k_rem_pio2.c revision 2116 
1/* @(#)k_rem_pio2.c 5.1 93/09/24 */
2/*
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5 *
6 * Developed at SunPro, a Sun Microsystems, Inc. business.
7 * Permission to use, copy, modify, and distribute this
8 * software is freely granted, provided that this notice
9 * is preserved.
10 * ====================================================
11 */
12
13#ifndef lint
14static char rcsid[] = "$Id: k_rem_pio2.c,v 1.5 1994/08/18 23:06:11 jtc Exp $";
15#endif
16
17/*
18 * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
19 * double x[],y[]; int e0,nx,prec; int ipio2[];
20 *
21 * __kernel_rem_pio2 return the last three digits of N with
22 *		y = x - N*pi/2
23 * so that |y| < pi/2.
24 *
25 * The method is to compute the integer (mod 8) and fraction parts of
26 * (2/pi)*x without doing the full multiplication. In general we
27 * skip the part of the product that are known to be a huge integer (
28 * more accurately, = 0 mod 8 ). Thus the number of operations are
29 * independent of the exponent of the input.
30 *
31 * (2/pi) is represented by an array of 24-bit integers in ipio2[].
32 *
33 * Input parameters:
34 * 	x[]	The input value (must be positive) is broken into nx
35 *		pieces of 24-bit integers in double precision format.
36 *		x[i] will be the i-th 24 bit of x. The scaled exponent
37 *		of x[0] is given in input parameter e0 (i.e., x[0]*2^e0
38 *		match x's up to 24 bits.
39 *
40 *		Example of breaking a double positive z into x[0]+x[1]+x[2]:
41 *			e0 = ilogb(z)-23
42 *			z  = scalbn(z,-e0)
43 *		for i = 0,1,2
44 *			x[i] = floor(z)
45 *			z    = (z-x[i])*2**24
46 *
47 *
48 *	y[]	ouput result in an array of double precision numbers.
49 *		The dimension of y[] is:
50 *			24-bit  precision	1
51 *			53-bit  precision	2
52 *			64-bit  precision	2
53 *			113-bit precision	3
54 *		The actual value is the sum of them. Thus for 113-bit
55 *		precison, one may have to do something like:
56 *
57 *		long double t,w,r_head, r_tail;
58 *		t = (long double)y[2] + (long double)y[1];
59 *		w = (long double)y[0];
60 *		r_head = t+w;
61 *		r_tail = w - (r_head - t);
62 *
63 *	e0	The exponent of x[0]
64 *
65 *	nx	dimension of x[]
66 *
67 *  	prec	an integer indicating the precision:
68 *			0	24  bits (single)
69 *			1	53  bits (double)
70 *			2	64  bits (extended)
71 *			3	113 bits (quad)
72 *
73 *	ipio2[]
74 *		integer array, contains the (24*i)-th to (24*i+23)-th
75 *		bit of 2/pi after binary point. The corresponding
76 *		floating value is
77 *
78 *			ipio2[i] * 2^(-24(i+1)).
79 *
80 * External function:
81 *	double scalbn(), floor();
82 *
83 *
84 * Here is the description of some local variables:
85 *
86 * 	jk	jk+1 is the initial number of terms of ipio2[] needed
87 *		in the computation. The recommended value is 2,3,4,
88 *		6 for single, double, extended,and quad.
89 *
90 * 	jz	local integer variable indicating the number of
91 *		terms of ipio2[] used.
92 *
93 *	jx	nx - 1
94 *
95 *	jv	index for pointing to the suitable ipio2[] for the
96 *		computation. In general, we want
97 *			( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8
98 *		is an integer. Thus
99 *			e0-3-24*jv >= 0 or (e0-3)/24 >= jv
100 *		Hence jv = max(0,(e0-3)/24).
101 *
102 *	jp	jp+1 is the number of terms in PIo2[] needed, jp = jk.
103 *
104 * 	q[]	double array with integral value, representing the
105 *		24-bits chunk of the product of x and 2/pi.
106 *
107 *	q0	the corresponding exponent of q[0]. Note that the
108 *		exponent for q[i] would be q0-24*i.
109 *
110 *	PIo2[]	double precision array, obtained by cutting pi/2
111 *		into 24 bits chunks.
112 *
113 *	f[]	ipio2[] in floating point
114 *
115 *	iq[]	integer array by breaking up q[] in 24-bits chunk.
116 *
117 *	fq[]	final product of x*(2/pi) in fq[0],..,fq[jk]
118 *
119 *	ih	integer. If >0 it indicates q[] is >= 0.5, hence
120 *		it also indicates the *sign* of the result.
121 *
122 */
123
124
125/*
126 * Constants:
127 * The hexadecimal values are the intended ones for the following
128 * constants. The decimal values may be used, provided that the
129 * compiler will convert from decimal to binary accurately enough
130 * to produce the hexadecimal values shown.
