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_tanl( long double x; long 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 = [0x3ffc4000, 0] = 0.15625 , then
42 * if x < 2^-57 (hx < 0x3fc40000 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 + 0x400)&0x7ffff800 (round x to a break point t)
49 * lt = 0
50 * i = (hy-0x3ffc4000)>>11; (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 + ...+ pp5*x^11
67 * cos(x') = 1 + qq1*x^2 + ...+ qq5*x^10
68 */
69
70 #include "libm.h"
71
72 #include <sys/isa_defs.h>
73
74 extern const long double _TBL_tanl_hi[], _TBL_tanl_lo[];
75 static const long double one = 1.0;
76
77 /*
78 * |sin(x) - (x+pp1*x^3+...+ pp5*x^11)| <= 2^-122.32 for |x|<1/64
79 */
80 static const long double
81 pp1 = -1.666666666666666666666666666586782940810e-0001L,
82 pp2 = 8.333333333333333333333003723660929317540e-0003L,
83 pp3 = -1.984126984126984076045903483778337804470e-0004L,
84 pp4 = 2.755731922361906641319723106210900949413e-0006L,
85 pp5 = -2.505198398570947019093998469135012057673e-0008L;
86
87 /*
88 * 2 10 -123.84
89 * |cos(x) - (1+qq1*x +...+ qq5*x )| <= 2 for |x|<=1/128
90 */
91 static const long double
92 qq1 = -4.999999999999999999999999999999378373641e-0001L,
93 qq2 = 4.166666666666666666666665478399327703130e-0002L,
94 qq3 = -1.388888888888888888058211230618051613494e-0003L,
95 qq4 = 2.480158730156105377771585658905303111866e-0005L,
96 qq5 = -2.755728099762526325736488376695157008736e-0007L;
97
98 /*
99 * |tan(x) - (x+t1*x^3+...+t6*x^13)|
100 * |------------------------------ | <= 2^-59.73 for |x|<0.15625
101 * | x |
102 */
103 static const long double
104 t1 = 3.333333333333333333333333333333423342490e-0001L,
105 t2 = 1.333333333333333333333333333093838744537e-0001L,
106 t3 = 5.396825396825396825396827906318682662250e-0002L,
107 t4 = 2.186948853615520282185576976994418486911e-0002L,
108 t5 = 8.863235529902196573354554519991152936246e-0003L,
109 t6 = 3.592128036572480064652191427543994878790e-0003L,
110 t7 = 1.455834387051455257856833807581901305474e-0003L,
111 t8 = 5.900274409318599857829983256201725587477e-0004L,
112 t9 = 2.391291152117265181501116961901122362937e-0004L,
113 t10 = 9.691533169382729742394024173194981882375e-0005L,
114 t11 = 3.927994733186415603228178184225780859951e-0005L,
115 t12 = 1.588300018848323824227640064883334101288e-0005L,
116 t13 = 6.916271223396808311166202285131722231723e-0006L;
117
118
119 long double
120 __k_tanl(long double x, long double y, int k)
121 {
122 long double a, t, z, w = 0.0, s, c;
123 int *pt = (int *)&t, *px = (int *)&x;
124 int i, j, hx, ix;
125
126 t = 1.0;
127 #if defined(__i386) || defined(__amd64)
128 XTOI(px, hx);
129 #else
130 hx = px[0];
131 #endif
132 ix = hx & 0x7fffffff;
133
134 if (ix < 0x3ffc4000) {
135 if (ix < 0x3fc60000) {
136 if ((i = (int)x) == 0) /* generate inexact */
137 w = x;
138 } else {
139 z = x * x;
140
141 if (ix < 0x3ff30000) { /* 2**-12 */
142 t = z * (t1 + z * (t2 + z * (t3 + z * t4)));
143 } else {
144 t = z * (t1 + z * (t2 + z * (t3 + z * (t4 + z *
145 (t5 + z * (t6 + z * (t7 + z * (t8 + z *
146 (t9 + z * (t10 + z * (t11 + z * (t12 + z *
147 t13))))))))))));
148 }
149
150 t = y + x * t;
151 w = x + t;
152 }
153
154 return (k == 0 ? w : -one / w);
155 }
156
157 j = (ix + 0x400) & 0x7ffff800;
158 i = (j - 0x3ffc4000) >> 11;
159 #if defined(__i386) || defined(__amd64)
160 ITOX(j, pt);
161 #else
162 pt[0] = j;
163 #endif
164
165 if (hx > 0)
166 x = y - (t - x);
167 else
168 x = (-y) - (t + x);
169
170 a = _TBL_tanl_hi[i];
171 z = x * x;
172 /* cos(x)-1 */
173 t = z * (qq1 + z * (qq2 + z * (qq3 + z * (qq4 + z * qq5))));
174 /* sin(x) */
175 s = x * (one + z * (pp1 + z * (pp2 + z * (pp3 + z * (pp4 + z * pp5)))));
176
177 if (k == 0) {
178 w = a * s;
179 t = _TBL_tanl_lo[i] + (s + a * w) / (one - (w - t));
180 return (hx < 0 ? -a - t : a + t);
181 } else {
182 w = s + a * t;
183 c = w + _TBL_tanl_lo[i];
184 z = (one - (a * s - t));
185 return (hx >= 0 ? z / (-a - c) : z / (a + c));
186 }
187 }