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/* INDENT OFF */
31/*
32 * __k_tan( double x;  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 = [0x3fc40000, 0] = 0.15625 , then
41 *	     if x < 2^-27 (hx < 0x3e400000 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 + 0x4000)&0x7fff8000	(round x to a break point t)
48 *		lt = 0
49 *		i  = (hy-0x3fc40000)>>15;	(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 + pp2*x^5
66 *			cos(x') = 1 + qq1*x^2 + qq2*x^4
67 */
68
69#include "libm.h"
70
71extern const double _TBL_tan_hi[], _TBL_tan_lo[];
72static const double q[] = {
73/* one  = */  1.0,
74/*
75 *                       2       2       -59.56
76 * |sin(x) - pp1*x*(pp2+x *(pp3+x )| <= 2        for |x|<1/64
77 */
78/* pp1  = */  8.33326120969096230395312119298978359438478946686e-0003,
79/* pp2  = */  1.20001038589438965215025680596868692381425944526e+0002,
80/* pp3  = */ -2.00001730975089451192161504877731204032897949219e+0001,
81
82/*
83 *                   2      2        -56.19
84 * |cos(x) - (1+qq1*x (qq2+x ))| <= 2        for |x|<=1/128
85 */
86/* qq1  = */  4.16665486385721928197511942926212213933467864990e-0002,
87/* qq2  = */ -1.20000339921340035687080671777948737144470214844e+0001,
88
89/*
90 * |tan(x) - PF(x)|
91 * |--------------| <= 2^-58.57 for |x|<0.15625
92 * |      x       |
93 *
94 * where (let z = x*x)
95 *	PF(x) = x + (t1*x*z)(t2 + z(t3 + z))(t4 + z)(t5 + z(t6 + z))
96 */
97/* t1 = */  3.71923358986516816929168705030406272271648049355e-0003,
98/* t2 = */  6.02645120354857866118436504621058702468872070312e+0000,
99/* t3 = */  2.42627327587398156083509093150496482849121093750e+0000,
100/* t4 = */  2.44968983934252770851003333518747240304946899414e+0000,
101/* t5 = */  6.07089252571767978849948121933266520500183105469e+0000,
102/* t6 = */ -2.49403756995593761658369658107403665781021118164e+0000,
103};
104
105
106#define	one q[0]
107#define	pp1 q[1]
108#define	pp2 q[2]
109#define	pp3 q[3]
110#define	qq1 q[4]
111#define	qq2 q[5]
112#define	t1  q[6]
113#define	t2  q[7]
114#define	t3  q[8]
115#define	t4  q[9]
116#define	t5  q[10]
117#define	t6  q[11]
118
119/* INDENT ON */
120
121
122double
123__k_tan(double x, double y, int k) {
124	double a, t, z, w = 0.0L, s, c, r, rh, xh, xl;
125	int i, j, hx, ix;
126
127	t = one;
128	hx = ((int *) &x)[HIWORD];
129	ix = hx & 0x7fffffff;
130	if (ix < 0x3fc40000) {		/* 0.15625 */
131		if (ix < 0x3e400000) {	/* 2^-27 */
132			if ((i = (int) x) == 0)		/* generate inexact */
133				w = x;
134			t = y;
135		} else {
136			z = x * x;
137			t = y + (((t1 * x) * z) * (t2 + z * (t3 + z))) *
138				((t4 + z) * (t5 + z * (t6 + z)));
139			w = x + t;
140		}
141		if (k == 0)
142			return (w);
143		/*
144		 * Compute -1/(x+T) with great care
145		 * Let r = -1/(x+T), rh = r chopped to 20 bits.
146		 * Also let xh	= x+T chopped to 20 bits, xl = (x-xh)+T. Then
147		 *   -1/(x+T)	= rh + (-1/(x+T)-rh) = rh + r*(1+rh*(x+T))
148		 *		= rh + r*((1+rh*xh)+rh*xl).
149		 */
150		rh = r = -one / w;
151		((int *) &rh)[LOWORD] = 0;
152		xh = w;
153		((int *) &xh)[LOWORD] = 0;
154		xl = (x - xh) + t;
155		return (rh + r * ((one + rh * xh) + rh * xl));
156	}
157	j = (ix + 0x4000) & 0x7fff8000;
158	i = (j - 0x3fc40000) >> 15;
159	((int *) &t)[HIWORD] = j;
160	if (hx > 0)
161		x = y - (t - x);
162	else
163		x = -y - (t + x);
164	a = _TBL_tan_hi[i];
165	z = x * x;
166	s = (pp1 * x) * (pp2 + z * (pp3 + z));	/* sin(x) */
167	t = (qq1 * z) * (qq2 + z);		/* cos(x) - 1 */
168	if (k == 0) {
169		w = a * s;
170		t = _TBL_tan_lo[i] + (s + a * w) / (one - (w - t));
171		return (hx < 0 ? -a - t : a + t);
172	} else {
173		w = s + a * t;
174		c = w + _TBL_tan_lo[i];
175		t = a * s - t;
176		/*
177		 * Now try to compute [(1-T)/(a+c)] accurately
178		 *
179		 * Let r = 1/(a+c), rh = (1-T)*r chopped to 20 bits.
180		 * Also let xh = a+c chopped to 20 bits, xl = (a-xh)+c. Then
181		 *	(1-T)/(a+c) = rh + ((1-T)/(a+c)-rh)
182		 *		= rh + r*(1-T-rh*(a+c))
183		 *		= rh + r*((1-T-rh*xh)-rh*xl)
184		 *		= rh + r*(((1-rh*xh)-T)-rh*xl)
185		 */
186		r = one / (a + c);
187		rh = (one - t) * r;
188		((int *) &rh)[LOWORD] = 0;
189		xh = a + c;
190		((int *) &xh)[LOWORD] = 0;
191		xl = (a - xh) + c;
192		z = rh + r * (((one - rh * xh) - t) - rh * xl);
193		return (hx >= 0 ? -z : z);
194	}
195}
196