1 
2 /* @(#)k_rem_pio2.c 5.1 93/09/24 */
3 /*
4  * ====================================================
5  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
6  *
7  * Developed at SunPro, 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
__ieee754_remainder(double x,double p)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 = ilogb(z)-23
39  *			z  = scalbn(z,-e0)
40  *		for i = 0,1,2
41  *			x[i] = 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 scalbn(), 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 #ifndef _DOUBLE_IS_32BITS
133 
134 #ifdef __STDC__
135 static const int init_jk[] = {2,3,4,6}; /* initial value for jk */
136 #else
137 static int init_jk[] = {2,3,4,6};
138 #endif
139 
140 #ifdef __STDC__
141 static const double PIo2[] = {
142 #else
143 static double PIo2[] = {
144 #endif
145   1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */
146   7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */
147   5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */
148   3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */
149   1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */
150   1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */
151   2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */
152   2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */
153 };
154 
155 #ifdef __STDC__
156 static const double
157 #else
158 static double
159 #endif
160 zero   = 0.0,
161 one    = 1.0,
162 two24   =  1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
163 twon24  =  5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */
164 
165 #ifdef __STDC__
166 	int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, const int32_t *ipio2)
167 #else
168 	int __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
169 	double x[], y[]; int e0,nx,prec; int32_t ipio2[];
170 #endif
171 {
172 	int32_t jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih;
173 	double z,fw,f[20],fq[20],q[20];
174 
175     /* initialize jk*/
176 	jk = init_jk[prec];
177 	jp = jk;
178 
179     /* determine jx,jv,q0, note that 3>q0 */
180 	jx =  nx-1;
181 	jv = (e0-3)/24; if(jv<0) jv=0;
182 	q0 =  e0-24*(jv+1);
183 
184     /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
185 	j = jv-jx; m = jx+jk;
186 	for(i=0;i<=m;i++,j++) f[i] = (j<0)? zero : (double) ipio2[j];
187 
188     /* compute q[0],q[1],...q[jk] */
189 	for (i=0;i<=jk;i++) {
190 	    for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; q[i] = fw;
191 	}
192 
193 	jz = jk;
194 recompute:
195     /* distill q[] into iq[] reversingly */
196 	for(i=0,j=jz,z=q[jz];j>0;i++,j--) {
197 	    fw    =  (double)((int32_t)(twon24* z));
198 	    iq[i] =  (int32_t)(z-two24*fw);
199 	    z     =  q[j-1]+fw;
200 	}
201 
202     /* compute n */
203 	z  = scalbn(z,(int)q0);		/* actual value of z */
204 	z -= 8.0*floor(z*0.125);		/* trim off integer >= 8 */
205 	n  = (int32_t) z;
206 	z -= (double)n;
207 	ih = 0;
208 	if(q0>0) {	/* need iq[jz-1] to determine n */
209 	    i  = (iq[jz-1]>>(24-q0)); n += i;
210 	    iq[jz-1] -= i<<(24-q0);
211 	    ih = iq[jz-1]>>(23-q0);
212 	}
213 	else if(q0==0) ih = iq[jz-1]>>23;
214 	else if(z>=0.5) ih=2;
215 
216 	if(ih>0) {	/* q > 0.5 */
217 	    n += 1; carry = 0;
218 	    for(i=0;i<jz ;i++) {	/* compute 1-q */
219 		j = iq[i];
220 		if(carry==0) {
221 		    if(j!=0) {
222 			carry = 1; iq[i] = 0x1000000- j;
223 		    }
224 		} else  iq[i] = 0xffffff - j;
225 	    }
226 	    if(q0>0) {		/* rare case: chance is 1 in 12 */
227 	        switch(q0) {
228 	        case 1:
229 	    	   iq[jz-1] &= 0x7fffff; break;
230 	    	case 2:
231 	    	   iq[jz-1] &= 0x3fffff; break;
232 	        }
233 	    }
234 	    if(ih==2) {
235 		z = one - z;
236 		if(carry!=0) z -= scalbn(one,(int)q0);
237 	    }
238 	}
239 
240     /* check if recomputation is needed */
241 	if(z==zero) {
242 	    j = 0;
243 	    for (i=jz-1;i>=jk;i--) j |= iq[i];
244 	    if(j==0) { /* need recomputation */
245 		for(k=1;iq[jk-k]==0;k++);   /* k = no. of terms needed */
246 
247 		for(i=jz+1;i<=jz+k;i++) {   /* add q[jz+1] to q[jz+k] */
248 		    f[jx+i] = (double) ipio2[jv+i];
249 		    for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j];
250 		    q[i] = fw;
251 		}
252 		jz += k;
253 		goto recompute;
254 	    }
255 	}
256 
257     /* chop off zero terms */
258 	if(z==0.0) {
259 	    jz -= 1; q0 -= 24;
260 	    while(iq[jz]==0) { jz--; q0-=24;}
261 	} else { /* break z into 24-bit if necessary */
262 	    z = scalbn(z,-(int)q0);
263 	    if(z>=two24) {
264 		fw = (double)((int32_t)(twon24*z));
265 		iq[jz] = (int32_t)(z-two24*fw);
266 		jz += 1; q0 += 24;
267 		iq[jz] = (int32_t) fw;
268 	    } else iq[jz] = (int32_t) z ;
269 	}
270 
271     /* convert integer "bit" chunk to floating-point value */
272 	fw = scalbn(one,(int)q0);
273 	for(i=jz;i>=0;i--) {
274 	    q[i] = fw*(double)iq[i]; fw*=twon24;
275 	}
276 
277     /* compute PIo2[0,...,jp]*q[jz,...,0] */
278 	for(i=jz;i>=0;i--) {
279 	    for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k];
280 	    fq[jz-i] = fw;
281 	}
282 
283     /* compress fq[] into y[] */
284 	switch(prec) {
285 	    case 0:
286 		fw = 0.0;
287 		for (i=jz;i>=0;i--) fw += fq[i];
288 		y[0] = (ih==0)? fw: -fw;
289 		break;
290 	    case 1:
291 	    case 2:
292 		fw = 0.0;
293 		for (i=jz;i>=0;i--) fw += fq[i];
294 		y[0] = (ih==0)? fw: -fw;
295 		fw = fq[0]-fw;
296 		for (i=1;i<=jz;i++) fw += fq[i];
297 		y[1] = (ih==0)? fw: -fw;
298 		break;
299 	    case 3:	/* painful */
300 		for (i=jz;i>0;i--) {
301 		    fw      = fq[i-1]+fq[i];
302 		    fq[i]  += fq[i-1]-fw;
303 		    fq[i-1] = fw;
304 		}
305 		for (i=jz;i>1;i--) {
306 		    fw      = fq[i-1]+fq[i];
307 		    fq[i]  += fq[i-1]-fw;
308 		    fq[i-1] = fw;
309 		}
310 		for (fw=0.0,i=jz;i>=2;i--) fw += fq[i];
311 		if(ih==0) {
312 		    y[0] =  fq[0]; y[1] =  fq[1]; y[2] =  fw;
313 		} else {
314 		    y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw;
315 		}
316 	}
317 	return n&7;
318 }
319 
320 #endif /* defined(_DOUBLE_IS_32BITS) */
321