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author Sam Lantinga <slouken@libsdl.org>
date Mon, 15 Sep 2008 06:46:23 +0000
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2756:a98604b691c8 2757:0581f49c9294
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 #if defined(LIBM_SCCS) && !defined(lint)
14 static char rcsid[] =
15 "$NetBSD: k_rem_pio2.c,v 1.7 1995/05/10 20:46:25 jtc Exp $";
16 #endif
17
18 /*
19 * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
20 * double x[],y[]; int e0,nx,prec; int ipio2[];
21 *
22 * __kernel_rem_pio2 return the last three digits of N with
23 * y = x - N*pi/2
24 * so that |y| < pi/2.
25 *
26 * The method is to compute the integer (mod 8) and fraction parts of
27 * (2/pi)*x without doing the full multiplication. In general we
28 * skip the part of the product that are known to be a huge integer (
29 * more accurately, = 0 mod 8 ). Thus the number of operations are
30 * independent of the exponent of the input.
31 *
32 * (2/pi) is represented by an array of 24-bit integers in ipio2[].
33 *
34 * Input parameters:
35 * x[] The input value (must be positive) is broken into nx
36 * pieces of 24-bit integers in double precision format.
37 * x[i] will be the i-th 24 bit of x. The scaled exponent
38 * of x[0] is given in input parameter e0 (i.e., x[0]*2^e0
39 * match x's up to 24 bits.
40 *
41 * Example of breaking a double positive z into x[0]+x[1]+x[2]:
42 * e0 = ilogb(z)-23
43 * z = scalbn(z,-e0)
44 * for i = 0,1,2
45 * x[i] = floor(z)
46 * z = (z-x[i])*2**24
47 *
48 *
49 * y[] ouput result in an array of double precision numbers.
50 * The dimension of y[] is:
51 * 24-bit precision 1
52 * 53-bit precision 2
53 * 64-bit precision 2
54 * 113-bit precision 3
55 * The actual value is the sum of them. Thus for 113-bit
56 * precison, one may have to do something like:
57 *
58 * long double t,w,r_head, r_tail;
59 * t = (long double)y[2] + (long double)y[1];
60 * w = (long double)y[0];
61 * r_head = t+w;
62 * r_tail = w - (r_head - t);
63 *
64 * e0 The exponent of x[0]
65 *
66 * nx dimension of x[]
67 *
68 * prec an integer indicating the precision:
69 * 0 24 bits (single)
70 * 1 53 bits (double)
71 * 2 64 bits (extended)
72 * 3 113 bits (quad)
73 *
74 * ipio2[]
75 * integer array, contains the (24*i)-th to (24*i+23)-th
76 * bit of 2/pi after binary point. The corresponding
77 * floating value is
78 *
79 * ipio2[i] * 2^(-24(i+1)).
80 *
81 * External function:
82 * double scalbn(), floor();
83 *
84 *
85 * Here is the description of some local variables:
86 *
87 * jk jk+1 is the initial number of terms of ipio2[] needed
88 * in the computation. The recommended value is 2,3,4,
89 * 6 for single, double, extended,and quad.
90 *
91 * jz local integer variable indicating the number of
92 * terms of ipio2[] used.
93 *
94 * jx nx - 1
95 *
96 * jv index for pointing to the suitable ipio2[] for the
97 * computation. In general, we want
98 * ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8
99 * is an integer. Thus
100 * e0-3-24*jv >= 0 or (e0-3)/24 >= jv
101 * Hence jv = max(0,(e0-3)/24).
102 *
103 * jp jp+1 is the number of terms in PIo2[] needed, jp = jk.
104 *
105 * q[] double array with integral value, representing the
106 * 24-bits chunk of the product of x and 2/pi.
107 *
108 * q0 the corresponding exponent of q[0]. Note that the
109 * exponent for q[i] would be q0-24*i.
110 *
111 * PIo2[] double precision array, obtained by cutting pi/2
112 * into 24 bits chunks.
