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
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
|
/* mpfr_atan -- arc-tangent of a floating-point number
Copyright 2001-2016 Free Software Foundation, Inc.
Contributed by the AriC and Caramel projects, INRIA.
This file is part of the GNU MPFR Library.
The GNU MPFR Library is free software; you can redistribute it and/or modify
it under the terms of the GNU Lesser General Public License as published by
the Free Software Foundation; either version 3 of the License, or (at your
option) any later version.
The GNU MPFR Library is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public
License for more details.
You should have received a copy of the GNU Lesser General Public License
along with the GNU MPFR Library; see the file COPYING.LESSER. If not, see
http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */
#define MPFR_NEED_LONGLONG_H
#include "mpfr-impl.h"
/* If x = p/2^r, put in y an approximation of atan(x)/x using 2^m terms
for the series expansion, with an error of at most 1 ulp.
Assumes |x| < 1.
If X=x^2, we want 1 - X/3 + X^2/5 - ... + (-1)^k*X^k/(2k+1) + ...
Assume p is non-zero.
When we sum terms up to x^k/(2k+1), the denominator Q[0] is
3*5*7*...*(2k+1) ~ (2k/e)^k.
*/
static void
mpfr_atan_aux (mpfr_ptr y, mpz_ptr p, long r, int m, mpz_t *tab)
{
mpz_t *S, *Q, *ptoj;
unsigned long n, i, k, j, l;
mpfr_exp_t diff, expo;
int im, done;
mpfr_prec_t mult, *accu, *log2_nb_terms;
mpfr_prec_t precy = MPFR_PREC(y);
MPFR_ASSERTD(mpz_cmp_ui (p, 0) != 0);
accu = (mpfr_prec_t*) (*__gmp_allocate_func) ((2 * m + 2) * sizeof (mpfr_prec_t));
log2_nb_terms = accu + m + 1;
/* Set Tables */
S = tab; /* S */
ptoj = S + 1*(m+1); /* p^2^j Precomputed table */
Q = S + 2*(m+1); /* Product of Odd integer table */
/* From p to p^2, and r to 2r */
mpz_mul (p, p, p);
MPFR_ASSERTD (2 * r > r);
r = 2 * r;
/* Normalize p */
n = mpz_scan1 (p, 0);
mpz_tdiv_q_2exp (p, p, n); /* exact */
MPFR_ASSERTD (r > n);
r -= n;
/* since |p/2^r| < 1, and p is a non-zero integer, necessarily r > 0 */
MPFR_ASSERTD (mpz_sgn (p) > 0);
MPFR_ASSERTD (m > 0);
/* check if p=1 (special case) */
l = 0;
/*
We compute by binary splitting, with X = x^2 = p/2^r:
P(a,b) = p if a+1=b, P(a,c)*P(c,b) otherwise
Q(a,b) = (2a+1)*2^r if a+1=b [except Q(0,1)=1], Q(a,c)*Q(c,b) otherwise
S(a,b) = p*(2a+1) if a+1=b, Q(c,b)*S(a,c)+Q(a,c)*P(a,c)*S(c,b) otherwise
Then atan(x)/x ~ S(0,i)/Q(0,i) for i so that (p/2^r)^i/i is small enough.
The factor 2^(r*(b-a)) in Q(a,b) is implicit, thus we have to take it
into account when we compute with Q.
