SDL  2.0
k_rem_pio2.c
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1 /*
2  * ====================================================
3  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
4  *
5  * Developed at SunPro, a Sun Microsystems, Inc. business.
6  * Permission to use, copy, modify, and distribute this
7  * software is freely granted, provided that this notice
8  * is preserved.
9  * ====================================================
10  */
11 
12 /*
13  * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
14  * double x[],y[]; int e0,nx,prec; int ipio2[];
15  *
16  * __kernel_rem_pio2 return the last three digits of N with
17  * y = x - N*pi/2
18  * so that |y| < pi/2.
19  *
20  * The method is to compute the integer (mod 8) and fraction parts of
21  * (2/pi)*x without doing the full multiplication. In general we
22  * skip the part of the product that are known to be a huge integer (
23  * more accurately, = 0 mod 8 ). Thus the number of operations are
24  * independent of the exponent of the input.
25  *
26  * (2/pi) is represented by an array of 24-bit integers in ipio2[].
27  *
28  * Input parameters:
29  * x[] The input value (must be positive) is broken into nx
30  * pieces of 24-bit integers in double precision format.
31  * x[i] will be the i-th 24 bit of x. The scaled exponent
32  * of x[0] is given in input parameter e0 (i.e., x[0]*2^e0
33  * match x's up to 24 bits.
34  *
35  * Example of breaking a double positive z into x[0]+x[1]+x[2]:
36  * e0 = ilogb(z)-23
37  * z = scalbn(z,-e0)
38  * for i = 0,1,2
39  * x[i] = floor(z)
40  * z = (z-x[i])*2**24
41  *
42  *
43  * y[] ouput result in an array of double precision numbers.
44  * The dimension of y[] is:
45  * 24-bit precision 1
46  * 53-bit precision 2
47  * 64-bit precision 2
48  * 113-bit precision 3
49  * The actual value is the sum of them. Thus for 113-bit
50  * precison, one may have to do something like:
51  *
52  * long double t,w,r_head, r_tail;
53  * t = (long double)y[2] + (long double)y[1];
54  * w = (long double)y[0];
55  * r_head = t+w;
56  * r_tail = w - (r_head - t);
57  *
58  * e0 The exponent of x[0]
59  *
60  * nx dimension of x[]
61  *
62  * prec an integer indicating the precision:
63  * 0 24 bits (single)
64  * 1 53 bits (double)
65  * 2 64 bits (extended)
66  * 3 113 bits (quad)
67  *
68  * ipio2[]
69  * integer array, contains the (24*i)-th to (24*i+23)-th
70  * bit of 2/pi after binary point. The corresponding
71  * floating value is
72  *
73  * ipio2[i] * 2^(-24(i+1)).
74  *
75  * External function:
76  * double scalbn(), floor();
77  *
78  *
79  * Here is the description of some local variables:
80  *
81  * jk jk+1 is the initial number of terms of ipio2[] needed
82  * in the computation. The recommended value is 2,3,4,
83  * 6 for single, double, extended,and quad.
84  *
85  * jz local integer variable indicating the number of
86  * terms of ipio2[] used.
87  *
88  * jx nx - 1
89  *
90  * jv index for pointing to the suitable ipio2[] for the
91  * computation. In general, we want
92  * ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8
93  * is an integer. Thus
94  * e0-3-24*jv >= 0 or (e0-3)/24 >= jv
95  * Hence jv = max(0,(e0-3)/24).
96  *
97  * jp jp+1 is the number of terms in PIo2[] needed, jp = jk.
98  *
99  * q[] double array with integral value, representing the
100  * 24-bits chunk of the product of x and 2/pi.
101  *
102  * q0 the corresponding exponent of q[0]. Note that the
103  * exponent for q[i] would be q0-24*i.
104  *
105  * PIo2[] double precision array, obtained by cutting pi/2
106  * into 24 bits chunks.
107  *
108  * f[] ipio2[] in floating point
109  *
110  * iq[] integer array by breaking up q[] in 24-bits chunk.
111  *
112  * fq[] final product of x*(2/pi) in fq[0],..,fq[jk]
113  *
114  * ih integer. If >0 it indicates q[] is >= 0.5, hence
115  * it also indicates the *sign* of the result.
116  *
117  */
118 
119 
120 /*
121  * Constants:
122  * The hexadecimal values are the intended ones for the following
123  * constants. The decimal values may be used, provided that the
124  * compiler will convert from decimal to binary accurately enough
125  * to produce the hexadecimal values shown.
