SDL  2.0
k_rem_pio2.c
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1 /* @(#)k_rem_pio2.c 5.1 93/09/24 */
2 /*
3  * ====================================================
4  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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];
61  * r_head = t+w;
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 
137 #include "SDL_assert.h"
138 
141 #ifdef __STDC__
142  static const int init_jk[] = { 2, 3, 4, 6 }; /* initial value for jk */
143 #else
144  static int init_jk[] = { 2, 3, 4, 6 };
145 #endif
146 
147 #ifdef __STDC__
148 static const double PIo2[] = {
149 #else
150 static double PIo2[] = {
151 #endif
152  1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */
153  7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */
154  5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */
155  3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */
156  1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */
157  1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */
158  2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */
159  2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */
160 };
161 
162 #ifdef __STDC__
163 static const double
164 #else
165 static double
166 #endif
167  zero = 0.0, one = 1.0, two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
168  twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */
169 
170 #ifdef __STDC__
172 __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec,
173  const int32_t * ipio2)
174 #else
176 __kernel_rem_pio2(x, y, e0, nx, prec, ipio2)
177  double x[], y[];
178  int e0, nx, prec;
179  int32_t ipio2[];
180 #endif
181 {
182  int32_t jz, jx, jv, jp, jk, carry, n, iq[20], i, j, k, m, q0, ih;
183  double z, fw, f[20], fq[20], q[20];
184 
185  /* initialize jk */
186  SDL_assert((prec >= 0) && (prec < SDL_arraysize(init_jk)));
187  jk = init_jk[prec];
188  SDL_assert((jk >= 2) && (jk <= 6));
189  jp = jk;
190 
191  /* determine jx,jv,q0, note that 3>q0 */
192  SDL_assert(nx > 0);
193  jx = nx - 1;
194  jv = (e0 - 3) / 24;
195  if (jv < 0)
196  jv = 0;
197  q0 = e0 - 24 * (jv + 1);
198 
199  /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
200  j = jv - jx;
201  m = jx + jk;
202  for (i = 0; i <= m; i++, j++)
203  f[i] = (j < 0) ? zero : (double) ipio2[j];
204 
205  /* compute q[0],q[1],...q[jk] */
206  for (i = 0; i <= jk; i++) {
207  for (j = 0, fw = 0.0; j <= jx; j++)
208  fw += x[j] * f[jx + i - j];
209  q[i] = fw;
210  }
211 
212  jz = jk;
213  recompute:
214  /* distill q[] into iq[] reversingly */
215  for (i = 0, j = jz, z = q[jz]; j > 0; i++, j--) {
216  fw = (double) ((int32_t) (twon24 * z));
217  iq[i] = (int32_t) (z - two24 * fw);
218  z = q[j - 1] + fw;
219  }
220 
221  /* compute n */
222  z = scalbn(z, q0); /* actual value of z */
223  z -= 8.0 * floor(z * 0.125); /* trim off integer >= 8 */
224  n = (int32_t) z;
225  z -= (double) n;
226  ih = 0;
227  if (q0 > 0) { /* need iq[jz-1] to determine n */
228  i = (iq[jz - 1] >> (24 - q0));
229  n += i;
230  iq[jz - 1] -= i << (24 - q0);
231  ih = iq[jz - 1] >> (23 - q0);
232  } else if (q0 == 0)
233  ih = iq[jz - 1] >> 23;
234  else if (z >= 0.5)
235  ih = 2;
236 
237  if (ih > 0) { /* q > 0.5 */
238  n += 1;
239  carry = 0;
240  for (i = 0; i < jz; i++) { /* compute 1-q */
241  j = iq[i];
242  if (carry == 0) {
243  if (j != 0) {
244  carry = 1;
245  iq[i] = 0x1000000 - j;
246  }
247  } else
248  iq[i] = 0xffffff - j;
249  }
250  if (q0 > 0) { /* rare case: chance is 1 in 12 */
251  switch (q0) {
252  case 1:
253  iq[jz - 1] &= 0x7fffff;
254  break;
255  case 2:
256  iq[jz - 1] &= 0x3fffff;
257  break;
258  }
259  }
260  if (ih == 2) {
261  z = one - z;
262  if (carry != 0)
263  z -= scalbn(one, q0);
264  }
265  }
266 
267  /* check if recomputation is needed */
268  if (z == zero) {
269  j = 0;
270  for (i = jz - 1; i >= jk; i--)
