sysdeps/ieee754/ldbl-128/k_tanl.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, see
  32     <http://www.gnu.org/licenses/>.  */
  33
  34 /* __kernel_tanl( 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 <math.h>
  60 #include <math_private.h>
  61 static const long double
  62   one = 1.0L,
  63   pio4hi = 7.8539816339744830961566084581987569936977E-1L,
  64   pio4lo = 2.1679525325309452561992610065108379921906E-35L,
  65
  66   /* tan x = x + x^3 / 3 + x^5 T(x^2)/U(x^2)
  67      0 <= x <= 0.6743316650390625
  68      Peak relative error 8.0e-36  */
  69  TH =  3.333333333333333333333333333333333333333E-1L,
  70  T0 = -1.813014711743583437742363284336855889393E7L,
  71  T1 =  1.320767960008972224312740075083259247618E6L,
  72  T2 = -2.626775478255838182468651821863299023956E4L,
  73  T3 =  1.764573356488504935415411383687150199315E2L,
  74  T4 = -3.333267763822178690794678978979803526092E-1L,
  75
  76  U0 = -1.359761033807687578306772463253710042010E8L,
  77  U1 =  6.494370630656893175666729313065113194784E7L,
  78  U2 = -4.180787672237927475505536849168729386782E6L,
  79  U3 =  8.031643765106170040139966622980914621521E4L,
  80  U4 = -5.323131271912475695157127875560667378597E2L;
  81   /* 1.000000000000000000000000000000000000000E0 */
  82
  83
  84 long double
  85 __kernel_tanl (long double x, long double y, int iy)
  86 {
  87   long double z, r, v, w, s;
  88   int32_t ix, sign;
  89   ieee854_long_double_shape_type u, u1;
  90
  91   u.value = x;
  92   ix = u.parts32.w0 & 0x7fffffff;
  93   if (ix < 0x3fc60000)          /* x < 2**-57 */
  94     {
  95       if ((int) x == 0)
  96         {                       /* generate inexact */
  97           if ((ix | u.parts32.w1 | u.parts32.w2 | u.parts32.w3
  98                | (iy + 1)) == 0)
  99             return one / fabs (x);
 100           else
 101             return (iy == 1) ? x : -one / x;
 102         }
 103     }
 104   if (ix >= 0x3ffe5942) /* |x| >= 0.6743316650390625 */
 105     {
 106       if ((u.parts32.w0 & 0x80000000) != 0)
 107         {
 108           x = -x;
 109           y = -y;
 110           sign = -1;
 111         }
 112       else
 113         sign = 1;
 114       z = pio4hi - x;
 115       w = pio4lo - y;
 116       x = z + w;
 117       y = 0.0;
 118     }
 119   z = x * x;
 120   r = T0 + z * (T1 + z * (T2 + z * (T3 + z * T4)));
 121   v = U0 + z * (U1 + z * (U2 + z * (U3 + z * (U4 + z))));
 122   r = r / v;
 123
 124   s = z * x;
 125   r = y + z * (s * r + y);
 126   r += TH * s;
 127   w = x + r;
 128   if (ix >= 0x3ffe5942)
 129     {
 130       v = (long double) iy;
 131       w = (v - 2.0 * (x - (w * w / (w + v) - r)));
 132       if (sign < 0)
 133         w = -w;
 134       return w;
 135     }
 136   if (iy == 1)
 137     return w;
 138   else
 139     {                           /* if allow error up to 2 ulp,
 140                                    simply return -1.0/(x+r) here */
 141       /*  compute -1.0/(x+r) accurately */
 142       u1.value = w;
 143       u1.parts32.w2 = 0;
 144       u1.parts32.w3 = 0;
 145       v = r - (u1.value - x);           /* u1+v = r+x */
 146       z = -1.0 / w;
 147       u.value = z;
 148       u.parts32.w2 = 0;
 149       u.parts32.w3 = 0;
 150       s = 1.0 + u.value * u1.value;
 151       return u.value + z * (s + u.value * v);
 152     }
 153 }