libquadmath/math/tanq.c

   1 /*
   2  * ====================================================
   3  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   4  *
   5  * Developed at SunPro, a Sun Microsystems, Inc. business.
   6  * Permission to use, copy, modify, and distribute this
   7  * software is freely granted, provided that this notice
   8  * is preserved.
   9  * ====================================================
  10  */
  11
  12 /*
  13   Long double expansions are
  14   Copyright (C) 2001 Stephen L. Moshier <moshier@na-net.ornl.gov>
  15   and are incorporated herein by permission of the author.  The author
  16   reserves the right to distribute this material elsewhere under different
  17   copying permissions.  These modifications are distributed here under
  18   the following terms:
  19
  20     This library is free software; you can redistribute it and/or
  21     modify it under the terms of the GNU Lesser General Public
  22     License as published by the Free Software Foundation; either
  23     version 2.1 of the License, or (at your option) any later version.
  24
  25     This library is distributed in the hope that it will be useful,
  26     but WITHOUT ANY WARRANTY; without even the implied warranty of
  27     MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
  28     Lesser General Public License for more details.
  29
  30     You should have received a copy of the GNU Lesser General Public
  31     License along with this library; if not, write to the Free Software
  32     Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307  USA */
  33
  34 /* __quadmath_kernel_tanq( x, y, k )
  35  * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854
  36  * Input x is assumed to be bounded by ~pi/4 in magnitude.
  37  * Input y is the tail of x.
  38  * Input k indicates whether tan (if k=1) or
  39  * -1/tan (if k= -1) is returned.
  40  *
  41  * Algorithm
  42  *      1. Since tan(-x) = -tan(x), we need only to consider positive x.
  43  *      2. if x < 2^-57, return x with inexact if x!=0.
  44  *      3. tan(x) is approximated by a rational form x + x^3 / 3 + x^5 R(x^2)
  45  *          on [0,0.67433].
  46  *
  47  *         Note: tan(x+y) = tan(x) + tan'(x)*y
  48  *                        ~ tan(x) + (1+x*x)*y
  49  *         Therefore, for better accuracy in computing tan(x+y), let
  50  *              r = x^3 * R(x^2)
  51  *         then
  52  *              tan(x+y) = x + (x^3 / 3 + (x^2 *(r+y)+y))
  53  *
  54  *      4. For x in [0.67433,pi/4],  let y = pi/4 - x, then
  55  *              tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
  56  *                     = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
  57  */
  58
  59 #include "quadmath-imp.h"
  60
  61
  62
  63 static const __float128
  64   one = 1.0Q,
  65   pio4hi = 7.8539816339744830961566084581987569936977E-1Q,
  66   pio4lo = 2.1679525325309452561992610065108379921906E-35Q,
  67
  68   /* tan x = x + x^3 / 3 + x^5 T(x^2)/U(x^2)
  69      0 <= x <= 0.6743316650390625
  70      Peak relative error 8.0e-36  */
  71  TH =  3.333333333333333333333333333333333333333E-1Q,
  72  T0 = -1.813014711743583437742363284336855889393E7Q,
  73  T1 =  1.320767960008972224312740075083259247618E6Q,
  74  T2 = -2.626775478255838182468651821863299023956E4Q,
  75  T3 =  1.764573356488504935415411383687150199315E2Q,
  76  T4 = -3.333267763822178690794678978979803526092E-1Q,
  77
  78  U0 = -1.359761033807687578306772463253710042010E8Q,
  79  U1 =  6.494370630656893175666729313065113194784E7Q,
  80  U2 = -4.180787672237927475505536849168729386782E6Q,
  81  U3 =  8.031643765106170040139966622980914621521E4Q,
  82  U4 = -5.323131271912475695157127875560667378597E2Q;
  83   /* 1.000000000000000000000000000000000000000E0 */
  84
  85
  86 static __float128
  87 __quadmath_kernel_tanq (__float128 x, __float128 y, int iy)
  88 {
  89   __float128 z, r, v, w, s;
  90   int32_t ix, sign = 1;
  91   ieee854_float128 u, u1;
  92
  93   u.value = x;
  94   ix = u.words32.w0 & 0x7fffffff;
  95   if (ix < 0x3fc60000)          /* x < 2**-57 */
  96     {
  97       if ((int) x == 0)
  98         {                       /* generate inexact */
  99           if ((ix | u.words32.w1 | u.words32.w2 | u.words32.w3
 100                | (iy + 1)) == 0)
 101             return one / fabsq (x);