113 *
114 * f[] ipio2[] in floating point
115 *
116 * iq[] integer array by breaking up q[] in 24-bits chunk.
117 *
118 * fq[] final product of x*(2/pi) in fq[0],..,fq[jk]
119 *
120 * ih integer. If >0 it indicates q[] is >= 0.5, hence
121 * it also indicates the *sign* of the result.
122 *
123 */
124
125
126 /*
127 * Constants:
128 * The hexadecimal values are the intended ones for the following
129 * constants. The decimal values may be used, provided that the
130 * compiler will convert from decimal to binary accurately enough
131 * to produce the hexadecimal values shown.
132 */
133
134 #include "math.h"
135 #include "math_private.h"
136
137 libm_hidden_proto(scalbn)
138 libm_hidden_proto(floor)
139 #ifdef __STDC__
140 static const int init_jk[] = { 2, 3, 4, 6 }; /* initial value for jk */
141 #else
142 static int init_jk[] = { 2, 3, 4, 6 };
143 #endif
144
145 #ifdef __STDC__
146 static const double PIo2[] = {
147 #else
148 static double PIo2[] = {
149 #endif
150 1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */
151 7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */
152 5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */
153 3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */
154 1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */
155 1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */
156 2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */
157 2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */
158 };
159
160 #ifdef __STDC__
161 static const double
162 #else
163 static double
164 #endif
165 zero = 0.0, one = 1.0, two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
166 twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */
167
168 #ifdef __STDC__
169 int attribute_hidden
170 __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec,
171 const int32_t * ipio2)
172 #else
173 int attribute_hidden
174 __kernel_rem_pio2(x, y, e0, nx, prec, ipio2)
175 double x[], y[];
176 int e0, nx, prec;
177 int32_t ipio2[];
178 #endif
179 {
180 int32_t jz, jx, jv, jp, jk, carry, n, iq[20], i, j, k, m, q0, ih;
181 double z, fw, f[20], fq[20], q[20];
182
183 /* initialize jk */
184 jk = init_jk[prec];
185 jp = jk;
186
187 /* determine jx,jv,q0, note that 3>q0 */
188 jx = nx - 1;
189 jv = (e0 - 3) / 24;
190 if (jv < 0)
191 jv = 0;
192 q0 = e0 - 24 * (jv + 1);
193
194 /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
195 j = jv - jx;
196 m = jx + jk;
197 for (i = 0; i <= m; i++, j++)
198 f[i] = (j < 0) ? zero : (double) ipio2[j];
199
200 /* compute q[0],q[1],...q[jk] */
201 for (i = 0; i <= jk; i++) {
202 for (j = 0, fw = 0.0; j <= jx; j++)
203 fw += x[j] * f[jx + i - j];
204 q[i] = fw;
205 }
206
207 jz = jk;
208 recompute:
209 /* distill q[] into iq[] reversingly */
210 for (i = 0, j = jz, z = q[jz]; j > 0; i++, j--) {
211 fw = (double) ((int32_t) (twon24 * z));
212 iq[i] = (int32_t) (z - two24 * fw);
213 z = q[j - 1] + fw;
214 }
215
216 /* compute n */
217 z = scalbn(z, q0); /* actual value of z */
218 z -= 8.0 * floor(z * 0.125); /* trim off integer >= 8 */
