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 * Copyright 2006 Sun Microsystems, Inc. All rights reserved.
27 * Use is subject to license terms.
28 */
29
30 /*
31 * long double __k_tanl(long double x; long double y, int k);
32 * kernel tan/cotan function on [-pi/4, pi/4], pi/4 ~ 0.785398164
33 * Input x is assumed to be bounded by ~pi/4 in magnitude.
34 * Input y is the tail of x.
35 * Input k indicate -- tan if k=0; else -1/tan
36 *
37 * Table look up algorithm
38 * 1. by tan(-x) = -tan(x), need only to consider positive x
39 * 2. if x < 5/32 = [0x3ffc4000, 0] = 0.15625 , then
40 * if x < 2^-57 (hx < 0x3fc40000 0), set w=x with inexact if x!= 0
41 * else
42 * z = x*x;
43 * w = x + (y+(x*z)*(t1+z*(t2+z*(t3+z*(t4+z*(t5+z*t6))))))
44 * return (k==0)? w: 1/w;
45 * 3. else
46 * ht = (hx + 0x400)&0x7ffff800 (round x to a break point t)
47 * lt = 0
48 * i = (hy-0x3ffc4000)>>11; (i<=64)
49 * x' = (x - t)+y (|x'| ~<= 2^-7)
50 * By
51 * tan(t+x')
52 * = (tan(t)+tan(x'))/(1-tan(x')tan(t))
53 * We have
54 * sin(x')+tan(t)*(tan(t)*sin(x'))
55 * = tan(t) + ------------------------------- for k=0
56 * cos(x') - tan(t)*sin(x')
57 *
58 * cos(x') - tan(t)*sin(x')
59 * = - -------------------------------------- for k=1
60 * tan(t) + tan(t)*(cos(x')-1) + sin(x')
61 *
62 *
63 * where tan(t) is from the table,
64 * sin(x') = x + pp1*x^3 + ...+ pp5*x^11
65 * cos(x') = 1 + qq1*x^2 + ...+ qq5*x^10
66 */
67
68 #include "libm.h"
69
70 extern const long double _TBL_tanl_hi[], _TBL_tanl_lo[];
71 static const long double
72 one = 1.0L,
73 /*
74 * 3 11 -122.32
75 * |sin(x) - (x+pp1*x +...+ pp5*x )| <= 2 for |x|<1/64
76 */
77 pp1 = -1.666666666666666666666666666586782940810e-0001L,
78 pp2 = +8.333333333333333333333003723660929317540e-0003L,
79 pp3 = -1.984126984126984076045903483778337804470e-0004L,
80 pp4 = +2.755731922361906641319723106210900949413e-0006L,
81 pp5 = -2.505198398570947019093998469135012057673e-0008L,
82 /*
83 * 2 10 -123.84
84 * |cos(x) - (1+qq1*x +...+ qq5*x )| <= 2 for |x|<=1/128
85 */
86 qq1 = -4.999999999999999999999999999999378373641e-0001L,
87 qq2 = +4.166666666666666666666665478399327703130e-0002L,
88 qq3 = -1.388888888888888888058211230618051613494e-0003L,
89 qq4 = +2.480158730156105377771585658905303111866e-0005L,
90 qq5 = -2.755728099762526325736488376695157008736e-0007L,
91 /*
92 * |tan(x) - (x+t1*x^3+...+t6*x^13)|
93 * |------------------------------ | <= 2^-59.73 for |x|<0.15625
94 * | x |
95 */
96 t1 = +3.333333333333333333333333333333423342490e-0001L,
97 t2 = +1.333333333333333333333333333093838744537e-0001L,
98 t3 = +5.396825396825396825396827906318682662250e-0002L,
99 t4 = +2.186948853615520282185576976994418486911e-0002L,
100 t5 = +8.863235529902196573354554519991152936246e-0003L,
101 t6 = +3.592128036572480064652191427543994878790e-0003L,
102 t7 = +1.455834387051455257856833807581901305474e-0003L,
103 t8 = +5.900274409318599857829983256201725587477e-0004L,
104 t9 = +2.391291152117265181501116961901122362937e-0004L,
105 t10 = +9.691533169382729742394024173194981882375e-0005L,
106 t11 = +3.927994733186415603228178184225780859951e-0005L,
107 t12 = +1.588300018848323824227640064883334101288e-0005L,
108 t13 = +6.916271223396808311166202285131722231723e-0006L;
109
110 #define i0 0
111
112 long double
113 __k_tanl(long double x, long double y, int k) {
114 long double a, t, z, w = 0, s, c;
115 int *pt = (int *) &t, *px = (int *) &x;
116 int i, j, hx, ix;
117
118 t = 1.0L;
119 hx = px[i0];
120 ix = hx & 0x7fffffff;
121 if (ix < 0x3ffc4000) {
122 *(3 - i0 + (int *) &t) = 1; /* make t = one+ulp */
123 if (ix < 0x3fc60000) {
124 if (((int) (x * t)) < 1) /* generate inexact */
125 w = x; /* generate underflow if subnormal */
126 } else {
127 z = x * x;
128 if (ix < 0x3ff30000) /* 2**-12 */
129 t = z * (t1 + z * (t2 + z * (t3 + z * t4)));
130 else
131 t = z * (t1 + z * (t2 + z * (t3 + z * (t4 +
132 z * (t5 + z * (t6 + z * (t7 + z * (t8 +
133 z * (t9 + z * (t10 + z * (t11 +
134 z * (t12 + z * t13))))))))))));
135 t = y + x * t;
136 w = x + t;
137 }
138 return (k == 0 ? w : -one / w);
139 }
140 j = (ix + 0x400) & 0x7ffff800;
141 i = (j - 0x3ffc4000) >> 11;
142 pt[i0] = j;
143 if (hx > 0)
144 x = y - (t - x);
145 else
146 x = (-y) - (t + x);
147 a = _TBL_tanl_hi[i];
148 z = x * x;
149 /* cos(x)-1 */
150 t = z * (qq1 + z * (qq2 + z * (qq3 + z * (qq4 + z * qq5))));
151 /* sin(x) */
152 s = x * (one + z * (pp1 + z * (pp2 + z * (pp3 + z * (pp4 + z * pp5)))));
153 if (k == 0) {
154 w = a * s;
155 t = _TBL_tanl_lo[i] + (s + a * w) / (one - (w - t));
156 return (hx < 0 ? -a - t : a + t);
157 } else {
158 w = s + a * t;
159 c = w + _TBL_tanl_lo[i];
160 z = one - (a * s - t);
161 return (hx >= 0 ? z / (-a - c) : z / (a + c));
162 }
163 }