1 /* mpfr_sin -- sine of a floating-point number
2
3 Copyright 2001-2023 Free Software Foundation, Inc.
4 Contributed by the AriC and Caramba projects, INRIA.
5
6 This file is part of the GNU MPFR Library.
7
8 The GNU MPFR Library is free software; you can redistribute it and/or modify
9 it under the terms of the GNU Lesser General Public License as published by
10 the Free Software Foundation; either version 3 of the License, or (at your
11 option) any later version.
12
13 The GNU MPFR Library is distributed in the hope that it will be useful, but
14 WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
15 or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public
16 License for more details.
17
18 You should have received a copy of the GNU Lesser General Public License
19 along with the GNU MPFR Library; see the file COPYING.LESSER. If not, see
20 https://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
21 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */
22
23 #define MPFR_NEED_LONGLONG_H
24 #include "mpfr-impl.h"
25
26 static int
27 mpfr_sin_fast (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
28 {
29 int inex;
30
31 inex = mpfr_sincos_fast (y, NULL, x, rnd_mode);
32 inex = inex & 3; /* 0: exact, 1: rounded up, 2: rounded down */
33 return (inex == 2) ? -1 : inex;
34 }
35
36 int
37 mpfr_sin (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
38 {
39 mpfr_t c, xr;
40 mpfr_srcptr xx;
41 mpfr_exp_t expx, err1, err;
42 mpfr_prec_t precy, m;
43 int inexact, sign, reduce;
44 MPFR_ZIV_DECL (loop);
45 MPFR_SAVE_EXPO_DECL (expo);
46
47 MPFR_LOG_FUNC
48 (("x[%Pd]=%.*Rg rnd=%d", mpfr_get_prec (x), mpfr_log_prec, x, rnd_mode),
49 ("y[%Pd]=%.*Rg inexact=%d", mpfr_get_prec (y), mpfr_log_prec, y,
50 inexact));
51
52 if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
53 {
54 if (MPFR_IS_NAN (x) || MPFR_IS_INF (x))
55 {
56 MPFR_SET_NAN (y);
57 MPFR_RET_NAN;
58 }
59 else /* x is zero */
60 {
61 MPFR_ASSERTD (MPFR_IS_ZERO (x));
62 MPFR_SET_ZERO (y);
63 MPFR_SET_SAME_SIGN (y, x);
64 MPFR_RET (0);
65 }
66 }
67
68 expx = MPFR_GET_EXP (x);
69 err1 = -2 * expx;
70
71 /* sin(x) = x - x^3/6 + ... so the error is < 2^(3*EXP(x)-2) */
72 MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, x, err1, 2, 0, rnd_mode, {});
73
74 MPFR_SAVE_EXPO_MARK (expo);
75
76 /* Compute initial precision */
77 precy = MPFR_PREC (y);
78
79 if (precy >= MPFR_SINCOS_THRESHOLD)
80 {
81 inexact = mpfr_sin_fast (y, x, rnd_mode);
82 goto end;
83 }
84
85 /* for x large, since argument reduction is expensive, we want to avoid
86 any failure in Ziv's strategy, thus we take into account expx too */
87 m = precy + MPFR_INT_CEIL_LOG2 (MAX(precy,expx)) + 8;
88
89 /* since we compute sin(x) as sqrt(1-cos(x)^2), and for x small we have
90 cos(x)^2 ~ 1 - x^2, when subtracting cos(x)^2 from 1 we will lose
91 about -2*expx bits if expx < 0 */
92 if (expx < 0)
93 {
94 /* The following assertion includes a check for integer overflow.
