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, write to the Free Software
32 Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA */
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
62
63static const __float128
64 one = 1.0Q,
65 pio4hi = 7.8539816339744830961566084581987569936977E-1Q,
66 pio4lo = 2.1679525325309452561992610065108379921906E-35Q,
67
68 /* tan x = x + x^3 / 3 + x^5 T(x^2)/U(x^2)
69 0 <= x <= 0.6743316650390625
70 Peak relative error 8.0e-36 */
71 TH = 3.333333333333333333333333333333333333333E-1Q,
72 T0 = -1.813014711743583437742363284336855889393E7Q,
73 T1 = 1.320767960008972224312740075083259247618E6Q,
74 T2 = -2.626775478255838182468651821863299023956E4Q,
75 T3 = 1.764573356488504935415411383687150199315E2Q,
76 T4 = -3.333267763822178690794678978979803526092E-1Q,
77
78 U0 = -1.359761033807687578306772463253710042010E8Q,
79 U1 = 6.494370630656893175666729313065113194784E7Q,
80 U2 = -4.180787672237927475505536849168729386782E6Q,
81 U3 = 8.031643765106170040139966622980914621521E4Q,
82 U4 = -5.323131271912475695157127875560667378597E2Q;
83 /* 1.000000000000000000000000000000000000000E0 */
84
85
86static __float128
87__quadmath_kernel_tanq (__float128 x, __float128 y, int iy)
88{
89 __float128 z, r, v, w, s;
90 int32_t ix, sign = 1;
91 ieee854_float128 u, u1;
92
93 u.value = x;
94 ix = u.words32.w0 & 0x7fffffff;
95 if (ix < 0x3fc60000) /* x < 2**-57 */
96 {
97 if ((int) x == 0)
98 { /* generate inexact */
99 if ((ix | u.words32.w1 | u.words32.w2 | u.words32.w3
100 | (iy + 1)) == 0)
101 return one / fabsq (x);
102 else if (iy == 1)
103 {
104 math_check_force_underflow (x);
105 return x;
106 }
107 else
108 return -one / x;
109 }
110 }
111 if (ix >= 0x3ffe5942) /* |x| >= 0.6743316650390625 */
112 {
113 if ((u.words32.w0 & 0x80000000) != 0)
114 {
115 x = -x;
116 y = -y;
117 sign = -1;
118 }
119 else
120 sign = 1;
121 z = pio4hi - x;
122 w = pio4lo - y;
123 x = z + w;
124 y = 0.0;
125 }
126 z = x * x;
127 r = T0 + z * (T1 + z * (T2 + z * (T3 + z * T4)));
128 v = U0 + z * (U1 + z * (U2 + z * (U3 + z * (U4 + z))));
129 r = r / v;
130
131 s = z * x;
132 r = y + z * (s * r + y);
133 r += TH * s;
134 w = x + r;
135 if (ix >= 0x3ffe5942)
136 {
137 v = (__float128) iy;
138 w = (v - 2.0Q * (x - (w * w / (w + v) - r)));
139 if (sign < 0)
140 w = -w;
141 return w;
142 }
143 if (iy == 1)
144 return w;
145 else
146 { /* if allow error up to 2 ulp,
147 simply return -1.0/(x+r) here */
148 /* compute -1.0/(x+r) accurately */
149 u1.value = w;
150 u1.words32.w2 = 0;
151 u1.words32.w3 = 0;
152 v = r - (u1.value - x); /* u1+v = r+x */
153 z = -1.0 / w;
154 u.value = z;
155 u.words32.w2 = 0;
156 u.words32.w3 = 0;
157 s = 1.0 + u.value * u1.value;
158 return u.value + z * (s + u.value * v);
159 }
160}
161
162
163
164
165
166
167
168/* tanq.c -- __float128 version of s_tan.c.
169 * Conversion to IEEE quad long double by Jakub Jelinek, jj@ultra.linux.cz.
170 */
171
172/* @(#)s_tan.c 5.1 93/09/24 */
173/*
174 * ====================================================
175 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
176 *
177 * Developed at SunPro, a Sun Microsystems, Inc. business.
178 * Permission to use, copy, modify, and distribute this
179 * software is freely granted, provided that this notice
180 * is preserved.
181 * ====================================================
182 */
183
184/* tanl(x)
185 * Return tangent function of x.
186 *
187 * kernel function:
188 * __quadmath_kernel_tanq ... tangent function on [-pi/4,pi/4]
189 * __quadmath_rem_pio2q ... argument reduction routine
190 *
191 * Method.
192 * Let S,C and T denote the sin, cos and tan respectively on
193 * [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2
194 * in [-pi/4 , +pi/4], and let n = k mod 4.
195 * We have
196 *
197 * n sin(x) cos(x) tan(x)
198 * ----------------------------------------------------------
199 * 0 S C T
200 * 1 C -S -1/T
201 * 2 -S -C T
202 * 3 -C S -1/T
203 * ----------------------------------------------------------
204 *
205 * Special cases:
206 * Let trig be any of sin, cos, or tan.
207 * trig(+-INF) is NaN, with signals;
208 * trig(NaN) is that NaN;
209 *
210 * Accuracy:
211 * TRIG(x) returns trig(x) nearly rounded
212 */
213
214
215__float128
216tanq (__float128 x)
217{
218 __float128 y[2],z=0.0Q;
219 int64_t n, ix;
220
221 /* High word of x. */
222 GET_FLT128_MSW64(ix,x);
223
224 /* |x| ~< pi/4 */
225 ix &= 0x7fffffffffffffffLL;
226 if(ix <= 0x3ffe921fb54442d1LL) return __quadmath_kernel_tanq(x,z,1);
227
228 /* tanl(Inf or NaN) is NaN */
229 else if (ix>=0x7fff000000000000LL) {
230 if (ix == 0x7fff000000000000LL) {
231 GET_FLT128_LSW64(n,x);
232 }
233 return x-x; /* NaN */
234 }
235
236 /* argument reduction needed */
237 else {
238 n = __quadmath_rem_pio2q(x,y);
239 /* 1 -- n even, -1 -- n odd */
240 return __quadmath_kernel_tanq(y[0],y[1],1-((n&1)<<1));
241 }
242}
243