/[cmucl]/src/lisp/k_rem_pio2.c
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Contents of /src/lisp/k_rem_pio2.c

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Revision 1.2 - (show annotations)
Fri Apr 11 07:49:46 2008 UTC (6 years ago) by cshapiro
Branch: MAIN
CVS Tags: sparc-tramp-assem-base, post-merge-intl-branch, merged-unicode-utf16-extfmt-2009-06-11, unicode-utf16-extfmt-2009-03-27, snapshot-2008-08, snapshot-2008-09, sse2-packed-2008-11-12, snapshot-2008-05, snapshot-2008-06, snapshot-2008-07, intl-branch-working-2010-02-19-1000, unicode-string-buffer-impl-base, sse2-base, release-20b-pre1, release-20b-pre2, unicode-string-buffer-base, sse2-packed-base, sparc-tramp-assem-2010-07-19, amd64-dd-start, release-19f-pre1, snapshot-2008-12, snapshot-2008-11, intl-2-branch-base, GIT-CONVERSION, unicode-utf16-sync-2008-12, cross-sol-x86-merged, label-2009-03-16, release-19f-base, merge-sse2-packed, merge-with-19f, intl-branch-working-2010-02-11-1000, unicode-snapshot-2009-05, unicode-snapshot-2009-06, unicode-utf16-sync-2008-07, unicode-utf16-sync-2008-09, unicode-utf16-extfmts-sync-2008-12, RELEASE_20b, unicode-utf16-sync-label-2009-03-16, RELEASE_19f, release-20a-base, cross-sol-x86-base, unicode-utf16-char-support-2009-03-26, unicode-utf16-char-support-2009-03-25, unicode-utf16-extfmts-pre-sync-2008-11, snapshot-2008-10, snapshot-2010-12, snapshot-2010-11, unicode-utf16-sync-2008-11, snapshot-2011-09, snapshot-2011-06, snapshot-2011-07, snapshot-2011-04, snapshot-2011-02, snapshot-2011-03, snapshot-2011-01, pre-merge-intl-branch, snapshot-2010-05, snapshot-2010-04, snapshot-2010-07, snapshot-2010-06, snapshot-2010-01, snapshot-2010-03, snapshot-2010-02, snapshot-2010-08, label-2009-03-25, cross-sol-x86-2010-12-20, sse2-checkpoint-2008-10-01, intl-branch-2010-03-18-1300, sse2-merge-with-2008-11, sse2-merge-with-2008-10, RELEASE_20a, release-20a-pre1, snapshot-2009-11, snapshot-2009-12, unicode-utf16-extfmt-2009-06-11, portable-clx-import-2009-06-16, unicode-utf16-string-support, cross-sparc-branch-base, intl-branch-base, unicode-utf16-base, portable-clx-base, snapshot-2009-08, snapshot-2009-02, snapshot-2009-01, snapshot-2009-07, snapshot-2009-05, snapshot-2009-04, HEAD
Branch point for: RELEASE-19F-BRANCH, portable-clx-branch, cross-sparc-branch, RELEASE-20B-BRANCH, unicode-string-buffer-branch, sparc-tramp-assem-branch, sse2-packed-branch, RELEASE-20A-BRANCH, amd64-dd-branch, unicode-string-buffer-impl-branch, intl-branch, unicode-utf16-branch, cross-sol-x86-branch, sse2-branch, intl-2-branch, unicode-utf16-extfmt-branch
Changes since 1.1: +1 -18 lines
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Retrieve the constituent words of a double through a union instead of
casts and pointer arithmetic.  This code now compiles correctly on GCC
version 4 without disabling the default assumption of strict aliasing
rules.  Also, remove lots of dead code associated with parts of fdlibm
that have not been imported into the runtime.
