aboutsummaryrefslogtreecommitdiff
path: root/sysdeps/ieee754/dbl-64/k_rem_pio2.c
blob: 047c6c2886db474227c11058c371a49b6dd50f7b (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
/* @(#)k_rem_pio2.c 5.1 93/09/24 */
/*
 * ====================================================
 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
 *
 * Developed at SunPro, a Sun Microsystems, Inc. business.
 * Permission to use, copy, modify, and distribute this
 * software is freely granted, provided that this notice
 * is preserved.
 * ====================================================
 */

#if defined(LIBM_SCCS) && !defined(lint)
static char rcsid[] = "$NetBSD: k_rem_pio2.c,v 1.7 1995/05/10 20:46:25 jtc Exp $";
#endif

/*
 * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
 * double x[],y[]; int e0,nx,prec; int ipio2[];
 *
 * __kernel_rem_pio2 return the last three digits of N with
 *		y = x - N*pi/2
 * so that |y| < pi/2.
 *
 * The method is to compute the integer (mod 8) and fraction parts of
 * (2/pi)*x without doing the full multiplication. In general we
 * skip the part of the product that are known to be a huge integer (
 * more accurately, = 0 mod 8 ). Thus the number of operations are
 * independent of the exponent of the input.
 *
 * (2/pi) is represented by an array of 24-bit integers in ipio2[].
 *
 * Input parameters:
 * 	x[]	The input value (must be positive) is broken into nx
 *		pieces of 24-bit integers in double precision format.
 *		x[i] will be the i-th 24 bit of x. The scaled exponent
 *		of x[0] is given in input parameter e0 (i.e., x[0]*2^e0
 *		match x's up to 24 bits.
 *
 *		Example of breaking a double positive z into x[0]+x[1]+x[2]:
 *			e0 = ilogb(z)-23
 *			z  = scalbn(z,-e0)
 *		for i = 0,1,2
 *			x[i] = floor(z)
 *			z    = (z-x[i])*2**24
 *
 *
 *	y[]	ouput result in an array of double precision numbers.
 *		The dimension of y[] is:
 *			24-bit  precision	1
 *			53-bit  precision	2
 *			64-bit  precision	2
 *			113-bit precision	3
 *		The actual value is the sum of them. Thus for 113-bit
 *		precision, one may have to do something like:
 *
 *		long double t,w,r_head, r_tail;
 *		t = (long double)y[2] + (long double)y[1];
 *		w = (long double)y[0];
 *		r_head = t+w;
 *		r_tail = w - (r_head - t);
 *
 *	e0	The exponent of x[0]
 *
 *	nx	dimension of x[]
 *
 *  	prec	an integer indicating the precision:
 *			0	24  bits (single)
 *			1	53  bits (double)
 *			2	64  bits (extended)
 *			3	113 bits (quad)
 *
 *	ipio2[]
 *		integer array, contains the (24*i)-th to (24*i+23)-th
 *		bit of 2/pi after binary point. The corresponding
 *		floating value is
 *
 *			ipio2[i] * 2^(-24(i+1)).
 *
 * External function:
 *	double scalbn(), floor();
 *
 *
 * Here is the description of some local variables:
 *
 * 	jk	jk+1 is the initial number of terms of ipio2[] needed
 *		in the computation. The recommended value is 2,3,4,
 *		6 for single, double, extended,and quad.
 *
 * 	jz	local integer variable indicating the number of
 *		terms of ipio2[] used.
 *
 *	jx	nx - 1
 *
 *	jv	index for pointing to the suitable ipio2[] for the
 *		computation. In general, we want
 *			( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8
 *		is an integer. Thus
 *			e0-3-24*jv >= 0 or (e0-3)/24 >= jv
 *		Hence jv = max(0,(e0-3)/24).
 *
 *	jp	jp+1 is the number of terms in PIo2[] needed, jp = jk.
 *
 * 	q[]	double array with integral value, representing the
 *		24-bits chunk of the product of x and 2/pi.
 *
 *	q0	the corresponding exponent of q[0]. Note that the
 *		exponent for q[i] would be q0-24*i.
 *
 *	PIo2[]	double precision array, obtained by cutting pi/2
 *		into 24 bits chunks.
 *
 *	f[]	ipio2[] in floating point
 *
 *	iq[]	integer array by breaking up q[] in 24-bits chunk.
 *
 *	fq[]	final product of x*(2/pi) in fq[0],..,fq[jk]
 *
 *	ih	integer. If >0 it indicates q[] is >= 0.5, hence
 *		it also indicates the *sign* of the result.
 *
 */


/*
 * Constants:
 * The hexadecimal values are the intended ones for the following
 * constants. The decimal values may be used, provided that the
 * compiler will convert from decimal to binary accurately enough
 * to produce the hexadecimal values shown.
 */

#include <math.h>
#include <math_private.h>

static const int init_jk[] = {2,3,4,6}; /* initial value for jk */

static const double PIo2[] = {
  1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */
  7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */
  5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */
  3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */
  1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */
  1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */
  2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */
  2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */
};

static const double
  zero   = 0.0,
  one    = 1.0,
  two24  = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
  twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */

int
__kernel_rem_pio2 (double *x, double *y, int e0, int nx, int prec,
                   const int32_t *ipio2)
{
  int32_t jz, jx, jv, jp, jk, carry, n, iq[20], i, j, k, m, q0, ih;
  double z, fw, f[20], fq[20], q[20];

  /* initialize jk*/
  jk = init_jk[prec];
  jp = jk;

  /* determine jx,jv,q0, note that 3>q0 */
  jx = nx - 1;
  jv = (e0 - 3) / 24; if (jv < 0)
    jv = 0;
  q0 = e0 - 24 * (jv + 1);

