1 |
740 |
jeremybenn |
#include "quadmath-imp.h"
|
2 |
|
|
#include <math.h>
|
3 |
|
|
|
4 |
|
|
|
5 |
|
|
/* @(#)k_rem_pio2.c 5.1 93/09/24 */
|
6 |
|
|
/*
|
7 |
|
|
* ====================================================
|
8 |
|
|
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
9 |
|
|
*
|
10 |
|
|
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
11 |
|
|
* Permission to use, copy, modify, and distribute this
|
12 |
|
|
* software is freely granted, provided that this notice
|
13 |
|
|
* is preserved.
|
14 |
|
|
* ====================================================
|
15 |
|
|
*/
|
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 |
|
|
* Constants:
|
126 |
|
|
* The hexadecimal values are the intended ones for the following
|
127 |
|
|
* constants. The decimal values may be used, provided that the
|
128 |
|
|
* compiler will convert from decimal to binary accurately enough
|
129 |
|
|
* to produce the hexadecimal values shown.
|
130 |
|
|
*/
|
131 |
|
|
|
132 |
|
|
|
133 |
|
|
static const int init_jk[] = {2,3,4,6}; /* initial value for jk */
|
134 |
|
|
|
135 |
|
|
static const double PIo2[] = {
|
136 |
|
|
1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */
|
137 |
|
|
7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */
|
138 |
|
|
5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */
|
139 |
|
|
3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */
|
140 |
|
|
1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */
|
141 |
|
|
1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */
|
142 |
|
|
2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */
|
143 |
|
|
2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */
|
144 |
|
|
};
|
145 |
|
|
|
146 |
|
|
static const double
|
147 |
|
|
zero = 0.0,
|
148 |
|
|
one = 1.0,
|
149 |
|
|
two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
|
150 |
|
|
twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */
|
151 |
|
|
|
152 |
|
|
|
153 |
|
|
static int
|
154 |
|
|
__quadmath_kernel_rem_pio2 (double *x, double *y, int e0, int nx, int prec, const int32_t *ipio2)
|
155 |
|
|
{
|
156 |
|
|
int32_t jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih;
|
157 |
|
|
double z,fw,f[20],fq[20],q[20];
|
158 |
|
|
|
159 |
|
|
/* initialize jk*/
|
160 |
|
|
jk = init_jk[prec];
|
161 |
|
|
jp = jk;
|
162 |
|
|
|
163 |
|
|
/* determine jx,jv,q0, note that 3>q0 */
|
164 |
|
|
jx = nx-1;
|
165 |
|
|
jv = (e0-3)/24; if(jv<0) jv=0;
|
166 |
|
|
q0 = e0-24*(jv+1);
|
167 |
|
|
|
168 |
|
|
/* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
|
169 |
|
|
j = jv-jx; m = jx+jk;
|
170 |
|
|
for(i=0;i<=m;i++,j++) f[i] = (j<0)? zero : (double) ipio2[j];
|
171 |
|
|
|
172 |
|
|
/* compute q[0],q[1],...q[jk] */
|
173 |
|
|
for (i=0;i<=jk;i++) {
|
174 |
|
|
for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; q[i] = fw;
|
175 |
|
|
}
|
176 |
|
|
|
177 |
|
|
jz = jk;
|
178 |
|
|
recompute:
|
179 |
|
|
/* distill q[] into iq[] reversingly */
|
180 |
|
|
for(i=0,j=jz,z=q[jz];j>0;i++,j--) {
|
181 |
|
|
fw = (double)((int32_t)(twon24* z));
|
182 |
|
|
iq[i] = (int32_t)(z-two24*fw);
|
183 |
|
|
z = q[j-1]+fw;
|
184 |
|
|
}
|
185 |
|
|
|
186 |
|
|
/* compute n */
|
187 |
|
|
z = scalbn(z,q0); /* actual value of z */
|
188 |
|
|
z -= 8.0*floor(z*0.125); /* trim off integer >= 8 */
|
189 |
|
|
n = (int32_t) z;
|
190 |
|
|
z -= (double)n;
|
191 |
|
|
ih = 0;
|
192 |
|
|
if(q0>0) { /* need iq[jz-1] to determine n */
|
193 |
|
|
i = (iq[jz-1]>>(24-q0)); n += i;
|
194 |
|
|
iq[jz-1] -= i<<(24-q0);
|
195 |
|
|
ih = iq[jz-1]>>(23-q0);
|
196 |
|
|
}
|
197 |
|
|
else if(q0==0) ih = iq[jz-1]>>23;
