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 <http://www.gnu.org/licenses/>. */
33
34 /* __quadmath_kernel_tanq( 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 "quadmath-imp.h"
60
61 static const __float128
62 one = 1,
63 pio4hi = 7.8539816339744830961566084581987569936977E-1Q,
64 pio4lo = 2.1679525325309452561992610065108379921906E-35Q,
65
66 /* tan x = x + x^3 / 3 + x^5 T(x^2)/U(x^2)
67 0 <= x <= 0.6743316650390625
68 Peak relative error 8.0e-36 */
69 TH = 3.333333333333333333333333333333333333333E-1Q,
70 T0 = -1.813014711743583437742363284336855889393E7Q,
71 T1 = 1.320767960008972224312740075083259247618E6Q,
72 T2 = -2.626775478255838182468651821863299023956E4Q,
73 T3 = 1.764573356488504935415411383687150199315E2Q,
74 T4 = -3.333267763822178690794678978979803526092E-1Q,
75
76 U0 = -1.359761033807687578306772463253710042010E8Q,
77 U1 = 6.494370630656893175666729313065113194784E7Q,
78 U2 = -4.180787672237927475505536849168729386782E6Q,
79 U3 = 8.031643765106170040139966622980914621521E4Q,
80 U4 = -5.323131271912475695157127875560667378597E2Q;
81 /* 1.000000000000000000000000000000000000000E0 */
82
83
84 __float128
85 __quadmath_kernel_tanq (__float128 x, __float128 y, int iy)
86 {
87 __float128 z, r, v, w, s;
88 int32_t ix, sign;
89 ieee854_float128 u, u1;
90
91 u.value = x;
92 ix = u.words32.w0 & 0x7fffffff;
93 if (ix < 0x3fc60000) /* x < 2**-57 */
94 {
95 if ((int) x == 0)
96 { /* generate inexact */
97 if ((ix | u.words32.w1 | u.words32.w2 | u.words32.w3
98 | (iy + 1)) == 0)
99 return one / fabsq (x);
100 else if (iy == 1)
101 {
102 math_check_force_underflow (x);
103 return x;
104 }
105 else
106 return -one / x;
107 }
108 }
109 if (ix >= 0x3ffe5942) /* |x| >= 0.6743316650390625 */
110 {
111 if ((u.words32.w0 & 0x80000000) != 0)
112 {
113 x = -x;
114 y = -y;
115 sign = -1;
116 }
117 else
118 sign = 1;
119 z = pio4hi - x;
120 w = pio4lo - y;
121 x = z + w;
122 y = 0.0;
123 }
124 z = x * x;
125 r = T0 + z * (T1 + z * (T2 + z * (T3 + z * T4)));
126 v = U0 + z * (U1 + z * (U2 + z * (U3 + z * (U4 + z))));
127 r = r / v;
128
129 s = z * x;
130 r = y + z * (s * r + y);
131 r += TH * s;
132 w = x + r;
133 if (ix >= 0x3ffe5942)
134 {
135 v = (__float128) iy;
136 w = (v - 2.0 * (x - (w * w / (w + v) - r)));
137 /* SIGN is set for arguments that reach this code, but not
138 otherwise, resulting in warnings that it may be used
139 uninitialized although in the cases where it is used it has
140 always been set. */
141
142
143 if (sign < 0)
144 w = -w;
145
146 return w;
147 }
148 if (iy == 1)
149 return w;
150 else
151 { /* if allow error up to 2 ulp,
152 simply return -1.0/(x+r) here */
153 /* compute -1.0/(x+r) accurately */
154 u1.value = w;
155 u1.words32.w2 = 0;
156 u1.words32.w3 = 0;
157 v = r - (u1.value - x); /* u1+v = r+x */
158 z = -1.0 / w;
159 u.value = z;
160 u.words32.w2 = 0;
161 u.words32.w3 = 0;
162 s = 1.0 + u.value * u1.value;
163 return u.value + z * (s + u.value * v);
164 }
165 }