1 /*
2 * CDDL HEADER START
3 *
4 * The contents of this file are subject to the terms of the
5 * Common Development and Distribution License (the "License").
6 * You may not use this file except in compliance with the License.
7 *
8 * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
9 * or http://www.opensolaris.org/os/licensing.
10 * See the License for the specific language governing permissions
11 * and limitations under the License.
12 *
13 * When distributing Covered Code, include this CDDL HEADER in each
14 * file and include the License file at usr/src/OPENSOLARIS.LICENSE.
15 * If applicable, add the following below this CDDL HEADER, with the
16 * fields enclosed by brackets "[]" replaced with your own identifying
17 * information: Portions Copyright [yyyy] [name of copyright owner]
18 *
19 * CDDL HEADER END
20 */
21
22 /*
23 * Copyright 2011 Nexenta Systems, Inc. All rights reserved.
24 */
25
26 /*
27 * Copyright 2006 Sun Microsystems, Inc. All rights reserved.
28 * Use is subject to license terms.
29 */
30
31
32 /*
33 * __k_tan( double x; double y; int k )
34 * kernel tan/cotan function on [-pi/4, pi/4], pi/4 ~ 0.785398164
35 * Input x is assumed to be bounded by ~pi/4 in magnitude.
36 * Input y is the tail of x.
37 * Input k indicate -- tan if k=0; else -1/tan
38 *
39 * Table look up algorithm
40 * 1. by tan(-x) = -tan(x), need only to consider positive x
41 * 2. if x < 5/32 = [0x3fc40000, 0] = 0.15625 , then
42 * if x < 2^-27 (hx < 0x3e400000 0), set w=x with inexact if x != 0
43 * else
44 * z = x*x;
45 * w = x + (y+(x*z)*(t1+z*(t2+z*(t3+z*(t4+z*(t5+z*t6))))))
46 * return (k == 0)? w: 1/w;
47 * 3. else
48 * ht = (hx + 0x4000)&0x7fff8000 (round x to a break point t)
49 * lt = 0
50 * i = (hy-0x3fc40000)>>15; (i<=64)
51 * x' = (x - t)+y (|x'| ~<= 2^-7)
52 * By
53 * tan(t+x')
54 * = (tan(t)+tan(x'))/(1-tan(x')tan(t))
55 * We have
56 * sin(x')+tan(t)*(tan(t)*sin(x'))
57 * = tan(t) + ------------------------------- for k=0
58 * cos(x') - tan(t)*sin(x')
59 *
60 * cos(x') - tan(t)*sin(x')
61 * = - -------------------------------------- for k=1
62 * tan(t) + tan(t)*(cos(x')-1) + sin(x')
63 *
64 *
65 * where tan(t) is from the table,
66 * sin(x') = x + pp1*x^3 + pp2*x^5
67 * cos(x') = 1 + qq1*x^2 + qq2*x^4
68 */
69
70 #include "libm.h"
71
72 extern const double _TBL_tan_hi[], _TBL_tan_lo[];
73
74 static const double q[] = {
75 /* one = */
76 1.0,
77
78 /*
79 * 2 2 -59.56
80 * |sin(x) - pp1*x*(pp2+x *(pp3+x )| <= 2 for |x|<1/64
81 */
82 /* pp1 = */ 8.33326120969096230395312119298978359438478946686e-0003,
83 /* pp2 = */ 1.20001038589438965215025680596868692381425944526e+0002,
84 /* pp3 = */ -2.00001730975089451192161504877731204032897949219e+0001,
85
86 /*
87 * 2 2 -56.19
88 * |cos(x) - (1+qq1*x (qq2+x ))| <= 2 for |x|<=1/128
89 */
90 /* qq1 = */ 4.16665486385721928197511942926212213933467864990e-0002,
91 /* qq2 = */ -1.20000339921340035687080671777948737144470214844e+0001,
92
93 /*
94 * |tan(x) - PF(x)|