131 */
132
133#include "math.h"
134#include "math_private.h"
135
136#ifdef __STDC__
137static const int init_jk[] = {2,3,4,6}; /* initial value for jk */
138#else
139static int init_jk[] = {2,3,4,6};
140#endif
141
142#ifdef __STDC__
143static const double PIo2[] = {
144#else
145static double PIo2[] = {
146#endif
147  1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */
148  7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */
149  5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */
150  3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */
151  1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */
152  1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */
153  2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */
154  2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */
155};
156
157#ifdef __STDC__
158static const double
159#else
160static double
161#endif
162zero   = 0.0,
163one    = 1.0,
164two24   =  1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
165twon24  =  5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */
166
167#ifdef __STDC__
168	int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, const int32_t *ipio2)
169#else
170	int __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
171	double x[], y[]; int e0,nx,prec; int32_t ipio2[];
172#endif
173{
174	int32_t jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih;
175	double z,fw,f[20],fq[20],q[20];
176
177    /* initialize jk*/
178	jk = init_jk[prec];
179	jp = jk;
180
181    /* determine jx,jv,q0, note that 3>q0 */
182	jx =  nx-1;
183	jv = (e0-3)/24; if(jv<0) jv=0;
184	q0 =  e0-24*(jv+1);
185
186    /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
187	j = jv-jx; m = jx+jk;
188	for(i=0;i<=m;i++,j++) f[i] = (j<0)? zero : (double) ipio2[j];
189
190    /* compute q[0],q[1],...q[jk] */
191	for (i=0;i<=jk;i++) {
192	    for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; q[i] = fw;
193	}
194
195	jz = jk;
196recompute:
197    /* distill q[] into iq[] reversingly */
198	for(i=0,j=jz,z=q[jz];j>0;i++,j--) {
199	    fw    =  (double)((int32_t)(twon24* z));
200	    iq[i] =  (int32_t)(z-two24*fw);
201	    z     =  q[j-1]+fw;
202	}
203
204    /* compute n */
205	z  = scalbn(z,q0);		/* actual value of z */
206	z -= 8.0*floor(z*0.125);		/* trim off integer >= 8 */
207	n  = (int32_t) z;
208	z -= (double)n;
209	ih = 0;
210	if(q0>0) {	/* need iq[jz-1] to determine n */
211	    i  = (iq[jz-1]>>(24-q0)); n += i;
212	    iq[jz-1] -= i<<(24-q0);
213	    ih = iq[jz-1]>>(23-q0);
214	}
215	else if(q0==0) ih = iq[jz-1]>>23;
216	else if(z>=0.5) ih=2;
217
218	if(ih>0) {	/* q > 0.5 */
219	    n += 1; carry = 0;
220	    for(i=0;i<jz ;i++) {	/* compute 1-q */
221		j = iq[i];
222		if(carry==0) {
223		    if(j!=0) {
224			carry = 1; iq[i] = 0x1000000- j;
225		    }
226		} else  iq[i] = 0xffffff - j;
227	    }
228	    if(q0>0) {		/* rare case: chance is 1 in 12 */
229	        switch(q0) {
230	        case 1:
231	    	   iq[jz-1] &= 0x7fffff; break;
232	    	case 2:
233	    	   iq[jz-1] &= 0x3fffff; break;
234	        }
235	    }
236	    if(ih==2) {
237		z = one - z;
238		if(carry!=0) z -= scalbn(one,q0);
239	    }
240	}
241
242    /* check if recomputation is needed */
243	if(z==zero) {
244	    j = 0;
245	    for (i=jz-1;i>=jk;i--) j |= iq[i];
246	    if(j==0) { /* need recomputation */
247		for(k=1;iq[jk-k]==0;k++);   /* k = no. of terms needed */
248
249		for(i=jz+1;i<=jz+k;i++) {   /* add q[jz+1] to q[jz+k] */
250		    f[jx+i] = (double) ipio2[jv+i];
251		    for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j];
252		    q[i] = fw;
253		}
254		jz += k;
255		goto recompute;
256	    }
257	}
258
259    /* chop off zero terms */
260	if(z==0.0) {
261	    jz -= 1; q0 -= 24;
262	    while(iq[jz]==0) { jz--; q0-=24;}
263	} else { /* break z into 24-bit if necessary */
264	    z = scalbn(z,-q0);
265	    if(z>=two24) {
266		fw = (double)((int32_t)(twon24*z));
267		iq[jz] = (int32_t)(z-two24*fw);
268		jz += 1; q0 += 24;
269		iq[jz] = (int32_t) fw;
270	    } else iq[jz] = (int32_t) z ;
271	}
272
273    /* convert integer "bit" chunk to floating-point value */
274	fw = scalbn(one,q0);
275	for(i=jz;i>=0;i--) {
276	    q[i] = fw*(double)iq[i]; fw*=twon24;
277	}
278
279    /* compute PIo2[0,...,jp]*q[jz,...,0] */
280	for(i=jz;i>=0;i--) {
281	    for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k];
282	    fq[jz-i] = fw;
283	}
284
285    /* compress fq[] into y[] */
286	switch(prec) {
287	    case 0:
288		fw = 0.0;
289		for (i=jz;i>=0;i--) fw += fq[i];
290		y[0] = (ih==0)? fw: -fw;
291		break;
292	    case 1:
293	    case 2:
294		fw = 0.0;
295		for (i=jz;i>=0;i--) fw += fq[i];
296		y[0] = (ih==0)? fw: -fw;
297		fw = fq[0]-fw;
298		for (i=1;i<=jz;i++) fw += fq[i];
299		y[1] = (ih==0)? fw: -fw;
300		break;
301	    case 3:	/* painful */
302		for (i=jz;i>0;i--) {
303		    fw      = fq[i-1]+fq[i];
304		    fq[i]  += fq[i-1]-fw;
305		    fq[i-1] = fw;
306		}
307		for (i=jz;i>1;i--) {
308		    fw      = fq[i-1]+fq[i];
309		    fq[i]  += fq[i-1]-fw;
310		    fq[i-1] = fw;
311		}
312		for (fw=0.0,i=jz;i>=2;i--) fw += fq[i];
313		if(ih==0) {
314		    y[0] =  fq[0]; y[1] =  fq[1]; y[2] =  fw;
315		} else {
316		    y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw;
317		}
318	}
319	return n&7;
320}
321