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
63 static 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
86 static __float128
__quadmath_kernel_tanq(__float128 x,__float128 y,int iy)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
103 return (iy == 1) ? x : -one / x;
104 }
105 }
106 if (ix >= 0x3ffe5942) /* |x| >= 0.6743316650390625 */
107 {
108 if ((u.words32.w0 & 0x80000000) != 0)
109 {
110 x = -x;
111 y = -y;
112 sign = -1;
113 }
114 else
115 sign = 1;
116 z = pio4hi - x;
117 w = pio4lo - y;
118 x = z + w;
119 y = 0.0;
120 }
121 z = x * x;
122 r = T0 + z * (T1 + z * (T2 + z * (T3 + z * T4)));
123 v = U0 + z * (U1 + z * (U2 + z * (U3 + z * (U4 + z))));
124 r = r / v;
125
126 s = z * x;
127 r = y + z * (s * r + y);
128 r += TH * s;
129 w = x + r;
130 if (ix >= 0x3ffe5942)
131 {
132 v = (__float128) iy;
133 w = (v - 2.0Q * (x - (w * w / (w + v) - r)));
134 if (sign < 0)
135 w = -w;
136 return w;
137 }
138 if (iy == 1)
139 return w;
140 else
141 { /* if allow error up to 2 ulp,
142 simply return -1.0/(x+r) here */
143 /* compute -1.0/(x+r) accurately */
144 u1.value = w;
145 u1.words32.w2 = 0;
146 u1.words32.w3 = 0;
147 v = r - (u1.value - x); /* u1+v = r+x */
148 z = -1.0 / w;
149 u.value = z;
150 u.words32.w2 = 0;
151 u.words32.w3 = 0;
152 s = 1.0 + u.value * u1.value;
153 return u.value + z * (s + u.value * v);
154 }
155 }
156
157
158
159
160
161
162
163 /* tanq.c -- __float128 version of s_tan.c.
164 * Conversion to IEEE quad long double by Jakub Jelinek, jj@ultra.linux.cz.
165 */
166
167 /* @(#)s_tan.c 5.1 93/09/24 */
168 /*
169 * ====================================================
170 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
171 *
172 * Developed at SunPro, a Sun Microsystems, Inc. business.
173 * Permission to use, copy, modify, and distribute this
174 * software is freely granted, provided that this notice
175 * is preserved.
176 * ====================================================
177 */
178
179 /* tanl(x)
180 * Return tangent function of x.
181 *
182 * kernel function:
183 * __quadmath_kernel_tanq ... tangent function on [-pi/4,pi/4]
184 * __quadmath_rem_pio2q ... argument reduction routine
185 *
186 * Method.
187 * Let S,C and T denote the sin, cos and tan respectively on
188 * [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2
189 * in [-pi/4 , +pi/4], and let n = k mod 4.
190 * We have
191 *
192 * n sin(x) cos(x) tan(x)
193 * ----------------------------------------------------------
194 * 0 S C T
195 * 1 C -S -1/T
196 * 2 -S -C T
197 * 3 -C S -1/T
198 * ----------------------------------------------------------
199 *
200 * Special cases:
201 * Let trig be any of sin, cos, or tan.
202 * trig(+-INF) is NaN, with signals;
203 * trig(NaN) is that NaN;
204 *
205 * Accuracy:
206 * TRIG(x) returns trig(x) nearly rounded
207 */
208
209
210 __float128
tanq(__float128 x)211 tanq (__float128 x)
212 {
213 __float128 y[2],z=0.0Q;
214 int64_t n, ix;
215
216 /* High word of x. */
217 GET_FLT128_MSW64(ix,x);
218
219 /* |x| ~< pi/4 */
220 ix &= 0x7fffffffffffffffLL;
221 if(ix <= 0x3ffe921fb54442d1LL) return __quadmath_kernel_tanq(x,z,1);
222
223 /* tanl(Inf or NaN) is NaN */
224 else if (ix>=0x7fff000000000000LL) {
225 if (ix == 0x7fff000000000000LL) {
226 GET_FLT128_LSW64(n,x);
227 }
228 return x-x; /* NaN */
229 }
230
231 /* argument reduction needed */
232 else {
233 n = __quadmath_rem_pio2q(x,y);
234 /* 1 -- n even, -1 -- n odd */
235 return __quadmath_kernel_tanq(y[0],y[1],1-((n&1)<<1));
236 }
237 }
238