[uclibc-ng.git] / libm / k_rem_pio2.c

/*
 * ====================================================
 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
 *
 * Developed at SunPro, 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 "math.h"
#include "math_private.h"

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 attribute_hidden __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, const int32_t *ipio2)
{
	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];

    /* 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)((int32_t)(twon24* z));
	    iq[i] =  (int32_t)(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  = (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] = 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)((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 = 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;
}
Commit	Line	Data
7ce331c0 EA	1	/*
	2	* ====================================================
	3	* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
	4	*
	5	* Developed at SunPro, a Sun Microsystems, Inc. business.
	6	* Permission to use, copy, modify, and distribute this
c4e44e97	7	* software is freely granted, provided that this notice
7ce331c0 EA	8	* is preserved.
	9	* ====================================================
	10	*/
	11
7ce331c0 EA	12	/*
	13	* __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
	14	* double x[],y[]; int e0,nx,prec; int ipio2[];
c4e44e97 EA	15	*
c4e44e97 EA	16	* __kernel_rem_pio2 return the last three digits of N with
7ce331c0 EA	17	* y = x - N*pi/2
	18	* so that \|y\| < pi/2.
	19	*
c4e44e97	20	* The method is to compute the integer (mod 8) and fraction parts of
7ce331c0 EA	21	* (2/pi)*x without doing the full multiplication. In general we
	22	* skip the part of the product that are known to be a huge integer (
	23	* more accurately, = 0 mod 8 ). Thus the number of operations are
	24	* independent of the exponent of the input.
	25	*
	26	* (2/pi) is represented by an array of 24-bit integers in ipio2[].
	27	*
	28	* Input parameters:
c4e44e97	29	* x[] The input value (must be positive) is broken into nx
7ce331c0	30	* pieces of 24-bit integers in double precision format.
c4e44e97 EA	31	* x[i] will be the i-th 24 bit of x. The scaled exponent
c4e44e97 EA	32	* of x[0] is given in input parameter e0 (i.e., x[0]*2^e0
7ce331c0 EA	33	* match x's up to 24 bits.
	34	*
	35	* Example of breaking a double positive z into x[0]+x[1]+x[2]:
	36	* e0 = ilogb(z)-23
	37	* z = scalbn(z,-e0)
	38	* for i = 0,1,2
	39	* x[i] = floor(z)
	40	* z = (z-x[i])2*24
	41	*
	42	*
	43	* y[] ouput result in an array of double precision numbers.
	44	* The dimension of y[] is:
	45	* 24-bit precision 1
	46	* 53-bit precision 2
	47	* 64-bit precision 2
	48	* 113-bit precision 3
	49	* The actual value is the sum of them. Thus for 113-bit
	50	* precison, one may have to do something like:
	51	*
	52	* long double t,w,r_head, r_tail;
	53	* t = (long double)y[2] + (long double)y[1];
	54	* w = (long double)y[0];
	55	* r_head = t+w;
	56	* r_tail = w - (r_head - t);
	57	*
	58	* e0 The exponent of x[0]
	59	*
	60	* nx dimension of x[]
	61	*
	62	* prec an integer indicating the precision:
	63	* 0 24 bits (single)
	64	* 1 53 bits (double)
	65	* 2 64 bits (extended)
	66	* 3 113 bits (quad)
	67	*
	68	* ipio2[]
c4e44e97 EA	69	* integer array, contains the (24i)-th to (24i+23)-th
c4e44e97 EA	70	* bit of 2/pi after binary point. The corresponding
7ce331c0 EA	71	* floating value is
	72	*
	73	* ipio2[i] * 2^(-24(i+1)).
	74	*
	75	* External function:
	76	* double scalbn(), floor();
	77	*
	78	*
	79	* Here is the description of some local variables:
	80	*
	81	* jk jk+1 is the initial number of terms of ipio2[] needed
	82	* in the computation. The recommended value is 2,3,4,
	83	* 6 for single, double, extended,and quad.
	84	*
c4e44e97 EA	85	* jz local integer variable indicating the number of
c4e44e97 EA	86	* terms of ipio2[] used.
