/* * ==================================================== * 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 are Copyright (C) 2001 Stephen L. Moshier <moshier@na-net.ornl.gov> and are incorporated herein by permission of the author. The author reserves the right to distribute this material elsewhere under different copying permissions. These modifications are distributed here under the following terms: This library is free software; you can redistribute it and/or modify it under the terms of the GNU Lesser General Public License as published by the Free Software Foundation; either version 2.1 of the License, or (at your option) any later version. This library is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License for more details. You should have received a copy of the GNU Lesser General Public License along with this library; if not, see <http://www.gnu.org/licenses/>. */ /* __quadmath_kernel_tanq( 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))) */ #include "quadmath-imp.h" static const __float128 one = 1, pio4hi = 7.8539816339744830961566084581987569936977E-1Q, pio4lo = 2.1679525325309452561992610065108379921906E-35Q, /* 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-1Q, T0 = -1.813014711743583437742363284336855889393E7Q, T1 = 1.320767960008972224312740075083259247618E6Q, T2 = -2.626775478255838182468651821863299023956E4Q, T3 = 1.764573356488504935415411383687150199315E2Q, T4 = -3.333267763822178690794678978979803526092E-1Q, U0 = -1.359761033807687578306772463253710042010E8Q, U1 = 6.494370630656893175666729313065113194784E7Q, U2 = -4.180787672237927475505536849168729386782E6Q, U3 = 8.031643765106170040139966622980914621521E4Q, U4 = -5.323131271912475695157127875560667378597E2Q; /* 1.000000000000000000000000000000000000000E0 */ __float128__quadmath_kernel_tanq (__float128 x, __float128 y, int iy){ __float128 z, r, v, w, s; int32_t ix, sign; ieee854_float128 u, u1; u.value = x; ix = u.words32.w0 & 0x7fffffff; if (ix < 0x3fc60000) /* x < 2**-57 */ { if ((int) x == 0) { /* generate inexact */ if ((ix | u.words32.w1 | u.words32.w2 | u.words32.w3 | (iy + 1)) == 0) return one / fabsq (x); else if (iy == 1) { math_check_force_underflow (x); return x; } else return -one / x; } } if (ix >= 0x3ffe5942) /* |x| >= 0.6743316650390625 */ { if ((u.words32.w0 & 0x80000000) != 0) { x = -x; y = -y; sign = -1; } else sign = 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 (ix >= 0x3ffe5942) { v = (__float128) iy; w = (v - 2.0 * (x - (w * w / (w + v) - r))); /* SIGN is set for arguments that reach this code, but not otherwise, resulting in warnings that it may be used uninitialized although in the cases where it is used it has always been set. */ 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.value = w; u1.words32.w2 = 0; u1.words32.w3 = 0; v = r - (u1.value - x); /* u1+v = r+x */ z = -1.0 / w; u.value = z; u.words32.w2 = 0; u.words32.w3 = 0; s = 1.0 + u.value * u1.value; return u.value + z * (s + u.value * v); }}
/*
* ====================================================
* 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 are
Copyright (C) 2001 Stephen L. Moshier <moshier@na-net.ornl.gov>
and are incorporated herein by permission of the author. The author
reserves the right to distribute this material elsewhere under different
copying permissions. These modifications are distributed here under
the following terms:
This library is free software; you can redistribute it and/or
modify it under the terms of the GNU Lesser General Public
License as published by the Free Software Foundation; either
version 2.1 of the License, or (at your option) any later version.
This library is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
Lesser General Public License for more details.
You should have received a copy of the GNU Lesser General Public
License along with this library; if not, see
<http://www.gnu.org/licenses/>. */
/* __quadmath_kernel_tanq( 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)))
*/
#include "quadmath-imp.h"
static const __float128
one = 1,
pio4hi = 7.8539816339744830961566084581987569936977E-1Q,
pio4lo = 2.1679525325309452561992610065108379921906E-35Q,
/* 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-1Q,
T0 = -1.813014711743583437742363284336855889393E7Q,
T1 = 1.320767960008972224312740075083259247618E6Q,
T2 = -2.626775478255838182468651821863299023956E4Q,
T3 = 1.764573356488504935415411383687150199315E2Q,
T4 = -3.333267763822178690794678978979803526092E-1Q,
U0 = -1.359761033807687578306772463253710042010E8Q,
U1 = 6.494370630656893175666729313065113194784E7Q,
U2 = -4.180787672237927475505536849168729386782E6Q,
U3 = 8.031643765106170040139966622980914621521E4Q,
U4 = -5.323131271912475695157127875560667378597E2Q;
/* 1.000000000000000000000000000000000000000E0 */
__float128
__quadmath_kernel_tanq (__float128 x, __float128 y, int iy)
{
__float128 z, r, v, w, s;
int32_t ix, sign;
ieee854_float128 u, u1;
u.value = x;
ix = u.words32.w0 & 0x7fffffff;
if (ix < 0x3fc60000) /* x < 2**-57 */
{
if ((int) x == 0)
{ /* generate inexact */
if ((ix | u.words32.w1 | u.words32.w2 | u.words32.w3
| (iy + 1)) == 0)
return one / fabsq (x);
else if (iy == 1)
{
math_check_force_underflow (x);
return x;
}
else
return -one / x;
}
}
if (ix >= 0x3ffe5942) /* |x| >= 0.6743316650390625 */
{
if ((u.words32.w0 & 0x80000000) != 0)
{
x = -x;
y = -y;
sign = -1;
}
else
sign = 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 (ix >= 0x3ffe5942)
{
v = (__float128) iy;
w = (v - 2.0 * (x - (w * w / (w + v) - r)));
/* SIGN is set for arguments that reach this code, but not
otherwise, resulting in warnings that it may be used
uninitialized although in the cases where it is used it has
always been set. */
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.value = w;
u1.words32.w2 = 0;
u1.words32.w3 = 0;
v = r - (u1.value - x); /* u1+v = r+x */
z = -1.0 / w;
u.value = z;
u.words32.w2 = 0;
u.words32.w3 = 0;
s = 1.0 + u.value * u1.value;
return u.value + z * (s + u.value * v);
}
}