# k_rem_pio2_freeBSD.c   [plain text]

```
/* @(#)k_rem_pio2.c 1.3 95/01/18 */
/*
* ====================================================
*
* 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:
*
*		t = (long double)y[2] + (long double)y[1];
*		w = (long double)y[0];
*		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)
*
*	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;
}
```