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
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 * Constants:
123 * The hexadecimal values are the intended ones for the following
124 * constants. The decimal values may be used, provided that the
125 * compiler will convert from decimal to binary accurately enough
126 * to produce the hexadecimal values shown.
127 */
128
129 #include "fdlibm.h"
130
131 #ifdef _NEED_FLOAT64
132
133 static const int init_jk[] = { 2, 3, 4, 6 }; /* initial value for jk */
134
135 static const __float64 PIo2[] = {
136 _F_64(1.57079625129699707031e+00), /* 0x3FF921FB, 0x40000000 */
137 _F_64(7.54978941586159635335e-08), /* 0x3E74442D, 0x00000000 */
138 _F_64(5.39030252995776476554e-15), /* 0x3CF84698, 0x80000000 */
139 _F_64(3.28200341580791294123e-22), /* 0x3B78CC51, 0x60000000 */
140 _F_64(1.27065575308067607349e-29), /* 0x39F01B83, 0x80000000 */
141 _F_64(1.22933308981111328932e-36), /* 0x387A2520, 0x40000000 */
142 _F_64(2.73370053816464559624e-44), /* 0x36E38222, 0x80000000 */
143 _F_64(2.16741683877804819444e-51), /* 0x3569F31D, 0x00000000 */
144 };
145
146 static const __float64
147 zero = _F_64(0.0), one = _F_64(1.0),
148 two24 = _F_64(1.67772160000000000000e+07), /* 0x41700000, 0x00000000 */
149 twon24 = _F_64(5.96046447753906250000e-08); /* 0x3E700000, 0x00000000 */
150
151 #ifdef __GNUC__
152 #pragma GCC diagnostic ignored "-Wpragmas"
153 #pragma GCC diagnostic ignored "-Wunknown-warning-option"
154 #pragma GCC diagnostic ignored "-Wmaybe-uninitialized"
155 /* GCC analyzer gets confused about the use of 'iq' here */
156 #pragma GCC diagnostic ignored "-Wanalyzer-use-of-uninitialized-value"
157 #endif
158
159 int
__kernel_rem_pio2(__float64 * x,__float64 * y,int e0,int nx,int prec,const __int32_t * ipio2)160 __kernel_rem_pio2(__float64 *x, __float64 *y, int e0, int nx, int prec,
161 const __int32_t *ipio2)
162 {
163 __int32_t jz, jx, jv, jp, jk, carry, n, iq[20], i, j, k, m, q0, ih;
164 __float64 z, fw, f[20], fq[20], q[20];
165
166 /* initialize jk*/
167 jk = init_jk[prec];
168 jp = jk;
169
170 /* determine jx,jv,q0, note that 3>q0 */
171 jx = nx - 1;
172 jv = (e0 - 3) / 24;
173 if (jv < 0)
174 jv = 0;
175 q0 = e0 - 24 * (jv + 1);
176
177 /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
178 j = jv - jx;
179 m = jx + jk;
180 for (i = 0; i <= m; i++, j++)
181 f[i] = (j < 0) ? zero : (__float64)ipio2[j];
182
183 /* compute q[0],q[1],...q[jk] */
184 for (i = 0; i <= jk; i++) {
185 for (j = 0, fw = _F_64(0.0); j <= jx; j++)
186 fw += x[j] * f[jx + i - j];
187 q[i] = fw;
188 }
189
190 jz = jk;
191 recompute:
192 /* distill q[] into iq[] reversingly */
193 for (i = 0, j = jz, z = q[jz]; j > 0; i++, j--) {
194 fw = (__float64)((__int32_t)(twon24 * z));
195 iq[i] = (__int32_t)(z - two24 * fw);
196 z = q[j - 1] + fw;
197 }
198
199 /* compute n */
200 z = scalbn(z, (int)q0); /* actual value of z */
201 z -= _F_64(8.0) * floor(z * _F_64(0.125)); /* trim off integer >= 8 */