219 n = (int32_t) z;
220 z -= (double) n;
221 ih = 0;
222 if (q0 > 0) { /* need iq[jz-1] to determine n */
223 i = (iq[jz - 1] >> (24 - q0));
224 n += i;
225 iq[jz - 1] -= i << (24 - q0);
226 ih = iq[jz - 1] >> (23 - q0);
227 } else if (q0 == 0)
228 ih = iq[jz - 1] >> 23;
229 else if (z >= 0.5)
230 ih = 2;
231
232 if (ih > 0) { /* q > 0.5 */
233 n += 1;
234 carry = 0;
235 for (i = 0; i < jz; i++) { /* compute 1-q */
236 j = iq[i];
237 if (carry == 0) {
238 if (j != 0) {
239 carry = 1;
240 iq[i] = 0x1000000 - j;
241 }
242 } else
243 iq[i] = 0xffffff - j;
244 }
245 if (q0 > 0) { /* rare case: chance is 1 in 12 */
246 switch (q0) {
247 case 1:
248 iq[jz - 1] &= 0x7fffff;
249 break;
250 case 2:
251 iq[jz - 1] &= 0x3fffff;
252 break;
253 }
254 }
255 if (ih == 2) {
256 z = one - z;
257 if (carry != 0)
258 z -= scalbn(one, q0);
259 }
260 }
261
262 /* check if recomputation is needed */
263 if (z == zero) {
264 j = 0;
265 for (i = jz - 1; i >= jk; i--)
266 j |= iq[i];
267 if (j == 0) { /* need recomputation */
268 for (k = 1; iq[jk - k] == 0; k++); /* k = no. of terms needed */
269
270 for (i = jz + 1; i <= jz + k; i++) { /* add q[jz+1] to q[jz+k] */
271 f[jx + i] = (double) ipio2[jv + i];
272 for (j = 0, fw = 0.0; j <= jx; j++)
273 fw += x[j] * f[jx + i - j];
274 q[i] = fw;
275 }
276 jz += k;
277 goto recompute;
278 }
279 }
280
281 /* chop off zero terms */
282 if (z == 0.0) {
283 jz -= 1;
284 q0 -= 24;
285 while (iq[jz] == 0) {
286 jz--;
287 q0 -= 24;
288 }
289 } else { /* break z into 24-bit if necessary */
290 z = scalbn(z, -q0);
291 if (z >= two24) {
292 fw = (double) ((int32_t) (twon24 * z));
293 iq[jz] = (int32_t) (z - two24 * fw);
294 jz += 1;
295 q0 += 24;
296 iq[jz] = (int32_t) fw;
297 } else
298 iq[jz] = (int32_t) z;
299 }
300
301 /* convert integer "bit" chunk to floating-point value */
302 fw = scalbn(one, q0);
303 for (i = jz; i >= 0; i--) {
304 q[i] = fw * (double) iq[i];
305 fw *= twon24;
306 }
307
308 /* compute PIo2[0,...,jp]*q[jz,...,0] */
309 for (i = jz; i >= 0; i--) {
310 for (fw = 0.0, k = 0; k <= jp && k <= jz - i; k++)
311 fw += PIo2[k] * q[i + k];
312 fq[jz - i] = fw;
313 }
314
315 /* compress fq[] into y[] */
316 switch (prec) {
317 case 0:
318 fw = 0.0;
319 for (i = jz; i >= 0; i--)
320 fw += fq[i];
321 y[0] = (ih == 0) ? fw : -fw;
322 break;
323 case 1:
324 case 2:
325 fw = 0.0;
326 for (i = jz; i >= 0; i--)
327 fw += fq[i];
328 y[0] = (ih == 0) ? fw : -fw;
329 fw = fq[0] - fw;
330 for (i = 1; i <= jz; i++)
331 fw += fq[i];
332 y[1] = (ih == 0) ? fw : -fw;
333 break;
334 case 3: /* painful */
335 for (i = jz; i > 0; i--) {
336 fw = fq[i - 1] + fq[i];
337 fq[i] += fq[i - 1] - fw;
338 fq[i - 1] = fw;
339 }
340 for (i = jz; i > 1; i--) {
341 fw = fq[i - 1] + fq[i];
342 fq[i] += fq[i - 1] - fw;
343 fq[i - 1] = fw;
344 }
345 for (fw = 0.0, i = jz; i >= 2; i--)
346 fw += fq[i];
347 if (ih == 0) {
348 y[0] = fq[0];
349 y[1] = fq[1];
350 y[2] = fw;
351 } else {
352 y[0] = -fq[0];
353 y[1] = -fq[1];
354 y[2] = -fw;
355 }
356 }
357 return n & 7;
358 }