*/
accu[0] = 0; /* accu[k] = Mult[0] + ... + Mult[k], where Mult[j] is the
number of bits of the corresponding term S[j]/Q[j] */
if (mpz_cmp_ui (p, 1) != 0)
{
/* p <> 1: precompute ptoj table */
mpz_set (ptoj[0], p);
for (im = 1 ; im <= m ; im ++)
mpz_mul (ptoj[im], ptoj[im - 1], ptoj[im - 1]);
/* main loop */
n = 1UL << m;
/* the ith term being X^i/(2i+1) with X=p/2^r, we can stop when
p^i/2^(r*i) < 2^(-precy), i.e. r*i > precy + log2(p^i) */
for (i = k = done = 0; (i < n) && (done == 0); i += 2, k ++)
{
/* initialize both S[k],Q[k] and S[k+1],Q[k+1] */
mpz_set_ui (Q[k+1], 2 * i + 3); /* Q(i+1,i+2) */
mpz_mul_ui (S[k+1], p, 2 * i + 1); /* S(i+1,i+2) */
mpz_mul_2exp (S[k], Q[k+1], r);
mpz_sub (S[k], S[k], S[k+1]); /* S(i,i+2) */
mpz_mul_ui (Q[k], Q[k+1], 2 * i + 1); /* Q(i,i+2) */
log2_nb_terms[k] = 1; /* S[k]/Q[k] corresponds to 2 terms */
for (j = (i + 2) >> 1, l = 1; (j & 1) == 0; l ++, j >>= 1, k --)
{
/* invariant: S[k-1]/Q[k-1] and S[k]/Q[k] correspond
to 2^l terms each. We combine them into S[k-1]/Q[k-1] */
MPFR_ASSERTD (k > 0);
mpz_mul (S[k], S[k], Q[k-1]);
mpz_mul (S[k], S[k], ptoj[l]);
mpz_mul (S[k-1], S[k-1], Q[k]);
mpz_mul_2exp (S[k-1], S[k-1], r << l);
mpz_add (S[k-1], S[k-1], S[k]);
mpz_mul (Q[k-1], Q[k-1], Q[k]);
log2_nb_terms[k-1] = l + 1;
/* now S[k-1]/Q[k-1] corresponds to 2^(l+1) terms */
MPFR_MPZ_SIZEINBASE2(mult, ptoj[l+1]);
/* FIXME: precompute bits(ptoj[l+1]) outside the loop? */
mult = (r << (l + 1)) - mult - 1;
accu[k-1] = (k == 1) ? mult : accu[k-2] + mult;
if (accu[k-1] > precy)
done = 1;
}
}
}
else /* special case p=1: the ith term being X^i/(2i+1) with X=1/2^r,
we can stop when r*i > precy i.e. i > precy/r */
{
n = 1UL << m;
for (i = k = 0; (i < n) && (i <= precy / r); i += 2, k ++)
{
mpz_set_ui (Q[k + 1], 2 * i + 3);
mpz_mul_2exp (S[k], Q[k+1], r);
mpz_sub_ui (S[k], S[k], 1 + 2 * i);
mpz_mul_ui (Q[k], Q[k + 1], 1 + 2 * i);
log2_nb_terms[k] = 1; /* S[k]/Q[k] corresponds to 2 terms */
for (j = (i + 2) >> 1, l = 1; (j & 1) == 0; l++, j >>= 1, k --)
{
MPFR_ASSERTD (k > 0);
mpz_mul (S[k], S[k], Q[k-1]);
mpz_mul (S[k-1], S[k-1], Q[k]);
mpz_mul_2exp (S[k-1], S[k-1], r << l);
mpz_add (S[k-1], S[k-1], S[k]);
mpz_mul (Q[k-1], Q[k-1], Q[k]);
log2_nb_terms[k-1] = l + 1;
}
}
}
/* we need to combine S[0]/Q[0]...S[k-1]/Q[k-1] */
l = 0; /* number of terms accumulated in S[k]/Q[k] */
while (k > 1)
{
k --;
/* combine S[k-1]/Q[k-1] and S[k]/Q[k] */
j = log2_nb_terms[k-1];
mpz_mul (S[k], S[k], Q[k-1]);
if (mpz_cmp_ui (p, 1) != 0)
mpz_mul (S[k], S[k], ptoj[j]);
mpz_mul (S[k-1], S[k-1], Q[k]);
l += 1 << log2_nb_terms[k];
mpz_mul_2exp (S[k-1], S[k-1], r * l);
mpz_add (S[k-1], S[k-1], S[k]);
