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k_rem_pio2.c (8327B)
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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
14 static char rcsid[] = "$FreeBSD: src/lib/msun/src/k_rem_pio2.c,v 1.6 2002/05/28 17:51:46 alfred 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 static const int init_jk[] = {2,3,4,6}; /* initial value for jk */
137
138 static const double PIo2[] = {
139 1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */
140 7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */
141 5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */
142 3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */
143 1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */
144 1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */
145 2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */
146 2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */
147 };
148
149 static const double
150 zero = 0.0,
151 one = 1.0,
152 two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
153 twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */
154
155 int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, const int32_t *ipio2)
156 {
157 int32_t jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih;
158 double z,fw,f[20],fq[20],q[20];
159
160 /* initialize jk*/
161 jk = init_jk[prec];
162 jp = jk;
163
164 /* determine jx,jv,q0, note that 3>q0 */
165 jx = nx-1;
166 jv = (e0-3)/24; if(jv<0) jv=0;
167 q0 = e0-24*(jv+1);
168
169 /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
170 j = jv-jx; m = jx+jk;
171 for(i=0;i<=m;i++,j++) f[i] = (j<0)? zero : (double) ipio2[j];
172
173 /* compute q[0],q[1],...q[jk] */
174 for (i=0;i<=jk;i++) {
175 for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; q[i] = fw;
176 }
177
178 jz = jk;
179 recompute:
180 /* distill q[] into iq[] reversingly */
181 for(i=0,j=jz,z=q[jz];j>0;i++,j--) {
182 fw = (double)((int32_t)(twon24* z));
183 iq[i] = (int32_t)(z-two24*fw);
184 z = q[j-1]+fw;
185 }
186
187 /* compute n */
188 z = scalbn(z,q0); /* actual value of z */
189 z -= 8.0*floor(z*0.125); /* trim off integer >= 8 */
190 n = (int32_t) z;
191 z -= (double)n;
192 ih = 0;
193 if(q0>0) { /* need iq[jz-1] to determine n */
194 i = (iq[jz-1]>>(24-q0)); n += i;
195 iq[jz-1] -= i<<(24-q0);
196 ih = iq[jz-1]>>(23-q0);
197 }
198 else if(q0==0) ih = iq[jz-1]>>23;
199 else if(z>=0.5) ih=2;
200
201 if(ih>0) { /* q > 0.5 */
202 n += 1; carry = 0;
203 for(i=0;i<jz ;i++) { /* compute 1-q */
204 j = iq[i];
205 if(carry==0) {
206 if(j!=0) {
207 carry = 1; iq[i] = 0x1000000- j;
208 }
209 } else iq[i] = 0xffffff - j;
210 }
211 if(q0>0) { /* rare case: chance is 1 in 12 */
212 switch(q0) {
213 case 1:
214 iq[jz-1] &= 0x7fffff; break;
215 case 2:
216 iq[jz-1] &= 0x3fffff; break;
217 }
218 }
219 if(ih==2) {
220 z = one - z;
221 if(carry!=0) z -= scalbn(one,q0);
222 }
223 }
224
225 /* check if recomputation is needed */
226 if(z==zero) {
227 j = 0;
228 for (i=jz-1;i>=jk;i--) j |= iq[i];
229 if(j==0) { /* need recomputation */
230 for(k=1;iq[jk-k]==0;k++); /* k = no. of terms needed */
231
232 for(i=jz+1;i<=jz+k;i++) { /* add q[jz+1] to q[jz+k] */
233 f[jx+i] = (double) ipio2[jv+i];
234 for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j];
235 q[i] = fw;
236 }
237 jz += k;
238 goto recompute;
239 }
240 }
241
242 /* chop off zero terms */
243 if(z==0.0) {
244 jz -= 1; q0 -= 24;
245 while(iq[jz]==0) { jz--; q0-=24;}
246 } else { /* break z into 24-bit if necessary */
247 z = scalbn(z,-q0);
248 if(z>=two24) {
249 fw = (double)((int32_t)(twon24*z));
250 iq[jz] = (int32_t)(z-two24*fw);
251 jz += 1; q0 += 24;
252 iq[jz] = (int32_t) fw;
253 } else iq[jz] = (int32_t) z ;
254 }
255
256 /* convert integer "bit" chunk to floating-point value */
257 fw = scalbn(one,q0);
258 for(i=jz;i>=0;i--) {
259 q[i] = fw*(double)iq[i]; fw*=twon24;
260 }
261
262 /* compute PIo2[0,...,jp]*q[jz,...,0] */
263 for(i=jz;i>=0;i--) {
264 for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k];
265 fq[jz-i] = fw;
266 }
267
268 /* compress fq[] into y[] */
269 switch(prec) {
270 case 0:
271 fw = 0.0;
272 for (i=jz;i>=0;i--) fw += fq[i];
273 y[0] = (ih==0)? fw: -fw;
274 break;
275 case 1:
276 case 2:
277 fw = 0.0;
278 for (i=jz;i>=0;i--) fw += fq[i];
279 y[0] = (ih==0)? fw: -fw;
280 fw = fq[0]-fw;
281 for (i=1;i<=jz;i++) fw += fq[i];
282 y[1] = (ih==0)? fw: -fw;
283 break;
284 case 3: /* painful */
285 for (i=jz;i>0;i--) {
286 fw = fq[i-1]+fq[i];
287 fq[i] += fq[i-1]-fw;
288 fq[i-1] = fw;
289 }
290 for (i=jz;i>1;i--) {
291 fw = fq[i-1]+fq[i];
292 fq[i] += fq[i-1]-fw;
293 fq[i-1] = fw;
294 }
295 for (fw=0.0,i=jz;i>=2;i--) fw += fq[i];
296 if(ih==0) {
297 y[0] = fq[0]; y[1] = fq[1]; y[2] = fw;
298 } else {
299 y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw;
300 }
301 }
302 return n&7;
303 }