|
198 |
|
|
else if(z>=0.5) ih=2;
|
199 |
|
|
|
200 |
|
|
if(ih>0) { /* q > 0.5 */
|
201 |
|
|
n += 1; carry = 0;
|
202 |
|
|
for(i=0;i<jz ;i++) { /* compute 1-q */
|
203 |
|
|
j = iq[i];
|
204 |
|
|
if(carry==0) {
|
205 |
|
|
if(j!=0) {
|
206 |
|
|
carry = 1; iq[i] = 0x1000000- j;
|
207 |
|
|
}
|
208 |
|
|
} else iq[i] = 0xffffff - j;
|
209 |
|
|
}
|
210 |
|
|
if(q0>0) { /* rare case: chance is 1 in 12 */
|
211 |
|
|
switch(q0) {
|
212 |
|
|
case 1:
|
213 |
|
|
iq[jz-1] &= 0x7fffff; break;
|
214 |
|
|
case 2:
|
215 |
|
|
iq[jz-1] &= 0x3fffff; break;
|
216 |
|
|
}
|
217 |
|
|
}
|
218 |
|
|
if(ih==2) {
|
219 |
|
|
z = one - z;
|
220 |
|
|
if(carry!=0) z -= scalbn(one,q0);
|
221 |
|
|
}
|
222 |
|
|
}
|
223 |
|
|
|
224 |
|
|
/* check if recomputation is needed */
|
225 |
|
|
if(z==zero) {
|
226 |
|
|
j = 0;
|
227 |
|
|
for (i=jz-1;i>=jk;i--) j |= iq[i];
|
228 |
|
|
if(j==0) { /* need recomputation */
|
229 |
|
|
for(k=1;iq[jk-k]==0;k++); /* k = no. of terms needed */
|
230 |
|
|
|
231 |
|
|
for(i=jz+1;i<=jz+k;i++) { /* add q[jz+1] to q[jz+k] */
|
232 |
|
|
f[jx+i] = (double) ipio2[jv+i];
|
233 |
|
|
for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j];
|
234 |
|
|
q[i] = fw;
|
235 |
|
|
}
|
236 |
|
|
jz += k;
|
237 |
|
|
goto recompute;
|
238 |
|
|
}
|
239 |
|
|
}
|
240 |
|
|
|
241 |
|
|
/* chop off zero terms */
|
242 |
|
|
if(z==0.0) {
|
243 |
|
|
jz -= 1; q0 -= 24;
|
244 |
|
|
while(iq[jz]==0) { jz--; q0-=24;}
|
245 |
|
|
} else { /* break z into 24-bit if necessary */
|
246 |
|
|
z = scalbn(z,-q0);
|
247 |
|
|
if(z>=two24) {
|
248 |
|
|
fw = (double)((int32_t)(twon24*z));
|
249 |
|
|
iq[jz] = (int32_t)(z-two24*fw);
|
250 |
|
|
jz += 1; q0 += 24;
|
251 |
|
|
iq[jz] = (int32_t) fw;
|
252 |
|
|
} else iq[jz] = (int32_t) z ;
|
253 |
|
|
}
|
254 |
|
|
|
255 |
|
|
/* convert integer "bit" chunk to floating-point value */
|
256 |
|
|
fw = scalbn(one,q0);
|
257 |
|
|
for(i=jz;i>=0;i--) {
|
258 |
|
|
q[i] = fw*(double)iq[i]; fw*=twon24;
|
259 |
|
|
}
|
260 |
|
|
|
261 |
|
|
/* compute PIo2[0,...,jp]*q[jz,...,0] */
|
262 |
|
|
for(i=jz;i>=0;i--) {
|
263 |
|
|
for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k];
|
264 |
|
|
fq[jz-i] = fw;
|
265 |
|
|
}
|
266 |
|
|
|
267 |
|
|
/* compress fq[] into y[] */
|
268 |
|
|
switch(prec) {
|
269 |
|
|
case 0:
|
270 |
|
|
fw = 0.0;
|
271 |
|
|
for (i=jz;i>=0;i--) fw += fq[i];
|
272 |
|
|
y[0] = (ih==0)? fw: -fw;
|
273 |
|
|
break;
|
274 |
|
|
case 1:
|
275 |
|
|
case 2:
|
276 |
|
|
fw = 0.0;
|
277 |
|
|
for (i=jz;i>=0;i--) fw += fq[i];
|
278 |
|
|
y[0] = (ih==0)? fw: -fw;
|
279 |
|
|
fw = fq[0]-fw;
|
280 |
|
|
for (i=1;i<=jz;i++) fw += fq[i];
|
281 |
|
|
y[1] = (ih==0)? fw: -fw;
|
282 |
|
|
break;
|
283 |
|
|
case 3: /* painful */
|
284 |
|
|
for (i=jz;i>0;i--) {
|
285 |
|
|
#if __FLT_EVAL_METHOD__ != 0
|
286 |
|
|
volatile
|
287 |
|
|
#endif
|
288 |
|
|
double fv = (double)(fq[i-1]+fq[i]);
|
289 |
|
|
fq[i] += fq[i-1]-fv;
|
290 |
|
|
fq[i-1] = fv;
|
291 |
|
|
}
|
292 |
|
|
for (i=jz;i>1;i--) {
|
293 |
|
|
#if __FLT_EVAL_METHOD__ != 0
|
294 |
|
|
volatile
|
295 |
|
|
#endif
|
296 |
|
|
double fv = (double)(fq[i-1]+fq[i]);
|
297 |
|
|
fq[i] += fq[i-1]-fv;
|
298 |
|
|
fq[i-1] = fv;
|
299 |
|
|
}
|
300 |
|
|
for (fw=0.0,i=jz;i>=2;i--) fw += fq[i];
|
301 |
|
|
if(ih==0) {
|
302 |
|
|
y[0] = fq[0]; y[1] = fq[1]; y[2] = fw;
|
303 |
|
|
} else {
|
304 |
|
|
y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw;
|
305 |
|
|
}
|
306 |
|
|
}
|
307 |
|
|
return n&7;
|
308 |
|
|
}
|
309 |
|
|
|
310 |
|
|
|
311 |
|
|
|
312 |
|
|
|
313 |
|
|
|
314 |
|
|
/* Quad-precision floating point argument reduction.
