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)
14static char rcsid[] = "$NetBSD: k_rem_pio2.c,v 1.7 1995/05/10 20:46:25 jtc 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[] output 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 * precision, 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-narrow-eval.h>
135#include <math_private.h>
136#include <libc-diag.h>
137
138static const int init_jk[] = {2,3,4,6}; /* initial value for jk */
139
140static const double PIo2[] = {
141 1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */
142 7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */
143 5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */
144 3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */
145 1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */
146 1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */
147 2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */
148 2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */
149};
150
151static const double
152 zero = 0.0,
153 one = 1.0,
154 two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
155 twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */
156
157int
158__kernel_rem_pio2 (double *x, double *y, int e0, int nx, int prec,
159 const int32_t *ipio2)
160{
161 int32_t jz, jx, jv, jp, jk, carry, n, iq[20], i, j, k, m, q0, ih;
162 double z, fw, f[20], fq[20], q[20];
163
164 /* initialize jk*/
165 jk = init_jk[prec];
166 jp = jk;
167
168 /* determine jx,jv,q0, note that 3>q0 */
169 jx = nx - 1;
170 jv = (e0 - 3) / 24; if (jv < 0)
171 jv = 0;
172 q0 = e0 - 24 * (jv + 1);
173
174 /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
175 j = jv - jx; m = jx + jk;
176 for (i = 0; i <= m; i++, j++)
177 f[i] = (j < 0) ? zero : (double) ipio2[j];
178
179 /* compute q[0],q[1],...q[jk] */
180 for (i = 0; i <= jk; i++)
181 {
182 for (j = 0, fw = 0.0; j <= jx; j++)
183 fw += x[j] * f[jx + i - j];
184 q[i] = fw;
185 }
186
187 jz = jk;
188recompute:
189 /* distill q[] into iq[] reversingly */
190 for (i = 0, j = jz, z = q[jz]; j > 0; i++, j--)
191 {
192 fw = (double) ((int32_t) (twon24 * z));
193 iq[i] = (int32_t) (z - two24 * fw);
194 z = q[j - 1] + fw;
195 }
196
197 /* compute n */
198 z = __scalbn (x: z, n: q0); /* actual value of z */
199 z -= 8.0 * floor (z * 0.125); /* trim off integer >= 8 */
200 n = (int32_t) z;
201 z -= (double) n;
202 ih = 0;
203 if (q0 > 0) /* need iq[jz-1] to determine n */
204 {
205 i = (iq[jz - 1] >> (24 - q0)); n += i;
206 iq[jz - 1] -= i << (24 - q0);
207 ih = iq[jz - 1] >> (23 - q0);
208 }
209 else if (q0 == 0)
210 ih = iq[jz - 1] >> 23;
211 else if (z >= 0.5)
212 ih = 2;
213
214 if (ih > 0) /* q > 0.5 */
215 {
216 n += 1; carry = 0;
217 for (i = 0; i < jz; i++) /* compute 1-q */
218 {
219 j = iq[i];
220 if (carry == 0)
221 {
222 if (j != 0)
223 {
224 carry = 1; iq[i] = 0x1000000 - j;
225 }
226 }
227 else
228 iq[i] = 0xffffff - j;
229 }
230 if (q0 > 0) /* rare case: chance is 1 in 12 */
231 {
232 switch (q0)
233 {
234 case 1:
235 iq[jz - 1] &= 0x7fffff; break;
236 case 2:
237 iq[jz - 1] &= 0x3fffff; break;
238 }
239 }
240 if (ih == 2)
241 {
242 z = one - z;
243 if (carry != 0)
244 z -= __scalbn (x: one, n: q0);
245 }
246 }
247
248 /* check if recomputation is needed */
249 if (z == zero)
250 {
251 j = 0;
252 for (i = jz - 1; i >= jk; i--)
253 j |= iq[i];
254 if (j == 0) /* need recomputation */
255 {
256 /* On s390x gcc 6.1 -O3 produces the warning "array subscript is below
257 array bounds [-Werror=array-bounds]". Only __ieee754_rem_pio2l
258 calls __kernel_rem_pio2 for normal numbers and |x| > pi/4 in case
259 of ldbl-96 and |x| > 3pi/4 in case of ldbl-128[ibm].
