986
|
1 |
|
|
2 |
/* @(#)k_rem_pio2.c 1.3 95/01/18 */
|
|
3 |
/*
|
|
4 |
* ====================================================
|
|
5 |
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
|
6 |
*
|
|
7 |
* Developed at SunSoft, 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 |
/*
|
|
123 |
* Constants:
|
|
124 |
* The hexadecimal values are the intended ones for the following
|
|
125 |
* constants. The decimal values may be used, provided that the
|
|
126 |
* compiler will convert from decimal to binary accurately enough
|
|
127 |
* to produce the hexadecimal values shown.
|
|
128 |
*/
|
|
129 |
|
|
130 |
#include "fdlibm.h"
|
|
131 |
|
|
132 |
#ifdef __STDC__
|
|
133 |
static const int init_jk[] = {2,3,4,6}; /* initial value for jk */
|
|
134 |
#else
|
|
135 |
static int init_jk[] = {2,3,4,6};
|
|
136 |
#endif
|
|
137 |
|
|
138 |
#ifdef __STDC__
|
|
139 |
static const double PIo2[] = {
|
|
140 |
#else
|
|
141 |
static double PIo2[] = {
|
|
142 |
#endif
|
|
143 |
1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */
|
|
144 |
7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */
|
|
145 |
5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */
|
|
146 |
3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */
|
|
147 |
1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */
|
|
148 |
1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */
|
|
149 |
2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */
|
|
150 |
2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */
|
|
151 |
};
|
|
152 |
|
|
153 |
#ifdef __STDC__
|
|
154 |
static const double
|
|
155 |
#else
|
|
156 |
static double
|
|
157 |
#endif
|
|
158 |
zero = 0.0,
|
|
159 |
one = 1.0,
|
|
160 |
two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
|
|
161 |
twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */
|
|
162 |
|
|
163 |
#ifdef __STDC__
|
|
164 |
int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, const int *ipio2)
|
|
165 |
#else
|
|
166 |
int __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
|
|
167 |
double x[], y[]; int e0,nx,prec; int ipio2[];
|
|
168 |
#endif
|
|
169 |
{
|
|
170 |
int jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih;
|
|
171 |
double z,fw,f[20],fq[20],q[20];
|
|
172 |
|
|
173 |
/* initialize jk*/
|
|
174 |
jk = init_jk[prec];
|
|
175 |
jp = jk;
|
|
176 |
|
|
177 |
/* determine jx,jv,q0, note that 3>q0 */
|
|
178 |
jx = nx-1;
|
|
179 |
jv = (e0-3)/24; if(jv<0) jv=0;
|
|
180 |
q0 = e0-24*(jv+1);
|
|
181 |
|
|
182 |
/* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
|
|
183 |
j = jv-jx; m = jx+jk;
|
|
184 |
for(i=0;i<=m;i++,j++) f[i] = (j<0)? zero : (double) ipio2[j];
|
|
185 |
|
|
186 |
/* compute q[0],q[1],...q[jk] */
|
|
187 |
for (i=0;i<=jk;i++) {
|
|
188 |
for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; q[i] = fw;
|
|
189 |
}
|
|
190 |
|
|
191 |
jz = jk;
|
|
192 |
recompute:
|
|
193 |
/* distill q[] into iq[] reversingly */
|
|
194 |
for(i=0,j=jz,z=q[jz];j>0;i++,j--) {
|
|
195 |
fw = (double)((int)(twon24* z));
|
|
196 |
iq[i] = (int)(z-two24*fw);
|
|
197 |
z = q[j-1]+fw;
|
|
198 |
}
|
|
199 |
|
|
200 |
/* compute n */
|
|
201 |
z = scalbn(z,q0); /* actual value of z */
|
|
202 |
z -= 8.0*floor(z*0.125); /* trim off integer >= 8 */
|
|
203 |
n = (int) z;
|
|
204 |
z -= (double)n;
|
