sysdeps/ieee754/ldbl-96/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^-33, 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 <float.h>
  60 #include <math.h>
  61 #include <math_private.h>
  62 static const long double
  63   one = 1.0L,
  64   pio4hi = 0xc.90fdaa22168c235p-4L,
  65   pio4lo = -0x3.b399d747f23e32ecp-68L,
  66
  67   /* tan x = x + x^3 / 3 + x^5 T(x^2)/U(x^2)
  68      0 <= x <= 0.6743316650390625
  69      Peak relative error 8.0e-36  */
  70  TH =  3.333333333333333333333333333333333333333E-1L,
  71  T0 = -1.813014711743583437742363284336855889393E7L,
  72  T1 =  1.320767960008972224312740075083259247618E6L,
  73  T2 = -2.626775478255838182468651821863299023956E4L,
  74  T3 =  1.764573356488504935415411383687150199315E2L,
  75  T4 = -3.333267763822178690794678978979803526092E-1L,
  76
  77  U0 = -1.359761033807687578306772463253710042010E8L,
  78  U1 =  6.494370630656893175666729313065113194784E7L,
  79  U2 = -4.180787672237927475505536849168729386782E6L,
  80  U3 =  8.031643765106170040139966622980914621521E4L,
  81  U4 = -5.323131271912475695157127875560667378597E2L;
  82   /* 1.000000000000000000000000000000000000000E0 */
  83
  84
  85 long double
  86 __kernel_tanl (long double x, long double y, int iy)
  87 {
  88   long double z, r, v, w, s;
  89   long double absx = fabsl (x);
  90   int sign;
  91
  92   if (absx < 0x1p-33)
  93     {
  94       if ((int) x == 0)
  95         {                       /* generate inexact */
  96           if (x == 0 && iy == -1)
  97             return one / fabsl (x);
  98           else if (iy == 1)
  99             {
 100               if (absx < LDBL_MIN)
 101                 {
 102                   long double force_underflow = x * x;
 103                   math_force_eval (force_underflow);
 104                 }
 105               return x;
 106             }
 107           else
 108             return -one / x;
 109         }
 110     }
 111   if (absx >= 0.6743316650390625L)
 112     {
 113       if (signbit (x))
 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 (absx >= 0.6743316650390625L)
 136     {
 137       v = (long double) iy;
 138       w = (v - 2.0 * (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     return -1.0 / (x + r);
 147 }