126  */
127 
128 #include "math_libm.h"
129 #include "math_private.h"
130 
131 #include "SDL_assert.h"
132 
133 static const int init_jk[] = {2,3,4,6}; /* initial value for jk */
134 
135 static const double PIo2[] = {
136  1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */
137  7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */
138  5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */
139  3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */
140  1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */
141  1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */
142  2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */
143  2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */
144 };
145 
146 static const double
147 zero = 0.0,
148 one = 1.0,
149 two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
150 twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */
151 
152 int32_t attribute_hidden __kernel_rem_pio2(double *x, double *y, int e0, int nx, const unsigned int prec, const int32_t *ipio2)
153 {
154  int32_t jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih;
155  double z,fw,f[20],fq[20],q[20];
156 
157  if (nx < 1) {
158  return 0;
159  }
160 
161  /* initialize jk*/
162  SDL_assert(prec < SDL_arraysize(init_jk));
163  jk = init_jk[prec];
164  SDL_assert(jk > 0);
165  jp = jk;
166 
167  /* determine jx,jv,q0, note that 3>q0 */
168  jx = nx-1;
169  jv = (e0-3)/24; if(jv<0) jv=0;
170  q0 = e0-24*(jv+1);
171 
172  /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
173  j = jv-jx; m = jx+jk;
174  for(i=0;i<=m;i++,j++) f[i] = (j<0)? zero : (double) ipio2[j];
175  if ((m+1) < SDL_arraysize(f)) {
176  SDL_memset(&f[m+1], 0, sizeof (f) - ((m+1) * sizeof (f[0])));
177  }
178 
179  /* compute q[0],q[1],...q[jk] */
180  for (i=0;i<=jk;i++) {
181  for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j];
182  q[i] = fw;
183  }
184 
185  jz = jk;
186 recompute:
187  /* distill q[] into iq[] reversingly */
188  for(i=0,j=jz,z=q[jz];j>0;i++,j--) {
189  fw = (double)((int32_t)(twon24* z));
190  iq[i] = (int32_t)(z-two24*fw);
191  z = q[j-1]+fw;
192  }
193  if (jz < SDL_arraysize(iq)) {
194  SDL_memset(&iq[jz], 0, sizeof (q) - (jz * sizeof (iq[0])));
195  }
196 
197  /* compute n */
198  z = scalbn(z,q0); /* actual value of z */
199  z -= 8.0*floor(z*0.125); /* trim off integer >= 8 */
200  n = (int32_t) z;
201  z -= (double)n;
202  ih = 0;
203  if(q0>0) { /* need iq[jz-1] to determine n */
204  i = (iq[jz-1]>>(24-q0)); n += i;
205  iq[jz-1] -= i<<(24-q0);
206  ih = iq[jz-1]>>(23-q0);
207  }
208  else if(q0==0) ih = iq[jz-1]>>23;
209  else if(z>=0.5) ih=2;
210 
211  if(ih>0) { /* q > 0.5 */
212  n += 1; carry = 0;
213  for(i=0;i<jz ;i++) { /* compute 1-q */
214  j = iq[i];
215  if(carry==0) {
216  if(j!=0) {
217  carry = 1; iq[i] = 0x1000000- j;
218  }
219  } else iq[i] = 0xffffff - j;
220  }
221  if(q0>0) { /* rare case: chance is 1 in 12 */
222  switch(q0) {
223  case 1:
224  iq[jz-1] &= 0x7fffff; break;
225  case 2:
226  iq[jz-1] &= 0x3fffff; break;
227  }
228  }
229  if(ih==2) {
230  z = one - z;
231  if(carry!=0) z -= scalbn(one,q0);
232  }
233  }
234 
235  /* check if recomputation is needed */
236  if(z==zero) {
237  j = 0;
238  for (i=jz-1;i>=jk;i--) j |= iq[i];
239  if(j==0) { /* need recomputation */
240  for(k=1;iq[jk-k]==0;k++); /* k = no. of terms needed */
241 
242  for(i=jz+1;i<=jz+k;i++) { /* add q[jz+1] to q[jz+k] */
243  f[jx+i] = (double) ipio2[jv+i];
244  for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j];
245  q[i] = fw;
246  }
247  jz += k;
248  goto recompute;
249  }
250  }
251 
252  /* chop off zero terms */
253  if(z==0.0) {
254  jz -= 1; q0 -= 24;
255  SDL_assert(jz >= 0);
256  while(iq[jz]==0) { jz--; SDL_assert(jz >= 0); q0-=24;}
257  } else { /* break z into 24-bit if necessary */
258  z = scalbn(z,-q0);
259  if(z>=two24) {
260  fw = (double)((int32_t)(twon24*z));
261  iq[jz] = (int32_t)(z-two24*fw);
262  jz += 1; q0 += 24;
263  iq[jz] = (int32_t) fw;
264  } else iq[jz] = (int32_t) z ;
265  }
266 
267  /* convert integer "bit" chunk to floating-point value */
268  fw = scalbn(one,q0);
269  for(i=jz;i>=0;i--) {
270  q[i] = fw*(double)iq[i]; fw*=twon24;
271  }
272 
273  /* compute PIo2[0,...,jp]*q[jz,...,0] */
274  for(i=jz;i>=0;i--) {
275  for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k];
276  fq[jz-i] = fw;
277  }
278  if ((jz+1) < SDL_arraysize(f)) {
279  SDL_memset(&fq[jz+1], 0, sizeof (fq) - ((jz+1) * sizeof (fq[0])));
280  }
281 
282  /* compress fq[] into y[] */
283  switch(prec) {
284  case 0:
285  fw = 0.0;
286  for (i=jz;i>=0;i--) fw += fq[i];
287  y[0] = (ih==0)? fw: -fw;
288  break;
289  case 1:
290  case 2:
291  fw = 0.0;
292  for (i=jz;i>=0;i--) fw += fq[i];
293  y[0] = (ih==0)? fw: -fw;
294  fw = fq[0]-fw;
295  for (i=1;i<=jz;i++) fw += fq[i];
296  y[1] = (ih==0)? fw: -fw;
297  break;
298  case 3: /* painful */
299  for (i=jz;i>0;i--) {
300  fw = fq[i-1]+fq[i];
301  fq[i] += fq[i-1]-fw;
302  fq[i-1] = fw;
303  }
304  for (i=jz;i>1;i--) {
305  fw = fq[i-1]+fq[i];
306  fq[i] += fq[i-1]-fw;
307  fq[i-1] = fw;
308  }
309  for (fw=0.0,i=jz;i>=2;i--) fw += fq[i];
310  if(ih==0) {
311  y[0] = fq[0]; y[1] = fq[1]; y[2] = fw;
312  } else {
313  y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw;
314  }
315  }
316  return n&7;
317 }