|
315 |
|
|
Copyright (C) 1999 Free Software Foundation, Inc.
|
316 |
|
|
This file is part of the GNU C Library.
|
317 |
|
|
Contributed by Jakub Jelinek <jj@ultra.linux.cz>
|
318 |
|
|
|
319 |
|
|
The GNU C Library is free software; you can redistribute it and/or
|
320 |
|
|
modify it under the terms of the GNU Lesser General Public
|
321 |
|
|
License as published by the Free Software Foundation; either
|
322 |
|
|
version 2.1 of the License, or (at your option) any later version.
|
323 |
|
|
|
324 |
|
|
The GNU C Library is distributed in the hope that it will be useful,
|
325 |
|
|
but WITHOUT ANY WARRANTY; without even the implied warranty of
|
326 |
|
|
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
|
327 |
|
|
Lesser General Public License for more details.
|
328 |
|
|
|
329 |
|
|
You should have received a copy of the GNU Lesser General Public
|
330 |
|
|
License along with the GNU C Library; if not, write to the Free
|
331 |
|
|
Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA
|
332 |
|
|
02111-1307 USA. */
|
333 |
|
|
|
334 |
|
|
/*
|
335 |
|
|
* Table of constants for 2/pi, 5628 hexadecimal digits of 2/pi
|
336 |
|
|
*/
|
337 |
|
|
static const int32_t two_over_pi[] = {
|
338 |
|
|
0xa2f983, 0x6e4e44, 0x1529fc, 0x2757d1, 0xf534dd, 0xc0db62,
|
339 |
|
|
0x95993c, 0x439041, 0xfe5163, 0xabdebb, 0xc561b7, 0x246e3a,
|
340 |
|
|
0x424dd2, 0xe00649, 0x2eea09, 0xd1921c, 0xfe1deb, 0x1cb129,
|
341 |
|
|
0xa73ee8, 0x8235f5, 0x2ebb44, 0x84e99c, 0x7026b4, 0x5f7e41,
|
342 |
|
|
0x3991d6, 0x398353, 0x39f49c, 0x845f8b, 0xbdf928, 0x3b1ff8,
|
343 |
|
|
0x97ffde, 0x05980f, 0xef2f11, 0x8b5a0a, 0x6d1f6d, 0x367ecf,
|
344 |
|
|
0x27cb09, 0xb74f46, 0x3f669e, 0x5fea2d, 0x7527ba, 0xc7ebe5,
|
345 |
|
|
0xf17b3d, 0x0739f7, 0x8a5292, 0xea6bfb, 0x5fb11f, 0x8d5d08,
|
346 |
|
|
0x560330, 0x46fc7b, 0x6babf0, 0xcfbc20, 0x9af436, 0x1da9e3,
|
347 |
|
|
0x91615e, 0xe61b08, 0x659985, 0x5f14a0, 0x68408d, 0xffd880,
|
348 |
|
|
0x4d7327, 0x310606, 0x1556ca, 0x73a8c9, 0x60e27b, 0xc08c6b,
|
349 |
|
|
0x47c419, 0xc367cd, 0xdce809, 0x2a8359, 0xc4768b, 0x961ca6,
|
350 |
|
|
0xddaf44, 0xd15719, 0x053ea5, 0xff0705, 0x3f7e33, 0xe832c2,
|
351 |
|
|
0xde4f98, 0x327dbb, 0xc33d26, 0xef6b1e, 0x5ef89f, 0x3a1f35,
|
352 |
|
|
0xcaf27f, 0x1d87f1, 0x21907c, 0x7c246a, 0xfa6ed5, 0x772d30,
|
353 |
|
|
0x433b15, 0xc614b5, 0x9d19c3, 0xc2c4ad, 0x414d2c, 0x5d000c,
|
354 |
|
|
0x467d86, 0x2d71e3, 0x9ac69b, 0x006233, 0x7cd2b4, 0x97a7b4,
|
355 |
|
|
0xd55537, 0xf63ed7, 0x1810a3, 0xfc764d, 0x2a9d64, 0xabd770,
|
356 |
|
|
0xf87c63, 0x57b07a, 0xe71517, 0x5649c0, 0xd9d63b, 0x3884a7,
|