7ce331c0 EA	87	*
	88	* jx nx - 1
	89	*
	90	* jv index for pointing to the suitable ipio2[] for the
	91	* computation. In general, we want
	92	* ( 2^e0x[0] ipio2[jv-1]*2^(-24jv) )/8
	93	* is an integer. Thus
	94	* e0-3-24*jv >= 0 or (e0-3)/24 >= jv
	95	* Hence jv = max(0,(e0-3)/24).
	96	*
	97	* jp jp+1 is the number of terms in PIo2[] needed, jp = jk.
	98	*
	99	* q[] double array with integral value, representing the
	100	* 24-bits chunk of the product of x and 2/pi.
	101	*
	102	* q0 the corresponding exponent of q[0]. Note that the
	103	* exponent for q[i] would be q0-24*i.
	104	*
	105	* PIo2[] double precision array, obtained by cutting pi/2
c4e44e97	106	* into 24 bits chunks.
7ce331c0	107	*
c4e44e97	108	* f[] ipio2[] in floating point
7ce331c0 EA	109	*
	110	* iq[] integer array by breaking up q[] in 24-bits chunk.
	111	*
	112	* fq[] final product of x*(2/pi) in fq[0],..,fq[jk]
	113	*
	114	* ih integer. If >0 it indicates q[] is >= 0.5, hence
	115	* it also indicates the sign of the result.
	116	*
	117	*/
	118
	119
	120	/*
	121	* Constants:
c4e44e97 EA	122	* The hexadecimal values are the intended ones for the following
	123	* constants. The decimal values may be used, provided that the
	124	* compiler will convert from decimal to binary accurately enough
7ce331c0 EA	125	* to produce the hexadecimal values shown.
	126	*/
	127
	128	#include "math.h"
	129	#include "math_private.h"
	130
7ce331c0	131	static const int init_jk[] = {2,3,4,6}; /* initial value for jk */
7ce331c0	132
7ce331c0	133	static const double PIo2[] = {
7ce331c0 EA	134	1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */
	135	7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */
	136	5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */
	137	3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */
	138	1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */
	139	1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */
	140	2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */
	141	2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */
	142	};
	143
c4e44e97	144	static const double
7ce331c0 EA	145	zero = 0.0,
	146	one = 1.0,
	147	two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
	148	twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */
	149
38b7304e	150	int attribute_hidden __kernel_rem_pio2(double x, double y, int e0, int nx, int prec, const int32_t *ipio2)
7ce331c0 EA	151	{
	152	int32_t jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih;
	153	double z,fw,f[20],fq[20],q[20];
	154
	155	/* initialize jk*/
	156	jk = init_jk[prec];
	157	jp = jk;
	158
	159	/* determine jx,jv,q0, note that 3>q0 */
	160	jx = nx-1;
	161	jv = (e0-3)/24; if(jv<0) jv=0;
	162	q0 = e0-24*(jv+1);
	163
	164	/* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
	165	j = jv-jx; m = jx+jk;
	166	for(i=0;i<=m;i++,j++) f[i] = (j<0)? zero : (double) ipio2[j];
	167
	168	/* compute q[0],q[1],...q[jk] */
	169	for (i=0;i<=jk;i++) {
	170	for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; q[i] = fw;
	171	}
	172
	173	jz = jk;
	174	recompute:
	175	/* distill q[] into iq[] reversingly */
	176	for(i=0,j=jz,z=q[jz];j>0;i++,j--) {
	177	fw = (double)((int32_t)(twon24* z));
