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[] = "$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[] 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 | * 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_private.h> |
135 | #include <libc-diag.h> |
136 | |
137 | static const int init_jk[] = {2,3,4,6}; /* initial value for jk */ |
138 | |
139 | static const double PIo2[] = { |
140 | 1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */ |
141 | 7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */ |
142 | 5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */ |
143 | 3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */ |
144 | 1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */ |
145 | 1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */ |
146 | 2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */ |
147 | 2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */ |
148 | }; |
149 | |
150 | static const double |
151 | zero = 0.0, |
152 | one = 1.0, |
153 | two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */ |
154 | twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */ |
155 | |
156 | int |
157 | __kernel_rem_pio2 (double *x, double *y, int e0, int nx, int prec, |
158 | const int32_t *ipio2) |
159 | { |
160 | int32_t jz, jx, jv, jp, jk, carry, n, iq[20], i, j, k, m, q0, ih; |
161 | double z, fw, f[20], fq[20], q[20]; |
162 | |
163 | /* initialize jk*/ |
164 | jk = init_jk[prec]; |
165 | jp = jk; |
166 | |
167 | /* determine jx,jv,q0, note that 3>q0 */ |
168 | jx = nx - 1; |
169 | jv = (e0 - 3) / 24; if (jv < 0) |
170 | jv = 0; |
171 | q0 = e0 - 24 * (jv + 1); |
172 | |
173 | /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */ |
174 | j = jv - jx; m = jx + jk; |
175 | for (i = 0; i <= m; i++, j++) |
176 | f[i] = (j < 0) ? zero : (double) ipio2[j]; |
177 | |
178 | /* compute q[0],q[1],...q[jk] */ |
179 | for (i = 0; i <= jk; i++) |
180 | { |
181 | for (j = 0, fw = 0.0; j <= jx; j++) |
182 | fw += x[j] * f[jx + i - j]; |
183 | q[i] = fw; |
184 | } |
185 | |
186 | jz = jk; |
187 | recompute: |
188 | /* distill q[] into iq[] reversingly */ |
189 | for (i = 0, j = jz, z = q[jz]; j > 0; i++, j--) |
190 | { |
191 | fw = (double) ((int32_t) (twon24 * z)); |
192 | iq[i] = (int32_t) (z - two24 * fw); |
193 | z = q[j - 1] + fw; |
194 | } |
195 | |
196 | /* compute n */ |
197 | z = __scalbn (z, q0); /* actual value of z */ |
198 | z -= 8.0 * __floor (z * 0.125); /* trim off integer >= 8 */ |
199 | n = (int32_t) z; |
200 | z -= (double) n; |
201 | ih = 0; |
202 | if (q0 > 0) /* need iq[jz-1] to determine n */ |
203 | { |
204 | i = (iq[jz - 1] >> (24 - q0)); n += i; |
205 | iq[jz - 1] -= i << (24 - q0); |
206 | ih = iq[jz - 1] >> (23 - q0); |
207 | } |
208 | else if (q0 == 0) |
209 | ih = iq[jz - 1] >> 23; |
210 | else if (z >= 0.5) |
211 | ih = 2; |
212 | |
213 | if (ih > 0) /* q > 0.5 */ |
214 | { |
215 | n += 1; carry = 0; |
216 | for (i = 0; i < jz; i++) /* compute 1-q */ |
217 | { |
218 | j = iq[i]; |
219 | if (carry == 0) |
220 | { |
221 | if (j != 0) |
222 | { |
223 | carry = 1; iq[i] = 0x1000000 - j; |
224 | } |
225 | } |
226 | else |
227 | iq[i] = 0xffffff - j; |
228 | } |
229 | if (q0 > 0) /* rare case: chance is 1 in 12 */ |
230 | { |
231 | switch (q0) |
232 | { |
233 | case 1: |
234 | iq[jz - 1] &= 0x7fffff; break; |
235 | case 2: |
236 | iq[jz - 1] &= 0x3fffff; break; |
