1
2/* @(#)k_rem_pio2.c 1.3 95/01/18 */
3/*
4 * ====================================================
5 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
6 *
7 * Developed at SunSoft, a Sun Microsystems, Inc. business.
8 * Permission to use, copy, modify, and distribute this
9 * software is freely granted, provided that this notice
10 * is preserved.
11 * ====================================================
12 */
13
14/*
15 * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
16 * double x[],y[]; int e0,nx,prec; int ipio2[];
17 *
18 * __kernel_rem_pio2 return the last three digits of N with
19 *		y = x - N*pi/2
20 * so that |y| < pi/2.
21 *
22 * The method is to compute the integer (mod 8) and fraction parts of
23 * (2/pi)*x without doing the full multiplication. In general we
24 * skip the part of the product that are known to be a huge integer (
25 * more accurately, = 0 mod 8 ). Thus the number of operations are
26 * independent of the exponent of the input.
27 *
28 * (2/pi) is represented by an array of 24-bit integers in ipio2[].
29 *
30 * Input parameters:
31 * 	x[]	The input value (must be positive) is broken into nx
32 *		pieces of 24-bit integers in double precision format.
33 *		x[i] will be the i-th 24 bit of x. The scaled exponent
34 *		of x[0] is given in input parameter e0 (i.e., x[0]*2^e0
35 *		match x's up to 24 bits.
36 *
37 *		Example of breaking a double positive z into x[0]+x[1]+x[2]:
38 *			e0 = ieee_ilogb(z)-23
39 *			z  = ieee_scalbn(z,-e0)
40 *		for i = 0,1,2
41 *			x[i] = ieee_floor(z)
42 *			z    = (z-x[i])*2**24
43 *
44 *
45 *	y[]	ouput result in an array of double precision numbers.
46 *		The dimension of y[] is:
47 *			24-bit  precision	1
48 *			53-bit  precision	2
49 *			64-bit  precision	2
50 *			113-bit precision	3
51 *		The actual value is the sum of them. Thus for 113-bit
52 *		precison, one may have to do something like:
53 *
54 *		long double t,w,r_head, r_tail;
55 *		t = (long double)y[2] + (long double)y[1];
56 *		w = (long double)y[0];
57 *		r_head = t+w;
58 *		r_tail = w - (r_head - t);
59 *
60 *	e0	The exponent of x[0]
61 *
62 *	nx	dimension of x[]
63 *
64 *  	prec	an integer indicating the precision:
65 *			0	24  bits (single)
66 *			1	53  bits (double)
67 *			2	64  bits (extended)
68 *			3	113 bits (quad)
69 *
70 *	ipio2[]
71 *		integer array, contains the (24*i)-th to (24*i+23)-th
72 *		bit of 2/pi after binary point. The corresponding
73 *		floating value is
74 *
75 *			ipio2[i] * 2^(-24(i+1)).
76 *
77 * External function:
78 *	double ieee_scalbn(), ieee_floor();
79 *
80 *
81 * Here is the description of some local variables:
82 *
83 * 	jk	jk+1 is the initial number of terms of ipio2[] needed
84 *		in the computation. The recommended value is 2,3,4,
85 *		6 for single, double, extended,and quad.
86 *
87 * 	jz	local integer variable indicating the number of
88 *		terms of ipio2[] used.
89 *
90 *	jx	nx - 1
91 *
92 *	jv	index for pointing to the suitable ipio2[] for the
93 *		computation. In general, we want
94 *			( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8
95 *		is an integer. Thus
96 *			e0-3-24*jv >= 0 or (e0-3)/24 >= jv
97 *		Hence jv = max(0,(e0-3)/24).
98 *
99 *	jp	jp+1 is the number of terms in PIo2[] needed, jp = jk.
100 *
101 * 	q[]	double array with integral value, representing the
102 *		24-bits chunk of the product of x and 2/pi.
103 *
104 *	q0	the corresponding exponent of q[0]. Note that the
105 *		exponent for q[i] would be q0-24*i.
106 *
107 *	PIo2[]	double precision array, obtained by cutting pi/2
108 *		into 24 bits chunks.
109 *
110 *	f[]	ipio2[] in floating point
111 *
112 *	iq[]	integer array by breaking up q[] in 24-bits chunk.
113 *
114 *	fq[]	final product of x*(2/pi) in fq[0],..,fq[jk]
115 *
116 *	ih	integer. If >0 it indicates q[] is >= 0.5, hence
117 *		it also indicates the *sign* of the result.
118 *
119 */
120
121
122/*
123 * Constants:
124 * The hexadecimal values are the intended ones for the following
125 * constants. The decimal values may be used, provided that the
126 * compiler will convert from decimal to binary accurately enough
127 * to produce the hexadecimal values shown.
