 0; i <= m; i++, j++)
		f[i] = (j < 0) ? zero : (double) two_over_pi[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) ((int32_t) (twon24 * z));
		iq[i] = (int32_t) (z - two24 * fw);
		z = q[j - 1] + fw;
	}

	/* compute n */
	z = __ieee754_scalbn(z, (int)q0);				/* actual value of z */
	z -= 8.0 * __ieee754_floor(z * 0.125);		/* trim off integer >= 8 */
	n = (int32_t) 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] = IC(0x1000000) - j;
				}
			} else
				iq[i] = IC(0xffffff) - j;
		}
		if (q0 > 0)
		{								/* rare case: chance is 1 in 12 */
			switch ((int)q0)
			{
			case 1:
				iq[jz - 1] &= IC(0x7fffff);
				break;
			case 2:
				iq[jz - 1] &= IC(0x3fffff);
				break;
			}
		}
		if (ih == 2)
		{
			z = one - z;
			if (carry != 0)
				z -= __ieee754_scalbn(one, (int)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) two_over_pi[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 = __ieee754_scalbn(z, (int)-q0);
		if (z >= two24)
		{
			fw = (double) ((int32_t) (twon24 * z));
			iq[jz] = (int32_t) (z - two24 * fw);
			jz += 1;
			q0 += 24;
			iq[jz] = (int32_t) fw;
		} else
		{
			iq[jz] = (int32_t) z;
		}
	}

	/* convert integer "bit" chunk to floating-point value */
	fw = __ieee754_scalbn(one, (int)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:
		{
			volatile double fv = 0.0;

			for (i = jz; i >= 0; i--)
				fv += fq[i];
			y[0] = (ih == 0) ? fv : -fv;
			fv = fq[0] - fv;
			for (i = 1; i <= jz; i++)
				fv += fq[i];
			y[1] = (ih == 0) ? fv : -fv;
		}
		break;
	case 3:							/* painful */
		for (i = jz; i > 0; i--)
		{
			volatile double fv = (double) (fq[i - 1] + fq[i]);

			fq[i] += fq[i - 1] - fv;
			fq[i - 1] = fv;
		}
		for (i = jz; i > 1; i--)
		{
			volatile double fv = (double) (fq[i - 1] + fq[i]);

			fq[i] += fq[i - 1] - fv;
			fq[i - 1] = fv;
		}
		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;
}
