e_rem_pio2.c

/*
 * ====================================================
 * 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.
 * ====================================================
 */

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

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


int32_t __ieee754_rem_pio2(double x, double *y)
{
	double z, w, t, r, fn;
	double tx[3];
	int32_t e0, i, j, nx, n, ix, hx;
	uint32_t low;

	static const int32_t npio2_hw[] = {
		IC(0x3FF921FB), IC(0x400921FB), IC(0x4012D97C), IC(0x401921FB), IC(0x401F6A7A), IC(0x4022D97C),
		IC(0x4025FDBB), IC(0x402921FB), IC(0x402C463A), IC(0x402F6A7A), IC(0x4031475C), IC(0x4032D97C),
		IC(0x40346B9C), IC(0x4035FDBB), IC(0x40378FDB), IC(0x403921FB), IC(0x403AB41B), IC(0x403C463A),
		IC(0x403DD85A), IC(0x403F6A7A), IC(0x40407E4C), IC(0x4041475C), IC(0x4042106C), IC(0x4042D97C),
		IC(0x4043A28C), IC(0x40446B9C), IC(0x404534AC), IC(0x4045FDBB), IC(0x4046C6CB), IC(0x40478FDB),
		IC(0x404858EB), IC(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)
	 */

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

	GET_HIGH_WORD(hx, x);				/* high word of x */
	ix = hx & IC(0x7fffffff);
	if (ix <= IC(0x3fe921fb))			/* |x| ~<= pi/4 , no need for reduction */
	{
		y[0] = x;
		y[1] = 0;
		return 0;
	}

	if (ix < IC(0x4002d97c))
	{									/* |x| < 3pi/4, special case with n=+-1 */
		if (hx > 0)
		{
			z = x - pio2_1;
			if (ix != IC(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 != IC(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 <= IC(0x413921fb))
	{									/* |x| ~<= 2^19*(pi/2), medium size */
		t = __ieee754_fabs(x);
		n = (int32_t) (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
		{
			uint32_t high;

			j = ix >> IEEE754_DOUBLE_SHIFT;
			y[0] = r - w;
			GET_HIGH_WORD(high, y[0]);
			i = j - ((high >> IEEE754_DOUBLE_SHIFT) & IEEE754_DOUBLE_MAXEXP);
			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;
				GET_HIGH_WORD(high, y[0]);
				i = j - ((high >> IEEE754_DOUBLE_SHIFT) & IEEE754_DOUBLE_MAXEXP);
				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 >= IC(0x7ff00000))
	{									/* x is inf or NaN */
		y[0] = y[1] = x - x;
		return 0;
	}
	/* set z = scalbn(|x|,ilogb(x)-23) */
	GET_LOW_WORD(low, x);
	e0 = (ix >> IEEE754_DOUBLE_SHIFT) - (IEEE754_DOUBLE_BIAS + 23);			/* e0 = ilogb(z)-23; */
	INSERT_WORDS(z, ix - (e0 << IEEE754_DOUBLE_SHIFT), low);
	for (i = 0; i < 2; i++)
	{
		tx[i] = (double) ((int32_t) (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);
	if (hx < 0)
	{
		y[0] = -y[0];
		y[1] = -y[1];
		return -n;
	}
	return n;
}