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 Long double expansions are
14 Copyright (C) 2001 Stephen L. Moshier <moshier@na-net.ornl.gov>
15 and are incorporated herein by permission of the author. The author
16 reserves the right to distribute this material elsewhere under different
17 copying permissions. These modifications are distributed here under
18 the following terms:
19
20 This library is free software; you can redistribute it and/or
21 modify it under the terms of the GNU Lesser General Public
22 License as published by the Free Software Foundation; either
23 version 2.1 of the License, or (at your option) any later version.
24
25 This library is distributed in the hope that it will be useful,
26 but WITHOUT ANY WARRANTY; without even the implied warranty of
27 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
28 Lesser General Public License for more details.
29
30 You should have received a copy of the GNU Lesser General Public
31 License along with this library; if not, see
32 <https://www.gnu.org/licenses/>. */
33
34 /* __kernel_tanl( x, y, k )
35 * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854
36 * Input x is assumed to be bounded by ~pi/4 in magnitude.
37 * Input y is the tail of x.
38 * Input k indicates whether tan (if k=1) or
39 * -1/tan (if k= -1) is returned.
40 *
41 * Algorithm
42 * 1. Since tan(-x) = -tan(x), we need only to consider positive x.
43 * 2. if x < 2^-57, return x with inexact if x!=0.
44 * 3. tan(x) is approximated by a rational form x + x^3 / 3 + x^5 R(x^2)
45 * on [0,0.67433].
46 *
47 * Note: tan(x+y) = tan(x) + tan'(x)*y
48 * ~ tan(x) + (1+x*x)*y
49 * Therefore, for better accuracy in computing tan(x+y), let
50 * r = x^3 * R(x^2)
51 * then
52 * tan(x+y) = x + (x^3 / 3 + (x^2 *(r+y)+y))
53 *
54 * 4. For x in [0.67433,pi/4], let y = pi/4 - x, then
55 * tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
56 * = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
57 */
58
59 #include <float.h>
60 #include <math.h>
61 #include <math_private.h>
62 #include <math-underflow.h>
63 #include <libc-diag.h>
64
65 static const long double
66 one = 1.0L,
67 pio4hi = 7.8539816339744830961566084581987569936977E-1L,
68 pio4lo = 2.1679525325309452561992610065108379921906E-35L,
69
70 /* tan x = x + x^3 / 3 + x^5 T(x^2)/U(x^2)
71 0 <= x <= 0.6743316650390625
72 Peak relative error 8.0e-36 */
73 TH = 3.333333333333333333333333333333333333333E-1L,
74 T0 = -1.813014711743583437742363284336855889393E7L,
75 T1 = 1.320767960008972224312740075083259247618E6L,
76 T2 = -2.626775478255838182468651821863299023956E4L,
77 T3 = 1.764573356488504935415411383687150199315E2L,
78 T4 = -3.333267763822178690794678978979803526092E-1L,
79
80 U0 = -1.359761033807687578306772463253710042010E8L,
81 U1 = 6.494370630656893175666729313065113194784E7L,
82 U2 = -4.180787672237927475505536849168729386782E6L,
83 U3 = 8.031643765106170040139966622980914621521E4L,
84 U4 = -5.323131271912475695157127875560667378597E2L;
85 /* 1.000000000000000000000000000000000000000E0 */
86
87
88 long double
89 __kernel_tanl (long double x, long double y, int iy)
90 {
91 long double z, r, v, w, s;
92 int32_t ix, sign, hx, lx;
93 double xhi;
94
95 xhi = ldbl_high (x);
96 EXTRACT_WORDS (hx, lx, xhi);
97 ix = hx & 0x7fffffff;
98 if (ix < 0x3c600000) /* x < 2**-57 */
99 {
100 if ((int) x == 0) /* generate inexact */
101 {
102 if ((ix | lx | (iy + 1)) == 0)
103 return one / fabs (x);
104 else if (iy == 1)
105 {
106 math_check_force_underflow (x);
107 return x;
108 }
109 else
110 return -one / x;
111 }
112 }
113 if (ix >= 0x3fe59420) /* |x| >= 0.6743316650390625 */
114 {
115 if ((hx & 0x80000000) != 0)
116 {
117 x = -x;
118 y = -y;
119 sign = -1;
120 }
121 else
122 sign = 1;
123 z = pio4hi - x;
124 w = pio4lo - y;
125 x = z + w;
126 y = 0.0;
127 }
128 z = x * x;
129 r = T0 + z * (T1 + z * (T2 + z * (T3 + z * T4)));
130 v = U0 + z * (U1 + z * (U2 + z * (U3 + z * (U4 + z))));
131 r = r / v;
132
133 s = z * x;
134 r = y + z * (s * r + y);
135 r += TH * s;
136 w = x + r;
137 if (ix >= 0x3fe59420)
138 {
139 v = (long double) iy;
140 w = (v - 2.0 * (x - (w * w / (w + v) - r)));
141 /* SIGN is set for arguments that reach this code, but not
142 otherwise, resulting in warnings that it may be used
143 uninitialized although in the cases where it is used it has
144 always been set. */
145 DIAG_PUSH_NEEDS_COMMENT;
146 DIAG_IGNORE_NEEDS_COMMENT (5, "-Wmaybe-uninitialized");
147 if (sign < 0)
148 w = -w;
149 DIAG_POP_NEEDS_COMMENT;
150 return w;
151 }
152 if (iy == 1)
153 return w;
154 else
155 { /* if allow error up to 2 ulp,
156 simply return -1.0/(x+r) here */
157 /* compute -1.0/(x+r) accurately */
158 long double u1, z1;
159
160 u1 = ldbl_high (w);
161 v = r - (u1 - x); /* u1+v = r+x */
162 z = -1.0 / w;
163 z1 = ldbl_high (z);
164 s = 1.0 + z1 * u1;
165 return z1 + z * (s + z1 * v);
166 }
167 }