libm/src/k_rem_pio2.c - platform/bionic - Git at Google


 /* @(#)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.
  * ====================================================
  */

 #ifndef lint
 static char rcsid[] = "$FreeBSD: src/lib/msun/src/k_rem_pio2.c,v 1.7 2005/02/04 18:26:06 das Exp $";
 #endif

 /*
  * __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 __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;
 }

	/* @(#)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.
	* ====================================================
	*/

	#ifndef lint
	static char rcsid[] = "$FreeBSD: src/lib/msun/src/k_rem_pio2.c,v 1.7 2005/02/04 18:26:06 das Exp $";
	#endif

	/*
	* __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 (24i)-th to (24i+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^e0x[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 __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 -= 8.0floor(z0.125); /* 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;
	}