zenilib  0.5.3.0
k_rem_pio2.c
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1 /* @(#)k_rem_pio2.c 5.1 93/09/24 */
2 /*
3  * ====================================================
5  *
6  * Developed at SunPro, a Sun Microsystems, Inc. business.
7  * Permission to use, copy, modify, and distribute this
8  * software is freely granted, provided that this notice
9  * is preserved.
10  * ====================================================
11  */
12
13 #if defined(LIBM_SCCS) && !defined(lint)
14 static const char rcsid[] =
15  "\$NetBSD: k_rem_pio2.c,v 1.7 1995/05/10 20:46:25 jtc Exp \$";
16 #endif
17
18 /*
19  * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
20  * double x[],y[]; int e0,nx,prec; int ipio2[];
21  *
22  * __kernel_rem_pio2 return the last three digits of N with
23  * y = x - N*pi/2
24  * so that |y| < pi/2.
25  *
26  * The method is to compute the integer (mod 8) and fraction parts of
27  * (2/pi)*x without doing the full multiplication. In general we
28  * skip the part of the product that are known to be a huge integer (
29  * more accurately, = 0 mod 8 ). Thus the number of operations are
30  * independent of the exponent of the input.
31  *
32  * (2/pi) is represented by an array of 24-bit integers in ipio2[].
33  *
34  * Input parameters:
35  * x[] The input value (must be positive) is broken into nx
36  * pieces of 24-bit integers in double precision format.
37  * x[i] will be the i-th 24 bit of x. The scaled exponent
38  * of x[0] is given in input parameter e0 (i.e., x[0]*2^e0
39  * match x's up to 24 bits.
40  *
41  * Example of breaking a double positive z into x[0]+x[1]+x[2]:
42  * e0 = ilogb(z)-23
43  * z = scalbn(z,-e0)
44  * for i = 0,1,2
45  * x[i] = floor(z)
46  * z = (z-x[i])*2**24
47  *
48  *
49  * y[] ouput result in an array of double precision numbers.
50  * The dimension of y[] is:
51  * 24-bit precision 1
52  * 53-bit precision 2
53  * 64-bit precision 2
54  * 113-bit precision 3
55  * The actual value is the sum of them. Thus for 113-bit
56  * precison, one may have to do something like:
57  *
58  * long double t,w,r_head, r_tail;
59  * t = (long double)y[2] + (long double)y[1];
60  * w = (long double)y[0];
62  * r_tail = w - (r_head - t);
63  *
64  * e0 The exponent of x[0]
65  *
66  * nx dimension of x[]
67  *
68  * prec an integer indicating the precision:
69  * 0 24 bits (single)
70  * 1 53 bits (double)
71  * 2 64 bits (extended)
72  * 3 113 bits (quad)
73  *
74  * ipio2[]
75  * integer array, contains the (24*i)-th to (24*i+23)-th
76  * bit of 2/pi after binary point. The corresponding
77  * floating value is
78  *
79  * ipio2[i] * 2^(-24(i+1)).
80  *
81  * External function:
82  * double scalbn(), floor();
83  *
84  *
85  * Here is the description of some local variables:
86  *
87  * jk jk+1 is the initial number of terms of ipio2[] needed
88  * in the computation. The recommended value is 2,3,4,
89  * 6 for single, double, extended,and quad.
90  *
91  * jz local integer variable indicating the number of
92  * terms of ipio2[] used.
93  *
94  * jx nx - 1
95  *
96  * jv index for pointing to the suitable ipio2[] for the
97  * computation. In general, we want
98  * ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8
99  * is an integer. Thus
100  * e0-3-24*jv >= 0 or (e0-3)/24 >= jv
101  * Hence jv = max(0,(e0-3)/24).
102  *
103  * jp jp+1 is the number of terms in PIo2[] needed, jp = jk.
104  *
105  * q[] double array with integral value, representing the
106  * 24-bits chunk of the product of x and 2/pi.
107  *
108  * q0 the corresponding exponent of q[0]. Note that the
109  * exponent for q[i] would be q0-24*i.
110  *
111  * PIo2[] double precision array, obtained by cutting pi/2
112  * into 24 bits chunks.
113  *
114  * f[] ipio2[] in floating point
115  *
116  * iq[] integer array by breaking up q[] in 24-bits chunk.
117  *
118  * fq[] final product of x*(2/pi) in fq[0],..,fq[jk]
119  *
120  * ih integer. If >0 it indicates q[] is >= 0.5, hence
121  * it also indicates the *sign* of the result.
122  *
123  */
124
125
126 /*
127  * Constants:
128  * The hexadecimal values are the intended ones for the following
129  * constants. The decimal values may be used, provided that the
130  * compiler will convert from decimal to binary accurately enough
131  * to produce the hexadecimal values shown.