128 */
129
130#include "fdlibm.h"
131
132#ifdef __STDC__
133static const int init_jk[] = {2,3,4,6}; /* initial value for jk */
134#else
135static int init_jk[] = {2,3,4,6};
136#endif
137
138#ifdef __STDC__
139static const double PIo2[] = {
140#else
141static double PIo2[] = {
142#endif
143  1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */
144  7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */
145  5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */
146  3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */
147  1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */
148  1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */
149  2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */
150  2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */
151};
152
153#ifdef __STDC__
154static const double
155#else
156static double
157#endif
158zero   = 0.0,
159one    = 1.0,
160two24   =  1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
161twon24  =  5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */
162
163#ifdef __STDC__
164	int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, const int *ipio2)
165#else
166	int __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
167	double x[], y[]; int e0,nx,prec; int ipio2[];
168#endif
169{
170	int jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih;
171	double z,fw,f[20],fq[20],q[20];
172
173    /* initialize jk*/
174	jk = init_jk[prec];
175	jp = jk;
176
177    /* determine jx,jv,q0, note that 3>q0 */
178	jx =  nx-1;
179	jv = (e0-3)/24; if(jv<0) jv=0;
180	q0 =  e0-24*(jv+1);
181
182    /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
183	j = jv-jx; m = jx+jk;
184	for(i=0;i<=m;i++,j++) f[i] = (j<0)? zero : (double) ipio2[j];
185
186    /* compute q[0],q[1],...q[jk] */
187	for (i=0;i<=jk;i++) {
188	    for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; q[i] = fw;
189	}
190
191	jz = jk;
192recompute:
193    /* distill q[] into iq[] reversingly */
194	for(i=0,j=jz,z=q[jz];j>0;i++,j--) {
195	    fw    =  (double)((int)(twon24* z));
196	    iq[i] =  (int)(z-two24*fw);
197	    z     =  q[j-1]+fw;
198	}
199
200    /* compute n */
201	z  = ieee_scalbn(z,q0);		/* actual value of z */
202	z -= 8.0*ieee_floor(z*0.125);		/* trim off integer >= 8 */
203	n  = (int) z;
204	z -= (double)n;
205	ih = 0;
206	if(q0>0) {	/* need iq[jz-1] to determine n */
207	    i  = (iq[jz-1]>>(24-q0)); n += i;
208	    iq[jz-1] -= i<<(24-q0);
209	    ih = iq[jz-1]>>(23-q0);
210	}
211	else if(q0==0) ih = iq[jz-1]>>23;
212	else if(z>=0.5) ih=2;
213
214	if(ih>0) {	/* q > 0.5 */
215	    n += 1; carry = 0;
216	    for(i=0;i<jz ;i++) {	/* compute 1-q */
217		j = iq[i];
218		if(carry==0) {
219		    if(j!=0) {
220			carry = 1; iq[i] = 0x1000000- j;
221		    }
222		} else  iq[i] = 0xffffff - j;
223	    }
224	    if(q0>0) {		/* rare case: chance is 1 in 12 */
225	        switch(q0) {
226	        case 1:
227	    	   iq[jz-1] &= 0x7fffff; break;
228	    	case 2:
229	    	   iq[jz-1] &= 0x3fffff; break;
230	        }
231	    }
232	    if(ih==2) {
233		z = one - z;
234		if(carry!=0) z -= ieee_scalbn(one,q0);
235	    }
236	}
237
238    /* check if recomputation is needed */
239	if(z==zero) {
240	    j = 0;
241	    for (i=jz-1;i>=jk;i--) j |= iq[i];
242	    if(j==0) { /* need recomputation */
243		for(k=1;iq[jk-k]==0;k++);   /* k = no. of terms needed */
244
245		for(i=jz+1;i<=jz+k;i++) {   /* add q[jz+1] to q[jz+k] */
246		    f[jx+i] = (double) ipio2[jv+i];
247		    for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j];
248		    q[i] = fw;
249		}
250		jz += k;
251		goto recompute;
252	    }
253	}
254
255    /* chop off zero terms */
256	if(z==0.0) {
257	    jz -= 1; q0 -= 24;
258	    while(iq[jz]==0) { jz--; q0-=24;}
259	} else { /* break z into 24-bit if necessary */
260	    z = ieee_scalbn(z,-q0);
261	    if(z>=two24) {
262		fw = (double)((int)(twon24*z));
263		iq[jz] = (int)(z-two24*fw);
264		jz += 1; q0 += 24;
265		iq[jz] = (int) fw;
266	    } else iq[jz] = (int) z ;
267	}
268
269    /* convert integer "bit" chunk to floating-point value */
270	fw = ieee_scalbn(one,q0);
271	for(i=jz;i>=0;i--) {
272	    q[i] = fw*(double)iq[i]; fw*=twon24;
273	}
274
275    /* compute PIo2[0,...,jp]*q[jz,...,0] */
276	for(i=jz;i>=0;i--) {
277	    for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k];
278	    fq[jz-i] = fw;
279	}
280
281    /* compress fq[] into y[] */
282	switch(prec) {
283	    case 0:
284		fw = 0.0;
285		for (i=jz;i>=0;i--) fw += fq[i];
286		y[0] = (ih==0)? fw: -fw;
287		break;
288	    case 1:
289	    case 2:
290		fw = 0.0;
291		for (i=jz;i>=0;i--) fw += fq[i];
292		y[0] = (ih==0)? fw: -fw;
293		fw = fq[0]-fw;
294		for (i=1;i<=jz;i++) fw += fq[i];
295		y[1] = (ih==0)? fw: -fw;
296		break;
297	    case 3:	/* painful */
298		for (i=jz;i>0;i--) {
299		    fw      = fq[i-1]+fq[i];
300		    fq[i]  += fq[i-1]-fw;
301		    fq[i-1] = fw;
302		}
303		for (i=jz;i>1;i--) {
304		    fw      = fq[i-1]+fq[i];
305		    fq[i]  += fq[i-1]-fw;
306		    fq[i-1] = fw;
307		}
308		for (fw=0.0,i=jz;i>=2;i--) fw += fq[i];
309		if(ih==0) {
310		    y[0] =  fq[0]; y[1] =  fq[1]; y[2] =  fw;
311		} else {
312		    y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw;
313		}
314	}
315	return n&7;
316}
317