__tan.c revision 25c28e83beb90e7c80452a7c818c5e6f73a07dc8
/*
* CDDL HEADER START
*
* The contents of this file are subject to the terms of the
* Common Development and Distribution License (the "License").
* You may not use this file except in compliance with the License.
*
* You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
* or http://www.opensolaris.org/os/licensing.
* See the License for the specific language governing permissions
* and limitations under the License.
*
* When distributing Covered Code, include this CDDL HEADER in each
* file and include the License file at usr/src/OPENSOLARIS.LICENSE.
* If applicable, add the following below this CDDL HEADER, with the
* fields enclosed by brackets "[]" replaced with your own identifying
* information: Portions Copyright [yyyy] [name of copyright owner]
*
* CDDL HEADER END
*/
/*
* Copyright 2011 Nexenta Systems, Inc. All rights reserved.
*/
/*
* Copyright 2006 Sun Microsystems, Inc. All rights reserved.
* Use is subject to license terms.
*/
/* INDENT OFF */
/*
* __k_tan( double x; double y; int k )
* kernel tan/cotan function on [-pi/4, pi/4], pi/4 ~ 0.785398164
* Input x is assumed to be bounded by ~pi/4 in magnitude.
* Input y is the tail of x.
* Input k indicate -- tan if k=0; else -1/tan
*
* Table look up algorithm
* 1. by tan(-x) = -tan(x), need only to consider positive x
* 2. if x < 5/32 = [0x3fc40000, 0] = 0.15625 , then
* if x < 2^-27 (hx < 0x3e400000 0), set w=x with inexact if x != 0
* else
* z = x*x;
* w = x + (y+(x*z)*(t1+z*(t2+z*(t3+z*(t4+z*(t5+z*t6))))))
* return (k == 0)? w: 1/w;
* 3. else
* ht = (hx + 0x4000)&0x7fff8000 (round x to a break point t)
* lt = 0
* i = (hy-0x3fc40000)>>15; (i<=64)
* x' = (x - t)+y (|x'| ~<= 2^-7)
* By
* tan(t+x')
* = (tan(t)+tan(x'))/(1-tan(x')tan(t))
* We have
* sin(x')+tan(t)*(tan(t)*sin(x'))
* = tan(t) + ------------------------------- for k=0
* cos(x') - tan(t)*sin(x')
*
* cos(x') - tan(t)*sin(x')
* = - -------------------------------------- for k=1
* tan(t) + tan(t)*(cos(x')-1) + sin(x')
*
*
* where tan(t) is from the table,
* sin(x') = x + pp1*x^3 + pp2*x^5
* cos(x') = 1 + qq1*x^2 + qq2*x^4
*/
#include "libm.h"
extern const double _TBL_tan_hi[], _TBL_tan_lo[];
static const double q[] = {
/* one = */ 1.0,
/*
* 2 2 -59.56
* |sin(x) - pp1*x*(pp2+x *(pp3+x )| <= 2 for |x|<1/64
*/
/* pp1 = */ 8.33326120969096230395312119298978359438478946686e-0003,
/* pp2 = */ 1.20001038589438965215025680596868692381425944526e+0002,
/* pp3 = */ -2.00001730975089451192161504877731204032897949219e+0001,
/*
* 2 2 -56.19
* |cos(x) - (1+qq1*x (qq2+x ))| <= 2 for |x|<=1/128
*/
/* qq1 = */ 4.16665486385721928197511942926212213933467864990e-0002,
/* qq2 = */ -1.20000339921340035687080671777948737144470214844e+0001,
/*
* |tan(x) - PF(x)|
* |--------------| <= 2^-58.57 for |x|<0.15625
* | x |
*
* where (let z = x*x)
* PF(x) = x + (t1*x*z)(t2 + z(t3 + z))(t4 + z)(t5 + z(t6 + z))
*/
/* t1 = */ 3.71923358986516816929168705030406272271648049355e-0003,
/* t2 = */ 6.02645120354857866118436504621058702468872070312e+0000,
/* t3 = */ 2.42627327587398156083509093150496482849121093750e+0000,
/* t4 = */ 2.44968983934252770851003333518747240304946899414e+0000,
/* t5 = */ 6.07089252571767978849948121933266520500183105469e+0000,
/* t6 = */ -2.49403756995593761658369658107403665781021118164e+0000,
};
#define one q[0]
#define pp1 q[1]
#define pp2 q[2]
#define pp3 q[3]
#define qq1 q[4]
#define qq2 q[5]
#define t1 q[6]
#define t2 q[7]
#define t3 q[8]
#define t4 q[9]
#define t5 q[10]
#define t6 q[11]
/* INDENT ON */
double
__k_tan(double x, double y, int k) {
double a, t, z, w = 0.0L, s, c, r, rh, xh, xl;
int i, j, hx, ix;
t = one;
hx = ((int *) &x)[HIWORD];
ix = hx & 0x7fffffff;
if (ix < 0x3fc40000) { /* 0.15625 */
if (ix < 0x3e400000) { /* 2^-27 */
if ((i = (int) x) == 0) /* generate inexact */
w = x;
t = y;
} else {
z = x * x;
t = y + (((t1 * x) * z) * (t2 + z * (t3 + z))) *
((t4 + z) * (t5 + z * (t6 + z)));
w = x + t;
}
if (k == 0)
return (w);
/*
* Compute -1/(x+T) with great care
* Let r = -1/(x+T), rh = r chopped to 20 bits.
* Also let xh = x+T chopped to 20 bits, xl = (x-xh)+T. Then
* -1/(x+T) = rh + (-1/(x+T)-rh) = rh + r*(1+rh*(x+T))
* = rh + r*((1+rh*xh)+rh*xl).
*/
rh = r = -one / w;
((int *) &rh)[LOWORD] = 0;
xh = w;
((int *) &xh)[LOWORD] = 0;
xl = (x - xh) + t;
return (rh + r * ((one + rh * xh) + rh * xl));
}
j = (ix + 0x4000) & 0x7fff8000;
i = (j - 0x3fc40000) >> 15;
((int *) &t)[HIWORD] = j;
if (hx > 0)
x = y - (t - x);
else
x = -y - (t + x);
a = _TBL_tan_hi[i];
z = x * x;
s = (pp1 * x) * (pp2 + z * (pp3 + z)); /* sin(x) */
t = (qq1 * z) * (qq2 + z); /* cos(x) - 1 */
if (k == 0) {
w = a * s;
t = _TBL_tan_lo[i] + (s + a * w) / (one - (w - t));
return (hx < 0 ? -a - t : a + t);
} else {
w = s + a * t;
c = w + _TBL_tan_lo[i];
t = a * s - t;
/*
* Now try to compute [(1-T)/(a+c)] accurately
*
* Let r = 1/(a+c), rh = (1-T)*r chopped to 20 bits.
* Also let xh = a+c chopped to 20 bits, xl = (a-xh)+c. Then
* (1-T)/(a+c) = rh + ((1-T)/(a+c)-rh)
* = rh + r*(1-T-rh*(a+c))
* = rh + r*((1-T-rh*xh)-rh*xl)
* = rh + r*(((1-rh*xh)-T)-rh*xl)
*/
r = one / (a + c);
rh = (one - t) * r;
((int *) &rh)[LOWORD] = 0;
xh = a + c;
((int *) &xh)[LOWORD] = 0;
xl = (a - xh) + c;
z = rh + r * (((one - rh * xh) - t) - rh * xl);
return (hx >= 0 ? -z : z);
}
}