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
|
/* mpfr_tanu -- tanu(x) = tan(2*pi*x/u)
mpfr_tanpi -- tanpi(x) = tan(pi*x)
Copyright 2020-2021 Free Software Foundation, Inc.
Contributed by the AriC and Caramba 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
https://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"
/* put in y the correctly rounded value of tan(2*pi*x/u) */
int
mpfr_tanu (mpfr_ptr y, mpfr_srcptr x, unsigned long u, mpfr_rnd_t rnd_mode)
{
mpfr_srcptr xp;
mpfr_prec_t precy, prec;
mpfr_exp_t expx, expt, err;
mpfr_t t, xr;
int inexact = 0, nloops = 0, underflow = 0;
MPFR_ZIV_DECL (loop);
MPFR_SAVE_EXPO_DECL (expo);
MPFR_LOG_FUNC (
("x[%Pu]=%.*Rg u=%lu rnd=%d", mpfr_get_prec (x), mpfr_log_prec, x, u,
rnd_mode),
("y[%Pu]=%.*Rg inexact=%d", mpfr_get_prec (y), mpfr_log_prec, y,
inexact));
if (u == 0 || MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
{
/* for u=0, return NaN */
if (u == 0 || MPFR_IS_NAN (x) || MPFR_IS_INF (x))
{
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
else /* x is zero */
{
MPFR_ASSERTD (MPFR_IS_ZERO (x));
MPFR_SET_ZERO (y);
MPFR_SET_SAME_SIGN (y, x);
MPFR_RET (0);
}
}
MPFR_SAVE_EXPO_MARK (expo);
/* Range reduction. We do not need to reduce the argument if it is
already reduced (|x| < u).
Note that the case |x| = u is better in the "else" branch as it
will give xr = 0. */
if (mpfr_cmpabs_ui (x, u) < 0)
{
xp = x;
}
else
{
mpfr_exp_t p = MPFR_GET_PREC (x) - MPFR_GET_EXP (x);
int inex;
/* Let's compute xr = x mod u, with signbit(xr) = signbit(x), which
may be important when x is a multiple of u, in which case xr = 0
(but this property is actually not needed in the code below).
The precision of xr is chosen to ensure that x mod u is exactly
representable in xr, e.g., the maximum size of u + the length of
the fractional part of x. Note that since |x| >= u in this branch,
the additional memory amount will not be more than the one of x.
Note that due to the rules on the special values, we needed to
consider a period of u instead of u/2. */
mpfr_init2 (xr, sizeof (unsigned long) * CHAR_BIT + (p < 0 ? 0 : p));
MPFR_DBGRES (inex = mpfr_fmod_ui (xr, x, u, MPFR_RNDN)); /* exact */
MPFR_ASSERTD (inex == 0);
if (MPFR_IS_ZERO (xr))
{
mpfr_clear (xr);
MPFR_SAVE_EXPO_FREE (expo);
MPFR_SET_ZERO (y);
MPFR_SET_SAME_SIGN (y, x);
MPFR_RET (0);
}
xp = xr;
}
/* now |xp/u| < 1 */
precy = MPFR_GET_PREC (y);
expx = MPFR_GET_EXP (xp);
/* For x large, since argument reduction is expensive, we want to avoid
any failure in Ziv's strategy, thus we take into account expx too. */
prec = precy + MAX(expx,MPFR_INT_CEIL_LOG2(precy)) + 8;
MPFR_ASSERTD(prec >= 2);
mpfr_init2 (t, prec);
MPFR_ZIV_INIT (loop, prec);
for (;;)
{
int inex;
nloops ++;
/* In the error analysis below, xp stands for x.
We first compute an approximation t of 2*pi*x/u, then call tan(t).