95 * |--------------| <= 2^-58.57 for |x|<0.15625
96 * | x |
97 *
98 * where (let z = x*x)
99 * PF(x) = x + (t1*x*z)(t2 + z(t3 + z))(t4 + z)(t5 + z(t6 + z))
100 */
101 /* t1 = */ 3.71923358986516816929168705030406272271648049355e-0003,
102 /* t2 = */ 6.02645120354857866118436504621058702468872070312e+0000,
103 /* t3 = */ 2.42627327587398156083509093150496482849121093750e+0000,
104 /* t4 = */ 2.44968983934252770851003333518747240304946899414e+0000,
105 /* t5 = */ 6.07089252571767978849948121933266520500183105469e+0000,
106 /* t6 = */ -2.49403756995593761658369658107403665781021118164e+0000,
107 };
108
109 #define one q[0]
110 #define pp1 q[1]
111 #define pp2 q[2]
112 #define pp3 q[3]
113 #define qq1 q[4]
114 #define qq2 q[5]
115 #define t1 q[6]
116 #define t2 q[7]
117 #define t3 q[8]
118 #define t4 q[9]
119 #define t5 q[10]
120 #define t6 q[11]
121
122
123 double
124 __k_tan(double x, double y, int k)
125 {
126 double a, t, z, w = 0.0L, s, c, r, rh, xh, xl;
127 int i, j, hx, ix;
128
129 t = one;
130 hx = ((int *)&x)[HIWORD];
131 ix = hx & 0x7fffffff;
132
133 if (ix < 0x3fc40000) { /* 0.15625 */
134 if (ix < 0x3e400000) { /* 2^-27 */
135 if ((i = (int)x) == 0) /* generate inexact */
136 w = x;
137
138 t = y;
139 } else {
140 z = x * x;
141 t = y + (((t1 * x) * z) * (t2 + z * (t3 + z))) * ((t4 +
142 z) * (t5 + z * (t6 + z)));
143 w = x + t;
144 }
145
146 if (k == 0)
147 return (w);
148
149 /*
150 * Compute -1/(x+T) with great care
151 * Let r = -1/(x+T), rh = r chopped to 20 bits.
152 * Also let xh = x+T chopped to 20 bits, xl = (x-xh)+T. Then
153 * -1/(x+T) = rh + (-1/(x+T)-rh) = rh + r*(1+rh*(x+T))
154 * = rh + r*((1+rh*xh)+rh*xl).
155 */
156 rh = r = -one / w;
157 ((int *)&rh)[LOWORD] = 0;
158 xh = w;
159 ((int *)&xh)[LOWORD] = 0;
160 xl = (x - xh) + t;
161 return (rh + r * ((one + rh * xh) + rh * xl));
162 }
163
164 j = (ix + 0x4000) & 0x7fff8000;
165 i = (j - 0x3fc40000) >> 15;
166 ((int *)&t)[HIWORD] = j;
167
168 if (hx > 0)
169 x = y - (t - x);
170 else
171 x = -y - (t + x);
172
173 a = _TBL_tan_hi[i];
174 z = x * x;
175 s = (pp1 * x) * (pp2 + z * (pp3 + z)); /* sin(x) */
176 t = (qq1 * z) * (qq2 + z); /* cos(x) - 1 */
177
178 if (k == 0) {
179 w = a * s;
180 t = _TBL_tan_lo[i] + (s + a * w) / (one - (w - t));
181 return (hx < 0 ? -a - t : a + t);
182 } else {
183 w = s + a * t;
184 c = w + _TBL_tan_lo[i];
185 t = a * s - t;
186
187 /*
188 * Now try to compute [(1-T)/(a+c)] accurately
189 *
190 * Let r = 1/(a+c), rh = (1-T)*r chopped to 20 bits.
191 * Also let xh = a+c chopped to 20 bits, xl = (a-xh)+c. Then
192 * (1-T)/(a+c) = rh + ((1-T)/(a+c)-rh)
193 * = rh + r*(1-T-rh*(a+c))
194 * = rh + r*((1-T-rh*xh)-rh*xl)
195 * = rh + r*(((1-rh*xh)-T)-rh*xl)
196 */
197 r = one / (a + c);
198 rh = (one - t) * r;
199 ((int *)&rh)[LOWORD] = 0;
200 xh = a + c;
201 ((int *)&xh)[LOWORD] = 0;
202 xl = (a - xh) + c;
203 z = rh + r * (((one - rh * xh) - t) - rh * xl);
204 return (hx >= 0 ? -z : z);
205 }
206 }