POK
k_rem_pio2.c
1 /*
2  * POK header
3  *
4  * The following file is a part of the POK project. Any modification should
5  * made according to the POK licence. You CANNOT use this file or a part of
6  * this file is this part of a file for your own project
7  *
8  * For more information on the POK licence, please see our LICENCE FILE
9  *
10  * Please follow the coding guidelines described in doc/CODING_GUIDELINES
11  *
12  * Copyright (c) 2007-2009 POK team
13  *
14  * Created by julien on Sat Jan 31 20:12:07 2009
15  */
16 
17 /* @(#)k_rem_pio2.c 5.1 93/09/24 */
18 /*
19  * ====================================================
20  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
21  *
22  * Developed at SunPro, a Sun Microsystems, Inc. business.
23  * Permission to use, copy, modify, and distribute this
24  * software is freely granted, provided that this notice
25  * is preserved.
26  * ====================================================
27  */
28 
29 #ifdef POK_NEEDS_LIBMATH
30 
31 /*
32  * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
33  * double x[],y[]; int e0,nx,prec; int ipio2[];
34  *
35  * __kernel_rem_pio2 return the last three digits of N with
36  * y = x - N*pi/2
37  * so that |y| < pi/2.
38  *
39  * The method is to compute the integer (mod 8) and fraction parts of
40  * (2/pi)*x without doing the full multiplication. In general we
41  * skip the part of the product that are known to be a huge integer (
42  * more accurately, = 0 mod 8 ). Thus the number of operations are
43  * independent of the exponent of the input.
44  *
45  * (2/pi) is represented by an array of 24-bit integers in ipio2[].
46  *
47  * Input parameters:
48  * x[] The input value (must be positive) is broken into nx
49  * pieces of 24-bit integers in double precision format.
50  * x[i] will be the i-th 24 bit of x. The scaled exponent
51  * of x[0] is given in input parameter e0 (i.e., x[0]*2^e0
52  * match x's up to 24 bits.
53  *
54  * Example of breaking a double positive z into x[0]+x[1]+x[2]:
55  * e0 = ilogb(z)-23
56  * z = scalbn(z,-e0)
57  * for i = 0,1,2
58  * x[i] = floor(z)
59  * z = (z-x[i])*2**24
60  *
61  *
62  * y[] output result in an array of double precision numbers.
63  * The dimension of y[] is:
64  * 24-bit precision 1
65  * 53-bit precision 2
66  * 64-bit precision 2
67  * 113-bit precision 3
68  * The actual value is the sum of them. Thus for 113-bit
69  * precison, one may have to do something like:
70  *
71  * long double t,w,r_head, r_tail;
72  * t = (long double)y[2] + (long double)y[1];
73  * w = (long double)y[0];
74  * r_head = t+w;
75  * r_tail = w - (r_head - t);
76  *
77  * e0 The exponent of x[0]
78  *
79  * nx dimension of x[]
80  *
81  * prec an integer indicating the precision:
82  * 0 24 bits (single)
83  * 1 53 bits (double)
84  * 2 64 bits (extended)
85  * 3 113 bits (quad)
86  *
87  * ipio2[]
88  * integer array, contains the (24*i)-th to (24*i+23)-th
89  * bit of 2/pi after binary point. The corresponding
90  * floating value is
91  *
92  * ipio2[i] * 2^(-24(i+1)).
93  *
94  * External function:
95  * double scalbn(), floor();
96  *
97  *
98  * Here is the description of some local variables:
99  *
100  * jk jk+1 is the initial number of terms of ipio2[] needed
101  * in the computation. The recommended value is 2,3,4,
102  * 6 for single, double, extended,and quad.
103  *
104  * jz local integer variable indicating the number of
105  * terms of ipio2[] used.
106  *
107  * jx nx - 1
108  *
109  * jv index for pointing to the suitable ipio2[] for the
110  * computation. In general, we want
111  * ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8
112  * is an integer. Thus
113  * e0-3-24*jv >= 0 or (e0-3)/24 >= jv
114  * Hence jv = max(0,(e0-3)/24).
115  *
116  * jp jp+1 is the number of terms in PIo2[] needed, jp = jk.
117  *
118  * q[] double array with integral value, representing the
119  * 24-bits chunk of the product of x and 2/pi.
120  *
121  * q0 the corresponding exponent of q[0]. Note that the
122  * exponent for q[i] would be q0-24*i.
123  *
124  * PIo2[] double precision array, obtained by cutting pi/2
125  * into 24 bits chunks.
126  *
127  * f[] ipio2[] in floating point
128  *
129  * iq[] integer array by breaking up q[] in 24-bits chunk.
130  *
131  * fq[] final product of x*(2/pi) in fq[0],..,fq[jk]
132  *
133  * ih integer. If >0 it indicates q[] is >= 0.5, hence
134  * it also indicates the *sign* of the result.
135  *
136  */
137 
138 
139 /*
140  * Constants:
141  * The hexadecimal values are the intended ones for the following
142  * constants. The decimal values may be used, provided that the
143  * compiler will convert from decimal to binary accurately enough
144  * to produce the hexadecimal values shown.