	178	iq[i] = (int32_t)(z-two24*fw);
	179	z = q[j-1]+fw;
	180	}
	181
	182	/* compute n */
	183	z = scalbn(z,q0); /* actual value of z */
	184	z -= 8.0floor(z0.125); /* trim off integer >= 8 */
	185	n = (int32_t) z;
	186	z -= (double)n;
	187	ih = 0;
	188	if(q0>0) { /* need iq[jz-1] to determine n */
	189	i = (iq[jz-1]>>(24-q0)); n += i;
	190	iq[jz-1] -= i<<(24-q0);
	191	ih = iq[jz-1]>>(23-q0);
c4e44e97	192	}
7ce331c0 EA	193	else if(q0==0) ih = iq[jz-1]>>23;
	194	else if(z>=0.5) ih=2;
	195
	196	if(ih>0) { /* q > 0.5 */
	197	n += 1; carry = 0;
	198	for(i=0;i<jz ;i++) { /* compute 1-q */
	199	j = iq[i];
	200	if(carry==0) {
	201	if(j!=0) {
	202	carry = 1; iq[i] = 0x1000000- j;
	203	}
	204	} else iq[i] = 0xffffff - j;
	205	}
	206	if(q0>0) { /* rare case: chance is 1 in 12 */
	207	switch(q0) {
	208	case 1:
	209	iq[jz-1] &= 0x7fffff; break;
	210	case 2:
	211	iq[jz-1] &= 0x3fffff; break;
	212	}
	213	}
	214	if(ih==2) {
	215	z = one - z;
	216	if(carry!=0) z -= scalbn(one,q0);
	217	}
	218	}
	219
	220	/* check if recomputation is needed */
	221	if(z==zero) {
	222	j = 0;
	223	for (i=jz-1;i>=jk;i--) j \|= iq[i];
	224	if(j==0) { /* need recomputation */
	225	for(k=1;iq[jk-k]==0;k++); /* k = no. of terms needed */
	226
	227	for(i=jz+1;i<=jz+k;i++) { /* add q[jz+1] to q[jz+k] */
	228	f[jx+i] = (double) ipio2[jv+i];
	229	for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j];
	230	q[i] = fw;
	231	}
	232	jz += k;
	233	goto recompute;
	234	}
	235	}
	236
	237	/* chop off zero terms */
	238	if(z==0.0) {
	239	jz -= 1; q0 -= 24;
	240	while(iq[jz]==0) { jz--; q0-=24;}
	241	} else { /* break z into 24-bit if necessary */
	242	z = scalbn(z,-q0);
c4e44e97	243	if(z>=two24) {
7ce331c0 EA	244	fw = (double)((int32_t)(twon24*z));
	245	iq[jz] = (int32_t)(z-two24*fw);
	246	jz += 1; q0 += 24;
	247	iq[jz] = (int32_t) fw;
	248	} else iq[jz] = (int32_t) z ;
	249	}
	250
	251	/* convert integer "bit" chunk to floating-point value */
	252	fw = scalbn(one,q0);
	253	for(i=jz;i>=0;i--) {
	254	q[i] = fw(double)iq[i]; fw=twon24;
	255	}
	256
	257	/* compute PIo2[0,...,jp]q[jz,...,0] /
	258	for(i=jz;i>=0;i--) {
	259	for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k];
	260	fq[jz-i] = fw;
	261	}
	262
	263	/* compress fq[] into y[] */
	264	switch(prec) {
	265	case 0:
	266	fw = 0.0;
	267	for (i=jz;i>=0;i--) fw += fq[i];
c4e44e97	268	y[0] = (ih==0)? fw: -fw;
7ce331c0 EA	269	break;
	270	case 1:
	271	case 2:
	272	fw = 0.0;
c4e44e97 EA	273	for (i=jz;i>=0;i--) fw += fq[i];
c4e44e97 EA	274	y[0] = (ih==0)? fw: -fw;
7ce331c0 EA	275	fw = fq[0]-fw;
7ce331c0 EA	276	for (i=1;i<=jz;i++) fw += fq[i];
c4e44e97	277	y[1] = (ih==0)? fw: -fw;
7ce331c0 EA	278	break;
	279	case 3: /* painful */
	280	for (i=jz;i>0;i--) {
c4e44e97	281	fw = fq[i-1]+fq[i];
7ce331c0 EA	282	fq[i] += fq[i-1]-fw;
	283	fq[i-1] = fw;
	284	}
	285	for (i=jz;i>1;i--) {
c4e44e97	286	fw = fq[i-1]+fq[i];
7ce331c0 EA	287	fq[i] += fq[i-1]-fw;
	288	fq[i-1] = fw;
	289	}
c4e44e97	290	for (fw=0.0,i=jz;i>=2;i--) fw += fq[i];
7ce331c0 EA	291	if(ih==0) {
	292	y[0] = fq[0]; y[1] = fq[1]; y[2] = fw;
	293	} else {
	294	y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw;
	295	}
	296	}
	297	return n&7;
	298	}