 102           else if (iy == 1)
 103             {
 104               math_check_force_underflow (x);
 105               return x;
 106             }
 107           else
 108             return -one / x;
 109         }
 110     }
 111   if (ix >= 0x3ffe5942) /* |x| >= 0.6743316650390625 */
 112     {
 113       if ((u.words32.w0 & 0x80000000) != 0)
 114         {
 115           x = -x;
 116           y = -y;
 117           sign = -1;
 118         }
 119       else
 120         sign = 1;
 121       z = pio4hi - x;
 122       w = pio4lo - y;
 123       x = z + w;
 124       y = 0.0;
 125     }
 126   z = x * x;
 127   r = T0 + z * (T1 + z * (T2 + z * (T3 + z * T4)));
 128   v = U0 + z * (U1 + z * (U2 + z * (U3 + z * (U4 + z))));
 129   r = r / v;
 130
 131   s = z * x;
 132   r = y + z * (s * r + y);
 133   r += TH * s;
 134   w = x + r;
 135   if (ix >= 0x3ffe5942)
 136     {
 137       v = (__float128) iy;
 138       w = (v - 2.0Q * (x - (w * w / (w + v) - r)));
 139       if (sign < 0)
 140         w = -w;
 141       return w;
 142     }
 143   if (iy == 1)
 144     return w;
 145   else
 146     {                           /* if allow error up to 2 ulp,
 147                                    simply return -1.0/(x+r) here */
 148       /*  compute -1.0/(x+r) accurately */
 149       u1.value = w;
 150       u1.words32.w2 = 0;
 151       u1.words32.w3 = 0;
 152       v = r - (u1.value - x);           /* u1+v = r+x */
 153       z = -1.0 / w;
 154       u.value = z;
 155       u.words32.w2 = 0;
 156       u.words32.w3 = 0;
 157       s = 1.0 + u.value * u1.value;
 158       return u.value + z * (s + u.value * v);
 159     }
 160 }
 161
 162
 163
 164 \f
 165
 166
 167
 168 /* tanq.c -- __float128 version of s_tan.c.
 169  * Conversion to IEEE quad long double by Jakub Jelinek, jj@ultra.linux.cz.
 170  */
 171
 172 /* @(#)s_tan.c 5.1 93/09/24 */
 173 /*
 174  * ====================================================
 175  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
 176  *
 177  * Developed at SunPro, a Sun Microsystems, Inc. business.
 178  * Permission to use, copy, modify, and distribute this
 179  * software is freely granted, provided that this notice
 180  * is preserved.
 181  * ====================================================
 182  */
 183
 184 /* tanl(x)
 185  * Return tangent function of x.
 186  *
 187  * kernel function:
 188  *      __quadmath_kernel_tanq  ... tangent function on [-pi/4,pi/4]
 189  *      __quadmath_rem_pio2q    ... argument reduction routine
 190  *
 191  * Method.
 192  *      Let S,C and T denote the sin, cos and tan respectively on
 193  *      [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2
 194  *      in [-pi/4 , +pi/4], and let n = k mod 4.
 195  *      We have
 196  *
 197  *          n        sin(x)      cos(x)        tan(x)
 198  *     ----------------------------------------------------------
 199  *          0          S           C             T
 200  *          1          C          -S            -1/T
 201  *          2         -S          -C             T
 202  *          3         -C           S            -1/T
 203  *     ----------------------------------------------------------
 204  *
 205  * Special cases:
 206  *      Let trig be any of sin, cos, or tan.
 207  *      trig(+-INF)  is NaN, with signals;
 208  *      trig(NaN)    is that NaN;
 209  *
 210  * Accuracy:
 211  *      TRIG(x) returns trig(x) nearly rounded
 212  */
 213
 214
 215 __float128
 216 tanq (__float128 x)
 217 {
 218         __float128 y[2],z=0.0Q;
 219         int64_t n, ix;
 220
 221     /* High word of x. */
 222         GET_FLT128_MSW64(ix,x);
 223
 224     /* |x| ~< pi/4 */
 225         ix &= 0x7fffffffffffffffLL;
 226         if(ix <= 0x3ffe921fb54442d1LL) return __quadmath_kernel_tanq(x,z,1);
 227
 228     /* tanl(Inf or NaN) is NaN */
 229         else if (ix>=0x7fff000000000000LL) {
 230             if (ix == 0x7fff000000000000LL) {
 231                 GET_FLT128_LSW64(n,x);
 232             }
 233             return x-x;         /* NaN */
 234         }
 235
 236     /* argument reduction needed */
 237         else {
 238             n = __quadmath_rem_pio2q(x,y);
 239                                         /*   1 -- n even, -1 -- n odd */
 240             return __quadmath_kernel_tanq(y[0],y[1],1-((n&1)<<1));
 241         }
 242 }