202 n = (__int32_t)z;
203 z -= (__float64)n;
204 ih = 0;
205 if (q0 > 0) { /* need iq[jz-1] to determine n */
206 i = (iq[jz - 1] >> (24 - q0));
207 n += i;
208 iq[jz - 1] -= i << (24 - q0);
209 ih = iq[jz - 1] >> (23 - q0);
210 } else if (q0 == 0)
211 ih = iq[jz - 1] >> 23;
212 else if (z >= _F_64(0.5))
213 ih = 2;
214
215 if (ih > 0) { /* q > 0.5 */
216 n += 1;
217 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;
223 iq[i] = 0x1000000 - j;
224 }
225 } else
226 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;
232 break;
233 case 2:
234 iq[jz - 1] &= 0x3fffff;
235 break;
236 }
237 }
238 if (ih == 2) {
239 z = one - z;
240 if (carry != 0)
241 z -= scalbn(one, (int)q0);
242 }
243 }
244
245 /* check if recomputation is needed */
246 if (z == zero) {
247 j = 0;
248 for (i = jz - 1; i >= jk; i--)
249 j |= iq[i];
250 if (j == 0) { /* need recomputation */
251 for (k = 1; iq[jk - k] == 0; k++)
252 ; /* k = no. of terms needed */
253
254 for (i = jz + 1; i <= jz + k; i++) { /* add q[jz+1] to q[jz+k] */
255 f[jx + i] = (__float64)ipio2[jv + i];
256 for (j = 0, fw = _F_64(0.0); j <= jx; j++)
257 fw += x[j] * f[jx + i - j];
258 q[i] = fw;
259 }
260 jz += k;
261 goto recompute;
262 }
263 }
264
265 /* chop off zero terms */
266 if (z == _F_64(0.0)) {
267 jz -= 1;
268 q0 -= 24;
269 while (iq[jz] == 0) {
270 jz--;
271 q0 -= 24;
272 }
273 } else { /* break z into 24-bit if necessary */
274 z = scalbn(z, -(int)q0);
275 if (z >= two24) {
276 fw = (__float64)((__int32_t)(twon24 * z));
277 iq[jz] = (__int32_t)(z - two24 * fw);
278 jz += 1;
279 q0 += 24;
280 iq[jz] = (__int32_t)fw;
281 } else
282 iq[jz] = (__int32_t)z;
283 }
284
285 /* convert integer "bit" chunk to floating-point value */
286 fw = scalbn(one, (int)q0);
287 for (i = jz; i >= 0; i--) {
288 q[i] = fw * (__float64)iq[i];
289 fw *= twon24;
290 }
291
292 /* compute PIo2[0,...,jp]*q[jz,...,0] */
293 for (i = jz; i >= 0; i--) {
294 for (fw = _F_64(0.0), k = 0; k <= jp && k <= jz - i; k++)
295 fw += PIo2[k] * q[i + k];
296 fq[jz - i] = fw;
297 }
298
299 /* compress fq[] into y[] */
300 switch (prec) {
301 case 0:
302 fw = _F_64(0.0);
303 for (i = jz; i >= 0; i--)
304 fw += fq[i];
305 y[0] = (ih == 0) ? fw : -fw;
306 break;
307 case 1:
308 case 2:
309 fw = _F_64(0.0);
310 for (i = jz; i >= 0; i--)
311 fw += fq[i];
312 y[0] = (ih == 0) ? fw : -fw;
313 fw = fq[0] - fw;
314 for (i = 1; i <= jz; i++)
315 fw += fq[i];
316 y[1] = (ih == 0) ? fw : -fw;
317 break;
318 case 3: /* painful */
319 for (i = jz; i > 0; i--) {
320 fw = fq[i - 1] + fq[i];
321 fq[i] += fq[i - 1] - fw;
322 fq[i - 1] = fw;
323 }
324 for (i = jz; i > 1; i--) {
325 fw = fq[i - 1] + fq[i];
326 fq[i] += fq[i - 1] - fw;
327 fq[i - 1] = fw;
328 }
329 for (fw = _F_64(0.0), i = jz; i >= 2; i--)
330 fw += fq[i];
331 if (ih == 0) {
332 y[0] = fq[0];
333 y[1] = fq[1];
334 y[2] = fw;
335 } else {
336 y[0] = -fq[0];
337 y[1] = -fq[1];
338 y[2] = -fw;
339 }
340 }
341 return n & 7;
342 }
343
344 #endif /* _NEED_FLOAT64 */
345
