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/* s_tanl.c -- long double version of s_tan.c.
* Conversion to IEEE quad long double by Jakub Jelinek, jj@ultra.linux.cz.
*/
/* @(#)s_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.
* ====================================================
*/
/* tanl(x)
* Return tangent function of x.
*
* kernel function:
* __kernel_tanl ... tangent function on [-pi/4,pi/4]
* __ieee754_rem_pio2l ... argument reduction routine
*
* Method.
* Let S,C and T denote the sin, cos and tan respectively on
* [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2
* in [-pi/4 , +pi/4], and let n = k mod 4.
* We have
*
* n sin(x) cos(x) tan(x)
* ----------------------------------------------------------
* 0 S C T
* 1 C -S -1/T
* 2 -S -C T
* 3 -C S -1/T
* ----------------------------------------------------------
*
* Special cases:
* Let trig be any of sin, cos, or tan.
* trig(+-INF) is NaN, with signals;
* trig(NaN) is that NaN;
*
* Accuracy:
* TRIG(x) returns trig(x) nearly rounded
*/
#include <math.h>
#include "mathl.h"
#include "trigl.h"
#ifdef HAVE_SINL
#ifdef HAVE_COSL
#include "trigl.c"
#endif
#endif
/*
* ====================================================
* 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.
* ====================================================
*/
/*
Long double expansions contributed by
Stephen L. Moshier <moshier@na-net.ornl.gov>
*/
/* __kernel_tanl( 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^-57, return x with inexact if x!=0.
* 3. tan(x) is approximated by a rational form x + x^3 / 3 + x^5 R(x^2)
* on [0,0.67433].
*
* 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
* r = x^3 * R(x^2)
* then
* tan(x+y) = x + (x^3 / 3 + (x^2 *(r+y)+y))
*
* 4. For x in [0.67433,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)))
*/
static const long double
pio4hi = 7.8539816339744830961566084581987569936977E-1L,
pio4lo = 2.1679525325309452561992610065108379921906E-35L,
/* tan x = x + x^3 / 3 + x^5 T(x^2)/U(x^2)
0 <= x <= 0.6743316650390625
Peak relative error 8.0e-36 */
TH = 3.333333333333333333333333333333333333333E-1L,
T0 = -1.813014711743583437742363284336855889393E7L,
T1 = 1.320767960008972224312740075083259247618E6L,
T2 = -2.626775478255838182468651821863299023956E4L,
T3 = 1.764573356488504935415411383687150199315E2L,
T4 = -3.333267763822178690794678978979803526092E-1L,
U0 = -1.359761033807687578306772463253710042010E8L,
U1 = 6.494370630656893175666729313065113194784E7L,
U2 = -4.180787672237927475505536849168729386782E6L,
U3 = 8.031643765106170040139966622980914621521E4L,
U4 = -5.323131271912475695157127875560667378597E2L;
/* 1.000000000000000000000000000000000000000E0 */
long double
kernel_tanl (long double x, long double y, int iy)
{
long double z, r, v, w, s, u, u1;
int invert = 0, sign;
sign = 1;
if (x < 0)
{
x = -x;
y = -y;
sign = -1;
}
if (x < 0.000000000000000006938893903907228377647697925567626953125L) /* x < 2**-57 */
{
if ((int) x == 0)
{ /* generate inexact */
if (iy == -1 && x == 0.0)
return 1.0L / fabs (x);
else
return (iy == 1) ? x : -1.0L / x;
}
}
if (x >= 0.6743316650390625) /* |x| >= 0.6743316650390625 */
{
invert = 1;
z = pio4hi - x;
w = pio4lo - y;
x = z + w;
y = 0.0;
}
z = x * x;
r = T0 + z * (T1 + z * (T2 + z * (T3 + z * T4)));
v = U0 + z * (U1 + z * (U2 + z * (U3 + z * (U4 + z))));
r = r / v;
s = z * x;
r = y + z * (s * r + y);
r += TH * s;
w = x + r;
if (invert)
{
v = (long double) iy;
w = (v - 2.0 * (x - (w * w / (w + v) - r)));
if (sign < 0)
w = -w;
return w;
}
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 */
u1 = (double) w;
v = r - (u1 - x);
z = -1.0 / w;
u = (double) z;
s = 1.0 + u * u1;
return u + z * (s + u * v);
}
}
long double
tanl (long double x)
{
long double y[2], z = 0.0L;
int n;
/* |x| ~< pi/4 */
if (x >= -0.7853981633974483096156608458198757210492 &&
x <= 0.7853981633974483096156608458198757210492)
return kernel_tanl (x, z, 1);
/* tanl(Inf or NaN) is NaN, tanl(0) is 0 */
else if (x + x == x || x != x)
return x - x; /* NaN */
/* argument reduction needed */
else
{
n = ieee754_rem_pio2l (x, y);
/* 1 -- n even, -1 -- n odd */
return kernel_tanl (y[0], y[1], 1 - ((n & 1) << 1));
}
}
#if 0
int
main ()
{
printf ("%.16Lg\n", tanl(0.7853981633974483096156608458198757210492));
printf ("%.16Lg\n", tanl(-0.7853981633974483096156608458198757210492));
printf ("%.16Lg\n", tanl(0.7853981633974483096156608458198757210492 *3));
printf ("%.16Lg\n", tanl(-0.7853981633974483096156608458198757210492 *31));
printf ("%.16Lg\n", tanl(0.7853981633974483096156608458198757210492 / 2));
printf ("%.16Lg\n", tanl(0.7853981633974483096156608458198757210492 * 3/2));
printf ("%.16Lg\n", tanl(0.7853981633974483096156608458198757210492 * 5/2));
}
#endif