mpz_mul (Q[k-1], Q[k-1], Q[k]);
}
(*__gmp_free_func) (accu, (2 * m + 2) * sizeof (mpfr_prec_t));
MPFR_MPZ_SIZEINBASE2 (diff, S[0]);
diff -= 2 * precy;
expo = diff;
if (diff >= 0)
mpz_tdiv_q_2exp (S[0], S[0], diff);
else
mpz_mul_2exp (S[0], S[0], -diff);
MPFR_MPZ_SIZEINBASE2 (diff, Q[0]);
diff -= precy;
expo -= diff;
if (diff >= 0)
mpz_tdiv_q_2exp (Q[0], Q[0], diff);
else
mpz_mul_2exp (Q[0], Q[0], -diff);
mpz_tdiv_q (S[0], S[0], Q[0]);
mpfr_set_z (y, S[0], MPFR_RNDD);
MPFR_SET_EXP (y, MPFR_EXP(y) + expo - r * (i - 1));
}
int
mpfr_atan (mpfr_ptr atan, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
{
mpfr_t xp, arctgt, sk, tmp, tmp2;
mpz_t ukz;
mpz_t *tabz;
mpfr_exp_t exptol;
mpfr_prec_t prec, realprec, est_lost, lost;
unsigned long twopoweri, log2p, red;
int comparaison, inexact;
int i, n0, oldn0;
MPFR_GROUP_DECL (group);
MPFR_SAVE_EXPO_DECL (expo);
MPFR_ZIV_DECL (loop);
MPFR_LOG_FUNC
(("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec (x), mpfr_log_prec, x, rnd_mode),
("atan[%Pu]=%.*Rg inexact=%d",
mpfr_get_prec (atan), mpfr_log_prec, atan, inexact));
/* Singular cases */
if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
{
if (MPFR_IS_NAN (x))
{
MPFR_SET_NAN (atan);
MPFR_RET_NAN;
}
else if (MPFR_IS_INF (x))
{
MPFR_SAVE_EXPO_MARK (expo);
if (MPFR_IS_POS (x)) /* arctan(+inf) = Pi/2 */
inexact = mpfr_const_pi (atan, rnd_mode);
else /* arctan(-inf) = -Pi/2 */
{
inexact = -mpfr_const_pi (atan,
MPFR_INVERT_RND (rnd_mode));
MPFR_CHANGE_SIGN (atan);
}
mpfr_div_2ui (atan, atan, 1, rnd_mode); /* exact (no exceptions) */
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (atan, inexact, rnd_mode);
}
else /* x is necessarily 0 */
{
MPFR_ASSERTD (MPFR_IS_ZERO (x));
MPFR_SET_ZERO (atan);
MPFR_SET_SAME_SIGN (atan, x);
MPFR_RET (0);
}
}
/* atan(x) = x - x^3/3 + x^5/5...
so the error is < 2^(3*EXP(x)-1)
so `EXP(x)-(3*EXP(x)-1)` = -2*EXP(x)+1 */
MPFR_FAST_COMPUTE_IF_SMALL_INPUT (atan, x, -2 * MPFR_GET_EXP (x), 1, 0,
rnd_mode, {});
/* Set x_p=|x| */
MPFR_TMP_INIT_ABS (xp, x);
MPFR_SAVE_EXPO_MARK (expo);
/* Other simple case arctan(-+1)=-+pi/4 */
comparaison = mpfr_cmp_ui (xp, 1);
if (MPFR_UNLIKELY (comparaison == 0))
{
int neg = MPFR_IS_NEG (x);
inexact = mpfr_const_pi (atan, MPFR_IS_POS (x) ? rnd_mode
: MPFR_INVERT_RND (rnd_mode));
if (neg)
{
inexact = -inexact;
MPFR_CHANGE_SIGN (atan);
}
mpfr_div_2ui (atan, atan, 2, rnd_mode); /* exact (no exceptions) */
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (atan, inexact, rnd_mode);
}
realprec = MPFR_PREC (atan) + MPFR_INT_CEIL_LOG2 (MPFR_PREC (atan)) + 4;
prec = realprec + GMP_NUMB_BITS;
/* Initialisation */
mpz_init (ukz);
MPFR_GROUP_INIT_4 (group, prec, sk, tmp, tmp2, arctgt);
