1/*
2 * Copyright (c) 1998, 2004, Oracle and/or its affiliates. All rights reserved.
3 * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.
4 *
5 * This code is free software; you can redistribute it and/or modify it
6 * under the terms of the GNU General Public License version 2 only, as
7 * published by the Free Software Foundation. Oracle designates this
8 * particular file as subject to the "Classpath" exception as provided
9 * by Oracle in the LICENSE file that accompanied this code.
10 *
11 * This code is distributed in the hope that it will be useful, but WITHOUT
12 * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
13 * FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
14 * version 2 for more details (a copy is included in the LICENSE file that
15 * accompanied this code).
16 *
17 * You should have received a copy of the GNU General Public License version
18 * 2 along with this work; if not, write to the Free Software Foundation,
19 * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA.
20 *
21 * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA
22 * or visit www.oracle.com if you need additional information or have any
23 * questions.
24 */
25
26/* __kernel_tan( x, y, k )
27 * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854
28 * Input x is assumed to be bounded by ~pi/4 in magnitude.
29 * Input y is the tail of x.
30 * Input k indicates whether tan (if k=1) or
31 * -1/tan (if k= -1) is returned.
32 *
33 * Algorithm
34 * 1. Since tan(-x) = -tan(x), we need only to consider positive x.
35 * 2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0.
36 * 3. tan(x) is approximated by a odd polynomial of degree 27 on
37 * [0,0.67434]
38 * 3 27
39 * tan(x) ~ x + T1*x + ... + T13*x
40 * where
41 *
42 * |tan(x) 2 4 26 | -59.2
43 * |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2
44 * | x |
45 *
46 * Note: tan(x+y) = tan(x) + tan'(x)*y
47 * ~ tan(x) + (1+x*x)*y
48 * Therefore, for better accuracy in computing tan(x+y), let
49 * 3 2 2 2 2
50 * r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
51 * then
52 * 3 2
53 * tan(x+y) = x + (T1*x + (x *(r+y)+y))
54 *
55 * 4. For x in [0.67434,pi/4], let y = pi/4 - x, then
56 * tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
57 * = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
58 */
59
60#include "fdlibm.h"
61#ifdef __STDC__
62static const double
63#else
64static double
65#endif
66one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
67pio4 = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
68pio4lo= 3.06161699786838301793e-17, /* 0x3C81A626, 0x33145C07 */
69T[] = {
70 3.33333333333334091986e-01, /* 0x3FD55555, 0x55555563 */
71 1.33333333333201242699e-01, /* 0x3FC11111, 0x1110FE7A */
72 5.39682539762260521377e-02, /* 0x3FABA1BA, 0x1BB341FE */
73 2.18694882948595424599e-02, /* 0x3F9664F4, 0x8406D637 */
74 8.86323982359930005737e-03, /* 0x3F8226E3, 0xE96E8493 */
75 3.59207910759131235356e-03, /* 0x3F6D6D22, 0xC9560328 */
76 1.45620945432529025516e-03, /* 0x3F57DBC8, 0xFEE08315 */
77 5.88041240820264096874e-04, /* 0x3F4344D8, 0xF2F26501 */
78 2.46463134818469906812e-04, /* 0x3F3026F7, 0x1A8D1068 */
79 7.81794442939557092300e-05, /* 0x3F147E88, 0xA03792A6 */
80 7.14072491382608190305e-05, /* 0x3F12B80F, 0x32F0A7E9 */
81 -1.85586374855275456654e-05, /* 0xBEF375CB, 0xDB605373 */
82 2.59073051863633712884e-05, /* 0x3EFB2A70, 0x74BF7AD4 */
83};
84
85#ifdef __STDC__
86 double __kernel_tan(double x, double y, int iy)
87#else
88 double __kernel_tan(x, y, iy)
89 double x,y; int iy;
90#endif
91{
92 double z,r,v,w,s;
93 int ix,hx;
94 hx = __HI(x); /* high word of x */
95 ix = hx&0x7fffffff; /* high word of |x| */
96 if(ix<0x3e300000) { /* x < 2**-28 */
97 if((int)x==0) { /* generate inexact */
98 if (((ix | __LO(x)) | (iy + 1)) == 0)
99 return one / fabs(x);
100 else {
101 if (iy == 1)
102 return x;
103 else { /* compute -1 / (x+y) carefully */
104 double a, t;
105
106 z = w = x + y;
107 __LO(z) = 0;
108 v = y - (z - x);
109 t = a = -one / w;
110 __LO(t) = 0;
111 s = one + t * z;
112 return t + a * (s + t * v);
113 }
114 }
115 }
116 }
117 if(ix>=0x3FE59428) { /* |x|>=0.6744 */
118 if(hx<0) {x = -x; y = -y;}
119 z = pio4-x;
120 w = pio4lo-y;
121 x = z+w; y = 0.0;
122 }
123 z = x*x;
124 w = z*z;
125 /* Break x^5*(T[1]+x^2*T[2]+...) into
126 * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
127 * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
128 */
129 r = T[1]+w*(T[3]+w*(T[5]+w*(T[7]+w*(T[9]+w*T[11]))));
130 v = z*(T[2]+w*(T[4]+w*(T[6]+w*(T[8]+w*(T[10]+w*T[12])))));
131 s = z*x;
132 r = y + z*(s*(r+v)+y);
133 r += T[0]*s;
134 w = x+r;
135 if(ix>=0x3FE59428) {
136 v = (double)iy;
137 return (double)(1-((hx>>30)&2))*(v-2.0*(x-(w*w/(w+v)-r)));
138 }
139 if(iy==1) return w;
140 else { /* if allow error up to 2 ulp,
141 simply return -1.0/(x+r) here */
142 /* compute -1.0/(x+r) accurately */
143 double a,t;
144 z = w;
145 __LO(z) = 0;
146 v = r-(z - x); /* z+v = r+x */
147 t = a = -1.0/w; /* a = -1.0/w */
148 __LO(t) = 0;
149 s = 1.0+t*z;
150 return t+a*(s+t*v);
151 }
152}
153