145  */
146 
147 #include <libm.h>
148 #include "math_private.h"
149 
150 static const int init_jk[] = {2,3,4,6}; /* initial value for jk */
151 
152 static const double PIo2[] = {
153  1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */
154  7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */
155  5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */
156  3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */
157  1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */
158  1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */
159  2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */
160  2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */
161 };
162 
163 static const double
164 zero = 0.0,
165 one = 1.0,
166 two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
167 twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */
168 
169 int
170 __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, const int *ipio2)
171 {
172  int32_t jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih;
173  double z,fw,f[20],fq[20],q[20];
174 
175  /* initialize jk*/
176  jk = init_jk[prec];
177  jp = jk;
178 
179  /* determine jx,jv,q0, note that 3>q0 */
180  jx = nx-1;
181  jv = (e0-3)/24; if(jv<0) jv=0;
182  q0 = e0-24*(jv+1);
183 
184  /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
185  j = jv-jx; m = jx+jk;
186  for(i=0;i<=m;i++,j++) f[i] = (j<0)? zero : (double) ipio2[j];
187 
188  /* compute q[0],q[1],...q[jk] */
189  for (i=0;i<=jk;i++) {
190  for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; q[i] = fw;
191  }
192 
193  jz = jk;
194 recompute:
195  /* distill q[] into iq[] reversingly */
196  for(i=0,j=jz,z=q[jz];j>0;i++,j--) {
197  fw = (double)((int32_t)(twon24* z));
198  iq[i] = (int32_t)(z-two24*fw);
199  z = q[j-1]+fw;
200  }
201 
202  /* compute n */
203  z = scalbn(z,q0); /* actual value of z */
204  z -= 8.0*floor(z*0.125); /* trim off integer >= 8 */
205  n = (int32_t) z;
206  z -= (double)n;
207  ih = 0;
208  if(q0>0) { /* need iq[jz-1] to determine n */
209  i = (iq[jz-1]>>(24-q0)); n += i;
210  iq[jz-1] -= i<<(24-q0);
211  ih = iq[jz-1]>>(23-q0);
212  }
213  else if(q0==0) ih = iq[jz-1]>>23;
214  else if(z>=0.5) ih=2;
215 
216  if(ih>0) { /* q > 0.5 */
217  n += 1; carry = 0;
218  for(i=0;i<jz ;i++) { /* compute 1-q */
219  j = iq[i];
220  if(carry==0) {
221  if(j!=0) {
222  carry = 1; iq[i] = 0x1000000- j;
223  }
224  } else iq[i] = 0xffffff - j;
225  }
226  if(q0>0) { /* rare case: chance is 1 in 12 */
227  switch(q0) {
228  case 1:
229  iq[jz-1] &= 0x7fffff; break;
230  case 2:
231  iq[jz-1] &= 0x3fffff; break;
232  }
233  }
234  if(ih==2) {
235  z = one - z;
236  if(carry!=0) z -= scalbn(one,q0);
237  }
238  }
239 
240  /* check if recomputation is needed */
241  if(z==zero) {
242  j = 0;
243  for (i=jz-1;i>=jk;i--) j |= iq[i];
244  if(j==0) { /* need recomputation */
245  for(k=1;iq[jk-k]==0;k++); /* k = no. of terms needed */
246 
247  for(i=jz+1;i<=jz+k;i++) { /* add q[jz+1] to q[jz+k] */
248  f[jx+i] = (double) ipio2[jv+i];
249  for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j];
250  q[i] = fw;
251  }
252  jz += k;
253  goto recompute;
254  }
255  }
256 
257  /* chop off zero terms */
258  if(z==0.0) {
259  jz -= 1; q0 -= 24;
260  while(iq[jz]==0) { jz--; q0-=24;}
261  } else { /* break z into 24-bit if necessary */
262  z = scalbn(z,-q0);
263  if(z>=two24) {
264  fw = (double)((int32_t)(twon24*z));
265  iq[jz] = (int32_t)(z-two24*fw);
266  jz += 1; q0 += 24;
267  iq[jz] = (int32_t) fw;
268  } else iq[jz] = (int32_t) z ;
269  }
270 
271  /* convert integer "bit" chunk to floating-point value */
272  fw = scalbn(one,q0);
273  for(i=jz;i>=0;i--) {
274  q[i] = fw*(double)iq[i]; fw*=twon24;
275  }
276 
277  /* compute PIo2[0,...,jp]*q[jz,...,0] */
278  for(i=jz;i>=0;i--) {
279  for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k];
280  fq[jz-i] = fw;
281  }
282 
283  /* compress fq[] into y[] */
284  switch(prec) {
285  case 0:
286  fw = 0.0;
287  for (i=jz;i>=0;i--) fw += fq[i];
288  y[0] = (ih==0)? fw: -fw;
289  break;
290  case 1:
291  case 2:
292  fw = 0.0;
293  for (i=jz;i>=0;i--) fw += fq[i];
294  y[0] = (ih==0)? fw: -fw;
295  fw = fq[0]-fw;
296  for (i=1;i<=jz;i++) fw += fq[i];
297  y[1] = (ih==0)? fw: -fw;
298  break;
299  case 3: /* painful */
300  for (i=jz;i>0;i--) {
301  fw = fq[i-1]+fq[i];
302  fq[i] += fq[i-1]-fw;
303  fq[i-1] = fw;
304  }
305  for (i=jz;i>1;i--) {
306  fw = fq[i-1]+fq[i];
307  fq[i] += fq[i-1]-fw;
308  fq[i-1] = fw;
309  }
310  for (fw=0.0,i=jz;i>=2;i--) fw += fq[i];
311  if(ih==0) {
312  y[0] = fq[0]; y[1] = fq[1]; y[2] = fw;
313  } else {
314  y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw;
315  }
316  }
317  return n&7;
318 }
319 
320 #endif