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

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Wed Jul 19 02:45:53 2006 UTC (7 years, 9 months ago) by rtoy
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CVS Tags: snapshot-2007-09, snapshot-2007-08, snapshot-2007-05, snapshot-2008-01, snapshot-2008-02, snapshot-2008-03, snapshot-2006-11, snapshot-2006-10, snapshot-2006-12, snapshot-2007-01, snapshot-2007-02, release-19e, release-19d, snapshot-2008-04, snapshot-2007-03, snapshot-2007-04, snapshot-2007-07, snapshot-2007-06, release-19d-base, release-19e-pre1, release-19e-pre2, release-19d-pre2, release-19d-pre1, release-19e-base, snapshot-2007-12, snapshot-2007-10, snapshot-2007-11, pre-telent-clx, snapshot-2006-08, snapshot-2006-09
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Initial import.
1 rtoy 1.1
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     }

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