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k_rem_pio2.c
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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 }