357 |
|
|
0xcb2324, 0x778ad6, 0x23545a, 0xb91f00, 0x1b0af1, 0xdfce19,
|
358 |
|
|
0xff319f, 0x6a1e66, 0x615799, 0x47fbac, 0xd87f7e, 0xb76522,
|
359 |
|
|
0x89e832, 0x60bfe6, 0xcdc4ef, 0x09366c, 0xd43f5d, 0xd7de16,
|
360 |
|
|
0xde3b58, 0x929bde, 0x2822d2, 0xe88628, 0x4d58e2, 0x32cac6,
|
361 |
|
|
0x16e308, 0xcb7de0, 0x50c017, 0xa71df3, 0x5be018, 0x34132e,
|
362 |
|
|
0x621283, 0x014883, 0x5b8ef5, 0x7fb0ad, 0xf2e91e, 0x434a48,
|
363 |
|
|
0xd36710, 0xd8ddaa, 0x425fae, 0xce616a, 0xa4280a, 0xb499d3,
|
364 |
|
|
0xf2a606, 0x7f775c, 0x83c2a3, 0x883c61, 0x78738a, 0x5a8caf,
|
365 |
|
|
0xbdd76f, 0x63a62d, 0xcbbff4, 0xef818d, 0x67c126, 0x45ca55,
|
366 |
|
|
0x36d9ca, 0xd2a828, 0x8d61c2, 0x77c912, 0x142604, 0x9b4612,
|
367 |
|
|
0xc459c4, 0x44c5c8, 0x91b24d, 0xf31700, 0xad43d4, 0xe54929,
|
368 |
|
|
0x10d5fd, 0xfcbe00, 0xcc941e, 0xeece70, 0xf53e13, 0x80f1ec,
|
369 |
|
|
0xc3e7b3, 0x28f8c7, 0x940593, 0x3e71c1, 0xb3092e, 0xf3450b,
|
370 |
|
|
0x9c1288, 0x7b20ab, 0x9fb52e, 0xc29247, 0x2f327b, 0x6d550c,
|
371 |
|
|
0x90a772, 0x1fe76b, 0x96cb31, 0x4a1679, 0xe27941, 0x89dff4,
|
372 |
|
|
0x9794e8, 0x84e6e2, 0x973199, 0x6bed88, 0x365f5f, 0x0efdbb,
|
373 |
|
|
0xb49a48, 0x6ca467, 0x427271, 0x325d8d, 0xb8159f, 0x09e5bc,
|
374 |
|
|
0x25318d, 0x3974f7, 0x1c0530, 0x010c0d, 0x68084b, 0x58ee2c,
|
375 |
|
|
0x90aa47, 0x02e774, 0x24d6bd, 0xa67df7, 0x72486e, 0xef169f,
|
376 |
|
|
0xa6948e, 0xf691b4, 0x5153d1, 0xf20acf, 0x339820, 0x7e4bf5,
|
377 |
|
|
0x6863b2, 0x5f3edd, 0x035d40, 0x7f8985, 0x295255, 0xc06437,
|
378 |
|
|
0x10d86d, 0x324832, 0x754c5b, 0xd4714e, 0x6e5445, 0xc1090b,
|
379 |
|
|
0x69f52a, 0xd56614, 0x9d0727, 0x50045d, 0xdb3bb4, 0xc576ea,
|
380 |
|
|
0x17f987, 0x7d6b49, 0xba271d, 0x296996, 0xacccc6, 0x5414ad,
|
381 |
|
|
0x6ae290, 0x89d988, 0x50722c, 0xbea404, 0x940777, 0x7030f3,
|
382 |
|
|
0x27fc00, 0xa871ea, 0x49c266, 0x3de064, 0x83dd97, 0x973fa3,
|
383 |
|
|
0xfd9443, 0x8c860d, 0xde4131, 0x9d3992, 0x8c70dd, 0xe7b717,
|
384 |
|
|
0x3bdf08, 0x2b3715, 0xa0805c, 0x93805a, 0x921110, 0xd8e80f,
|
385 |
|
|
0xaf806c, 0x4bffdb, 0x0f9038, 0x761859, 0x15a562, 0xbbcb61,
|
386 |
|
|
0xb989c7, 0xbd4010, 0x04f2d2, 0x277549, 0xf6b6eb, 0xbb22db,
|
387 |
|
|
0xaa140a, 0x2f2689, 0x768364, 0x333b09, 0x1a940e, 0xaa3a51,
|
388 |
|
|
0xc2a31d, 0xaeedaf, 0x12265c, 0x4dc26d, 0x9c7a2d, 0x9756c0,
|
389 |
|
|
0x833f03, 0xf6f009, 0x8c402b, 0x99316d, 0x07b439, 0x15200c,
|
390 |
|
|
0x5bc3d8, 0xc492f5, 0x4badc6, 0xa5ca4e, 0xcd37a7, 0x36a9e6,
|
391 |
|
|
0x9492ab, 0x6842dd, 0xde6319, 0xef8c76, 0x528b68, 0x37dbfc,
|
392 |
|
|
0xaba1ae, 0x3115df, 0xa1ae00, 0xdafb0c, 0x664d64, 0xb705ed,
