e_rem_pio2.c

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/*
 * @(#)e_rem_pio2.c	1.12 06/10/10
 *
 * Copyright  1990-2008 Sun Microsystems, Inc. All Rights Reserved.  
 * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER  
 *   
 * This program is free software; you can redistribute it and/or  
 * modify it under the terms of the GNU General Public License version  
 * 2 only, as published by the Free Software Foundation.   
 *   
 * This program is distributed in the hope that it will be useful, but  
 * WITHOUT ANY WARRANTY; without even the implied warranty of  
 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU  
 * General Public License version 2 for more details (a copy is  
 * included at /legal/license.txt).   
 *   
 * You should have received a copy of the GNU General Public License  
 * version 2 along with this work; if not, write to the Free Software  
 * Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA  
 * 02110-1301 USA   
 *   
 * Please contact Sun Microsystems, Inc., 4150 Network Circle, Santa  
 * Clara, CA 95054 or visit www.sun.com if you need additional  
 * information or have any questions. 
 *
 */

/* __ieee754_rem_pio2(x,y)
 *
 * return the remainder of x rem pi/2 in y[0]+y[1]
 * use __kernel_rem_pio2()
 */

#include "fdlibm.h"

/*
 * Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi
 */
#ifdef __STDC__
static const int two_over_pi[] = {
#else
static int two_over_pi[] = {
#endif
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,
};

#ifdef __STDC__
static const int npio2_hw[] = {
#else
static int npio2_hw[] = {
#endif
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,
};

/*
 * 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)
 */

#ifdef __STDC__
static const double
#else
static double
#endif
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 */

#ifdef __STDC__
	int __ieee754_rem_pio2(double x, double *y)
#else
	int __ieee754_rem_pio2(x,y)
	double x,y[];
#endif
{
	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; */
		z = CVMdoubleSub(x, pio2_1);
		if(ix!=0x3ff921fb) { 	/* 33+53 bit pi is good enough */
		    /* y[0] = z - pio2_1t; */
		    y[0] = CVMdoubleSub(z, pio2_1t);
		    /* y[1] = (z-y[0])-pio2_1t; */
		    y[1] = CVMdoubleSub(CVMdoubleSub(z, y[0]), pio2_1t);
		} else {		/* near pi/2, use 33+33+53 bit pi */
		    /* z -= pio2_2; */
		    z = CVMdoubleSub(z, pio2_2);
		    /* y[0] = z - pio2_2t; */
		    y[0] = CVMdoubleSub(z, pio2_2t);
		    /* y[1] = (z-y[0])-pio2_2t; */
		    y[1] = CVMdoubleSub(CVMdoubleSub(z, y[0]), pio2_2t);
		}
		return 1;
	    } else {	/* negative x */
		/* z = x + pio2_1; */
		z = CVMdoubleAdd(x, pio2_1);
		if(ix!=0x3ff921fb) { 	/* 33+53 bit pi is good enough */
		    /* y[0] = z + pio2_1t; */
		    y[0] = CVMdoubleAdd(z, pio2_1t);
		    /* y[1] = (z-y[0])+pio2_1t; */
		    y[1] = CVMdoubleAdd(CVMdoubleSub(z, y[0]), pio2_1t);
		} else {		/* near pi/2, use 33+33+53 bit pi */
		    /* z += pio2_2; */
		    z = CVMdoubleAdd(z, pio2_2);
		    /* y[0] = z + pio2_2t; */
		    y[0] = CVMdoubleAdd(z, pio2_2t);
		    /* y[1] = (z-y[0])+pio2_2t; */
		    y[1] = CVMdoubleAdd(CVMdoubleSub(z, y[0]), pio2_2t);
		}
		return -1;
	    }
	}
	if(ix<=0x413921fb) { /* |x| ~<= 2^19*(pi/2), medium size */
	    t  = fabs(x);
	    /* n  = (int) (t*invpio2+half); */
	    n  = (int) CVMdoubleAdd(CVMdoubleMul(t, invpio2), half);
	    fn = (double)n;
	    /* r  = t-fn*pio2_1; */
	    r  = CVMdoubleSub(t, CVMdoubleMul(fn, pio2_1));
	    /* w  = fn*pio2_1t; */ /* 1st round good to 85 bit */
	    w  = CVMdoubleMul(fn, pio2_1t);
	    if(n<32&&ix!=npio2_hw[n-1]) {
		/* y[0] = r-w; */ /* quick check no cancellation */
		y[0] = CVMdoubleSub(r, w);
	    } else {
	        j  = ix>>20;
	        /* y[0] = r-w; */
	        y[0] = CVMdoubleSub(r, w);
	        i = j-(((__HI(y[0]))>>20)&0x7ff);
	        if(i>16) {  /* 2nd iteration needed, good to 118 */
		    double temp, temp2;
		    t  = r;
		    /* w  = fn*pio2_2; */
		    w  = CVMdoubleMul(fn, pio2_2);
		    /* r  = t-w; */
		    r  = CVMdoubleSub(t, w);
		    /* w  = fn*pio2_2t-((t-r)-w); */
		    temp2 = CVMdoubleSub(t, r);
		    temp2 = CVMdoubleSub(temp2, w);
		    temp = CVMdoubleMul(fn, pio2_2t);
		    w = CVMdoubleSub(temp, temp2);
		    /* y[0] = r-w; */
		    y[0] = CVMdoubleSub(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; */
		    	w  = CVMdoubleMul(fn, pio2_3);
		    	/* r  = t-w; */
		    	r  = CVMdoubleSub(t, w);
		    	/* w  = fn*pio2_3t-((t-r)-w); */
		    	temp2 = CVMdoubleSub(t, r);
		    	temp2 = CVMdoubleSub(temp2, w);
		    	temp = CVMdoubleMul(fn, pio2_3t);
		    	w = CVMdoubleSub(temp, temp2);
		    	/* y[0] = r-w; */
		    	y[0] = CVMdoubleSub(r, w);
		    }
		}
	    }
	    /* y[1] = (r-y[0])-w; */
	    y[1] = CVMdoubleSub(CVMdoubleSub(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; */
	    y[0] = y[1] = CVMdoubleSub(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; */
		z = CVMdoubleMul(CVMdoubleSub(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;
}