260 Thus x can't be zero and ipio2 is not zero, too. Thus not all iq[]
261 values can't be zero. */
262 DIAG_PUSH_NEEDS_COMMENT;
263 DIAG_IGNORE_NEEDS_COMMENT (6.1, "-Warray-bounds");
264 for (k = 1; iq[jk - k] == 0; k++)
265 ; /* k = no. of terms needed */
266 DIAG_POP_NEEDS_COMMENT;
267
268 for (i = jz + 1; i <= jz + k; i++) /* add q[jz+1] to q[jz+k] */
269 {
270 f[jx + i] = (double) ipio2[jv + i];
271 for (j = 0, fw = 0.0; j <= jx; j++)
272 fw += x[j] * f[jx + i - j];
273 q[i] = fw;
274 }
275 jz += k;
276 goto recompute;
277 }
278 }
279
280 /* chop off zero terms */
281 if (z == 0.0)
282 {
283 jz -= 1; q0 -= 24;
284 while (iq[jz] == 0)
285 {
286 jz--; q0 -= 24;
287 }
288 }
289 else /* break z into 24-bit if necessary */
290 {
291 z = __scalbn (x: z, n: -q0);
292 if (z >= two24)
293 {
294 fw = (double) ((int32_t) (twon24 * z));
295 iq[jz] = (int32_t) (z - two24 * fw);
296 jz += 1; q0 += 24;
297 iq[jz] = (int32_t) fw;
298 }
299 else
300 iq[jz] = (int32_t) z;
301 }
302
303 /* jz is always nonnegative here, because the result is never zero to
304 full precision (this function is not called for zero arguments).
305 Help the compiler to know it. */
306 if (jz < 0) __builtin_unreachable ();
307
308 /* convert integer "bit" chunk to floating-point value */
309 fw = __scalbn (x: one, n: q0);
310 for (i = jz; i >= 0; i--)
311 {
312 q[i] = fw * (double) iq[i]; fw *= twon24;
313 }
314
315 /* compute PIo2[0,...,jp]*q[jz,...,0] */
316 for (i = jz; i >= 0; i--)
317 {
318 for (fw = 0.0, k = 0; k <= jp && k <= jz - i; k++)
319 fw += PIo2[k] * q[i + k];
320 fq[jz - i] = fw;
321 }
322
323 /* compress fq[] into y[] */
324 switch (prec)
325 {
326 case 0:
327 fw = 0.0;
328 for (i = jz; i >= 0; i--)
329 fw += fq[i];
330 y[0] = (ih == 0) ? fw : -fw;
331 break;
332 case 1:
333 case 2:;
334 double fv = 0.0;
335 for (i = jz; i >= 0; i--)
336 fv = math_narrow_eval (fv + fq[i]);
337 y[0] = (ih == 0) ? fv : -fv;
338 fv = math_narrow_eval (fq[0] - fv);
339 for (i = 1; i <= jz; i++)
340 fv = math_narrow_eval (fv + fq[i]);
341 y[1] = (ih == 0) ? fv : -fv;
342 break;
343 case 3: /* painful */
344 for (i = jz; i > 0; i--)
345 {
346 double fv = math_narrow_eval (fq[i - 1] + fq[i]);
347 fq[i] += fq[i - 1] - fv;
348 fq[i - 1] = fv;
349 }
350 for (i = jz; i > 1; i--)
351 {
352 double fv = math_narrow_eval (fq[i - 1] + fq[i]);
353 fq[i] += fq[i - 1] - fv;
354 fq[i - 1] = fv;
355 }
356 for (fw = 0.0, i = jz; i >= 2; i--)
357 fw += fq[i];
358 if (ih == 0)
359 {
360 y[0] = fq[0]; y[1] = fq[1]; y[2] = fw;
361 }
362 else
363 {
364 y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw;
365 }
366 }
367 return n & 7;
368}
369

source code of glibc/sysdeps/ieee754/dbl-64/k_rem_pio2.c