|
393 |
|
|
0x306529, 0xbf5657, 0x3aff47, 0xb9f96a, 0xf3be75, 0xdf9328,
|
394 |
|
|
0x3080ab, 0xf68c66, 0x15cb04, 0x0622fa, 0x1de4d9, 0xa4b33d,
|
395 |
|
|
0x8f1b57, 0x09cd36, 0xe9424e, 0xa4be13, 0xb52333, 0x1aaaf0,
|
396 |
|
|
0xa8654f, 0xa5c1d2, 0x0f3f0b, 0xcd785b, 0x76f923, 0x048b7b,
|
397 |
|
|
0x721789, 0x53a6c6, 0xe26e6f, 0x00ebef, 0x584a9b, 0xb7dac4,
|
398 |
|
|
0xba66aa, 0xcfcf76, 0x1d02d1, 0x2df1b1, 0xc1998c, 0x77adc3,
|
399 |
|
|
0xda4886, 0xa05df7, 0xf480c6, 0x2ff0ac, 0x9aecdd, 0xbc5c3f,
|
400 |
|
|
0x6dded0, 0x1fc790, 0xb6db2a, 0x3a25a3, 0x9aaf00, 0x9353ad,
|
401 |
|
|
0x0457b6, 0xb42d29, 0x7e804b, 0xa707da, 0x0eaa76, 0xa1597b,
|
402 |
|
|
0x2a1216, 0x2db7dc, 0xfde5fa, 0xfedb89, 0xfdbe89, 0x6c76e4,
|
403 |
|
|
0xfca906, 0x70803e, 0x156e85, 0xff87fd, 0x073e28, 0x336761,
|
404 |
|
|
0x86182a, 0xeabd4d, 0xafe7b3, 0x6e6d8f, 0x396795, 0x5bbf31,
|
405 |
|
|
0x48d784, 0x16df30, 0x432dc7, 0x356125, 0xce70c9, 0xb8cb30,
|
406 |
|
|
0xfd6cbf, 0xa200a4, 0xe46c05, 0xa0dd5a, 0x476f21, 0xd21262,
|
407 |
|
|
0x845cb9, 0x496170, 0xe0566b, 0x015299, 0x375550, 0xb7d51e,
|
408 |
|
|
0xc4f133, 0x5f6e13, 0xe4305d, 0xa92e85, 0xc3b21d, 0x3632a1,
|
409 |
|
|
0xa4b708, 0xd4b1ea, 0x21f716, 0xe4698f, 0x77ff27, 0x80030c,
|
410 |
|
|
0x2d408d, 0xa0cd4f, 0x99a520, 0xd3a2b3, 0x0a5d2f, 0x42f9b4,
|
411 |
|
|
0xcbda11, 0xd0be7d, 0xc1db9b, 0xbd17ab, 0x81a2ca, 0x5c6a08,
|
412 |
|
|
0x17552e, 0x550027, 0xf0147f, 0x8607e1, 0x640b14, 0x8d4196,
|
413 |
|
|
0xdebe87, 0x2afdda, 0xb6256b, 0x34897b, 0xfef305, 0x9ebfb9,
|
414 |
|
|
0x4f6a68, 0xa82a4a, 0x5ac44f, 0xbcf82d, 0x985ad7, 0x95c7f4,
|
415 |
|
|
0x8d4d0d, 0xa63a20, 0x5f57a4, 0xb13f14, 0x953880, 0x0120cc,
|
416 |
|
|
0x86dd71, 0xb6dec9, 0xf560bf, 0x11654d, 0x6b0701, 0xacb08c,
|
417 |
|
|
0xd0c0b2, 0x485551, 0x0efb1e, 0xc37295, 0x3b06a3, 0x3540c0,
|
418 |
|
|
0x7bdc06, 0xcc45e0, 0xfa294e, 0xc8cad6, 0x41f3e8, 0xde647c,
|
419 |
|
|
0xd8649b, 0x31bed9, 0xc397a4, 0xd45877, 0xc5e369, 0x13daf0,
|
420 |
|
|
0x3c3aba, 0x461846, 0x5f7555, 0xf5bdd2, 0xc6926e, 0x5d2eac,
|
421 |
|
|
0xed440e, 0x423e1c, 0x87c461, 0xe9fd29, 0xf3d6e7, 0xca7c22,
|
422 |
|
|
0x35916f, 0xc5e008, 0x8dd7ff, 0xe26a6e, 0xc6fdb0, 0xc10893,
|
423 |
|
|
0x745d7c, 0xb2ad6b, 0x9d6ecd, 0x7b723e, 0x6a11c6, 0xa9cff7,
|
424 |
|
|
0xdf7329, 0xbac9b5, 0x5100b7, 0x0db2e2, 0x24ba74, 0x607de5,
|
425 |
|
|
0x8ad874, 0x2c150d, 0x0c1881, 0x94667e, 0x162901, 0x767a9f,
|
426 |
|
|
0xbefdfd, 0xef4556, 0x367ed9, 0x13d9ec, 0xb9ba8b, 0xfc97c4,
|
427 |
|
|
0x27a831, 0xc36ef1, 0x36c594, 0x56a8d8, 0xb5a8b4, 0x0ecccf,
|
428 |
|
|
0x2d8912, 0x34576f, 0x89562c, 0xe3ce99, 0xb920d6, 0xaa5e6b,