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 static const int init_jk[] = {2,3,4,6}; /* initial value for jk */
133
134 static const double PIo2[] = {
135 1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */
136 7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */
137 5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */
138 3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */
139 1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */
140 1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */
141 2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */
142 2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */
143 };
144
145 static const double
146 zero = 0.0,
147 one = 1.0,
148 two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
149 twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */
150
151 int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, const int *ipio2)
152 {
153 int jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih;
154 double z,fw,f[20],fq[20],q[20];
155
156 /* initialize jk*/
157 jk = init_jk[prec];
158 jp = jk;
159
160 /* determine jx,jv,q0, note that 3>q0 */
161 jx = nx-1;
162 jv = (e0-3)/24; if(jv<0) jv=0;
163 q0 = e0-24*(jv+1);
164
165 /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
166 j = jv-jx; m = jx+jk;
167 for(i=0;i<=m;i++,j++) f[i] = (j<0)? zero : (double) ipio2[j];
168
169 /* compute q[0],q[1],...q[jk] */
170 for (i=0;i<=jk;i++) {
171 for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; q[i] = fw;
172 }
173
174 jz = jk;
175 recompute:
176 /* distill q[] into iq[] reversingly */
177 for(i=0,j=jz,z=q[jz];j>0;i++,j--) {
178 fw = (double)((int)(twon24* z));
179 iq[i] = (int)(z-two24*fw);
180 z = q[j-1]+fw;
181 }
182
183 /* compute n */
184 z = scalbn(z,q0); /* actual value of z */
185 z -= 8.0*floor(z*0.125); /* trim off integer >= 8 */
186 n = (int) z;
187 z -= (double)n;
188 ih = 0;
189 if(q0>0) { /* need iq[jz-1] to determine n */
190 i = (iq[jz-1]>>(24-q0)); n += i;
191 iq[jz-1] -= i<<(24-q0);
192 ih = iq[jz-1]>>(23-q0);
193 }
194 else if(q0==0) ih = iq[jz-1]>>23;
195 else if(z>=0.5) ih=2;
196
197 if(ih>0) { /* q > 0.5 */
198 n += 1; carry = 0;
199 for(i=0;i<jz ;i++) { /* compute 1-q */
200 j = iq[i];
201 if(carry==0) {
202 if(j!=0) {
203 carry = 1; iq[i] = 0x1000000- j;
204 }
205 } else iq[i] = 0xffffff - j;
206 }
207 if(q0>0) { /* rare case: chance is 1 in 12 */
208 switch(q0) {
209 case 1:
210 iq[jz-1] &= 0x7fffff; break;
211 case 2:
212 iq[jz-1] &= 0x3fffff; break;
213 }
214 }
215 if(ih==2) {
216 z = one - z;
217 if(carry!=0) z -= scalbn(one,q0);
218 }
219 }
220
221 /* check if recomputation is needed */
222 if(z==zero) {
223 j = 0;
224 for (i=jz-1;i>=jk;i--) j |= iq[i];
225 if(j==0) { /* need recomputation */
226 for(k=1;iq[jk-k]==0;k++); /* k = no. of terms needed */
227
228 for(i=jz+1;i<=jz+k;i++) { /* add q[jz+1] to q[jz+k] */
229 f[jx+i] = (double) ipio2[jv+i];
230 for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j];
231 q[i] = fw;
232 }
233 jz += k;
234 goto recompute;
235 }
236 }
237
238 /* chop off zero terms */
239 if(z==0.0) {
240 jz -= 1; q0 -= 24;
241 while(iq[jz]==0) { jz--; q0-=24;}
242 } else { /* break z into 24-bit if necessary */
243 z = scalbn(z,-q0);
244 if(z>=two24) {
245 fw = (double)((int)(twon24*z));
246 iq[jz] = (int)(z-two24*fw);
247 jz += 1; q0 += 24;
248 iq[jz] = (int) fw;
249 } else iq[jz] = (int) z ;
250 }
251
252 /* convert integer "bit" chunk to floating-point value */
253 fw = scalbn(one,q0);
254 for(i=jz;i>=0;i--) {
255 q[i] = fw*(double)iq[i]; fw*=twon24;
256 }
257
258 /* compute PIo2[0,...,jp]*q[jz,...,0] */
259 for(i=jz;i>=0;i--) {
260 for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k];
261 fq[jz-i] = fw;
262 }
263
264 /* compress fq[] into y[] */
265 switch(prec) {
266 case 0:
267 fw = 0.0;
268 for (i=jz;i>=0;i--) fw += fq[i];
269 y[0] = (ih==0)? fw: -fw;
270 break;
271 case 1:
272 case 2:
273 fw = 0.0;
274 for (i=jz;i>=0;i--) fw += fq[i];
275 y[0] = (ih==0)? fw: -fw;
276 fw = fq[0]-fw;
277 for (i=1;i<=jz;i++) fw += fq[i];
278 y[1] = (ih==0)? fw: -fw;
279 break;
280 case 3: /* painful */
281 for (i=jz;i>0;i--) {
282 fw = fq[i-1]+fq[i];
283 fq[i] += fq[i-1]-fw;
284 fq[i-1] = fw;
285 }
286 for (i=jz;i>1;i--) {
287 fw = fq[i-1]+fq[i];
288 fq[i] += fq[i-1]-fw;
289 fq[i-1] = fw;
290 }
291 for (fw=0.0,i=jz;i>=2;i--) fw += fq[i];
292 if(ih==0) {
293 y[0] = fq[0]; y[1] = fq[1]; y[2] = fw;
294 } else {
295 y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw;
296 }
297 }
298 return n&7;
299 }

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