sysdeps/ieee754/ldbl-128ibm/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, write to the Free Software
  32     Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307  USA */
  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 #ifdef __STDC__
  62 static const long double
  63 #else
  64 static long double
  65 #endif
  66   one = 1.0L,
  67   pio4hi = 7.8539816339744830961566084581987569936977E-1L,
  68   pio4lo = 2.1679525325309452561992610065108379921906E-35L,
  69
  70   /* tan x = x + x^3 / 3 + x^5 T(x^2)/U(x^2)
  71      0 <= x <= 0.6743316650390625
  72      Peak relative error 8.0e-36  */
  73  TH =  3.333333333333333333333333333333333333333E-1L,
  74  T0 = -1.813014711743583437742363284336855889393E7L,
  75  T1 =  1.320767960008972224312740075083259247618E6L,
  76  T2 = -2.626775478255838182468651821863299023956E4L,
  77  T3 =  1.764573356488504935415411383687150199315E2L,
  78  T4 = -3.333267763822178690794678978979803526092E-1L,
  79
  80  U0 = -1.359761033807687578306772463253710042010E8L,
  81  U1 =  6.494370630656893175666729313065113194784E7L,
  82  U2 = -4.180787672237927475505536849168729386782E6L,
  83  U3 =  8.031643765106170040139966622980914621521E4L,
  84  U4 = -5.323131271912475695157127875560667378597E2L;
  85   /* 1.000000000000000000000000000000000000000E0 */
  86
  87
  88 #ifdef __STDC__
  89 long double
  90 __kernel_tanl (long double x, long double y, int iy)
  91 #else
  92 long double
  93 __kernel_tanl (x, y, iy)
  94      long double x, y;
  95      int iy;
  96 #endif
  97 {
  98   long double z, r, v, w, s;
  99   int32_t ix, sign;
 100   ieee854_long_double_shape_type u, u1;
 101
 102   u.value = x;
 103   ix = u.parts32.w0 & 0x7fffffff;
 104   if (ix < 0x3c600000)          /* x < 2**-57 */
 105     {
 106       if ((int) x == 0)
 107         {                       /* generate inexact */
 108           if ((ix | u.parts32.w1 | (u.parts32.w2 & 0x7fffffff) | u.parts32.w3
 109                | (iy + 1)) == 0)
 110             return one / fabs (x);
 111           else
 112             return (iy == 1) ? x : -one / x;
 113         }
 114     }
 115   if (ix >= 0x3fe59420) /* |x| >= 0.6743316650390625 */
 116     {
 117       if ((u.parts32.w0 & 0x80000000) != 0)
 118         {
 119           x = -x;
 120           y = -y;
 121           sign = -1;
 122         }
 123       else
 124         sign = 1;
 125       z = pio4hi - x;
 126       w = pio4lo - y;
 127       x = z + w;
 128       y = 0.0;
 129     }
 130   z = x * x;
 131   r = T0 + z * (T1 + z * (T2 + z * (T3 + z * T4)));
 132   v = U0 + z * (U1 + z * (U2 + z * (U3 + z * (U4 + z))));
 133   r = r / v;
 134
 135   s = z * x;
 136   r = y + z * (s * r + y);
 137   r += TH * s;
 138   w = x + r;
 139   if (ix >= 0x3fe59420)
 140     {
 141       v = (long double) iy;
 142       w = (v - 2.0 * (x - (w * w / (w + v) - r)));
 143       if (sign < 0)
 144         w = -w;
 145       return w;
 146     }
 147   if (iy == 1)
 148     return w;
 149   else
 150     {                           /* if allow error up to 2 ulp,
 151                                    simply return -1.0/(x+r) here */
 152       /*  compute -1.0/(x+r) accurately */
 153       u1.value = w;
 154       u1.parts32.w2 = 0;
 155       u1.parts32.w3 = 0;
 156       v = r - (u1.value - x);           /* u1+v = r+x */
 157       z = -1.0 / w;
 158       u.value = z;
 159       u.parts32.w2 = 0;
 160       u.parts32.w3 = 0;
 161       s = 1.0 + u.value * u1.value;
 162       return u.value + z * (s + u.value * v);
 163     }
 164 }