|
429 |
|
|
0x9c2a3e, 0xcc5f11, 0x4a0bfd, 0xfbf4e1, 0x6d3b8e, 0x2c86e2,
|
430 |
|
|
0x84d4e9, 0xa9b4fc, 0xd1eeef, 0xc9352e, 0x61392f, 0x442138,
|
431 |
|
|
0xc8d91b, 0x0afc81, 0x6a4afb, 0xd81c2f, 0x84b453, 0x8c994e,
|
432 |
|
|
0xcc2254, 0xdc552a, 0xd6c6c0, 0x96190b, 0xb8701a, 0x649569,
|
433 |
|
|
0x605a26, 0xee523f, 0x0f117f, 0x11b5f4, 0xf5cbfc, 0x2dbc34,
|
434 |
|
|
0xeebc34, 0xcc5de8, 0x605edd, 0x9b8e67, 0xef3392, 0xb817c9,
|
435 |
|
|
0x9b5861, 0xbc57e1, 0xc68351, 0x103ed8, 0x4871dd, 0xdd1c2d,
|
436 |
|
|
0xa118af, 0x462c21, 0xd7f359, 0x987ad9, 0xc0549e, 0xfa864f,
|
437 |
|
|
0xfc0656, 0xae79e5, 0x362289, 0x22ad38, 0xdc9367, 0xaae855,
|
438 |
|
|
0x382682, 0x9be7ca, 0xa40d51, 0xb13399, 0x0ed7a9, 0x480569,
|
439 |
|
|
0xf0b265, 0xa7887f, 0x974c88, 0x36d1f9, 0xb39221, 0x4a827b,
|
440 |
|
|
0x21cf98, 0xdc9f40, 0x5547dc, 0x3a74e1, 0x42eb67, 0xdf9dfe,
|
441 |
|
|
0x5fd45e, 0xa4677b, 0x7aacba, 0xa2f655, 0x23882b, 0x55ba41,
|
442 |
|
|
0x086e59, 0x862a21, 0x834739, 0xe6e389, 0xd49ee5, 0x40fb49,
|
443 |
|
|
0xe956ff, 0xca0f1c, 0x8a59c5, 0x2bfa94, 0xc5c1d3, 0xcfc50f,
|
444 |
|
|
0xae5adb, 0x86c547, 0x624385, 0x3b8621, 0x94792c, 0x876110,
|
445 |
|
|
0x7b4c2a, 0x1a2c80, 0x12bf43, 0x902688, 0x893c78, 0xe4c4a8,
|
446 |
|
|
0x7bdbe5, 0xc23ac4, 0xeaf426, 0x8a67f7, 0xbf920d, 0x2ba365,
|
447 |
|
|
0xb1933d, 0x0b7cbd, 0xdc51a4, 0x63dd27, 0xdde169, 0x19949a,
|
448 |
|
|
0x9529a8, 0x28ce68, 0xb4ed09, 0x209f44, 0xca984e, 0x638270,
|
449 |
|
|
0x237c7e, 0x32b90f, 0x8ef5a7, 0xe75614, 0x08f121, 0x2a9db5,
|
450 |
|
|
0x4d7e6f, 0x5119a5, 0xabf9b5, 0xd6df82, 0x61dd96, 0x023616,
|
451 |
|
|
0x9f3ac4, 0xa1a283, 0x6ded72, 0x7a8d39, 0xa9b882, 0x5c326b,
|
452 |
|
|
0x5b2746, 0xed3400, 0x7700d2, 0x55f4fc, 0x4d5901, 0x8071e0,
|
453 |
|
|
0xe13f89, 0xb295f3, 0x64a8f1, 0xaea74b, 0x38fc4c, 0xeab2bb,
|
454 |
|
|
0x47270b, 0xabc3a7, 0x34ba60, 0x52dd34, 0xf8563a, 0xeb7e8a,
|
455 |
|
|
0x31bb36, 0x5895b7, 0x47f7a9, 0x94c3aa, 0xd39225, 0x1e7f3e,
|
456 |
|
|
0xd8974e, 0xbba94f, 0xd8ae01, 0xe661b4, 0x393d8e, 0xa523aa,
|
457 |
|
|
0x33068e, 0x1633b5, 0x3bb188, 0x1d3a9d, 0x4013d0, 0xcc1be5,
|
458 |
|
|
0xf862e7, 0x3bf28f, 0x39b5bf, 0x0bc235, 0x22747e, 0xa247c0,
|
459 |
|
|
0xd52d1f, 0x19add3, 0x9094df, 0x9311d0, 0xb42b25, 0x496db2,
|
460 |
|
|
0xe264b2, 0x5ef135, 0x3bc6a4, 0x1a4ad0, 0xaac92e, 0x64e886,
|
461 |
|
|
0x573091, 0x982cfb, 0x311b1a, 0x08728b, 0xbdcee1, 0x60e142,
|
462 |
|
|
0xeb641d, 0xd0bba3, 0xe559d4, 0x597b8c, 0x2a4483, 0xf332ba,
|
463 |
|
|
0xf84867, 0x2c8d1b, 0x2fa9b0, 0x50f3dd, 0xf9f573, 0xdb61b4,
|
464 |
|
|
0xfe233e, 0x6c41a6, 0xeea318, 0x775a26, 0xbc5e5c, 0xcea708,