  /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
  j = jv - jx; m = jx + jk;
  for (i = 0; i <= m; i++, j++)
    f[i] = (j < 0) ? zero : (double) ipio2[j];

  /* compute q[0],q[1],...q[jk] */
  for (i = 0; i <= jk; i++)
    {
      for (j = 0, fw = 0.0; j <= jx; j++)
	fw += x[j] * f[jx + i - j];
      q[i] = fw;
    }

  jz = jk;
recompute:
  /* distill q[] into iq[] reversingly */
  for (i = 0, j = jz, z = q[jz]; j > 0; i++, j--)
    {
      fw = (double) ((int32_t) (twon24 * z));
      iq[i] = (int32_t) (z - two24 * fw);
      z = q[j - 1] + fw;
    }

  /* compute n */
  z = __scalbn (z, q0);                 /* actual value of z */
  z -= 8.0 * __floor (z * 0.125);               /* trim off integer >= 8 */
  n = (int32_t) z;
  z -= (double) n;
  ih = 0;
  if (q0 > 0)           /* need iq[jz-1] to determine n */
    {
      i = (iq[jz - 1] >> (24 - q0)); n += i;
      iq[jz - 1] -= i << (24 - q0);
      ih = iq[jz - 1] >> (23 - q0);
    }
  else if (q0 == 0)
    ih = iq[jz - 1] >> 23;
  else if (z >= 0.5)
    ih = 2;

  if (ih > 0)           /* q > 0.5 */
    {
      n += 1; carry = 0;
      for (i = 0; i < jz; i++)          /* compute 1-q */
	{
	  j = iq[i];
	  if (carry == 0)
	    {
	      if (j != 0)
		{
		  carry = 1; iq[i] = 0x1000000 - j;
		}
	    }
	  else
	    iq[i] = 0xffffff - j;
	}
      if (q0 > 0)               /* rare case: chance is 1 in 12 */
	{
	  switch (q0)
	    {
	    case 1:
	      iq[jz - 1] &= 0x7fffff; break;
	    case 2:
	      iq[jz - 1] &= 0x3fffff; break;
	    }
	}
      if (ih == 2)
	{
	  z = one - z;
	  if (carry != 0)
	    z -= __scalbn (one, q0);
	}
    }

  /* check if recomputation is needed */
  if (z == zero)
    {
      j = 0;
      for (i = jz - 1; i >= jk; i--)
	j |= iq[i];
      if (j == 0)      /* need recomputation */
	{
	  for (k = 1; iq[jk - k] == 0; k++)
	    ;                               /* k = no. of terms needed */

	  for (i = jz + 1; i <= jz + k; i++) /* add q[jz+1] to q[jz+k] */
	    {
	      f[jx + i] = (double) ipio2[jv + i];
	      for (j = 0, fw = 0.0; j <= jx; j++)
		fw += x[j] * f[jx + i - j];
	      q[i] = fw;
	    }
	  jz += k;
	  goto recompute;
	}
    }

  /* chop off zero terms */
  if (z == 0.0)
    {
      jz -= 1; q0 -= 24;
      while (iq[jz] == 0)
	{
	  jz--; q0 -= 24;
	}
    }
  else           /* break z into 24-bit if necessary */
    {
      z = __scalbn (z, -q0);
      if (z >= two24)
	{
	  fw = (double) ((int32_t) (twon24 * z));
	  iq[jz] = (int32_t) (z - two24 * fw);
	  jz += 1; q0 += 24;
	  iq[jz] = (int32_t) fw;
	}
      else
	iq[jz] = (int32_t) z;
    }

  /* convert integer "bit" chunk to floating-point value */
  fw = __scalbn (one, q0);
  for (i = jz; i >= 0; i--)
    {
      q[i] = fw * (double) iq[i]; fw *= twon24;
    }

  /* compute PIo2[0,...,jp]*q[jz,...,0] */
  for (i = jz; i >= 0; i--)
    {
      for (fw = 0.0, k = 0; k <= jp && k <= jz - i; k++)
	fw += PIo2[k] * q[i + k];
      fq[jz - i] = fw;
    }

  /* compress fq[] into y[] */
  switch (prec)
    {
    case 0:
      fw = 0.0;
      for (i = jz; i >= 0; i--)
	fw += fq[i];
      y[0] = (ih == 0) ? fw : -fw;
      break;
    case 1:
    case 2:;
#if __FLT_EVAL_METHOD__ != 0
      volatile
#endif
      double fv = 0.0;
      for (i = jz; i >= 0; i--)
	fv += fq[i];
      y[0] = (ih == 0) ? fv : -fv;
      fv = fq[0] - fv;
      for (i = 1; i <= jz; i++)
	fv += fq[i];
      y[1] = (ih == 0) ? fv : -fv;
      break;
    case 3:             /* painful */
      for (i = jz; i > 0; i--)
	{
#if __FLT_EVAL_METHOD__ != 0
	  volatile
#endif
	  double fv = (double) (fq[i - 1] + fq[i]);
	  fq[i] += fq[i - 1] - fv;
	  fq[i - 1] = fv;
	}
      for (i = jz; i > 1; i--)
	{
#if __FLT_EVAL_METHOD__ != 0
	  volatile
#endif
	  double fv = (double) (fq[i - 1] + fq[i]);
	  fq[i] += fq[i - 1] - fv;
	  fq[i - 1] = fv;
	}
      for (fw = 0.0, i = jz; i >= 2; i--)
	fw += fq[i];
      if (ih == 0)
	{
	  y[0] = fq[0]; y[1] = fq[1]; y[2] = fw;
	}
      else
	{
	  y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw;
	}
    }
  return n & 7;
}