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 <libc-internal.h>
  61 #include <math.h>
  62 #include <math_private.h>
  63 static const long double
  64   one = 1.0L,
  65   pio4hi = 0xc.90fdaa22168c235p-4L,
  66   pio4lo = -0x3.b399d747f23e32ecp-68L,
  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-1L,
  72  T0 = -1.813014711743583437742363284336855889393E7L,
  73  T1 =  1.320767960008972224312740075083259247618E6L,
  74  T2 = -2.626775478255838182468651821863299023956E4L,
  75  T3 =  1.764573356488504935415411383687150199315E2L,
  76  T4 = -3.333267763822178690794678978979803526092E-1L,
  77
  78  U0 = -1.359761033807687578306772463253710042010E8L,
  79  U1 =  6.494370630656893175666729313065113194784E7L,
  80  U2 = -4.180787672237927475505536849168729386782E6L,
  81  U3 =  8.031643765106170040139966622980914621521E4L,
  82  U4 = -5.323131271912475695157127875560667378597E2L;
  83   /* 1.000000000000000000000000000000000000000E0 */
  84
  85
  86 long double
  87 __kernel_tanl (long double x, long double y, int iy)
  88 {
  89   long double z, r, v, w, s;
  90   long double absx = fabsl (x);
  91   int sign;
  92
  93   if (absx < 0x1p-33)
  94     {
  95       if ((int) x == 0)
  96         {                       /* generate inexact */
  97           if (x == 0 && iy == -1)
  98             return one / fabsl (x);
  99           else if (iy == 1)
 100             {
 101               if (absx < LDBL_MIN)
 102                 {
 103                   long double force_underflow = x * x;
 104                   math_force_eval (force_underflow);
 105                 }
 106               return x;
 107             }
 108           else
 109             return -one / x;
 110         }
 111     }
 112   if (absx >= 0.6743316650390625L)
 113     {
 114       if (signbit (x))
 115         {
 116           x = -x;
 117           y = -y;
 118           sign = -1;
 119         }
 120       else
 121         sign = 1;
 122       z = pio4hi - x;
 123       w = pio4lo - y;
 124       x = z + w;
 125       y = 0.0;
 126     }
 127   z = x * x;
 128   r = T0 + z * (T1 + z * (T2 + z * (T3 + z * T4)));
 129   v = U0 + z * (U1 + z * (U2 + z * (U3 + z * (U4 + z))));
 130   r = r / v;
 131
 132   s = z * x;
 133   r = y + z * (s * r + y);
 134   r += TH * s;
 135   w = x + r;
 136   if (absx >= 0.6743316650390625L)
 137     {
 138       v = (long double) iy;
 139       w = (v - 2.0 * (x - (w * w / (w + v) - r)));
 140       /* SIGN is set for arguments that reach this code, but not
 141         otherwise, resulting in warnings that it may be used
 142         uninitialized although in the cases where it is used it has
 143         always been set.  */
 144       DIAG_PUSH_NEEDS_COMMENT;
 145 #if __GNUC_PREREQ (4, 7)
 146       DIAG_IGNORE_NEEDS_COMMENT (4.8, "-Wmaybe-uninitialized");
 147 #else
 148       DIAG_IGNORE_NEEDS_COMMENT (4.8, "-Wuninitialized");
 149 #endif
 150       if (sign < 0)
 151         w = -w;
 152       DIAG_POP_NEEDS_COMMENT;
 153       return w;
 154     }
 155   if (iy == 1)
 156     return w;
 157   else
 158     return -1.0 / (x + r);
 159 }