|
465 |
|
|
0x94dc57, 0xe20196, 0xf1e839, 0xbe4851, 0x5d2d2f, 0x4e9555,
|
466 |
|
|
0xd96ec2, 0xe7d755, 0x6304e0, 0xc02e0e, 0xfc40a0, 0xbbf9b3,
|
467 |
|
|
0x7125a7, 0x222dfb, 0xf619d8, 0x838c1c, 0x6619e6, 0xb20d55,
|
468 |
|
|
0xbb5137, 0x79e809, 0xaf9149, 0x0d73de, 0x0b0da5, 0xce7f58,
|
469 |
|
|
0xac1934, 0x724667, 0x7a1a13, 0x9e26bc, 0x4555e7, 0x585cb5,
|
470 |
|
|
0x711d14, 0x486991, 0x480d60, 0x56adab, 0xd62f64, 0x96ee0c,
|
471 |
|
|
0x212ff3, 0x5d6d88, 0xa67684, 0x95651e, 0xab9e0a, 0x4ddefe,
|
472 |
|
|
0x571010, 0x836a39, 0xf8ea31, 0x9e381d, 0xeac8b1, 0xcac96b,
|
473 |
|
|
0x37f21e, 0xd505e9, 0x984743, 0x9fc56c, 0x0331b7, 0x3b8bf8,
|
474 |
|
|
0x86e56a, 0x8dc343, 0x6230e7, 0x93cfd5, 0x6a8f2d, 0x733005,
|
475 |
|
|
0x1af021, 0xa09fcb, 0x7415a1, 0xd56b23, 0x6ff725, 0x2f4bc7,
|
476 |
|
|
0xb8a591, 0x7fac59, 0x5c55de, 0x212c38, 0xb13296, 0x5cff50,
|
477 |
|
|
0x366262, 0xfa7b16, 0xf4d9a6, 0x2acfe7, 0xf07403, 0xd4d604,
|
478 |
|
|
0x6fd916, 0x31b1bf, 0xcbb450, 0x5bd7c8, 0x0ce194, 0x6bd643,
|
479 |
|
|
0x4fd91c, 0xdf4543, 0x5f3453, 0xe2b5aa, 0xc9aec8, 0x131485,
|
480 |
|
|
0xf9d2bf, 0xbadb9e, 0x76f5b9, 0xaf15cf, 0xca3182, 0x14b56d,
|
481 |
|
|
0xe9fe4d, 0x50fc35, 0xf5aed5, 0xa2d0c1, 0xc96057, 0x192eb6,
|
482 |
|
|
0xe91d92, 0x07d144, 0xaea3c6, 0x343566, 0x26d5b4, 0x3161e2,
|
483 |
|
|
0x37f1a2, 0x209eff, 0x958e23, 0x493798, 0x35f4a6, 0x4bdc02,
|
484 |
|
|
0xc2be13, 0xbe80a0, 0x0b72a3, 0x115c5f, 0x1e1bd1, 0x0db4d3,
|
485 |
|
|
0x869e85, 0x96976b, 0x2ac91f, 0x8a26c2, 0x3070f0, 0x041412,
|
486 |
|
|
0xfc9fa5, 0xf72a38, 0x9c6878, 0xe2aa76, 0x50cfe1, 0x559274,
|
487 |
|
|
0x934e38, 0x0a92f7, 0x5533f0, 0xa63db4, 0x399971, 0xe2b755,
|
488 |
|
|
0xa98a7c, 0x008f19, 0xac54d2, 0x2ea0b4, 0xf5f3e0, 0x60c849,
|
489 |
|
|
0xffd269, 0xae52ce, 0x7a5fdd, 0xe9ce06, 0xfb0ae8, 0xa50cce,
|
490 |
|
|
0xea9d3e, 0x3766dd, 0xb834f5, 0x0da090, 0x846f88, 0x4ae3d5,
|
491 |
|
|
0x099a03, 0x2eae2d, 0xfcb40a, 0xfb9b33, 0xe281dd, 0x1b16ba,
|
492 |
|
|
0xd8c0af, 0xd96b97, 0xb52dc9, 0x9c277f, 0x5951d5, 0x21ccd6,
|
493 |
|
|
0xb6496b, 0x584562, 0xb3baf2, 0xa1a5c4, 0x7ca2cf, 0xa9b93d,
|
494 |
|
|
0x7b7b89, 0x483d38,
|
495 |
|
|
};
|
496 |
|
|
|
497 |
|
|
static const __float128 c[] = {
|
498 |
|
|
/* 93 bits of pi/2 */
|
499 |
|
|
#define PI_2_1 c[0]
|
500 |
|
|
1.57079632679489661923132169155131424e+00Q, /* 3fff921fb54442d18469898cc5100000 */
|
501 |
|
|
|
502 |
|
|
/* pi/2 - PI_2_1 */
|
503 |
|
|
#define PI_2_1t c[1]
|
504 |
|
|
8.84372056613570112025531863263659260e-29Q, /* 3fa1c06e0e68948127044533e63a0106 */
|
505 |
|
|
};
|
506 |
|
|
|
507 |
|
|
|
508 |
