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 <float.h>
  60 #include <libc-internal.h>
  61 #include <math.h>
  62 #include <math_private.h>
  63 static const _Float128
  64   one = 1,
  65   pio4hi = L(7.8539816339744830961566084581987569936977E-1),
  66   pio4lo = L(2.1679525325309452561992610065108379921906E-35),
  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 =  L(3.333333333333333333333333333333333333333E-1),
  72  T0 = L(-1.813014711743583437742363284336855889393E7),
  73  T1 =  L(1.320767960008972224312740075083259247618E6),
  74  T2 = L(-2.626775478255838182468651821863299023956E4),
  75  T3 =  L(1.764573356488504935415411383687150199315E2),
  76  T4 = L(-3.333267763822178690794678978979803526092E-1),
  77
  78  U0 = L(-1.359761033807687578306772463253710042010E8),
  79  U1 =  L(6.494370630656893175666729313065113194784E7),
  80  U2 = L(-4.180787672237927475505536849168729386782E6),
  81  U3 =  L(8.031643765106170040139966622980914621521E4),
  82  U4 = L(-5.323131271912475695157127875560667378597E2);
  83   /* 1.000000000000000000000000000000000000000E0 */
  84
  85
  86 _Float128
  87 __kernel_tanl (_Float128 x, _Float128 y, int iy)
  88 {
  89   _Float128 z, r, v, w, s;
  90   int32_t ix, sign;
  91   ieee854_long_double_shape_type u, u1;
  92
  93   u.value = x;
  94   ix = u.parts32.w0 & 0x7fffffff;
  95   if (ix < 0x3fc60000)          /* x < 2**-57 */
  96     {
  97       if ((int) x == 0)
  98         {                       /* generate inexact */
  99           if ((ix | u.parts32.w1 | u.parts32.w2 | u.parts32.w3
 100                | (iy + 1)) == 0)
 101             return one / fabsl (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.parts32.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 - 2.0 * (x - (w * w / (w + v) - r)));
 139       /* SIGN is set for arguments that reach this code, but not
 140          otherwise, resulting in warnings that it may be used
 141          uninitialized although in the cases where it is used it has
 142          always been set.  */
 143       DIAG_PUSH_NEEDS_COMMENT;
 144       DIAG_IGNORE_NEEDS_COMMENT (5, "-Wmaybe-uninitialized");
 145       if (sign < 0)
 146         w = -w;
 147       DIAG_POP_NEEDS_COMMENT;
 148       return w;
 149     }
 150   if (iy == 1)
 151     return w;
 152   else
 153     {                           /* if allow error up to 2 ulp,
 154                                    simply return -1.0/(x+r) here */
 155       /*  compute -1.0/(x+r) accurately */
 156       u1.value = w;
 157       u1.parts32.w2 = 0;
 158       u1.parts32.w3 = 0;
 159       v = r - (u1.value - x);           /* u1+v = r+x */
 160       z = -1.0 / w;
 161       u.value = z;
 162       u.parts32.w2 = 0;
 163       u.parts32.w3 = 0;
 164       s = 1.0 + u.value * u1.value;
 165       return u.value + z * (s + u.value * v);
 166     }
 167 }