oldn0 = 0;
tabz = (mpz_t *) 0;
MPFR_ZIV_INIT (loop, prec);
for (;;)
{
/* First, if |x| < 1, we need to have more prec to be able to round (sup)
n0 = ceil(log(prec_requested + 2 + 1+ln(2.4)/ln(2))/log(2)) */
mpfr_prec_t sup;
sup = MPFR_GET_EXP (xp) < 0 ? 2 - MPFR_GET_EXP (xp) : 1; /* sup >= 1 */
n0 = MPFR_INT_CEIL_LOG2 ((realprec + sup) + 3);
/* since realprec >= 4, n0 >= ceil(log2(8)) >= 3, thus 3*n0 > 2 */
prec = (realprec + sup) + 1 + MPFR_INT_CEIL_LOG2 (3*n0-2);
/* the number of lost bits due to argument reduction is
9 - 2 * EXP(sk), which we estimate by 9 + 2*ceil(log2(p))
since we manage that sk < 1/p */
if (MPFR_PREC (atan) > 100)
{
log2p = MPFR_INT_CEIL_LOG2(prec) / 2 - 3;
est_lost = 9 + 2 * log2p;
prec += est_lost;
}
else
log2p = est_lost = 0; /* don't reduce the argument */
/* Initialisation */
MPFR_GROUP_REPREC_4 (group, prec, sk, tmp, tmp2, arctgt);
if (MPFR_LIKELY (oldn0 == 0))
{
oldn0 = 3 * (n0 + 1);
tabz = (mpz_t *) (*__gmp_allocate_func) (oldn0 * sizeof (mpz_t));
for (i = 0; i < oldn0; i++)
mpz_init (tabz[i]);
}
else if (MPFR_UNLIKELY (oldn0 < 3 * (n0 + 1)))
{
tabz = (mpz_t *) (*__gmp_reallocate_func)
(tabz, oldn0 * sizeof (mpz_t), 3 * (n0 + 1)*sizeof (mpz_t));
for (i = oldn0; i < 3 * (n0 + 1); i++)
mpz_init (tabz[i]);
oldn0 = 3 * (n0 + 1);
}
/* The mpfr_ui_div below mustn't underflow. This is guaranteed by
MPFR_SAVE_EXPO_MARK, but let's check that for maintainability. */
MPFR_ASSERTD (__gmpfr_emax <= 1 - __gmpfr_emin);
if (comparaison > 0) /* use atan(xp) = Pi/2 - atan(1/xp) */
mpfr_ui_div (sk, 1, xp, MPFR_RNDN);
else
mpfr_set (sk, xp, MPFR_RNDN);
/* now 0 < sk <= 1 */
/* Argument reduction: atan(x) = 2 atan((sqrt(1+x^2)-1)/x).
We want |sk| < k/sqrt(p) where p is the target precision. */
lost = 0;
for (red = 0; MPFR_GET_EXP(sk) > - (mpfr_exp_t) log2p; red ++)
{
lost = 9 - 2 * MPFR_EXP(sk);
mpfr_mul (tmp, sk, sk, MPFR_RNDN);
mpfr_add_ui (tmp, tmp, 1, MPFR_RNDN);
mpfr_sqrt (tmp, tmp, MPFR_RNDN);
mpfr_sub_ui (tmp, tmp, 1, MPFR_RNDN);
if (red == 0 && comparaison > 0)
/* use xp = 1/sk */
mpfr_mul (sk, tmp, xp, MPFR_RNDN);
else
mpfr_div (sk, tmp, sk, MPFR_RNDN);
}
/* we started from x0 = 1/|x| if |x| > 1, and |x| otherwise, thus
we had x0 = min(|x|, 1/|x|) <= 1, and applied 'red' times the
argument reduction x -> (sqrt(1+x^2)-1)/x, which keeps 0 < x < 1,
thus 0 < sk <= 1, and sk=1 can occur only if red=0 */
/* If sk=1, then if |x| < 1, we have 1 - 2^(-prec-1) <= |x| < 1,
or if |x| > 1, we have 1 - 2^(-prec-1) <= 1/|x| < 1, thus in all
cases ||x| - 1| <= 2^(-prec), from which it follows
|atan|x| - Pi/4| <= 2^(-prec), given the Taylor expansion
atan(1+x) = Pi/4 + x/2 - x^2/4 + ...
