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243 lines
6.8 KiB
C
243 lines
6.8 KiB
C
/*
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* ====================================================
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* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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*
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* Developed at SunPro, a Sun Microsystems, Inc. business.
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* Permission to use, copy, modify, and distribute this
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* software is freely granted, provided that this notice
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* is preserved.
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* ====================================================
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*/
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/*
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Long double expansions are
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Copyright (C) 2001 Stephen L. Moshier <moshier@na-net.ornl.gov>
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and are incorporated herein by permission of the author. The author
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reserves the right to distribute this material elsewhere under different
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copying permissions. These modifications are distributed here under
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the following terms:
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This library is free software; you can redistribute it and/or
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modify it under the terms of the GNU Lesser General Public
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License as published by the Free Software Foundation; either
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version 2.1 of the License, or (at your option) any later version.
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This library is distributed in the hope that it will be useful,
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but WITHOUT ANY WARRANTY; without even the implied warranty of
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MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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Lesser General Public License for more details.
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You should have received a copy of the GNU Lesser General Public
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License along with this library; if not, write to the Free Software
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Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA */
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/* __quadmath_kernel_tanq( x, y, k )
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* kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854
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* Input x is assumed to be bounded by ~pi/4 in magnitude.
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* Input y is the tail of x.
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* Input k indicates whether tan (if k=1) or
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* -1/tan (if k= -1) is returned.
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*
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* Algorithm
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* 1. Since tan(-x) = -tan(x), we need only to consider positive x.
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* 2. if x < 2^-57, return x with inexact if x!=0.
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* 3. tan(x) is approximated by a rational form x + x^3 / 3 + x^5 R(x^2)
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* on [0,0.67433].
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*
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* Note: tan(x+y) = tan(x) + tan'(x)*y
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* ~ tan(x) + (1+x*x)*y
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* Therefore, for better accuracy in computing tan(x+y), let
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* r = x^3 * R(x^2)
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* then
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* tan(x+y) = x + (x^3 / 3 + (x^2 *(r+y)+y))
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*
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* 4. For x in [0.67433,pi/4], let y = pi/4 - x, then
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* tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
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* = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
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*/
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#include "quadmath-imp.h"
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static const __float128
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one = 1.0Q,
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pio4hi = 7.8539816339744830961566084581987569936977E-1Q,
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pio4lo = 2.1679525325309452561992610065108379921906E-35Q,
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/* tan x = x + x^3 / 3 + x^5 T(x^2)/U(x^2)
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0 <= x <= 0.6743316650390625
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Peak relative error 8.0e-36 */
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TH = 3.333333333333333333333333333333333333333E-1Q,
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T0 = -1.813014711743583437742363284336855889393E7Q,
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T1 = 1.320767960008972224312740075083259247618E6Q,
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T2 = -2.626775478255838182468651821863299023956E4Q,
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T3 = 1.764573356488504935415411383687150199315E2Q,
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T4 = -3.333267763822178690794678978979803526092E-1Q,
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U0 = -1.359761033807687578306772463253710042010E8Q,
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U1 = 6.494370630656893175666729313065113194784E7Q,
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U2 = -4.180787672237927475505536849168729386782E6Q,
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U3 = 8.031643765106170040139966622980914621521E4Q,
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U4 = -5.323131271912475695157127875560667378597E2Q;
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/* 1.000000000000000000000000000000000000000E0 */
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static __float128
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__quadmath_kernel_tanq (__float128 x, __float128 y, int iy)
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{
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__float128 z, r, v, w, s;
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int32_t ix, sign = 1;
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ieee854_float128 u, u1;
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u.value = x;
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ix = u.words32.w0 & 0x7fffffff;
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if (ix < 0x3fc60000) /* x < 2**-57 */
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{
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if ((int) x == 0)
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{ /* generate inexact */
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if ((ix | u.words32.w1 | u.words32.w2 | u.words32.w3
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| (iy + 1)) == 0)
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return one / fabsq (x);
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else if (iy == 1)
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{
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math_check_force_underflow (x);
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return x;
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}
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else
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return -one / x;
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}
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}
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if (ix >= 0x3ffe5942) /* |x| >= 0.6743316650390625 */
