```   1 /*
3  *
4  * The contents of this file are subject to the terms of the
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
15  * If applicable, add the following below this CDDL HEADER, with the
16  * fields enclosed by brackets "[]" replaced with your own identifying
18  *
20  */
21
22 /*
24  */
25 /*
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 }
```