271  j |= iq[i];
272  if (j == 0) { /* need recomputation */
273  for (k = 1; iq[jk - k] == 0; k++); /* k = no. of terms needed */
274 
275  for (i = jz + 1; i <= jz + k; i++) { /* add q[jz+1] to q[jz+k] */
276  f[jx + i] = (double) ipio2[jv + i];
277  for (j = 0, fw = 0.0; j <= jx; j++)
278  fw += x[j] * f[jx + i - j];
279  q[i] = fw;
280  }
281  jz += k;
282  goto recompute;
283  }
284  }
285 
286  /* chop off zero terms */
287  if (z == 0.0) {
288  jz -= 1;
289  q0 -= 24;
290  while (iq[jz] == 0) {
291  jz--;
292  q0 -= 24;
293  }
294  } else { /* break z into 24-bit if necessary */
295  z = scalbn(z, -q0);
296  if (z >= two24) {
297  fw = (double) ((int32_t) (twon24 * z));
298  iq[jz] = (int32_t) (z - two24 * fw);
299  jz += 1;
300  q0 += 24;
301  iq[jz] = (int32_t) fw;
302  } else
303  iq[jz] = (int32_t) z;
304  }
305 
306  /* convert integer "bit" chunk to floating-point value */
307  fw = scalbn(one, q0);
308  for (i = jz; i >= 0; i--) {
309  q[i] = fw * (double) iq[i];
310  fw *= twon24;
311  }
312 
313  /* compute PIo2[0,...,jp]*q[jz,...,0] */
314  for (i = jz; i >= 0; i--) {
315  for (fw = 0.0, k = 0; k <= jp && k <= jz - i; k++)
316  fw += PIo2[k] * q[i + k];
317  fq[jz - i] = fw;
318  }
319 
320  /* compress fq[] into y[] */
321  switch (prec) {
322  case 0:
323  fw = 0.0;
324  for (i = jz; i >= 0; i--)
325  fw += fq[i];
326  y[0] = (ih == 0) ? fw : -fw;
327  break;
328  case 1:
329  case 2:
330  fw = 0.0;
331  for (i = jz; i >= 0; i--)
332  fw += fq[i];
333  y[0] = (ih == 0) ? fw : -fw;
334  fw = fq[0] - fw;
335  for (i = 1; i <= jz; i++)
336  fw += fq[i];
337  y[1] = (ih == 0) ? fw : -fw;
338  break;
339  case 3: /* painful */
340  for (i = jz; i > 0; i--) {
341  fw = fq[i - 1] + fq[i];
342  fq[i] += fq[i - 1] - fw;
343  fq[i - 1] = fw;
344  }
345  for (i = jz; i > 1; i--) {
346  fw = fq[i - 1] + fq[i];
347  fq[i] += fq[i - 1] - fw;
348  fq[i - 1] = fw;
349  }
350  for (fw = 0.0, i = jz; i >= 2; i--)
351  fw += fq[i];
352  if (ih == 0) {
353  y[0] = fq[0];
354  y[1] = fq[1];
355  y[2] = fw;
356  } else {
357  y[0] = -fq[0];
358  y[1] = -fq[1];
359  y[2] = -fw;
360  }
361  }
362  return n & 7;
363 }
static double PIo2[]
Definition: k_rem_pio2.c:150
GLdouble n
signed int int32_t
GLint GLint GLint GLint GLint x
Definition: SDL_opengl.h:1567
GLdouble GLdouble GLdouble GLdouble q
Definition: SDL_opengl.h:2080
const GLfloat * m
int attribute_hidden __kernel_rem_pio2(x, y, int e0, int nx, int prec, ipio2)
Definition: k_rem_pio2.c:176
GLfloat f
#define attribute_hidden
Definition: math_private.h:24
static double one
Definition: k_rem_pio2.c:167
#define scalbn
Definition: math_private.h:40
static double two24
Definition: k_rem_pio2.c:167
GLint GLint GLint GLint GLint GLint y
Definition: SDL_opengl.h:1567
GLbyte nx
static double twon24
Definition: k_rem_pio2.c:168
return Display return Display Bool Bool int int int return Display XEvent Bool(*) XPointer return Display return Display Drawable _Xconst char unsigned int unsigned int return Display Pixmap Pixmap XColor XColor unsigned int unsigned int return Display _Xconst char char int char return Display Visual unsigned int int int char unsigned int unsigned int in i)
Definition: SDL_x11sym.h:42
#define SDL_assert(condition)
Definition: SDL_assert.h:167
GLdouble GLdouble z
static double zero
Definition: k_rem_pio2.c:167
#define SDL_arraysize(array)
Definition: SDL_stdinc.h:93
#define floor
Definition: math_private.h:37
return Display return Display Bool Bool int int int return Display XEvent Bool(*) XPointer return Display return Display Drawable _Xconst char unsigned int unsigned int return Display Pixmap Pixmap XColor XColor unsigned int unsigned int return Display _Xconst char char int char return Display Visual unsigned int int int char unsigned int unsigned int int in j)
Definition: SDL_x11sym.h:42
libm_hidden_proto(scalbn)
Definition: k_rem_pio2.c:139