95 At this point, precy < MPFR_SINCOS_THRESHOLD, so that both m and
96 err1 should be small enough. But the assertion makes the code
97 safer (a smart compiler might be able to remove it). */
98 MPFR_ASSERTN (err1 <= MPFR_PREC_MAX - m);
99 m += err1;
100 }
101
102 if (expx >= 2)
103 {
104 mpfr_init2 (c, expx + m - 1);
105 mpfr_init2 (xr, m);
106 }
107 else
108 mpfr_init2 (c, m);
109
110 MPFR_ZIV_INIT (loop, m);
111 for (;;)
112 {
113 /* first perform argument reduction modulo 2*Pi (if needed),
114 also helps to determine the sign of sin(x) */
115 /* TODO: Perform range reduction in a way so that the sine can
116 be computed directly from the cosine with sin(x)=cos(pi/2-x),
117 without the need of sqrt(1 - x^2). */
118 if (expx >= 2) /* If Pi < x < 4, we need to reduce too, to determine
119 the sign of sin(x). For 2 <= |x| < Pi, we could avoid
120 the reduction. */
121 {
122 reduce = 1;
123 /* As expx + m - 1 will silently be converted into mpfr_prec_t
124 in the mpfr_set_prec call, the assert below may be useful to
125 avoid undefined behavior. */
126 MPFR_ASSERTN (expx + m - 1 <= MPFR_PREC_MAX);
127 mpfr_set_prec (c, expx + m - 1);
128 mpfr_set_prec (xr, m);
129 mpfr_const_pi (c, MPFR_RNDN);
130 mpfr_mul_2ui (c, c, 1, MPFR_RNDN);
131 mpfr_remainder (xr, x, c, MPFR_RNDN);
132 /* The analysis is similar to that of cos.c:
133 |xr - x - 2kPi| <= 2^(2-m). Thus we can decide the sign
134 of sin(x) if xr is at distance at least 2^(2-m) of both
135 0 and +/-Pi. */
136 mpfr_div_2ui (c, c, 1, MPFR_RNDN);
137 /* Since c approximates Pi with an error <= 2^(2-expx-m) <= 2^(-m),
138 it suffices to check that c - |xr| >= 2^(2-m). */
139 if (MPFR_IS_POS (xr))
140 mpfr_sub (c, c, xr, MPFR_RNDZ);
141 else
142 mpfr_add (c, c, xr, MPFR_RNDZ);
143 if (MPFR_IS_ZERO(xr)
144 || MPFR_GET_EXP(xr) < (mpfr_exp_t) 3 - (mpfr_exp_t) m
145 || MPFR_IS_ZERO(c)
146 || MPFR_GET_EXP(c) < (mpfr_exp_t) 3 - (mpfr_exp_t) m)
147 goto ziv_next;
148
149 /* |xr - x - 2kPi| <= 2^(2-m), thus |sin(xr) - sin(x)| <= 2^(2-m) */
150 xx = xr;
151 }
152 else /* the input argument is already reduced */
153 {
154 reduce = 0;
155 xx = x;
156 }
157
158 sign = MPFR_SIGN(xx);
159 /* now that the argument is reduced, precision m is enough */
160 mpfr_set_prec (c, m);
161 mpfr_cos (c, xx, MPFR_RNDA); /* c = cos(x) rounded away */
162 mpfr_sqr (c, c, MPFR_RNDU); /* away */
163 mpfr_ui_sub (c, 1, c, MPFR_RNDZ);
164 mpfr_sqrt (c, c, MPFR_RNDZ);
165 if (MPFR_IS_NEG_SIGN(sign))
166 MPFR_CHANGE_SIGN(c);
167
168 /* Warning: c may be 0! */
169 if (MPFR_UNLIKELY (MPFR_IS_ZERO (c)))
170 {
171 /* Huge cancellation: increase prec a lot! */
172 m = MAX (m, MPFR_PREC (x));
173 m = 2 * m;
174 }
175 else
176 {
177 /* the absolute error on c is at most 2^(3-m-EXP(c)),
178 plus 2^(2-m) if there was an argument reduction.
179 Since EXP(c) <= 1, 3-m-EXP(c) >= 2-m, thus the error
180 is at most 2^(3-m-EXP(c)) in case of argument reduction. */
181 err = 2 * MPFR_GET_EXP (c) + (mpfr_exp_t) m - 3 - (reduce != 0);
182 if (MPFR_CAN_ROUND (c, err, precy, rnd_mode))
183 break;
184
185 /* check for huge cancellation (Near 0) */
186 if (err < (mpfr_exp_t) MPFR_PREC (y))
187 m += MPFR_PREC (y) - err;
188 /* Check if near 1 */
189 if (MPFR_GET_EXP (c) == 1)
190 m += m;
191 }
192
193 ziv_next:
194 /* Else generic increase */
195 MPFR_ZIV_NEXT (loop, m);
196 }
197 MPFR_ZIV_FREE (loop);
198
199 inexact = mpfr_set (y, c, rnd_mode);
200 /* inexact cannot be 0, since this would mean that c was representable
201 within the target precision, but in that case mpfr_can_round will fail */
202
203 mpfr_clear (c);
204 if (expx >= 2)
205 mpfr_clear (xr);
206
207 end:
208 MPFR_SAVE_EXPO_FREE (expo);
209 return mpfr_check_range (y, inexact, rnd_mode);
210 }