|
205 |
ih = 0;
|
|
206 |
if(q0>0) { /* need iq[jz-1] to determine n */
|
|
207 |
i = (iq[jz-1]>>(24-q0)); n += i;
|
|
208 |
iq[jz-1] -= i<<(24-q0);
|
|
209 |
ih = iq[jz-1]>>(23-q0);
|
|
210 |
}
|
|
211 |
else if(q0==0) ih = iq[jz-1]>>23;
|
|
212 |
else if(z>=0.5) ih=2;
|
|
213 |
|
|
214 |
if(ih>0) { /* q > 0.5 */
|
|
215 |
n += 1; carry = 0;
|
|
216 |
for(i=0;i<jz ;i++) { /* compute 1-q */
|
|
217 |
j = iq[i];
|
|
218 |
if(carry==0) {
|
|
219 |
if(j!=0) {
|
|
220 |
carry = 1; iq[i] = 0x1000000- j;
|
|
221 |
}
|
|
222 |
} else iq[i] = 0xffffff - j;
|
|
223 |
}
|
|
224 |
if(q0>0) { /* rare case: chance is 1 in 12 */
|
|
225 |
switch(q0) {
|
|
226 |
case 1:
|
|
227 |
iq[jz-1] &= 0x7fffff; break;
|
|
228 |
case 2:
|
|
229 |
iq[jz-1] &= 0x3fffff; break;
|
|
230 |
}
|
|
231 |
}
|
|
232 |
if(ih==2) {
|
|
233 |
z = one - z;
|
|
234 |
if(carry!=0) z -= scalbn(one,q0);
|
|
235 |
}
|
|
236 |
}
|
|
237 |
|
|
238 |
/* check if recomputation is needed */
|
|
239 |
if(z==zero) {
|
|
240 |
j = 0;
|
|
241 |
for (i=jz-1;i>=jk;i--) j |= iq[i];
|
|
242 |
if(j==0) { /* need recomputation */
|
|
243 |
for(k=1;iq[jk-k]==0;k++); /* k = no. of terms needed */
|
|
244 |
|
|
245 |
for(i=jz+1;i<=jz+k;i++) { /* add q[jz+1] to q[jz+k] */
|
|
246 |
f[jx+i] = (double) ipio2[jv+i];
|
|
247 |
for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j];
|
|
248 |
q[i] = fw;
|
|
249 |
}
|
|
250 |
jz += k;
|
|
251 |
goto recompute;
|
|
252 |
}
|
|
253 |
}
|
|
254 |
|
|
255 |
/* chop off zero terms */
|
|
256 |
if(z==0.0) {
|
|
257 |
jz -= 1; q0 -= 24;
|
|
258 |
while(iq[jz]==0) { jz--; q0-=24;}
|
|
259 |
} else { /* break z into 24-bit if necessary */
|
|
260 |
z = scalbn(z,-q0);
|
|
261 |
if(z>=two24) {
|
|
262 |
fw = (double)((int)(twon24*z));
|
|
263 |
iq[jz] = (int)(z-two24*fw);
|
|
264 |
jz += 1; q0 += 24;
|
|
265 |
iq[jz] = (int) fw;
|
|
266 |
} else iq[jz] = (int) z ;
|
|
267 |
}
|
|
268 |
|
|
269 |
/* convert integer "bit" chunk to floating-point value */
|
|
270 |
fw = scalbn(one,q0);
|
|
271 |
for(i=jz;i>=0;i--) {
|
|
272 |
q[i] = fw*(double)iq[i]; fw*=twon24;
|
|
273 |
}
|
|
274 |
|
|
275 |
/* compute PIo2[0,...,jp]*q[jz,...,0] */
|
|
276 |
for(i=jz;i>=0;i--) {
|
|
277 |
for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k];
|
|
278 |
fq[jz-i] = fw;
|
|
279 |
}
|
|
280 |
|
|
281 |
/* compress fq[] into y[] */
|
|
282 |
switch(prec) {
|
|
283 |
case 0:
|
|
284 |
fw = 0.0;
|
|
285 |
for (i=jz;i>=0;i--) fw += fq[i];
|
|
286 |
y[0] = (ih==0)? fw: -fw;
|
|
287 |
break;
|
|
288 |
case 1:
|
|
289 |
case 2:
|
|
290 |
fw = 0.0;
|
|
291 |
for (i=jz;i>=0;i--) fw += fq[i];
|
|
292 |
y[0] = (ih==0)? fw: -fw;
|
|
293 |
fw = fq[0]-fw;
|
|
294 |
for (i=1;i<=jz;i++) fw += fq[i];
|
|
295 |
y[1] = (ih==0)? fw: -fw;
|
|
296 |
break;
|
|
297 |
case 3: /* painful */
|
|
298 |
for (i=jz;i>0;i--) {
|
|
299 |
fw = fq[i-1]+fq[i];
|
|
300 |
fq[i] += fq[i-1]-fw;
|
|
301 |
fq[i-1] = fw;
|
|
302 |
}
|
|
303 |
for (i=jz;i>1;i--) {
|
|
304 |
fw = fq[i-1]+fq[i];
|
|
305 |
fq[i] += fq[i-1]-fw;
|
|
306 |
fq[i-1] = fw;
|
|
307 |
}
|
|
308 |
for (fw=0.0,i=jz;i>=2;i--) fw += fq[i];
|
|
309 |
if(ih==0) {
|
|
310 |
y[0] = fq[0]; y[1] = fq[1]; y[2] = fw;
|
|
311 |
} else {
|
|
312 |
y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw;
|
|
313 |
}
|
|
314 |
}
|
|
315 |
return n&7;
|
|
316 |
}
|