132  */
133
134 #include "math_libm.h"
135 #include "math_private.h"
136
139 #ifdef __STDC__
140  static const int init_jk[] = { 2, 3, 4, 6 }; /* initial value for jk */
141 #else
142  static int init_jk[] = { 2, 3, 4, 6 };
143 #endif
144
145 #ifdef __STDC__
146 static const double PIo2[] = {
147 #else
148 static double PIo2[] = {
149 #endif
150  1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */
151  7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */
152  5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */
153  3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */
154  1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */
155  1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */
156  2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */
157  2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */
158 };
159
160 #ifdef __STDC__
161 static const double
162 #else
163 static double
164 #endif
165  zero = 0.0, one = 1.0, two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
166  twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */
167
168 #ifdef __STDC__
170 __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec,
171  const int32_t * ipio2)
172 #else
174 __kernel_rem_pio2(x, y, e0, nx, prec, ipio2)
175  double x[], y[];
176  int e0, nx, prec;
177  int32_t ipio2[];
178 #endif
179 {
180  int32_t jz, jx, jv, jp, jk, carry, n, iq[20], i, j, k, m, q0, ih;
181  double z, fw, f[20], fq[20], q[20];
182
183  /* initialize jk */
184  jk = init_jk[prec];
185  jp = jk;
186
187  /* determine jx,jv,q0, note that 3>q0 */
188  jx = nx - 1;
189  jv = (e0 - 3) / 24;
190  if (jv < 0)
191  jv = 0;
192  q0 = e0 - 24 * (jv + 1);
193
194  /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
195  j = jv - jx;
196  m = jx + jk;
197  for (i = 0; i <= m; i++, j++)
198  f[i] = (j < 0) ? zero : (double) ipio2[j];
199
200  /* compute q[0],q[1],...q[jk] */
201  for (i = 0; i <= jk; i++) {
202  for (j = 0, fw = 0.0; j <= jx; j++)
203  fw += x[j] * f[jx + i - j];
204  q[i] = fw;
205  }
206
207  jz = jk;
208  recompute:
209  /* distill q[] into iq[] reversingly */
210  for (i = 0, j = jz, z = q[jz]; j > 0; i++, j--) {
211  fw = (double) ((int32_t) (twon24 * z));
212  iq[i] = (int32_t) (z - two24 * fw);
213  z = q[j - 1] + fw;
214  }
215
216  /* compute n */
217  z = scalbn(z, q0); /* actual value of z */
218  z -= 8.0 * floor(z * 0.125); /* trim off integer >= 8 */
219  n = (int32_t) z;
220  z -= (double) n;
221  ih = 0;
222  if (q0 > 0) { /* need iq[jz-1] to determine n */
223  i = (iq[jz - 1] >> (24 - q0));
224  n += i;
225  iq[jz - 1] -= i << (24 - q0);
226  ih = iq[jz - 1] >> (23 - q0);
227  } else if (q0 == 0)
228  ih = iq[jz - 1] >> 23;
229  else if (z >= 0.5)
230  ih = 2;
231
232  if (ih > 0) { /* q > 0.5 */
233  n += 1;
234  carry = 0;
235  for (i = 0; i < jz; i++) { /* compute 1-q */
236  j = iq[i];
237  if (carry == 0) {
238  if (j != 0) {
239  carry = 1;
240  iq[i] = 0x1000000 - j;
241  }
242  } else
243  iq[i] = 0xffffff - j;
244  }
245  if (q0 > 0) { /* rare case: chance is 1 in 12 */
246  switch (q0) {
247  case 1:
248  iq[jz - 1] &= 0x7fffff;
249  break;
250  case 2:
251  iq[jz - 1] &= 0x3fffff;
252  break;
253  }
254  }
255  if (ih == 2) {
256  z = one - z;
257  if (carry != 0)
258  z -= scalbn(one, q0);
259  }
260  }
261
262  /* check if recomputation is needed */
263  if (z == zero) {
264  j = 0;
265  for (i = jz - 1; i >= jk; i--)
266  j |= iq[i];
267  if (j == 0) { /* need recomputation */
268  for (k = 1; iq[jk - k] == 0; k++); /* k = no. of terms needed */