|
|
int32_t
|
509 |
|
|
__quadmath_rem_pio2q (__float128 x, __float128 *y)
|
510 |
|
|
{
|
511 |
|
|
__float128 z, w, t;
|
512 |
|
|
double tx[8];
|
513 |
|
|
int64_t exp, n, ix, hx;
|
514 |
|
|
uint64_t lx;
|
515 |
|
|
|
516 |
|
|
GET_FLT128_WORDS64 (hx, lx, x);
|
517 |
|
|
ix = hx & 0x7fffffffffffffffLL;
|
518 |
|
|
if (ix <= 0x3ffe921fb54442d1LL) /* x in <-pi/4, pi/4> */
|
519 |
|
|
{
|
520 |
|
|
y[0] = x;
|
521 |
|
|
y[1] = 0;
|
522 |
|
|
return 0;
|
523 |
|
|
}
|
524 |
|
|
|
525 |
|
|
if (ix < 0x40002d97c7f3321dLL) /* |x| in <pi/4, 3pi/4) */
|
526 |
|
|
{
|
527 |
|
|
if (hx > 0)
|
528 |
|
|
{
|
529 |
|
|
/* 113 + 93 bit PI is ok */
|
530 |
|
|
z = x - PI_2_1;
|
531 |
|
|
y[0] = z - PI_2_1t;
|
532 |
|
|
y[1] = (z - y[0]) - PI_2_1t;
|
533 |
|
|
return 1;
|
534 |
|
|
}
|
535 |
|
|
else
|
536 |
|
|
{
|
537 |
|
|
/* 113 + 93 bit PI is ok */
|
538 |
|
|
z = x + PI_2_1;
|
539 |
|
|
y[0] = z + PI_2_1t;
|
540 |
|
|
y[1] = (z - y[0]) + PI_2_1t;
|
541 |
|
|
return -1;
|
542 |
|
|
}
|
543 |
|
|
}
|
544 |
|
|
|
545 |
|
|
if (ix >= 0x7fff000000000000LL) /* x is +=oo or NaN */
|
546 |
|
|
{
|
547 |
|
|
y[0] = x - x;
|
548 |
|
|
y[1] = y[0];
|
549 |
|
|
return 0;
|
550 |
|
|
}
|
551 |
|
|
|
552 |
|
|
/* Handle large arguments.
|
553 |
|
|
We split the 113 bits of the mantissa into 5 24bit integers
|
554 |
|
|
stored in a double array. */
|
555 |
|
|
exp = (ix >> 48) - 16383 - 23;
|
556 |
|
|
|
557 |
|
|
/* This is faster than doing this in floating point, because we
|
558 |
|
|
have to convert it to integers anyway and like this we can keep
|
559 |
|
|
both integer and floating point units busy. */
|
560 |
|
|
tx [0] = (double)(((ix >> 25) & 0x7fffff) | 0x800000);
|
561 |
|
|
tx [1] = (double)((ix >> 1) & 0xffffff);
|
562 |
|
|
tx [2] = (double)(((ix << 23) | (lx >> 41)) & 0xffffff);
|
563 |
|
|
tx [3] = (double)((lx >> 17) & 0xffffff);
|
564 |
|
|
tx [4] = (double)((lx << 7) & 0xffffff);
|
565 |
|
|
|
566 |
|
|
n = __quadmath_kernel_rem_pio2 (tx, tx + 5, exp,
|
567 |
|
|
((lx << 7) & 0xffffff) ? 5 : 4,
|
568 |
|
|
3, two_over_pi);
|
569 |
|
|
|
570 |
|
|
/* The result is now stored in 3 double values, we need to convert it into
|
571 |
|
|
two __float128 values. */
|
572 |
|
|
t = (__float128) tx [6] + (__float128) tx [7];
|
573 |
|
|
w = (__float128) tx [5];
|
574 |
|
|
|
575 |
|
|
if (hx >= 0)
|
576 |
|
|
{
|
577 |
|
|
y[0] = w + t;
|
578 |
|
|
y[1] = t - (y[0] - w);
|
579 |
|
|
return n;
|
580 |
|
|
}
|
581 |
|
|
else
|
582 |
|
|
{
|
583 |
|
|
y[0] = -(w + t);
|
584 |
|
|
y[1] = -t - (y[0] + w);
|
585 |
|
|
return -n;
|
586 |
|
|
}
|
587 |
|
|
}
|