If t = 2*pi*x/u + s, then
|tan(t) - tan(2*pi*x/u)| = |s| * (1 + tan(v)^2) where v is in the
interval [t, t+s]. If we ensure that |t| >= |2*pi*x/u|, since tan() is
increasing, we can bound tan(v)^2 by tan(t)^2. */
mpfr_set_prec (t, prec);
mpfr_const_pi (t, MPFR_RNDU); /* t = pi * (1 + theta1) where
|theta1| <= 2^(1-prec) */
mpfr_mul_2ui (t, t, 1, MPFR_RNDN); /* t = 2*pi * (1 + theta1) */
mpfr_mul (t, t, xp, MPFR_RNDA); /* t = 2*pi*x * (1 + theta2)^2 where
|theta2| <= 2^(1-prec) */
inex = mpfr_div_ui (t, t, u, MPFR_RNDN);
/* t = 2*pi*x/u * (1 + theta3)^3 where |theta3| <= 2^(1-prec) */
/* if t is zero here, it means the division by u underflows, then
tan(t) also underflows, since |tan(x)| <= |x|. */
if (MPFR_UNLIKELY (MPFR_IS_ZERO (t)))
{
inexact = mpfr_underflow (y, rnd_mode, MPFR_SIGN(t));
MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, MPFR_FLAGS_INEXACT
| MPFR_FLAGS_UNDERFLOW);
underflow = 1;
goto end;
}
/* emulate mpfr_div_ui (t, t, u, MPFR_RNDA) above, so that t is rounded
away from zero */
if (MPFR_SIGN(t) > 0 && inex < 0)
mpfr_nextabove (t);
else if (MPFR_SIGN(t) < 0 && inex > 0)
mpfr_nextbelow (t);
expt = MPFR_GET_EXP (t);
/* since prec >= 3, |(1 + theta3)^3 - 1| <= 4*theta3 <= 2^(3-prec)
thus |s| = |t - 2*pi*x/u| <= |t| * 2^(3-prec) */
mpfr_tan (t, t, MPFR_RNDA);
{
/* compute an upper bound for 1+tan(t)^2 */
mpfr_t z;
mpfr_init2 (z, 64);
mpfr_sqr (z, t, MPFR_RNDU);
mpfr_add_ui (z, z, 1, MPFR_RNDU);
expt += MPFR_GET_EXP (z);
/* now |t - tan(2*pi*x/u)| <= ulp(t) + 2^(expt + 3 - prec) */
mpfr_clear (z);
}
/* t cannot be zero here, since we excluded t=0 before, which is the
only exact case where tan(t)=0, and we round away from zero */
err = expt + 3 - prec;
expt = MPFR_GET_EXP (t); /* new exponent of t */
/* the total error is bounded by 2^err + ulp(t) = 2^err + 2^(expt-prec)
thus if err <= expt-prec, it is bounded by 2^(expt-prec+1),
otherwise it is bounded by 2^(err+1). */
err = (err <= expt - prec) ? expt - prec + 1 : err + 1;
/* normalize err for mpfr_can_round */
err = expt - err;
if (MPFR_CAN_ROUND (t, err, precy, rnd_mode))
break;
/* Check exact cases only after the first level of Ziv' strategy, to
avoid slowing down the average case. Exact cases are when 2*pi*x/u
is a multiple of pi/4, i.e., x/u a multiple of 1/8:
(a) x/u = {0,1/2} mod 1: return +0 or -0
(b) x/u = {1/4,3/4} mod 1: return +Inf or -Inf
(c) x/u = {1/8,3/8,5/8,7/8} mod 1: return 1 or -1 */
if (nloops == 1)
{
inexact = mpfr_div_ui (t, xp, u, MPFR_RNDA);
mpfr_mul_2ui (t, t, 3, MPFR_RNDA);
if (inexact == 0 && mpfr_integer_p (t))
{
mpz_t z;
unsigned long mod8;
mpz_init (z);
inexact = mpfr_get_z (z, t, MPFR_RNDZ);
MPFR_ASSERTN(inexact == 0);
mod8 = mpz_fdiv_ui (z, 8);
mpz_clear (z);
if (mod8 == 0 || mod8 == 4) /* case (a) */
mpfr_set_zero (y, ((mod8 == 0) ? +1 : -1) * MPFR_SIGN (x));
else if (mod8 == 2 || mod8 == 6) /* case (b) */
{
mpfr_set_inf (y, (mod8 == 2) ? +1 : -1);
MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, MPFR_FLAGS_DIVBY0);
}
else /* case (c) */
{
if (mod8 == 1 || mod8 == 5)
mpfr_set_ui (y, 1, rnd_mode);
else
mpfr_set_si (y, -1, rnd_mode);
}
goto end;
}
}
MPFR_ZIV_NEXT (loop, prec);
}
MPFR_ZIV_FREE (loop);
inexact = mpfr_set (y, t, rnd_mode);
end:
mpfr_clear (t);
if (xp != x)
{
MPFR_ASSERTD (xp == xr);
mpfr_clear (xr);
}
MPFR_SAVE_EXPO_FREE (expo);
return underflow ? inexact : mpfr_check_range (y, inexact, rnd_mode);
}
int
mpfr_tanpi (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
{
return mpfr_tanu (y, x, 2, rnd_mode);
}
|