269
270  for (i = jz + 1; i <= jz + k; i++) { /* add q[jz+1] to q[jz+k] */
271  f[jx + i] = (double) ipio2[jv + i];
272  for (j = 0, fw = 0.0; j <= jx; j++)
273  fw += x[j] * f[jx + i - j];
274  q[i] = fw;
275  }
276  jz += k;
277  goto recompute;
278  }
279  }
280
281  /* chop off zero terms */
282  if (z == 0.0) {
283  jz -= 1;
284  q0 -= 24;
285  while (iq[jz] == 0) {
286  jz--;
287  q0 -= 24;
288  }
289  } else { /* break z into 24-bit if necessary */
290  z = scalbn(z, -q0);
291  if (z >= two24) {
292  fw = (double) ((int32_t) (twon24 * z));
293  iq[jz] = (int32_t) (z - two24 * fw);
294  jz += 1;
295  q0 += 24;
296  iq[jz] = (int32_t) fw;
297  } else
298  iq[jz] = (int32_t) z;
299  }
300
301  /* convert integer "bit" chunk to floating-point value */
302  fw = scalbn(one, q0);
303  for (i = jz; i >= 0; i--) {
304  q[i] = fw * (double) iq[i];
305  fw *= twon24;
306  }
307
308  /* compute PIo2[0,...,jp]*q[jz,...,0] */
309  for (i = jz; i >= 0; i--) {
310  for (fw = 0.0, k = 0; k <= jp && k <= jz - i; k++)
311  fw += PIo2[k] * q[i + k];
312  fq[jz - i] = fw;
313  }
314
315  /* compress fq[] into y[] */
316  switch (prec) {
317  case 0:
318  fw = 0.0;
319  for (i = jz; i >= 0; i--)
320  fw += fq[i];
321  y[0] = (ih == 0) ? fw : -fw;
322  break;
323  case 1:
324  case 2:
325  fw = 0.0;
326  for (i = jz; i >= 0; i--)
327  fw += fq[i];
328  y[0] = (ih == 0) ? fw : -fw;
329  fw = fq[0] - fw;
330  for (i = 1; i <= jz; i++)
331  fw += fq[i];
332  y[1] = (ih == 0) ? fw : -fw;
333  break;
334  case 3: /* painful */
335  for (i = jz; i > 0; i--) {
336  fw = fq[i - 1] + fq[i];
337  fq[i] += fq[i - 1] - fw;
338  fq[i - 1] = fw;
339  }
340  for (i = jz; i > 1; i--) {
341  fw = fq[i - 1] + fq[i];
342  fq[i] += fq[i - 1] - fw;
343  fq[i - 1] = fw;
344  }
345  for (fw = 0.0, i = jz; i >= 2; i--)
346  fw += fq[i];
347  if (ih == 0) {
348  y[0] = fq[0];
349  y[1] = fq[1];
350  y[2] = fw;
351  } else {
352  y[0] = -fq[0];
353  y[1] = -fq[1];
354  y[2] = -fw;
355  }
356  }
357  return n & 7;
358 }
ih
Definition: k_rem_pio2.c:221
jv
Definition: k_rem_pio2.c:189
static double PIo2[]
Definition: k_rem_pio2.c:148
GLclampf f
Definition: glew.h:3390
int32_t k
Definition: e_log.c:102
int32_t e0
Definition: e_rem_pio2.c:100
GLclampd n
Definition: glew.h:7287
EGLSurface EGLint x
Definition: eglext.h:293
jk
Definition: k_rem_pio2.c:184
int32_t j
Definition: e_log.c:102
GLfloat GLfloat GLfloat GLfloat nx
Definition: glew.h:14904
static int init_jk[]
Definition: k_rem_pio2.c:142
long int32_t
Definition: types.h:9
static double one
Definition: k_rem_pio2.c:165
jz
Definition: k_rem_pio2.c:207
jx
Definition: k_rem_pio2.c:188
double fw
Definition: k_rem_pio2.c:179
static double two24
Definition: k_rem_pio2.c:165
int prec
Definition: k_rem_pio2.c:176
jp
Definition: k_rem_pio2.c:185
double fq[20]
Definition: k_rem_pio2.c:179
static double twon24
Definition: k_rem_pio2.c:166
double floor(double x)
Definition: s_floor.c:40
q0
Definition: k_rem_pio2.c:192
EGLSurface EGLint EGLint y
Definition: eglext.h:293
int __kernel_rem_pio2(double *, double *, int, int, int, const int *) attribute_hidden
int32_t ipio2[]
Definition: k_rem_pio2.c:177
double attribute_hidden
static double zero
Definition: k_rem_pio2.c:165
#define libm_hidden_proto(x)
Definition: math_private.h:25
GLdouble GLdouble GLdouble GLdouble q
Definition: glew.h:1400
iq[i]
Definition: k_rem_pio2.c:212
GLint GLint GLint GLint z
Definition: gl2ext.h:1214
double scalbn(double x, int n)
Definition: s_scalbn.c:42
int i
Definition: pngrutil.c:1377
#define m(i, j)