Since Pi/4 = 0.785..., the error is at most one ulp.
*/
if (MPFR_UNLIKELY(mpfr_cmp_ui (sk, 1) == 0))
{
mpfr_const_pi (arctgt, MPFR_RNDN); /* 1/2 ulp extra error */
mpfr_div_2ui (arctgt, arctgt, 2, MPFR_RNDN); /* exact */
realprec = prec - 2;
goto can_round;
}
/* Assignation */
MPFR_SET_ZERO (arctgt);
twopoweri = 1 << 0;
MPFR_ASSERTD (n0 >= 4);
for (i = 0 ; i < n0; i++)
{
if (MPFR_UNLIKELY (MPFR_IS_ZERO (sk)))
break;
/* Calculation of trunc(tmp) --> mpz */
mpfr_mul_2ui (tmp, sk, twopoweri, MPFR_RNDN);
mpfr_trunc (tmp, tmp);
if (!MPFR_IS_ZERO (tmp))
{
/* tmp = ukz*2^exptol */
exptol = mpfr_get_z_2exp (ukz, tmp);
/* since the s_k are decreasing (see algorithms.tex),
and s_0 = min(|x|, 1/|x|) < 1, we have sk < 1,
thus exptol < 0 */
MPFR_ASSERTD (exptol < 0);
mpz_tdiv_q_2exp (ukz, ukz, (unsigned long int) (-exptol));
/* since tmp is a non-zero integer, and tmp = ukzold*2^exptol,
we now have ukz = tmp, thus ukz is non-zero */
/* Calculation of arctan(Ak) */
mpfr_set_z (tmp, ukz, MPFR_RNDN);
mpfr_div_2ui (tmp, tmp, twopoweri, MPFR_RNDN);
mpfr_atan_aux (tmp2, ukz, twopoweri, n0 - i, tabz);
mpfr_mul (tmp2, tmp2, tmp, MPFR_RNDN);
/* Addition */
mpfr_add (arctgt, arctgt, tmp2, MPFR_RNDN);
/* Next iteration */
mpfr_sub (tmp2, sk, tmp, MPFR_RNDN);
mpfr_mul (sk, sk, tmp, MPFR_RNDN);
mpfr_add_ui (sk, sk, 1, MPFR_RNDN);
mpfr_div (sk, tmp2, sk, MPFR_RNDN);
}
twopoweri <<= 1;
}
/* Add last step (Arctan(sk) ~= sk */
mpfr_add (arctgt, arctgt, sk, MPFR_RNDN);
/* argument reduction */
mpfr_mul_2exp (arctgt, arctgt, red, MPFR_RNDN);
if (comparaison > 0)
{ /* atan(x) = Pi/2-atan(1/x) for x > 0 */
mpfr_const_pi (tmp, MPFR_RNDN);
mpfr_div_2ui (tmp, tmp, 1, MPFR_RNDN);
mpfr_sub (arctgt, tmp, arctgt, MPFR_RNDN);
}
MPFR_SET_POS (arctgt);
can_round:
if (MPFR_LIKELY (MPFR_CAN_ROUND (arctgt, realprec + est_lost - lost,
MPFR_PREC (atan), rnd_mode)))
break;
MPFR_ZIV_NEXT (loop, realprec);
}
MPFR_ZIV_FREE (loop);
inexact = mpfr_set4 (atan, arctgt, rnd_mode, MPFR_SIGN (x));
for (i = 0 ; i < oldn0 ; i++)
mpz_clear (tabz[i]);
mpz_clear (ukz);
(*__gmp_free_func) (tabz, oldn0 * sizeof (mpz_t));
MPFR_GROUP_CLEAR (group);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (atan, inexact, rnd_mode);
}
|