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/* @(#)k_tan.c 5.1 93/09/24 */
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunPro, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
#ifndef lint
static char rcsid
[] = "$\Id: k_tan.c,v 1.2 1995/05/30 05:49:14 rgrimes Exp $";
#endif
/* __kernel_tan( x, y, k )
* kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854
* Input x is assumed to be bounded by ~pi/4 in magnitude.
* Input y is the tail of x.
* Input k indicates whether tan (if k=1) or
* -1/tan (if k= -1) is returned.
*
* Algorithm
* 1. Since tan(-x) = -tan(x), we need only to consider positive x.
* 2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0.
* 3. tan(x) is approximated by a odd polynomial of degree 27 on
* [0,0.67434]
* 3 27
* tan(x) ~ x + T1*x + ... + T13*x
* where
*
* |tan(x) 2 4 26 | -59.2
* |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2
* | x |
*
* Note: tan(x+y) = tan(x) + tan'(x)*y
* ~ tan(x) + (1+x*x)*y
* Therefore, for better accuracy in computing tan(x+y), let
* 3 2 2 2 2
* r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
* then
* 3 2
* tan(x+y) = x + (T1*x + (x *(r+y)+y))
*
* 4. For x in [0.67434,pi/4], let y = pi/4 - x, then
* tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
* = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
*/
#include "math.h"
#include "math_private.h"
#ifdef __STDC__
static const double
#else
static double
#endif
one
= 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
pio4
= 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
pio4lo
= 3.06161699786838301793e-17, /* 0x3C81A626, 0x33145C07 */
T
[] = {
3.33333333333334091986e-01, /* 0x3FD55555, 0x55555563 */
1.33333333333201242699e-01, /* 0x3FC11111, 0x1110FE7A */
5.39682539762260521377e-02, /* 0x3FABA1BA, 0x1BB341FE */
2.18694882948595424599e-02, /* 0x3F9664F4, 0x8406D637 */
8.86323982359930005737e-03, /* 0x3F8226E3, 0xE96E8493 */
3.59207910759131235356e-03, /* 0x3F6D6D22, 0xC9560328 */
1.45620945432529025516e-03, /* 0x3F57DBC8, 0xFEE08315 */
5.88041240820264096874e-04, /* 0x3F4344D8, 0xF2F26501 */
2.46463134818469906812e-04, /* 0x3F3026F7, 0x1A8D1068 */
7.81794442939557092300e-05, /* 0x3F147E88, 0xA03792A6 */
7.14072491382608190305e-05, /* 0x3F12B80F, 0x32F0A7E9 */
-1.85586374855275456654e-05, /* 0xBEF375CB, 0xDB605373 */
2.59073051863633712884e-05, /* 0x3EFB2A70, 0x74BF7AD4 */
};
#ifdef __STDC__
double __kernel_tan
(double x
, double y
, int iy
)
#else
double __kernel_tan
(x
, y
, iy
)
double x
,y
; int iy
;
#endif
{
double z
,r
,v
,w
,s
;
int32_t ix
,hx
;
GET_HIGH_WORD
(hx
,x
);
ix
= hx
&0x7fffffff; /* high word of |x| */
if(ix
<0x3e300000) /* x < 2**-28 */
{if((int)x
==0) { /* generate inexact */
u_int32_t low
;
GET_LOW_WORD
(low
,x
);
if(((ix
|low
)|(iy
+1))==0) return one
/fabs(x
);
else return (iy
==1)? x
: -one
/x
;
}
}
if(ix
>=0x3FE59428) { /* |x|>=0.6744 */
if(hx
<0) {x
= -x
; y
= -y
;}
z
= pio4
-x
;
w
= pio4lo
-y
;
x
= z
+w
; y
= 0.0;
}
z
= x
*x
;
w
= z
*z
;
/* Break x^5*(T[1]+x^2*T[2]+...) into
* x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
* x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
*/
r
= T
[1]+w
*(T
[3]+w
*(T
[5]+w
*(T
[7]+w
*(T
[9]+w
*T
[11]))));
v
= z
*(T
[2]+w
*(T
[4]+w
*(T
[6]+w
*(T
[8]+w
*(T
[10]+w
*T
[12])))));
s
= z
*x
;
r
= y
+ z
*(s
*(r
+v
)+y
);
r
+= T
[0]*s
;
w
= x
+r
;
if(ix
>=0x3FE59428) {
v
= (double)iy
;
return (double)(1-((hx
>>30)&2))*(v
-2.0*(x
-(w
*w
/(w
+v
)-r
)));
}
if(iy
==1) return w
;
else { /* if allow error up to 2 ulp,
simply return -1.0/(x+r) here */
/* compute -1.0/(x+r) accurately */
double a
,t
;
z
= w
;
SET_LOW_WORD
(z
,0);
v
= r
-(z
- x
); /* z+v = r+x */
t
= a
= -1.0/w
; /* a = -1.0/w */
SET_LOW_WORD
(t
,0);
s
= 1.0+t
*z
;
return t
+a
*(s
+t
*v
);
}
}