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 * long double __k_tanl(long double x; long double y, int k);
33 * kernel tan/cotan function on [-pi/4, pi/4], pi/4 ~ 0.785398164
34 * Input x is assumed to be bounded by ~pi/4 in magnitude.
35 * Input y is the tail of x.
36 * Input k indicate -- tan if k=0; else -1/tan
37 *
38 * Table look up algorithm
39 * 1. by tan(-x) = -tan(x), need only to consider positive x
40 * 2. if x < 5/32 = [0x3ffc4000, 0] = 0.15625 , then
41 * if x < 2^-57 (hx < 0x3fc40000 0), set w=x with inexact if x != 0
42 * else
43 * z = x*x;
44 * w = x + (y+(x*z)*(t1+z*(t2+z*(t3+z*(t4+z*(t5+z*t6))))))
45 * return (k == 0)? w: 1/w;
46 * 3. else
47 * ht = (hx + 0x400)&0x7ffff800 (round x to a break point t)
48 * lt = 0
49 * i = (hy-0x3ffc4000)>>11; (i<=64)
50 * x' = (x - t)+y (|x'| ~<= 2^-7)
51 * By
52 * tan(t+x')
53 * = (tan(t)+tan(x'))/(1-tan(x')tan(t))
54 * We have
55 * sin(x')+tan(t)*(tan(t)*sin(x'))
56 * = tan(t) + ------------------------------- for k=0
57 * cos(x') - tan(t)*sin(x')
58 *
59 * cos(x') - tan(t)*sin(x')
60 * = - -------------------------------------- for k=1
61 * tan(t) + tan(t)*(cos(x')-1) + sin(x')
62 *
63 *
64 * where tan(t) is from the table,
65 * sin(x') = x + pp1*x^3 + ...+ pp5*x^11
66 * cos(x') = 1 + qq1*x^2 + ...+ qq5*x^10
67 */
68
69 #include "libm.h"
70
71 extern const long double _TBL_tanl_hi[], _TBL_tanl_lo[];
72 static const long double one = 1.0L;
73
74 /*
75 * 3 11 -122.32
76 * |sin(x) - (x+pp1*x +...+ pp5*x )| <= 2 for |x|<1/64
77 */
78 static const long double
79 pp1 = -1.666666666666666666666666666586782940810e-0001L,
80 pp2 = +8.333333333333333333333003723660929317540e-0003L,
81 pp3 = -1.984126984126984076045903483778337804470e-0004L,
82 pp4 = +2.755731922361906641319723106210900949413e-0006L,
83 pp5 = -2.505198398570947019093998469135012057673e-0008L;
84
85 /*
86 * 2 10 -123.84
87 * |cos(x) - (1+qq1*x +...+ qq5*x )| <= 2 for |x|<=1/128
88 */
89 static const long double
90 qq1 = -4.999999999999999999999999999999378373641e-0001L,
91 qq2 = +4.166666666666666666666665478399327703130e-0002L,
92 qq3 = -1.388888888888888888058211230618051613494e-0003L,
93 qq4 = +2.480158730156105377771585658905303111866e-0005L,
94 qq5 = -2.755728099762526325736488376695157008736e-0007L;
95
96 /*
97 * |tan(x) - (x+t1*x^3+...+t6*x^13)|
98 * |------------------------------ | <= 2^-59.73 for |x|<0.15625
99 * | x |
100 */
101 static const long double
102 t1 = +3.333333333333333333333333333333423342490e-0001L,
103 t2 = +1.333333333333333333333333333093838744537e-0001L,
104 t3 = +5.396825396825396825396827906318682662250e-0002L,
105 t4 = +2.186948853615520282185576976994418486911e-0002L,
106 t5 = +8.863235529902196573354554519991152936246e-0003L,
107 t6 = +3.592128036572480064652191427543994878790e-0003L,
108 t7 = +1.455834387051455257856833807581901305474e-0003L,
109 t8 = +5.900274409318599857829983256201725587477e-0004L,
110 t9 = +2.391291152117265181501116961901122362937e-0004L,
111 t10 = +9.691533169382729742394024173194981882375e-0005L,
112 t11 = +3.927994733186415603228178184225780859951e-0005L,
113 t12 = +1.588300018848323824227640064883334101288e-0005L,
114 t13 = +6.916271223396808311166202285131722231723e-0006L;
115
116 #define i0 0
117
118 long double
119 __k_tanl(long double x, long double y, int k)
120 {
121 long double a, t, z, w = 0, s, c;
122 int *pt = (int *)&t, *px = (int *)&x;
123 int i, j, hx, ix;
124
125 t = 1.0L;
126 hx = px[i0];
127 ix = hx & 0x7fffffff;
128
129 if (ix < 0x3ffc4000) {
130 *(3 - i0 + (int *)&t) = 1; /* make t = one+ulp */
131
132 if (ix < 0x3fc60000) {
133 if (((int)(x * t)) < 1) /* generate inexact */
134 w = x; /* generate underflow if subnormal */
135 } else {
136 z = x * x;
137
138 if (ix < 0x3ff30000) { /* 2**-12 */
139 t = z * (t1 + z * (t2 + z * (t3 + z * t4)));
140 } else {
141 t = z * (t1 + z * (t2 + z * (t3 + z * (t4 + z *
142 (t5 + z * (t6 + z * (t7 + z * (t8 + z *
143 (t9 + z * (t10 + z * (t11 + z * (t12 + z *
144 t13))))))))))));
145 }
146
147 t = y + x * t;
148 w = x + t;
149 }
150
151 return (k == 0 ? w : -one / w);
152 }
153
154 j = (ix + 0x400) & 0x7ffff800;
155 i = (j - 0x3ffc4000) >> 11;
156 pt[i0] = j;
157
158 if (hx > 0)
159 x = y - (t - x);
160 else
161 x = (-y) - (t + x);
162
163 a = _TBL_tanl_hi[i];
164 z = x * x;
165 /* cos(x)-1 */
166 t = z * (qq1 + z * (qq2 + z * (qq3 + z * (qq4 + z * qq5))));
167 /* sin(x) */
168 s = x * (one + z * (pp1 + z * (pp2 + z * (pp3 + z * (pp4 + z * pp5)))));
169
170 if (k == 0) {
171 w = a * s;
172 t = _TBL_tanl_lo[i] + (s + a * w) / (one - (w - t));
173 return (hx < 0 ? -a - t : a + t);
174 } else {
175 w = s + a * t;
176 c = w + _TBL_tanl_lo[i];
177 z = one - (a * s - t);
178 return (hx >= 0 ? z / (-a - c) : z / (a + c));
179 }
180 }