237 | } |
238 | } |
239 | if (ih == 2) |
240 | { |
241 | z = one - z; |
242 | if (carry != 0) |
243 | z -= __scalbn (one, q0); |
244 | } |
245 | } |
246 | |
247 | /* check if recomputation is needed */ |
248 | if (z == zero) |
249 | { |
250 | j = 0; |
251 | for (i = jz - 1; i >= jk; i--) |
252 | j |= iq[i]; |
253 | if (j == 0) /* need recomputation */ |
254 | { |
255 | /* On s390x gcc 6.1 -O3 produces the warning "array subscript is below |
256 | array bounds [-Werror=array-bounds]". Only __ieee754_rem_pio2l |
257 | calls __kernel_rem_pio2 for normal numbers and |x| > pi/4 in case |
258 | of ldbl-96 and |x| > 3pi/4 in case of ldbl-128[ibm]. |
259 | Thus x can't be zero and ipio2 is not zero, too. Thus not all iq[] |
260 | values can't be zero. */ |
261 | DIAG_PUSH_NEEDS_COMMENT; |
262 | DIAG_IGNORE_NEEDS_COMMENT (6.1, "-Warray-bounds" ); |
263 | for (k = 1; iq[jk - k] == 0; k++) |
264 | ; /* k = no. of terms needed */ |
265 | DIAG_POP_NEEDS_COMMENT; |
266 | |
267 | for (i = jz + 1; i <= jz + k; i++) /* add q[jz+1] to q[jz+k] */ |
268 | { |
269 | f[jx + i] = (double) ipio2[jv + i]; |
270 | for (j = 0, fw = 0.0; j <= jx; j++) |
271 | fw += x[j] * f[jx + i - j]; |
272 | q[i] = fw; |
273 | } |
274 | jz += k; |
275 | goto recompute; |
276 | } |
277 | } |
278 | |
279 | /* chop off zero terms */ |
280 | if (z == 0.0) |
281 | { |
282 | jz -= 1; q0 -= 24; |
283 | while (iq[jz] == 0) |
284 | { |
285 | jz--; q0 -= 24; |
286 | } |
287 | } |
288 | else /* break z into 24-bit if necessary */ |
289 | { |
290 | z = __scalbn (z, -q0); |
291 | if (z >= two24) |
292 | { |
293 | fw = (double) ((int32_t) (twon24 * z)); |
294 | iq[jz] = (int32_t) (z - two24 * fw); |
295 | jz += 1; q0 += 24; |
296 | iq[jz] = (int32_t) fw; |
297 | } |
298 | else |
299 | iq[jz] = (int32_t) z; |
300 | } |
301 | |
302 | /* convert integer "bit" chunk to floating-point value */ |
303 | fw = __scalbn (one, q0); |
304 | for (i = jz; i >= 0; i--) |
305 | { |
306 | q[i] = fw * (double) iq[i]; fw *= twon24; |
307 | } |
308 | |
309 | /* compute PIo2[0,...,jp]*q[jz,...,0] */ |
310 | for (i = jz; i >= 0; i--) |
311 | { |
312 | for (fw = 0.0, k = 0; k <= jp && k <= jz - i; k++) |
313 | fw += PIo2[k] * q[i + k]; |
314 | fq[jz - i] = fw; |
315 | } |
316 | |
317 | /* compress fq[] into y[] */ |
318 | switch (prec) |
319 | { |
320 | case 0: |
321 | fw = 0.0; |
322 | for (i = jz; i >= 0; i--) |
323 | fw += fq[i]; |
324 | y[0] = (ih == 0) ? fw : -fw; |
325 | break; |
326 | case 1: |
327 | case 2:; |
328 | double fv = 0.0; |
329 | for (i = jz; i >= 0; i--) |
330 | fv = math_narrow_eval (fv + fq[i]); |
331 | y[0] = (ih == 0) ? fv : -fv; |
332 | fv = math_narrow_eval (fq[0] - fv); |
333 | for (i = 1; i <= jz; i++) |
334 | fv = math_narrow_eval (fv + fq[i]); |
335 | y[1] = (ih == 0) ? fv : -fv; |
336 | break; |
337 | case 3: /* painful */ |
338 | for (i = jz; i > 0; i--) |
339 | { |
340 | double fv = math_narrow_eval (fq[i - 1] + fq[i]); |
341 | fq[i] += fq[i - 1] - fv; |
342 | fq[i - 1] = fv; |
343 | } |
344 | for (i = jz; i > 1; 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 (fw = 0.0, i = jz; i >= 2; i--) |
351 | fw += fq[i]; |
352 | if (ih == 0) |
353 | { |
354 | y[0] = fq[0]; y[1] = fq[1]; y[2] = fw; |
355 | } |
356 | else |
357 | { |
358 | y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw; |
359 | } |
360 | } |
361 | return n & 7; |
362 | } |
363 | |