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{
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if ((u.words32.w0 & 0x80000000) != 0)
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{
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x = -x;
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y = -y;
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sign = -1;
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}
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else
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sign = 1;
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z = pio4hi - x;
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w = pio4lo - y;
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x = z + w;
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y = 0.0;
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}
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z = x * x;
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r = T0 + z * (T1 + z * (T2 + z * (T3 + z * T4)));
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v = U0 + z * (U1 + z * (U2 + z * (U3 + z * (U4 + z))));
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r = r / v;
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s = z * x;
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r = y + z * (s * r + y);
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r += TH * s;
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w = x + r;
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if (ix >= 0x3ffe5942)
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{
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v = (__float128) iy;
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w = (v - 2.0Q * (x - (w * w / (w + v) - r)));
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if (sign < 0)
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w = -w;
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return w;
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}
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if (iy == 1)
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return w;
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else
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{ /* if allow error up to 2 ulp,
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simply return -1.0/(x+r) here */
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/* compute -1.0/(x+r) accurately */
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u1.value = w;
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u1.words32.w2 = 0;
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u1.words32.w3 = 0;
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v = r - (u1.value - x); /* u1+v = r+x */
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z = -1.0 / w;
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u.value = z;
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u.words32.w2 = 0;
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u.words32.w3 = 0;
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s = 1.0 + u.value * u1.value;
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return u.value + z * (s + u.value * v);
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}
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}
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/* tanq.c -- __float128 version of s_tan.c.
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* Conversion to IEEE quad long double by Jakub Jelinek, jj@ultra.linux.cz.
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*/
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/* @(#)s_tan.c 5.1 93/09/24 */
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/*
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* ====================================================
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* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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*
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* Developed at SunPro, a Sun Microsystems, Inc. business.
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* Permission to use, copy, modify, and distribute this
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* software is freely granted, provided that this notice
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* is preserved.
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* ====================================================
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*/
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/* tanl(x)
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* Return tangent function of x.
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*
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* kernel function:
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* __quadmath_kernel_tanq ... tangent function on [-pi/4,pi/4]
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* __quadmath_rem_pio2q ... argument reduction routine
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*
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* Method.
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* Let S,C and T denote the sin, cos and tan respectively on
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* [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2
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* in [-pi/4 , +pi/4], and let n = k mod 4.
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* We have
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*
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* n sin(x) cos(x) tan(x)
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* ----------------------------------------------------------
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* 0 S C T
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* 1 C -S -1/T
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* 2 -S -C T
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* 3 -C S -1/T
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* ----------------------------------------------------------
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*
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* Special cases:
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* Let trig be any of sin, cos, or tan.
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* trig(+-INF) is NaN, with signals;
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* trig(NaN) is that NaN;
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*
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* Accuracy:
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* TRIG(x) returns trig(x) nearly rounded
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*/
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__float128
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tanq (__float128 x)
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{
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__float128 y[2],z=0.0Q;
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int64_t n, ix;
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/* High word of x. */
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GET_FLT128_MSW64(ix,x);
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/* |x| ~< pi/4 */
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ix &= 0x7fffffffffffffffLL;
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if(ix <= 0x3ffe921fb54442d1LL) return __quadmath_kernel_tanq(x,z,1);
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/* tanl(Inf or NaN) is NaN */
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else if (ix>=0x7fff000000000000LL) {
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if (ix == 0x7fff000000000000LL) {
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GET_FLT128_LSW64(n,x);
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}
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return x-x; /* NaN */
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}
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/* argument reduction needed */
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else {
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n = __quadmath_rem_pio2q(x,y);
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/* 1 -- n even, -1 -- n odd */
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return __quadmath_kernel_tanq(y[0],y[1],1-((n&1)<<1));
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}
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}
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